Fourier Series and Transform
Overview
Why Fourier transform
Trigonometric functions
Who is Fourier
Fourier series
Fourier transform
Discrete Fourier transform
Fast Fourier transform
2D Fourier transform
Tips
Why Fourier transform
Fourier not being noble could not enter the artillery although he was a second Newton
⎯ Francois Jean Dominique Arago
For signal processing Fourier transform is the tool to connect the time domain and frequency domain
Why would we do the exchange between time domain and frequency domain Because we can do all kinds of useful analytical tricks in the frequency domain that are just too hard to do computationally with the original time series in the time domain Simplify the calculation
Why Fourier transform
)502sin()( ttf π=
Why Fourier transform
This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Overview
Why Fourier transform
Trigonometric functions
Who is Fourier
Fourier series
Fourier transform
Discrete Fourier transform
Fast Fourier transform
2D Fourier transform
Tips
Why Fourier transform
Fourier not being noble could not enter the artillery although he was a second Newton
⎯ Francois Jean Dominique Arago
For signal processing Fourier transform is the tool to connect the time domain and frequency domain
Why would we do the exchange between time domain and frequency domain Because we can do all kinds of useful analytical tricks in the frequency domain that are just too hard to do computationally with the original time series in the time domain Simplify the calculation
Why Fourier transform
)502sin()( ttf π=
Why Fourier transform
This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Why Fourier transform
Fourier not being noble could not enter the artillery although he was a second Newton
⎯ Francois Jean Dominique Arago
For signal processing Fourier transform is the tool to connect the time domain and frequency domain
Why would we do the exchange between time domain and frequency domain Because we can do all kinds of useful analytical tricks in the frequency domain that are just too hard to do computationally with the original time series in the time domain Simplify the calculation
Why Fourier transform
)502sin()( ttf π=
Why Fourier transform
This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Why Fourier transform
)502sin()( ttf π=
Why Fourier transform
This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Why Fourier transform
This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Trigonometric functions (ex1)
Trigonometric system is the periodic functions as
1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip
The properties of trigonometric system
Trigonometric system is the orthogonal system
2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=
(sin ) cos (cos ) sinθ θ θ θ= = minus
int =int =minusminus
π
π
π
π0sin0cos nxdxnxdx int =
minus
π
π0cossin nxdxmx
int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
coscos int⎩⎨⎧
=ne
=minus
π
π π nmnm
nxdxmx0
sinsin
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
In Matlab
Int(f)Int(fab)
if f is a symbolic expression
q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)
Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral
q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x
intb
adxxf )(
int=b
adxxfq )(
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Who is Fourier
Fourier is one of the Francersquos greatest administrators historians and mathematicians
He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817
Donrsquot believe it
Neither did Lagrange Laplace Poisson and other big wigs
Not translated into English until 1878
But itrsquos true
Jean Baptiste Joseph Fourier(France 1768~1830)
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourierrsquos basic idea
Trigonometric functions sin(x) and cos(x) has the period 2π
sin(nx) and cos(nx) have period 2πn
The linear combination of these functions or multiply each by a constant the adding result still has a period 2π
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series
For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines
To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get
mxdxxfam cos)(1int=minus
π
ππ
)sincos(2
)(1
0 sum ++=infin
=mmm mxbmxaaxf
mxdxxfbm sin)(1int=minus
π
ππ
dxxfa int=minus
π
ππ)(1
0
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (ex2)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
π
0
sin1sin1
cos)1(1cos1
0
0
0
01
=
minusminus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
xx
dxxdxxa
0
sin1sin1
cos)1(1cos1
0
0
0
0
=
+minus=
int minus+int=
minus
minus
π
π
π
π
ππ
ππ
m
m
m
mxm
mxm
dxmxdxmxa
Period function
The parameters are 0)1(111 0
00 =int minus+int=
minusdxdxa
π
π
ππ
5314== n
nbn π
6420 == kbk
mxdxxfbm sin)(1int=minus
π
ππ
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
=+
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
+ =
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example (conthellip)
⎩⎨⎧
ltltminusminusltlt
=01
01)(
xx
xfπ
πPeriod function
The Fourier series is
)55sin
33sin
1sin(4)( +++=
xxxxfπ
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series example
π20)( ltlt= xxxfPeriod function
The parameters are
)33sin
22sin
1sin(2)( +++minus=
xxxxf π
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier series general
For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as
The f(xrsquo) can be described by the Fourier series as
where
Replace ω 2πT and xrsquo t
22 dxT
dxxT
x ππ==
21))2sin()2cos((2
)(1
0 =sum ++=infin
=mx
Pmbx
Tmaaxf
mmm
ππ
)2cos()(2 2
2dxx
Tmxf
Ta
T
Tm
πint=
minus
)(2 2
20 dxxf
Ta
T
Tint=
minus
)2sin()(2 2
2dxx
Tmxf
Tb
T
Tm
πint=
minus
21)sincos(2
)(1
0 =sum ++=infin
=mtmbtmaatf
mmm ωω
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
The complex form of Fourier series
))(2
)(2
(2
)(1
0 sum minus+++=infin
=
minusminus
m
timtimmtimtimm eei
beeaatf ωωωω
Euler formulae
Fourier seriesieex
eexxixexixe
ixix
ixix
ix
ix
2)(sin2)(cos
sincossincos
minus
minus
minus minus=+=
rArr⎭⎬⎫
minus=+=
For certain m=k
Denoting that as
The complex form of Fourier series
tikkktikkk
tiktik
k
tiktik
kkk
eibaeibaieebeeatkbtka
ωω
ωωωω
ωω
minus
minusminus
++
minus=
minus+
+=+
22
22sincos
2
2
20
0kk
kkk
k
ibacibacac +=
minus== minus
int=
sumsum =++=
minus
minus
infin
minusinfin=
infin
=
minusminus
2
2
10
)(2
)()(
T
T
tikk
k
tikk
k
tikk
tikk
dtetfT
c
ecececctf
ω
ωωω
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform
For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get
Define
Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform
int=
int=
infin
infinminus
infin
infinminus
minus
ωωπ
ω
ω
ω
deFtf
dtetfF
ti
ti
)(21)(
)()(
intintrarr
sum intrarr
sum int=
infin
infinminus
minusinfin
infinminus
infin
minusinfin= minus
minus=
infin
minusinfin= minus
minus
ξξωπ
ξξωπ
ωξω
ξωωπ
ωω
defde
def
edtetfT
tf
iti
k
T
T
tikT
k
tikT
T
tik
)(21
)(1
))(2()(
2
2
)(2
2
2
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Parsevalrsquos law
The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula
int=intinfin
infinminus
infin
infinminusωω dFdttf 22 )()(
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(x) g(x)
- Similarity property g(x)=f(ax)
-Shift formula given g(x) = f(x+b)
- Derivative formula
)()())()(( ωω bGaFxbgxafFT +=+
)(1)(a
Fa
G ωω =
)())(( ωωFixfFT =
)()( ωω ω FeG bi=
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
In Matlab
F = fourier(f)
This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents
int=infin
infinminus
minus dxexfF xiωω )()(
f = ifourier(F)
This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents
int=infin
infinminusωω
πω deFxf xi)(
21)(
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Discrete Fourier Transform (DFT)
Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform
This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)
To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps
1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN
2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)
3) Define the signal values at each sampling points as fk=f(tk)
4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Discrete Fourier Transform (DFT)
5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is
6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT
int minus==infin
infinminus
minus )1(0)()( NndttfeF tnin
ωω
1210)()(1
0minus=sum=
minus
=
minus NnetfFN
k
ktnikn
ωω
This is the Discrete Fourier Transform
The inverse Discrete Fourier Transform is defined as
1
0
1( ) ( ) 012 1n k
Ni t
k nn
f t F e k NN
ωωminus
=
= = minussum
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fast Fourier Transform (FFT)
Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965
This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)
The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096
Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
In Matlab
Y = fft(X)
This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = fft(Xn)
This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated
Y = ifft(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft(Xn)
This command returns the n-point inverse DFT of X
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Delta function
Delta function
⎩⎨⎧ =
==otherst
ttf0
01)()( δ
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Uniform function
Unit function1)( =tf
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π3f t)
= +
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Sin function
example g(t) = sin(2πf t) + (13)sin(2π 3f t)
= +
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Fourier transform Cos function
cos functions)502cos()( ttf π=
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Applications convolution
The time domain recorded waveform is a convolution product
Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform
int minus=infin
infinminusτττ dstvtr )()()( 0
)()()( 0 ωωω SVRFFT
=rArr
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Applications convolution
The simulated DI water waveform and ethanol waveform at room temperature by FFT
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
2D Integral Fourier transform
2D integral Fourier transform is
Inverse 2D Fourier transform is
)()( )(int int=infin
infinminus
infin
infinminus
+minus dxdyeyxfvuF vyuxi
)(21)( )(int int=
infin
infinminus
infin
infinminus
+ dudvevuFyxf vyuxi
π
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
2D Discrete Fourier transformFor 2D function f(xy) DFT is
Rewrite above equation
We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NnMm
eyxfvuFM
xk
N
yk
yvxuiykxknm
yknxkm
sum=
sum ⎥⎦⎤
⎢⎣⎡ sum=
minus
=
minus
minus
=
minusminus
=
minus
1
0
1
0
1
0
)()(
)()(
M
m
xiunm
M
m
xiuN
n
yivykxknm
xkm
xkmykn
evxFvuF
eeyxfvuF
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is
Rewrite above equation
We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations
12101210
)()(1
0
1
0
)(
minus=minus=
sum sum=minus
=
minus
=
+minus
NykMxk
evuFyxfM
m
N
n
ykynvxkxmuinmykxk
[ ]sum=
sum sum=
minus
=
minus
minus
=
minus
=
1
0
1
0
1
0
)(1)(
)(1)(
M
m
xkxmiunxkykxk
M
m
xkxmiuN
n
ykynivnmykxk
evxFMN
yxf
eevuFMN
yxf
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
2D Fourier transform properties
The properties of Fourier transfrom
- Linearity property given f(xy) g(xy)
- Shift formula given g(xy) = f(x+ay+b)
- Similarity property g(xy)=f(axby)
- Convolution
)()())()(( vubGvuaFyxbgyxafFT +=+
)()( )( vuFevuG vbuai +=
)()()()( vuHvuGyxhyxg sdot=lowast
)(1)(bv
auF
abvuG =
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
In Matlab
Y = fft2(X)
This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm
Y = fft2(Xmn)
This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n
Y = ifft2(X)
This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm
Y = ifft2(Xmn)
This command returns the m-point of x and n-point of y inverse DFT of X(xy)
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Image Compression (JPEG)
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
JPEG compression comparison
89k 12k
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Tips Nyquist frequency
Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal
The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain
On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain
tfNF ∆
=21
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Tips Sampling ratesampling total time
Sampling rate is the sampling interval This will control the highest frequency band
Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band
Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Tips windows
Any sampling data range is limitedfinite
( ) sin(2 5 )f t tπ=
+=
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
Tips windows
( ) sin(2 5 )f t tπ=
The true sampling signal is frsquo(t) = f(t)win(t)
After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)
For the true transformed the signal we have to de-convolution of the transformed the results
)()()( ωωω WINFF lowast=
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-
The Homework of Fourier transform is using Matlab
Do the Fourier transform of one simple harmonic function
f1(x)=sin(2pi500t)
and
f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)
Please practice with
1) choosing two different sampling rate (how many points you sampled in total in time domain N)
2) choosing two different sampling interval (delta_t)
3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain
- Fourier Series and Transform
- Overview
- Why Fourier transform
- Why Fourier transform
- Trigonometric functions (ex1)
- In Matlab
- Who is Fourier
- Fourierrsquos basic idea
- Fourier series
- Fourier series example (ex2)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example (conthellip)
- Fourier series example
- Fourier series general
- The complex form of Fourier series
- Fourier transform
- Fourier transform Parsevalrsquos law
- Fourier transform properties
- In Matlab
- Discrete Fourier Transform (DFT)
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
- In Matlab
- Fourier transform Delta function
- Fourier transform Uniform function
- Fourier transform Sin function
- Fourier transform Sin function
- Fourier transform Cos function
- Applications convolution
- Applications convolution
- 2D Integral Fourier transform
- 2D Discrete Fourier transform
- 2D Inverse Discrete Fourier transform
- 2D Fourier transform properties
- In Matlab
- Image Compression (JPEG)
- JPEG compression comparison
- Tips Nyquist frequency
- Tips Sampling ratesampling total time
- Tips windows
- Tips windows
-