chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp
DESCRIPTION
dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dspTRANSCRIPT
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CHAPTER 5:
DISCRETE FOURIER TRANSFORM (DFT)
Lesson #19: DTFT of DT periodic signals
Lesson #20: DFT and Inverse DFT
Lesson #21: DFT properties
Lesson #22: Fast Fourier Transform (FFT)
Lesson #23: Applications of DFT/FFT
Lesson #24: Using of DFT/FFT
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Lecture #19
DTFT of periodic signals
1. Review of Fourier Transforms (FT)
2. Fourier Series expansions
3. DTFT of periodic signals
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FT of CT periodic signals
Tdtetx
Taeatx
tjk
T
k
tjk
k
k
2;)(
1;)( 0
00
Discrete and aperiodic in frequency domain
Continuous and periodic in time domain
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FT of CT aperiodic signals
Continuous and aperiodic in time domain
1
Continuous and aperiodic in frequency domain
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FT of DT aperiodic signals
Discrete and aperiodic in time domain
Continuous and periodic in frequency with period 2π
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Recapitulation
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Can guess?
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Lecture #19
DTFT of periodic signals
1. Review of Fourier Transforms (FT)
2. Fourier Series expansions
3. DTFT of periodic signals
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Nn
njk
k
njk
Nk
k enxN
aeanx 00 ][1
;][ N
20
dtetxT
aeatxtjk
T
k
tjk
k
k00 )(
1;)(
• CT periodic signals with period T:
• DT periodic signals with period N:
Note: finite sums over an interval length of one periodic N
n)Nk(jn
N
2)Nk(jn
N
2jk
njk 00 eeee
T
20
Fourier Series expansion
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Given a DT periodic signals with period N:
Example of Fourier Series expansion
[ ] [ ]k
p n n kN
N
1e]n[p
N
1a
1N
0n
N/n2jk
k
. . . -2N -N 0 N 2N . . . . .
1
0
2
][N
k
nN
jk
keanp
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Lecture #19
DTFT of periodic signals
1. Review of Fourier Transforms (FT)
2. Fourier Series expansions
3. DTFT of periodic signals
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• CT periodic signals with period T:
• DT periodic signals with period N:
DTFT of periodic signals
)k(a2)(Xea)t(x0
kk
Ftjk
kk
0
)k(a2)(X]n[x0
kk
F
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. . . -2N -N 0 N 2N . . . . . . . . n
Example of calculating DTFT of
periodic signals
kk0k
N
2k
N
2ka2)(P
2π/N 2π/N 2π/N 2π/N 2π/N
. . . -4π/N -2π/N 0 2π/N 4π/N . . . . Ω
Height = 1
Spacing = N
Height = 2π/N
Spacing = 2π/N
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Another approach to get DTFT of
periodic signals
0
[ ] 0 1[ ]
0 otherwise
x n n Nx n
0 N-1
x(n) is periodic signal; x0(n) is a part of x(n) that is repeated
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Another approach (cont)
0
[ ] 0 1[ ]
0 otherwise
x n n Nx n
0 0 0[ ] [ ] [ ] [ ] [ ] [ ]k k k
x n x n kN x n n kN x n n kN
p(n) in previous example
0 0[ ] [ ] [ ] ( ) ( ) ( )F
x n x n p n X P X
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Another approach (cont)
0 0[ ] [ ] [ ] ( ) ( ) ( )F
x n x n p n X P X
k
k
Nk
NkX
N
Nk
NXX
)2
()2
(2
)2
(2
)()(
0
0
It has N distinct values at k = 0, 1, …, N-1
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Inverse DTFT
1
0
2
0
2
0
0
2
0
0
2
0
1
)2
(1
)2
()2
(1
)2
()2
(2
2
1
)(2
1][)(
N
k
N
nkj
k
nj
nj
k
nj
eN
kXN
deN
kN
kXN
deN
kN
kXN
deXnxXDFT
This is obtained from this property of the impulse:
elsewhere
btatfdttttf
b
a0
)()()(
00
0
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Summary
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Procedure to calculate DTFT of
periodic signals
Step 1:
Start with x0(n) – one period of x(n), with zero everywhere else
Step 2:
Find the DTFT X0(Ω) of the signal x0[n] above
Step 3:
Find X0(Ω) at N equally spacing frequency points X0(k2π/N)
Step 4:
Obtain the DTFT of x(n): k N
kN
kXN
X )2
()2
(2
)( 0
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Example of calculating DTFT of
periodic signals
33
0
00 21)()( jj
n
nj eeenxX
k = 0 X0(0) = 4; k = 1 X0(1) = 1+j
k = 2 X0(2) = -2; k = 3 X0(3) = 1-j
0 1 2 3
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Example (cont)
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Example (cont)
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a) Draw the magnitude plot of the FT of a CT sinusoid x(t), that
has a frequency of 24 Hz
b) Sample this signal x(t) at 40 Hz. Draw the magnitude plot of
the FT of the sampled signal
c) Reconstruct a CT signal from the samples above with a LPF, it
will look like a CT sinusoid of what frequency?
d) Consider DT signal that is obtained by the sampling x(t) at a
sampling frequency of 40 Hz is x(n). Is x(n) a periodic signal?
If yes, what is its period?
e) Specify the DTFT of the DT signal x(n).
DTFT and DFS example
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a) Draw the magnitude plot of the FT of a CT sinusoid x(t), that has
a frequency of 24 Hz
The FT of a sinusoidal has nonzero values only at certain points
DTFT and DFS examples
-24 24 f [Hz]
-48π 48π ω [rad/s]
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b) Sample this signal x(t) at 40 Hz. Draw the magnitude plot of the
FT of the sampled signal
Sampled signal FT magnitude = copies of unsampled signal FT
magnitude; each copy is centered at k40(Hz) or k80π(rad/s)
DTFT and DFS examples
-64 -56 -24 -16 0 16 24 40 56 64 f[Hz]
-128π -48π -32π 32π 48π 80π 112π
ω[rad/s]
fs = 40Hz fs = 40Hz
8Hz
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c) Reconstruct a CT signal from the samples above with a LPF, it will
look like a CT sinusoid of what frequency?
The unit circle represents the scale of frequency range for a
signal sampled at 40Hz the reconstructed signal looks like
16Hz.
DTFT and DFS examples
20
24
16
0 16 24 40 56 64 f[Hz]
32π 48π 80π 112π ω[rad/s]
8Hz
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d) Consider DT signal that is obtained by the sampling x(t) at a
sampling frequency of 40 Hz is x(n). Is x(n) a periodic signal? If
yes, what is its period?
The unsampled signal: x(t) = cos(2π.24t)
The sampled signal:
x[n] = cos(2π.24.(1/40)t) = cos(6π/5)n = cos (4π/5)n
The sampled signal is periodic with N = 5
DTFT and DFS examples
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e) Specify the DTFT of the DT signal x(n).
DTFT and DFS examples
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Lecture #20
DFT and inverse DFT
1. DFT and inverse DFT formulas
2. DFT and inverse DFT examples
3. Frequency resolution of the DFT
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DFT to the rescue!
Both CTFT and DTFT produce continuous function of
frequency can’t calculate an infinite continuum of frequencies
using a computer
Most real-world data is not in the form like anu(n)
DFT can be used as a FT approximation that can calculate a finite
set of discrete-frequency spectrum values from a finite set of
discrete-time samples of an analog signal
Could we calculate the frequency spectrum of a signal
using a digital computer with CTFT/DTFT?
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Building the DFT formula
“Window” x(n) is like multiplying
the signal by the finite length
rectangular window
Discrete time
signal x(n)
Continuous time
signal x(t)
Discrete time
signal x0(n)
Finite length
sample
window
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Building the DFT formula (cont)
Discrete time
signal x(n)
Continuous time
signal x(t)
Discrete time
signal x0(n)
Finite length
sample
window
DTFT
1
0
000 ][][)(N
n
nj
n
nj enxenxX
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Building the DFT formula (cont)
Discrete time
signal x(n)
Continuous time
signal x(t)
Discrete time
signal x0(n)
Finite length
sample
window
DTFT
Sample
at N
values
around
the unit
circle
Discrete Fourier Transform DFT X(k)
Discrete + periodic with period N
X0(Ω)
Continuous + periodic
with period 2π
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Notation conventions
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Notation conventions (cont)
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DFT and inverse DFT formulas
Notation
You only have to store N points
N
2j
N eW
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Lecture #20
DFT and inverse DFT
1. DFT and inverse DFT formulas
2. DFT and inverse DFT examples
3. Frequency resolution of the DFT
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DFT and IDFT examples
Ex.1. Find the DFT of x(n) = 1, n = 0, 1, 2, …, (N-1)
k = 0 X(k) = X(0) = N
k ≠ 0 X(k) = 0
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DFT and IDFT examples
Ex.2. Given y(n) = δ(n-2) and N = 8, find Y(k)
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DFT and IDFT examples
Ex.3. Find the IDFT of X(k) = 1, k = 0, 1, …, 7.
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DFT and IDFT examples
Ex.4. Given x(n) = δ(n) + 2δ(n-1) +3δ(n-2) + δ(n-3) and
N = 4. Find X(k).
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DFT and IDFT examples
Ex.5. Given X(k) = 2δ(k) + 2δ(k-2) and N = 4. Find x(n).
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DFT in matrix forms
)1)(1()1(21
)1(21
1
0
1
1
1111
]1[
]1[
]0[
]1[
]1[
]0[
:'
1,...,1,0][][
NN
N
N
N
N
N
N
NNN
N
NN
N
n
kn
N
WWW
WWWW
NX
X
X
X
Nx
x
x
x
definesLet
NkWnxkX
The N-point DFT may be expressed in matrix form as:
XN = WNxN
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IDFT in matrix forms
)1)(1()1(21
)1(21
1
0
1
1
1111
]1[
]1[
]0[
]1[
]1[
]0[
:'
1,...,1,0][1
][
NN
N
N
N
N
N
N
NNN
N
NN
N
k
kn
N
WWW
WWWW
NX
X
X
X
Nx
x
x
x
definesLet
NnWkXN
nx
The N-point DFT may be expressed in matrix form as:
matrixidentityI
NIWW
XWN
XWx
N
NNN
NNNNN
:
1
*
*1
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Example of calculation of DFT in matrix form
Find DFT of the signal
x[n] =
δ[n-1] + 2δ[n-2] + 3δ[n-3]
j
jxWXThen
jj
jj
WWW
WWW
WWWW
x
definesLet
22
2
22
6
.
11
1111
11
1111
1
1
1
1111
3
2
1
0
:'
444
9
4
6
4
3
4
6
4
4
4
2
4
3
4
2
4
1
4
4
4
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Lecture #20
DFT and inverse DFT
1. DFT and inverse DFT formulas
2. DFT and inverse DFT examples
3. Frequency resolution of the DFT
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Discrete frequency spectrum computed from DFT has the
spacing between frequency samples of:
Frequency resolution of the DFT
N
ff s
N
2
The choice of N determines the resolution of the frequency
spectrum, or vice-versa
To obtain the adequate resolution:
1. Increasing of the duration of data input to the DFT
2. Zero padding, which adds no new information, but effects a better
interpolator
![Page 48: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/48.jpg)
Examples of N = 5 and N = 15
![Page 49: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/49.jpg)
Examples of N = 15 and N = 30
![Page 50: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/50.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 51: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/51.jpg)
DFT properties
x(n)
Finite length
xp(n)
Periodic
N periodic extension
truncate
N periodic extension
truncate
X(k)
Finite length Xp(k)
Periodic
DFT
DFT
-1
Xp(Ω)
Periodic
DTFT
DTFT-1
![Page 52: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/52.jpg)
DFT properties
x(n)
Finite length
xp(n)
Periodic
N periodic extension
truncate
N periodic extension
truncate
X(k)
Finite length Xp(k)
Periodic
DFT
DFT
-1
Xp(Ω)
Periodic
DTFT
DTFT-1
![Page 53: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/53.jpg)
Periodicity
x[n+N] = x[n] for all n
X[k+N] = X[k] for all k
If x[n] and X[k] are an N-point DFT pair, then
![Page 54: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/54.jpg)
Linearity
Note: The length of x1[n] is same with the length of x2[n]
][][][][ 22112211 kXakXanxanxaDFT
Proof:
Infer from the definition formula of DFT
![Page 55: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/55.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 56: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/56.jpg)
Circular time shift property
]k[XW]mn[x kmDFT
Proof:
Infer from the relation between DFT and DTFT
][][)(
:2
)(][
2
kXWkXeXe
Nk
Xemnx
kmm
Njk
mj
mjDTFT
![Page 57: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/57.jpg)
Circular
time
shift
Circular shift by m is the same as a shift by m modulo N
x(n)
xp(n)
xp(n+2)
x(n+2)
![Page 58: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/58.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 59: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/59.jpg)
Circular frequency shift property
][][ ln lkXWnxDFT
Proof:
Similar to the circular time shift property
![Page 60: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/60.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 61: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/61.jpg)
Circular convolution property
The non-zero length of x1(n) and x2(n) can be no longer
than N
The shift operation is circular shift
The flip operation is circular flip
1
0
212121 ][][][][][].[N
p
DFT
pnxpxnxnxkXkX
The circular convolution is not the linear convolution in chapter 2
![Page 62: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/62.jpg)
Direct method to calculate circular
convolution
1. Draw a circle with N values of x(n) with N equally spaced angles
in a counterclockwise direction.
2. Draw a smaller radius circle with N values of h(n) with equally
spaced angles in a clockwise direction. Superimpose the centers of 2
circles, and have h(0) in front of x(0).
3. Calculate y(0) by multiplying the corresponding values on each
radial line, and then adding the products.
4. Find succeeding values of y(n) in the same way after rotating the
inner disk counterclockwise through the angle 2πk/N
![Page 63: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/63.jpg)
Example to calculate circular convolution
Evaluate the circular convolution, y(n) of 2 signals:
x1(n) = [ 1 2 3 4 ]; x2(n) = [ 0 1 2 3 ]
1
2
3
4 0
1
2
3
y(0) = 16
1
2
3
4 1
2
3
0
y(1) = 18
![Page 64: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/64.jpg)
Example (cont)
Evaluate the circular convolution, y(n) of 2 signals:
x1(n) = [ 1 2 3 4 ]; x2(n) = [ 0 1 2 3 ]
1
2
3
4 2
3
0
1
y(2) = 16
1
2
3
4 3
0
1
2
y(3) = 10
![Page 65: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/65.jpg)
Another method to calculate circular convolution
x1(n)
x2(n)
X1(k)
X2(k)
DFT
DFT
IDFT y(n)
Ex. x1(n) = [ 1 2 3 4 ]; x2(n) = [ 0 1 2 3 ]
X1(k) = [ 10, -2+j2, -2, -2-j2 ];
X2(k) = [ 6, -2+j2, -2, -2-j2 ];
Y(k) = X1(k).X2(k) = [ 60, -j8, 4, j8 ]
y(n) = [ 16, 18, 16, 10 ]
![Page 66: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/66.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 67: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/67.jpg)
Multiplication
The non-zero length of X1(k) and X2(k) can be no longer
than N
The shift operation is circular shift
The flip operation is circular flip
1
0
212121 ][][1
][][1
][].[N
l
DFT
lkXlXN
kXkXN
nxnx
![Page 68: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/68.jpg)
Lecture #21
DFT properties
1. Periodicity and Linearity
2. Circular time shift
3. Circular frequency shift
4. Circular convolution
5. Multiplication
6. Parseval’s theorem
![Page 69: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/69.jpg)
Parseval’s theorem
The general form:
Special case:
1
0
1
0
][*][1
][*][N
k
N
n
kYkXN
nynx
1
0
21
0
2 |][|1
|][|N
k
N
n
kXN
nx
![Page 70: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/70.jpg)
Parseval’s theorem example
Given x(n) = δ(n) + 2δ(n-1) +3δ(n-2) + δ(n-3) and N = 4.
1
0
2222222
22221
0
2
154
60)121127(
4
1|][|
1
151321|][|
N
k
N
n
kXN
nx
![Page 71: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/71.jpg)
Multiple choice questions
Ex1:
It is desired to build a sampling system which takes a
recorded orchestra music signal s(t) and converts it to a
discretely-sampled form. In order to experimentally determine
the bandwidth (in frequency) of this signal the best transform
would be
a. The Fourier transform of a continuous function or a discrete
sequence
b. The Z-transform
c. The Fourier series
d. The discrete Fourier transform
![Page 72: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/72.jpg)
Multiple choice questions
Ex2:
A mathematical model of human speech puts out a perfectly
periodic signal for steady vowel sounds. In order to accurately
determine the harmonics of the fundamental frequency for
this mathematical model, the best transform should be
a. The Fourier transform of a continuous function or a discrete
sequence
b. The Z-transform
c. The Fourier series
d. The discrete Fourier transform
![Page 73: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/73.jpg)
Multiple choice questions
Ex3:
A student wishes to convolve the continuous-time mathematical
functions g(t) and h(t). She/he uniformly samples these
functions, and uses a computer to get, at best, an approximate
result. Another student claims to be able to get an exact result
via a transform. She/he must be thinking of which transform
a. The Fourier transform of a continuous function or a discrete
sequence
b. The Z-transform
c. The Fourier series
d. The discrete Fourier transform
![Page 74: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/74.jpg)
Multiple choice questions
Ex4:
A student wishes to convolve the discrete-time mathematical
functions g(n) and h(n). She/he claims to be able to get an
exact result via one of the transforms below. She/he must be
thinking of which transforms
a. The Fourier transform of a continuous function or a discrete
sequence
b. The Z-transform
c. The Fourier series
d. The discrete Fourier transform
![Page 75: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/75.jpg)
Lecture #22
Fast Fourier Transform (FFT)
1. What is FFT?
2. The decomposition-in-time Fast Fourier
Transform algorithm (DIT-FFT)
3. The decomposition-in-frequency Fast
Fourier Transform algorithm (DIF-FFT)
![Page 76: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/76.jpg)
Using some "tricks" and choosing the appropriate N value, the DFT
can be implemented in a very computationally efficient way – this is
the Fast Fourier Transform (FFT) - which is the basis of lots of
contemporary DSP hardware
N
2j
N eW1Nn0W]k[XN
1]n[x
1Nk0W]n[x]k[X
1N
0n
nk
N
1N
0n
nk
N
Recall DFT and IDFT definition
Nth complex root of unity:
![Page 77: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/77.jpg)
What does it take to compute DFT or IDFT?
N2 complex multiplications for all N points of X(k) or x[n]
FFT: the algorithm to optimize the computational process by
breaking the N-point DFT into smaller DFTs
Ex.: N is a power of 2: split N-point DFT into two N/2-point DFTs,
then split each of these into two of length N/4, etc., until we
have N/2 subsequences of length 2
Cooley and Tukey (1965):
Fast Fourier Transform (FFT)
tionsmultiplicacomplex 2
log2
2N
NN
![Page 78: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/78.jpg)
Some properties of can be exploited
Other useful properties:
Wnk
N
k andn in y periodicit ,
symmetry conjugatecomplex ,
)()(
*)(
WWW
WWWnNk
N
Nnk
N
kn
N
kn
N
nNk
N
knN
WW
WW
eWWWW
knN
kn
N
kn
N
kn
Njnkn
N
nN
N
kn
N
nN
k
N
2
2
2)
2(
oddn if ,
evenn if ,
FFT (cont)
![Page 79: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/79.jpg)
N Co
mp
lex m
ult
iplicati
on
nu
mb
ers
FFT
Comparison of DFT and FFT efficiency
![Page 80: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/80.jpg)
Comparison of DFT and FFT efficiency
Number of Complex multiplications Complex multiplications Speed improvement
points N in direct DFT in FFT factor
4
8
16
32
64
128
256
512
1024
16
64
256
1,024
4,096
16,384
65,536
262,144
1,048,576
4
12
32
80
192
448
1,024
2,304
5,120
4.0
5.3
8.0
12.8
21.3
36.6
64.0
113.8
204.8
![Page 81: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/81.jpg)
Lecture #22
Fast Fourier Transform (FFT)
1. What is FFT?
2. The decomposition-in-time Fast
Fourier Transform algorithm (DIT-FFT)
3. The decomposition-in-frequency Fast
Fourier Transform algorithm (DIF-FFT)
![Page 82: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/82.jpg)
G(k) is N/2 points DFT of the even numbered data: x(0),
x(2), x(4), …., x(N-2).
H(k) is the N/2 points DFT of the odd numbered data:
x(1), x(3), …, x(N-1).
even odd
[ ] [ ] [ ]kn kn
n n
X k x n W x n W
DIT-FFT with N as a 2-radix number
1N,...,1,0k ),k(H)k(G)k(X Wk
N
We divide X(k) into 2 parts:
![Page 83: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/83.jpg)
2 21 1
2 (2 1)
0 0
[ ] [2 ] [2 1]
N N
mk k m
m m
X k x m W x m W
2 21 1
2 2
0 0
[2 ]( ) [2 1]( )
N N
mk k mk
m m
x m W W x m W=
2/N
)2/N/(2j2N/2j2
NWeeW
G(k) and H(k) are of length N/2; X(k) is of length N
G(k)=G(k+N/2) and H(k)=H(k+N/2)
)()(][][)(
1
0
2/
1
0
2/
22
kHkGWmhWWmgkX Wk
Nm
mk
N
k
N
m
mk
N
NN
DIT-FFT (cont)
Note:
![Page 84: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/84.jpg)
4
k
848]k[HW]k[G]k[X
]3[HW]3[G]7[X]3[HW]3[G]3[X
]2[HW]2[G]6[X]2[HW]2[G]2[X
]1[HW]1[G]5[X]1[HW]1[G]1[X
]0[HW]0[G]4[X]0[HW]0[G]0[X
7
8
3
8
6
8
2
8
5
8
1
8
4
8
0
8
DIT-FFT of length N = 8
![Page 85: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/85.jpg)
DFT
N = 4
DFT
N = 4
# complex multiplications are reduced: 82 = 64 2x(4)2 + 8 = 40
Signal
flow
graph
for
8-point
FFT
![Page 86: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/86.jpg)
2424 ][][][ kHWkGkX k
DIT-FFT of length N = 4
]1[]1[]3[
]0[]0[]2[
]1[]1[]1[
]0[]0[]0[
3
4
2
4
1
4
0
4
HWGX
HWGX
HWGX
HWGX
]1[]1[]3[ 3
4 HWGX
]1[]1[]1[ 1
4 HWGX
]0[]0[]2[ 2
4 HWGX
]0[]0[]0[ 0
4 HWGX
DFT
N = 2
DFT
N = 2
W0
x[0]
x[2]
x[1]
x[3]
W1
W2
W3
# multiplications are reduced: 42 = 16 2x(2)2 + 4 = 12
![Page 87: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/87.jpg)
DIT-FFT of length N = 2
# multiplications are
reduced: 22 = 4 0;
just 2 additions!!! x[1] - 1
x[0] X[0]
X[1]
]1[]0[]1[]0[]1[
]1[]0[]1[]0[]0[
1,10,][][
1.11.0
0.10.0
2
21
0
xxWxWxX
xxWxWxX
eWkWnxkXj
n
nk
Butterfly diagram
![Page 88: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/88.jpg)
DFT
N = 4
DFT
N = 4
First step: split 8-point DFT into two 4-point DFTs
Example
of
8-point
FFT
![Page 89: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/89.jpg)
0W
0W
W4
W2
W0
W0
W6
W6
W2
W4
DFT N = 2
DFT N = 2
DFT N = 2
DFT N = 2
G[0]
G[1]
G[2]
G[3]
H[0]
H[1]
H[2]
H[3]
Example
of
8-point
FFT
(cont)
Second step: split each of 4-point DFT into two 2-point DFTs
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Example
of
8-point
FFT
(cont) 0W
0W
W4
W2
W0
W0
W6
W6
W2
W4
Third step: combine all signal flow graphs of step 1 and step 2
W0
W1
W2
W3
W4
W5
W6
W7
There are 3 = log28 stages; 4 butterfly diagrams in each stage
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Wr+N/2
Wr+N/2 = -Wr
Wr - 1
Butterfly diagram
![Page 92: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/92.jpg)
Now the overall computation is reduced to:
tionsmultiplicacomplex 2
log2
2N
NN
The
overall
signal
flow
graph
for
8-point
DIT-FFT
![Page 93: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/93.jpg)
Wr - 1
In-place computation
a
b
A
B
Each butterfly takes (a,b) to produce (A,B) no need to save
(a,b) can store (A,B) in the same locations as (a,b)
We need 2N store registers to store the results at each stage and
these registers are also used throughout the computation of the
N-point DFT
The order of the input: bit-reversed order
The order of the output: natural order
![Page 94: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/94.jpg)
Lecture #22
Fast Fourier Transform (FFT)
1. What is FFT?
2. The decomposition-in-time Fast Fourier
Transform algorithm (DIT-FFT)
3. The decomposition-in-frequency Fast
Fourier Transform algorithm (DIF-FFT)
![Page 95: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/95.jpg)
Rewrite the second half:
DIF-FFT with N as a 2-radix number
1)2/(
0
1)2/(
0
1
)2/( 2
2)( 22
N
n
kn
N
k
N
N
n
nk
N
N
Nn
kn
N WN
nxWWN
nxWnxNN
1,...,1,0 ,)()()(1)2/(
0
1
)2/(
NkWnxWnxkXN
n
N
Nn
kn
N
kn
N
kjNNkjk
N eeWN
)1()2/)(/2(2
We divide X(k) in half – first half and second half
Since:
then
1,...,1,0 ,)2/()1()()(1)2/(
0
NkWNnxnxkXN
n
kn
N
k
![Page 96: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/96.jpg)
DIF-FFT (cont)
12
,...,1,0 ,)2/()()12(
12
,...,1,0 ,)2/()()2(
1)2/(
0
2/
1)2/(
0
2/
NkWWNnxnxkX
NkWNnxnxkX
N
n
kn
N
n
N
N
n
kn
N
2
2/
2
NN WW
Decompose X(k) into two frequency sequences, even and odd:
Since:
then,
12
,...,1,0 ,)2/()()12(
12
,...,1,0 ,)2/()()2(
1)2/(
0
)12(
1)2/(
0
2
NkWNnxnxkX
NkWNnxnxkX
N
n
nk
N
N
n
kn
N
![Page 97: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/97.jpg)
DIF-FFT butterfly
x[n]
x[n+N/2]
x[n] + x[n+N/2]
(x[n] - x[n+N/2])WNn
-1 WN
n
![Page 98: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/98.jpg)
DIF-FFT of length N = 4
-1
-1
-1
-1 W41
W40
x[0] + x[2]
x[1] + x[3]
(x[0] - x[2])W40
(x[1] - x[3])W41
x[0]
x[1]
x[2]
x[3]
X[0]
X[2]
X[1]
X[3]
![Page 99: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/99.jpg)
Lecture #23
DFT applications
1. Approximation of the Fourier Transform
of analog signals
2. Linear convolution
![Page 100: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/100.jpg)
Approximation of Fourier Transform
Using DFT as a discrete-frequency approximation of the DTFT
and also an approximation of the CTFT
Step1. Determine the resolution ΔΩ = 2π/N required for the DFT to
be useful for its intended purpose determine N (N often = 2n)
Step 2. Determine the sampling frequency required to sample the
analog signal so as to avoid aliasing ωs ≥ 2ωM
Step 3. Accumulate N samples of the analog signal over N.T
seconds (T = 2π/ωs)
Step 4. Calculate DFT directly or using FFT algorithm
![Page 101: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/101.jpg)
0 0.5 1 1.5 2 2.5 30
0.5
1
n
f[n]
0 1 2 3 4 50
1
2
3
omega
mag
nitu
de
Example
f(t) = rect[(t-1)/2]
F(ω) = 2sinc(ω)e-jω
Ts = 1s
%Error of sample 1 = 50%
![Page 102: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/102.jpg)
Example
0 5 10 150
0.5
1
n
f[n]
0 5 10 15 20 25 300
1
2
3
omega
mag
nitu
de
f(t) = rect[(t-1)/2]
F(ω) = 2sinc(ω)e-jω
Ts = 0.2s
%Error of sample 1 = 10%
![Page 103: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/103.jpg)
0 5 10 15 20 25 30 350
0.5
1
n
f[n]
0 10 20 30 40 50 60 700
1
2
3
omega
mag
nitu
de
Example
f(t) = rect[(t-1)/2]
F(ω) = 2sinc(ω)e-jω
Ts = 0.1s
%Error of sample 1 = 5%
![Page 104: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/104.jpg)
Windowing
Source of error: truncation or “windowing” of the periodic
extension of the DT sequence implied in the DFT development
Windowing = multiplying the periodic extension of x[n] by a
rectangular function with duration N.T DFT involves a convolution
DFT = sampled(DTFT*sinc)
This multiplication causes spectrum-leakage distortion
To reduce the affect of spectrum-leakage distortion:
- Increase the sampling frequency
- Increase the number of samples
- Choose an appropriate windowing function (Hamming, Hanning)
![Page 105: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/105.jpg)
Lecture #23
DFT applications
1. Approximation of the Fourier Transform of
analog signals
2. Linear convolution
![Page 106: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/106.jpg)
Recall linear convolution
N1: the non-zero length of x1(n); N2: the non-zero
length of x2(n); Ny = N1 + N2 -1
The shift operation is the regular shift
The flip operation is the regular flip
p
pnxpxnxnxny ][][][*][][ 2121
![Page 107: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/107.jpg)
Recall circular convolution
The non-zero length of x1(n) and x2(n) can be no longer
than N
The shift operation is circular shift
The flip operation is circular flip
1
0
2121 ][][][][][N
p
pnxpxnxnxny
![Page 108: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/108.jpg)
Calculation of the linear convolution
The circular convolution of 2 sequences of length N1 and N2
can be made equal to the linear convolution of 2 sequences
by zero padding both sequences so that they both consists
of N1+N2-1 samples.
x1(n)
N1 samples
x2(n)
N2 samples
x’1(n)
x’2(n)
Zero padding
(N2-1) zeros IDFT
x1(n)*x2(n) Zero padding
(N1-1) zeros
DFT
DFT
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Example of calculation of the linear convolution
x1(n) = [ 1 2 3 4 ]; x2(n) = [ 0 1 2 3 ]
x’1(n) = [ 1 2 3 4 0 0 0 ]; x’2(n) = [ 0 1 2 3 0 0 0 ]
X’1(k) = [ 10, -2.0245-j6.2240, 0.3460+j2.4791, 0.1784-j2.4220, 0.1784+j2.4220, 0.3460-j2.4791, -2.0245-j6.2240 ];
X’2(k) = [ 6, -2.5245-j4.0333, -0.1540+j2.2383, -0.3216-j1.7950, -0.3216+j1.7950, -0.1540-j2.2383, -2.5245+j4.0333];
Y’(k) = [ 60, -19.9928+j23.8775, -5.6024+j0.3927, -5.8342-j0.8644, -4.4049+j0.4585, -5.6024-j0.3927, -19.9928 +j23.8775 ]
IDFTY’(k) = y’(n) = [ 0 1 4 10 16 17 12 ]
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Lecture #24
Using DFT/FFT
![Page 111: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/111.jpg)
The response, y(n) of a filter with the impulse response, h(n)
to an input, x(n) can be calculated by using the DFT
Limitations: All sample values of the signal must be
accumulated before the process begins
Great memory
Long time delay
Not suitable for long duration input
Solution: block filtering
Filtering
![Page 112: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/112.jpg)
Divide long input signal, x(n) into non-overlapped blocks,
xb(n) of appropriate length for FFT calculation.
Convolve each block xb(n) and h(n) to get the output,
yb(n)
Overlap-add outputs, yb(n) together to form the output
signal, y(n)
Overlap-add technique in block filtering
![Page 113: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/113.jpg)
Example of overlap-add technique
h(n)
x(n)
![Page 114: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/114.jpg)
Example of overlap-add technique
x0(n)
x1(n)
x2(n)
![Page 115: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/115.jpg)
Example (cont)
y0(n)
y1(n)
y2(n)
![Page 116: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/116.jpg)
Prob.1 Prob. 5
12.10 12.18
Prob. 2 Prob. 6
12.11 12.22
Prob. 3 Prob. 7
12.12 12.24
Prob. 4 Prob. 8
12.15 12.30
HW
![Page 117: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/117.jpg)
Prob.9
Given signals below and their DFT-5
HW
]n[]n[s)c(
]1n[]n[x)b(
]4n[4]3n[3]2n[2]1n[]n[x)a(
2
1
1. Find y[n] so that Y[k] = X1[k].X2[k]
2. Does x3[n] exist, if S[k] = X1[k].X3[k]?
![Page 118: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/118.jpg)
Prob.10
Prove the Parseval’s theorem
HW
1
0
1
0
][*][1
][*][N
k
N
n
kYkXN
nynx
![Page 119: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/119.jpg)
Lecture #20
DFT applications
1. Approximation of the Fourier Transform
of analog signals
2. Convolution
3. Filtering
4. Correlation
5. ESD estimation
![Page 120: Chapter 5_dsp1_co thuc_full, dsp1, co thuc, xu li tin hieu, bkdn, dsp](https://reader034.vdocuments.us/reader034/viewer/2022052310/545092e8b1af9f09098b4da3/html5/thumbnails/120.jpg)
Windowing
0 10 20 30 40 50 60 700
0.5
1
n
f[n]
0 5 10 15 20 25 30 350
1
2
3
omega
mag
nitu
de