the fourier transform - web.sonoma.edu · es 442 fourier transform 3 review: fourier trignometric...
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EE 442 Fourier Transform 1
The Fourier TransformEE 442 Analog & Digital Communication Systems
Lecture 4
Voice signal
time
frequency (Hz)
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ES 442 Fourier Transform 2
Jean Joseph Baptiste Fourier
March 21, 1768 to May 16, 1830
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ES 442 Fourier Transform 3
Review: Fourier Trignometric Series (for Periodic Waveforms)
Agbo & Sadiku;Section 2.5pp. 26-27
( )0 0 0
n 1
0 0
0
0
0
0
Equation (2.10) should read (time was missing in book):
( ) cos( ) sin( )
2 1where and
and (Equations 2.12a, b, & c)
1( ) (DC term)
2( )cos( ) for 1, 2, 3,
n n
T
T
n
t
e
t
f t a a n b n
fT T
a f t dtT
a f t n
t
n t dtT
=
= + +
= =
=
= =
0
0
.
2( )sin( ) for 1, 2, 3, .
T
n
tc
b f t n t dt n etcT
= =
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ES 442 Fourier Transform 4
Fourier Trigonometric Series in Amplitude-Phase Format
Agbo & Sadiku;Section 2.5
Page 27
( )0 0
n 1
0 0
2 2
1
Equations (2.13) and (2.14) should read:
( ) cos( )
tan
Also known as polar form of Fourier series.
n n
n n n
nn
n
f t A A n
a A
A a b and
b
a
t
=
−
= + +
=
= +
= −
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EE 442 Fourier Transform 5
Example: Periodic Square Wave as Sum of Sinusoids
Line Spectra
3f0
f0
5f0
7f0
Even or Odd?
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EE 442 Fourier Transform 6
Example: Periodic Square Wave (continued)
http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx
1 1 13 5 7
4( ) sin( ) sin(3 ) sin(5 ) sin(7 )f t t t t t
= + + + +
Fundamental only
Five terms
Eleven terms
Forty-nine terms
t
This is an odd function Question:What would make this an
even function?
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EE 442 Fourier Transform 7
Sinusoidal Waveforms are the Building Blocks in the Fourier Series
Simple Harmonic Motion Produces Sinusoidal Waveforms
Sheet of paper unrolls in this direction
FuturePast
Time t
MechanicalOscillation
ElectricalLC Circuit
Oscillation
LC Tank Circuit
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EE 442 Fourier Transform 8
Visualizing a Signal – Time Domain & Frequency Domain
Source: Agilent Technologies Application Note 150, “Spectrum Analyzer Basics”http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf
Amplitude
To go from time domainto frequency domain
we use Fourier Transform
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EE 442 Fourier Transform 9
Example: Where Both Sine & Cosine Terms are Required
http://www.peterstone.name/Maplepgs/fourier.html#anchor2315207
Period T0
Note phase shift in the fundamental frequency sine waveform.
Both even and odd parts to the waveform.
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EE 442 Fourier Transform 10
Fourier Series
DiscreteFourier
Transform
Fourier Transform
DiscreteFourier
Transform
Continuous time Discrete time
Periodic
Aperiodic
Fourier series for continuous-time periodic signals → discrete spectraFourier transform for continuous aperiodic signals → continuous spectra
Fourier Series versus Fourier Transform
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ES 442 Fourier Transform 11
Definition of Fourier Transform
1
The Fourier transform ( . ., spectrum) of is ( ):
( ) ( ) ( )
1( ) ( ) ( )
2
Therefore, ( ) ( ) is a Fourier Transform pair
j t
j t
i e f(t) F
F F f t f t e dt
f t F F F e d
f t F
−
−
−
−
= =
= =
Agbo & Sadiku;Section 2.7;
pp. 40-41
Note: Remember = 2 f
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EE 442 Fourier Transform 12
Fourier Transform Produces a Continuous Spectrum
{ f(t)} gives a spectra consisting of a continuous sum of
exponentials with frequencies ranging from - to + .
where |F()| is the continuous amplitude spectrum of f(t)and
() is the continuous phase spectrum of f(t).
,)()( )( jeFF =
Often only the magnitude of F() is displayed and the phaseis ignored.
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ES 442 Fourier Transform 13
Example: Impulse Function (t)
0
0( ) ( ) ( ) 1j t j t j
tF F t t e dt e e
− −
=−
= = = = =
0if0
0if
=
t
t=)(t
( )
1
F t
)(21
1)(
t
Delta function has unity area.
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EE 442 Fourier Transform14
Example: Fourier Transform of Single Rectangular Pulse
1
time t2
2
−2
0
Remember = 2f
1 for2 2
0 for all 2
t
t
−
( ) === )/(IIrect)( tttf
( ) )/(IIrect)( tttf ==
( ) ( )( )
=
−
−=
−
−=
−=
==
−
−
−
−
−
−
−
2
2sin2sin2
)()(
2/2/2/
2/
2/
2/
j
j
j
ee
j
e
dtedtetfF
jjtj
tjtj
Pulse ofwidth
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ES 442 Fourier Transform 15
( )( )
( )
fF sinc
2
2sin)( =
=
Fourier Transform of Single Rectangular Pulse (continued)
F()
0
3−
2−
2
3
−
1
time t2
2
−2
0
sinc function
Note the pulse is
time centered
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EE 442 Fourier Transform 16
Properties of the Sinc Function
Definition of the sinc function:
Sinc Properties:1. sinc(x) is an even function of x.
2. sinc(x) = 0 at points where sin(x) = 0, that is, sinc(x) = 0 when x = , 2, 3, … .
3. Using L’Hôpital’s rule, it can be shown that sinc(0) = 1.4. sinc(x) oscillates as sin(x) oscillates and monotonically
decreases as 1/ x decreases as | x | increases.5. sinc(x) is the Fourier transform of a single rectangular pulse.
sin( )sinc( )
xx
x=
= =
sin( ) sin( )sinc( ) sinc( )and
x xx x
x x
Warning:There are two definitions for the sinc(x) function. They are
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ES 442 Fourier Transform 17
Periodic Pulse Train Morphing Into a Single Pulse
https://www.quora.com/What-is-the-exact-difference-between-continuous-fourier-transform-discrete-Time-Fourier-Transform-DTFT-Discrete-Fourier-Transform-DFT-Fourier-series-and-Discrete-Fourier-Series-DFS-In-which-cases-is-which-one-used
Frequency resolutioninversely proportionalto the period.Ck
Ck
Ck
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EE 442 Fourier Transform 18
Sinc Function Tradeoff: Pulse Duration versus Bandwidth
1( )g t
t t
2 ( )g t
3( )g t
t
1( )G f
2 ( )G f
3( )G f
f
f
f
1
2
T1
2
T− 2
2
T2
2
T−
3
2
T3
2
T−
2T
1T
3T
1
1
T1
1
T−
2
1
T2
1
T
−
3
1
T3
1
T−
T1 > T2 > T3
Also called the
Time ScalingProperty
(Section 2.8.2)
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EE 442 Fourier Transform 19
Properties of Fourier Transforms
2.8.1 Linearity (Superposition) Property2.8.2 Time-Scaling Property2.8.3 Time-Shifting Property2.8.4 Frequency-Shifting Property2.8.5 Time Differentiation Property2.8.6 Frequency Differentiation Property2.8.7 Time Integration Property2.8.8 Time-Frequency Duality Property2.8.9 Convolution Property
Agbo & SadikuSection 2.8;pp. 46 to 58
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EE 442 Fourier Transform 20
2.8.1 Linearity (Superposition) Property
Given f(t) F() and g(t) G() ;
Then f(t) + g(t) F() + G() (additivity)
also kf(t) kF() and mg(t) mG() (homogeneity)Note: k and m are constants
Combining these we have,
kf(t) + mg(t) kF () + mG ()
Hence, the Fourier Transform is a linear transformation.
This is the same definition for linearity as used in your circuits and systems course, EE 400.
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ES 442 Fourier Transform 21
2.8.2 Time Scaling Property
1
( )f at Fa a
=
*
( ) ( )
Let & ,
1( ) ( )
Hence, ( ) ( ) ( )
j t
j t
f at f at e dt
at d adt
df at f e F
a a a
f - t F F
−
−
−
−
=
= =
= =
= − =
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EE 442 Fourier Transform 22
Time-Scaling Property (continued)
Time compression of a signal results in spectral expansion andtime expansion of a signal results in spectral compression.
1
( )f at Fa a
=
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EE 442 Fourier Transform 23
2.8.3 Time Shifting Property
( )0
0( )j t
f t t e F −
− =
( )
0
9 0
0 0
0 0
( )
0
( ) ( )
Let , &
( ) ( )
( )
j t
j t
j t j tj
f t t f t t e dt
t t d dt t t
f t t f e d
e f e d e F
−
−
− +
−
− −−
−
− = −
= − = = +
− = =
= =
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EE 442 Fourier Transform 24
Time-Shifting Property (continued)
t
t
This time shifted pulseis both even and odd.
Even function. Both must be
identical.
)(F)(tf
A time shiftproduces a phase
shift in itsspectrum.
Delaying a signal by t0 seconds does not change its amplitudespectrum, but the phase spectrum is changed by -2ft0.
Note that the phase spectrum shift changes linearly with frequency f.
( )0
0( )j t
f t t e F −
− =
( )( ) ( )( )2 2
( ) Re ImF F F = +
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ES 442 Fourier Transform 25
2.8.4 Frequency Shifting Property
( )0
0( )j t
f t e F = −
( )
( ) ( )
( ) ( )
0 0
0
0 0
( )
0
10 2
10 0 02
( ) ( )
( )
Special application:
Apply to cos ;
( )cos ( ) ( )
j t j t j t
j t
j t j t
f t e f t e e dt
f t e d F
t e e
f t t F F
−
−
− −
−
−
=
= = −
= +
= − + +
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ES 442 Fourier Transform 26
An Important Formula to Remember in EE 442
exp cos( ) sin( )j j =
( )
( )
1sin( ) exp exp
2
1cos( ) exp exp
2
j jj
j j
= − −
= + −
( )
22 1
/2 1 for
1 for
, tan
j jn
i
n evenj and
n odd
ba jb r a b
a
e e
e where r
−
= =
−
+ = + =
=jrejba =+
Euler's formula
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ES 442 Fourier Transform 27
Frequency Shifting Property is Very Useful in Communications
( )g t
( ) ( )g t G f
( )G f
fBB−
fCf
( )g t ( )cos(2 )Cg t f t
Cf−
2A
A
t
t
( )g t2B
Multiplication of a signal g(t) by the factor [cos(2fCt)]places G(f) centered at f = fC.
Carrier frequency is fc &g(t) is the message signal
USBLSB
USBLSB
2B
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EE 442 Fourier Transform 28
Frequency-Shifting Property (continued)
Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.
Note phase shift
cos(2 )( ) Ct f tg
sin(2 )( ) Ct f tg
( )g t( )G f
( )G f
( )g f
2
2−
( )a ( )b
( )d( )c
( )e ( )f
cos(2 )Cf t
Message signal
fC-fC
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29EE 442 Fourier Transform
Modulation Comes From Frequency Shifting Property
( )
( )0
0
Given FT pair: ( )
then, ( )j t
f t F
f t e F
−
Amplitude Modulation Example:
Audio tone:
sin(t)
Sinusoidal carrier signal:
Amplitude Modulated
Signal
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ES 442 Fourier Transform 30
Fourier Transform of AM Tone Modulated Signal
Only positive frequencies shown;Must include negative frequencies.
f(t)
F()Carrier signal
fC = 500 Hz
Modulated AM sidebands
http://www.ni.com/tutorial/5421/en/
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ES 442 Fourier Transform 31
Modulation of Baseband and Carrier Signals
fC
= f0
https://slideplayer.com/slide/10939951/
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EE 442 Fourier Transform 32
Transform Duality Property
Given ( ) ( ), then
( ) ( )
and
( ) ( )
g t G f
g t G f
G t g f
−
Note the minus sign!
Because of the minus sign they are not perfectly symmetrical – See the illustration on next slide.
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EE 442 Fourier Transform 33
Illustration of Fourier Transform Duality
2
2
−
1( )G f
2 ( )G f
1( )g t
2 ( )g t
t f
ft
1T
1
1
T−
What doesthis imply?
2
−
2
1−
1
2
2−
3
1
1−
2
4
4
−
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EE 442 Fourier Transform 34
Fourier Transform of Complex Exponentials
1
1
1
1
2
2 2
2
2
2 2
2
( ) ( )
Evaluate for
( )
( )
( ) ( )
Evaluate for
( )
( )
c
c
c C
c
c
f f
c
c C
c
c
f f
c
c c
c
c c
j f t
j f t j f t
j f t
j f t
j f t j f t
j f
F f f f f df
f f
F f f df
f f and
F f f f f df
f f
F f f df
f f
e
e e
e
e
e e
e
−
−
−
=
−
−
−
=−
−
− −
−
−
− = −
=
− = =
−
+ = +
= −
+ = =
+
ct
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EE 442 Fourier Transform 35
Fourier Transform of Sinusoidal Functions
( )
−
−
− + −
= +
2 2
2 212
!Taking ( ) and ( )
! !
We use these results to find of cos(2 ) and sin(2 )
Using the identities for cos(2 ) and sin(2 ),
cos(2 ) & cos(2 )
c cc c
c c
j f t j f t
j f t j f t
nf f f f
r n r
FT ft ft
ft ft
ft ft
e e
e e
− = −
+ + −
+ − −
2 212
1212
Therefore,
cos(2 ) ( ) ( ) , and
sin(2 ) ( ) ( )
c c
c c
c cj f t j f t
j
j
ft f f f f
ft f f f f
e e
ff fcfc
-fc
-fc
sin(2fct)cos(2fct)
*
ImRe
sin
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EE 442 Fourier Transform 36
Summary of Several Fourier
Transform Pairs See
Agbo & Sidiku;Table 2.5,Page 54;------------
See also theFourier
TransformPair Handout
http://media.cheggcdn.com/media/db0/db0fffe9-45f5-40a3-a05b-12a179139400/phpsu63he.png
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EE 442 Fourier Transform 37
Spectrum Analyzer Shows Frequency Domain
A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. It measures frequency, power, harmonics, distortion, noise, spurious signals and bandwidth.
➢ It is an electronic receiver➢Measure magnitude of signals➢ Does not measure phase of signals➢ Complements time domain
Bluetooth Spectrum
Courtesy: Keysight Technologies
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EE 442 Fourier Transform 38
Re
f
fc
-fc
2
A
2
A
cos(2 )A f t
Blue arrows indicatepositive phase directions
Fourier Transform of Cosine Signal
cos(2 ) ( ) ( )2
c c c
AA f t f f f f = + + −
3D View
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ES 442 Fourier Transform 39
Fourier Transform of Cosine Signal (as shown in textbooks)
t
0f
FT
f
Real axis
0( )f −cos( )t
0f−
0T− 0T
0
0
1f
T=
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EE 442 Fourier Transform 40
Re
f
fc
-fc
2
B
2
B
sin(2 )B f t
Fourier Transform of Sine Signal
sin(2 ) ( ) ( )2
c c c
BB f t j f f f f = + − −
We must subtract 90from cos(x) to get sin(x)
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EE 442 Fourier Transform 41
Fourier Transform of Sine Signal (as usually shown in textbooks)
t
0( )f −
0f0f−
2
Bj
−2
Bj
FT
Imaginary axis
f
B
Bsin(0t)
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ES 442 Fourier Transform 42
Visualizing Fourier Spectrum of Sinusoidal Signals
Multiplying byj is a phase
shift
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EE 442 Fourier Transform 43
Re
f
fc
-fc
2
B
2
B
2
A
2e j ftR
2
A
2e j ftR
−
-
− − +
2 2e e e ej j ft j j f tR R
2 2
1tan2 2
A B AR and
B −
= + = −
Fourier Transform of a Phase Shifted Sinusoidal Signal(with phase information shown)
http://www.ece.iit.edu/~biitcomm/research/references/Other/Tutorials%20in%20Communications%20Engineering/Tutorial%207%20-%20Hilbert%20Transform%20and%20the%20Complex%20Envelope.pdf
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EE 442 Fourier Transform 44
Selected References
1. Paul J. Nahin, The Science of Radio, 2nd edition, Springer, New York, 2001. A novel presentation of radio and the engineering behind it; it has some selected historical discussions that are very insightful.
2. Keysight Technologies, Application Note 243, The Fundamentals of Signal Analysis; http://literature.cdn.keysight.com/litweb/pdf/5952-8898E.pdf?id=1000000205:epsg:apn
3. Agilent Technologies, Application Note 150, Spectrum Analyzer Basics; http://cp.literature.agilent.com/litweb/pdf/5952-0292.pdf
4. Ronald Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill Book Company, New York, 1999. I think this is the best book covering the Fourier Transform (Bracewell gives many insightful views and discussions on the FT and it is considered a classic textbook).
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ES 442 Fourier Transform 45
Auxiliary Slides For Introducing Sampling
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EE 442 Fourier Transform 46
Fourier Transform of Impulse Train (t) (Shah Function)
0f 02 f0f−02 f− 0 f0T 02T0T−02T− 0 t
0Period T=0
1Period
T=
Ш(t) = 0 0( ) ( )n n
t nT t nT
=− =−
− = +
12
12
( ) 1
n
n
t dt
+
−
= Ш
Shah function (Ш(t)):
aka “Dirac Comb Function,” Shah Function & “Sampling Function”
Ш(t)
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EE 442 Fourier Transform 47
Shah Function (Impulse Train) Applications
0( ) ( ) ( )n
f t f n t nT
=−
= −Ш(t)
0( ) ( )n
f t f t nT
=−
= −Ш(t)
The sampling property is given by
The “replicating property” is given by the convolution operation:
Convolution
Convolution theorem:
1 2 1 2( ) ( ) ( ) ( )g t g t G f G f
1 2 1 2( ) ( ) ( ) ( )g t g t G f G f
and
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EE 442 Fourier Transform 48
Sampling Function in Operation
0( ) ( ) ( )n
f t f n t nT
=−
= −Ш(t)
0( )t nT −
( )f t
(0)f0( ) (1)f T f=
0(2 ) (2)f T f=
0T
t
t
t
0T
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ES 442 Fourier Transform 49
Speech, Trumpet and Street Traffic Signals
https://www.mdpi.com/2076-3417/6/5/143/htm
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ES 442 Fourier Transform 50
Building Expression for Sin(x) With Infinite Series
https://betterexplained.com/wp-content/uploads/sine/