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EE 442 Fourier Transform 1
The Fourier Transform
EE 442 Analog & Digital Communication Systems Spring 2017
Lecture 4
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Review: Fourier Trignometric Series (for Periodic Waveforms)
2 EE 442 Fourier Transform
1 1 0
0 0 01
0 0
0
0
is defined in time interval of
( ) cos( 2 ) sin( 2 )
1where ( ) is of period , and .
We can calculate the coefficients from th
Periodic function g(
e equations:
t
2
)
n nn
n
t t t T
g t a a n f t b n f t
g t T fT
a gT
1 0
1
1 0
1
0
0
0
( ) cos( 2 ) for n = 1, 2, 3, 4,
2( ) sin( 2 ) for n = 1, 2, 3, 4,
t T
t
t T
n
t
t n f t dt
b g t n f t dtT
t
g(t)
T0
Lathi & Ding; pp. 50-61
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3 EE 442 Fourier Transform
Review: Exponential Fourier Series (for Periodic Functions)
1 1 0
0 0
0
02
Again, is defined in time interval
( ) for 0, 1, 2, 3,
1where again ( ) is of period , and .
We can calculate the coefficients fr
periodic function
om the
g(t
eq i
)
uat o
nn
n
jn f t
t t t T
g t D n
g t T fT
D
e
00
02
n:
1( ) for ranging from - to .n
T
jn f tD g t dt n
Te
But, periodicity is too limited for our needs, so we will need to proceed on to the Fourier Transform.
The reason is that signals used in communication systems are not periodic (it is said they are aperiodic).
Lathi & Ding; pp. 50-61
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EE 442 Fourier Transform 4
Sinusoidal Waveforms are the Building Blocks in the Fourier Series
Simple Harmonic Motion Produces Sinusoidal Waveforms
Sheet of paper unrolls in this direction
Future Past
Time t
Mechanical Oscillation
Electrical LC Circuit
Oscillation
LC Tank Circuit
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EE 442 Fourier Transform 5
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 domain to frequency domain:
We use Fourier Transform
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EE 442 Fourier Transform 6
Example: Periodic Square Wave
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
Odd function
Question: What would make this an
even function?
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EE 442 Fourier Transform 7
Example: Periodic Square Wave (continued)
Line Spectra
3f0
f0
5f0
7f0
Even or Odd?
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EE 442 Fourier Transform 8
https://en.wikipedia.org/wiki/Fourier_series
1 1 13 5 7
4( ) sin( ) sin(3 ) sin(5 ) sin(7 )f t t t t t
t
Square Wave From Fundamental + 3rd + 5th & 7th Harmonics
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EE 442 Fourier Transform 9
Another Example: Both Sine & Cosine Functions Required
http://www.peterstone.name/Maplepgs/fourier.html#anchor2315207
Period T0
Note phase shift in the fundamental frequency sine waveform.
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EE 442 Fourier Transform 10
Gibb’s Phenomena at Discontinuities in Waveforms
About 9% overshoot at discontinuities of waveform. This is an artifact of the Fourier series representation.
time t
Am
plit
ud
e
The Gibbs phenomenon is an overshoot (or "ringing") of Fourier series occurring at simple discontinuities.
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EE 442 Fourier Transform 11
Fourier Series
Discrete Fourier
Transform
Continuous Fourier
Transform
Discrete Fourier
Transform
Continuous time Discrete time
Periodic
Aperiodic
Fourier series for continuous-time periodic signals → discrete spectra Fourier transform for continuous aperiodic signals → continuous spectra
Fourier Series versus Fourier Transform
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EE 442 Fourier Transform 12
Definition of Fourier Transform
2( )
j ftg t dte
2( )
j ftG f dfe
( )g t ( )G f
Time-frequency duality: ( ) ( ) ( ) ( )g t G f and G t g f
We say “near symmetry” because the signs in the exponentials are different between the Fourier transform and the inverse Fourier transform.
Figure 3.15 Lathi & Ding Cyclic relation:
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EE 442 Fourier Transform 13
Fourier Transform Produces a Continuous Spectrum
FT [g(t)] gives a spectra consisting of a continuous sum of exponentials with frequencies from - to + .
( ) ( ) exp ( )G f G f j f
where |G(f)| is the continuous amplitude spectrum of g(t) and (f) is the continuous phase spectrum of g(t).
Supplemental Reading: Lathi & Ding;
pp. 97-99
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EE 442 Fourier Transform 14
Example: Rectangular Pulse
1
G(f)
f time t
2
1
2
3
1
2
3
2
2
0 0
( ) ( ) ( / )g t rect t t
( ) ( ) ( / )g t rect t t 1 for
2 2
0 for all 2
t
t
sinc(f )
Remember = 2f
Lathi & Ding; pp. 101-102 Example 3.2
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EE 442 Fourier Transform 15
Two Definitions of the Sinc Function
sin( )sinc ( )
and
sin( )sinc ( )
xx
x
xx
x
Ref.: Lathi and Ding, page 100; uses the definition sinc (x) = sin( )x
x
Sometimes sinc (x)
is written as Sa (x).
(1)
(2)
x
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EE 442 Fourier Transform 16
Definition of the Sinc Function
Unfortunately, there are two definitions of the sinc function in use.
Format 1 (Lathi and Ding, 4th edition – See pp. 100 – 102) Format 2 (as used in many other textbooks)
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 with increasing |x| 5. sinc(x) is the Fourier transform of a single rectangular pulse
sin( )sinc( )
xx
x
sin( )sinc( )
xx
x
We will use this form
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EE 442 Fourier Transform 17
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
Lathi & Ding; pp. 110-111
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EE 442 Fourier Transform 18
Sinc Function Appears in Both Pulse Train & a Single Pulse
Pulse Train – Fourier Series Single pulse – Fourier Transform
Continuous Spectrum
Discrete Spectrum
What does this imply for a digital binary communication signal?
sin( )
( )
ftF f
ft
2 21( ) ( ) ( )f t u t u t
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EE 442 Fourier Transform 19
Some Insight into the Sinc Function
Lathi & Ding; pp. 100-101
sin( )sinc( )
xx
x 1
x
1
x
sinc( )x
x
sin( )x
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EE 442 Fourier Transform 20
Properties of Fourier Transforms
1. Linearity (Superposition) Property 2. Time-Frequency Duality Property 3. Transform Duality Property 4. Time-Scaling Property 5. Time-Shifting Property 6. Frequency-Shifting Property 7. Time Differentiation & Time Integration Property 8. Area Under g(t) Property 9. Area Under G(f) Property 10.Conjugate Functions Property
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EE 442 Fourier Transform 21
Linearity (Superposition) Property
Given g1(t) G1(f) and g2(t) G2(f) ; Then g1(t) + g2(t) G1(f) + G2(f) (additivity) also kg1(t) kG1(f) and kg2(t) kG2(f) (homogeneity) Combining these we have,
kg1(t) + mg2(t) kG1(f) + mG2(f) Hence, the Fourier Transform is a linear transformation.
This is the same definition for linearity as used in your circuits and systems EE400 course.
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EE 442 Fourier Transform 22
Time-Frequency Duality Property
Lathi & Ding; pp. 106-109
0
0
0
0
2
2
Given ( ) ( ), then
( - ) ( )
and
g(t) ( )
j ft
j ft
g t G f
g t t G f
G f f
e
e
This leads directly to the transform Duality Property
2( )
j ftg t dte
2( )
j ftG f dfe
( )g t ( )G f
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EE 442 Fourier Transform 23
Transform Duality Property
Lathi & Ding pp. 108-109
Given ( ) ( ), then
( ) ( )
and
( ) ( )
g t G f
g t G f
G t g f
See illustration on next page for anexample!
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EE 442 Fourier Transform 24
Illustration of Fourier Transform Duality
1( )G f
2( )G f
1( )g t
2( )g t
1
2
T1
2
T t
1
1
T1
1
T
f
ft
1T
1
1
T1
1
T
1T
1
2
T1
2
T
1
2
T1
2
T
1
1
1f
T
Lathi & Ding; Page 109
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EE 442 Fourier Transform 25
Time-Scaling Property
Given ( ) ( ),
then for a real constant a,
1( ) ( )
g t G f
fg at G
a a
Time compression of a signal results in spectral expansion and time expansion of a signal results in spectral compression.
Lathi & Ding; pp. 110-112
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EE 442 Fourier Transform 26
Time-Shifting Property
002
Given ( ) ( ),
then ( ) ( )j ft
g t G f
g t t G f e
Delaying a signal by t0 seconds does not change its amplitude spectrum, but the phase spectrum is changed by -2ft0.
Note that the phase spectrum shift changes linearly with frequency f.
Frequency f
Frequency 2f
=
= 2
Lathi & Ding; pp. 112-113
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27
( )g t
( ) ( )g t G f
( )G f
fBB
fCf
( )g t ( )cos(2 )Cg t f
Cf
2A
A
t
t
( )g t2B
Lathi & Ding; pp. 114-119
Frequency-Shifting Property
0
0
2
Given ( ) ( ),
then ( ) ( )j f t
g t G f
g t G f fe
aka Modulation Property
Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.
cos(2 )Cf
EE 442 Fourier Transform
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EE 442 Fourier Transform 28
Frequency-Shifting Property (continued)
Note phase shift
0( )cos(2 )g t f
0( )sin(2 )g t f
( )g t( )G f
( )G f
( )g f
2
2
a b
d c
e f
Lathi & Ding; Figure 3.21 Page 115
Multiplication of a signal g(t) by the factor places G(f) centered at f = fC.
0cos(2 )f
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EE 442 Fourier Transform 29
Time Differentiation & Time Integration Property
Lathi & Ding; pp. 121-123
12
Given ( ) ( )
For time differentiation:
( )2 ( )
( )and 2 ( )
For time integration:
( )( ) (0) ( )
2
nn
n
t
g t G f
dg tj f G f
dt
d g tj f G f
dt
G fg d G f
j f
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EE 442 Fourier Transform 30
Area Under g(t) & Area Under G(f) Properties
Given ( ) ( ),
Then ( ) (0)
g t G f
g t dt G
That is, the area under a function g(t) is equal to the value of its Fourier transform G(f) at f = 0.
Given ( ) ( ),
Then (0) ( )
g t G f
g G f df
That is, the value of a function g(t) at t = 0 is equal to the area under its Fourier transform G(f).
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EE 442 Fourier Transform 31
Conjugate Functions Property
Given ( ) ( ),
Then for a complex-valued time function g(t),
we have
* ( ) * ( )
where the star symbol (*) denotes the complex
conjugate operation.
The corollary to this property is
* ( ) * ( )
g t G f
g t G f
g t G f
--- End of Fourier Transform Property Slides ---
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EE 442 Fourier Transform 32
Some Important Fourier Transform Pairs
Impulse Function Impulse train Unit Rectangle Pulse Unit Triangle Pulse Exponential Pulse Signum Function
Refer to Handout
on FT Pairs
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EE 442 Fourier Transform 33
The Impulse Function (t) (Impulse Function)
( ) 0 for all t 0t
( ) 1t dt
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EE 442 Fourier Transform 34
Fourier Transform of the Impulse Function (t)
2 2 (0)( ) ( ) 1
because the exponential function to zero power is unity.
Therefor, we can write (t) 1
j ft j fF t t dte e
t f
g(t) =(t) G(f) =1 1
(t) is often called the Dirac delta function
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EE 442 Fourier Transform 35
Inverse Fourier Transform of the Impulse Function (t)
1 2 (0)( ) ( ) 1
because the exponential function to zero power is unity.
Therefor, we can write 1 (t)
j ft j tF f f dfe e
This is just a DC signal. A DC signal is zero frequency.
t f
g(t) G(f) = (f) 1
0
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EE 442 Fourier Transform 36
Fourier Transform of Impulse Train (t) (Shah Function)
0f 02 f0f02 f 0 f0T 02T0T02T 0 t
0Period T0
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 37
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 38
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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EE 442 Fourier Transform 39
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
Lathi & Ding; pp. 104
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EE 442 Fourier Transform 40
Fourier Transform of Sinusoidal Functions
2 2
2 2 21 12 2
Taking ( ) and ( )
We use these results to find FT 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 jj
f f f f
ft ft
ft ft
ft ft
e e
e e e
2
1212
Therefore,
cos(2 ) ( ) ( ) , and
sin(2 ) ( ) ( )
c c
c c
c cf t j f t
j
ft f f f f
ft f f f f
e
f f fc fc
-fc
-fc
sin(2fct) cos(2fct)
Lathi & Ding; pp. 105
*
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EE 442 Fourier Transform 41
Fourier Transform of Signum (Sign) Function
1, 0
sgn( ) 0, 0
1, 0
for t
t for t
for t
t
+1
-1
Sgn(t)
We approximate the signum function using
exp( ), 0
( ) 0, 0
exp( ), 0
at for t
g t for t
at for t
t
+1
-1
2 2
2 20
4 1( ) , because
(2 )
4 1lim
(2 )a
j fG f
a f j f
j f j
a f f j f
Lathi & Ding; pp. 105-106
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EE 442 Fourier Transform 42
Summary of Several Fourier Transform Pairs
Lathi & Ding; Table 3.1 Page 107;
For a more complete Table of Fourier
Transforms ------------
See also the Fourier
Transform Pair Handout
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EE 442 Fourier Transform 43
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
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EE 442 Fourier Transform 44
Re
f
fc
-fc
2
A
2
A
cos(2 )A f t
Blue arrows indicate positive phase directions
Fourier Transform of Cosine Signal
cos(2 ) ( ) ( )2
c c c
AA f t f f f f
3D View Not in
Lathi & Ding
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ES 442 Fourier Transform 45
Fourier Transform of Cosine Signal (as usually 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 46
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 90 from cos(x) to get sin(x)
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EE 442 Fourier Transform 47
Fourier Transform of Sine Signal (as usually shown in textbooks)
sin( )t
t
0( )f
0f0f
2
Bj
2
Bj
FT
Imaginary axis
f
B
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EE 442 Fourier Transform 48
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 ftR R
2 2
1tan2 2
A B AR and
B
Fourier Transform of a Phase Shifted Sinusoidal Signal (with phase information shown)
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EE 442 Fourier Transform 49
Even, Odd Relationships for Fourier Transform (1)
Re Re
Re Re
g(t)
g(t)
G(f)
G(f)
Im Im
Im Im
Real; Even Real; Even
Real; Odd Imaginary; Odd
f
f
t
t
From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.
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EE 442 Fourier Transform 50
Re Re
Re Re
g(t)
g(t)
G(f)
G(f)
Im Im
Im Im
Imaginary; Even Imaginary; Even
Imaginary; Odd Real; Odd
f
f
t
t
Even, Odd Relationships for Fourier Transform (2)
From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.
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EE 442 Fourier Transform 51
Effect of the Position of a Pulse on Fourier Transform
2 2
( ) Re( ( )) Im( ( ))G f G f G f
t
t
f
f
This time shifted pulse is both even and odd.
Even function.
Both must be
identical.
( )G f
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EE 442 Fourier Transform 52
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 interesting.
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 text).
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ES 442 Fourier Transform 53
Jean Joseph Baptiste Fourier
March 21, 1768 to May 16, 1830
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ES 442 Fourier Transform 54