The Fourier Transform
ES 442 Analog & Digital Communication Systems
Spring 2016
Lecture 3
ES 442 Fourier Transform
1
Review: Fourier Trignometric Series (for Periodic Waveforms)
Periodic function g(t) is defined in time interval of t1  t  t1  T0

g(t )  a0    an cos(n2 f 0t )  bn sin(n2 f 0t )
n1
1
where g(t ) is of period T0 , and f 0  .
T0
We can calculate the coefficients from the equations:
2
an 
T0
t1 T0
2
bn 
T0
t1 T0

Lathi & Ding
pp. 50-61
g(t )  cos(n2 f 0t )dt for n = 1, 2, 3, 4,
t1

t1
g(t )  sin(n2 f 0t )dt for n = 1, 2, 3, 4,
g(t)
T0
ES 442 Fourier Transform
t
2
Review: Exponential Fourier Series (for Periodic Functions)
Again, periodic function g(t) is defined in time interval t1  t  t1  T0
g(t ) 


n 
Dn e
jn 2 f 0t
for n  0, 1, 2, 3,
where again g(t ) is of period T0 , and
f0 
1
.
T0
Lathi & Ding
pp. 50-61
We can calculate the coefficients Dn  from the equation:
Dn 
1
jn 2 f 0t
g
(
t
)

e
dt for n ranging from - to .

T0 T0
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).
ES 442 Fourier Transform
3
Review: Time Domain and Frequency Domain
The Fourier series expresses any time-periodic function in terms of
sinusoidal functions (only need to specify the frequency, amplitude and
phase of each sinusoid). When you think about that it is remarkable that
any periodic function can be simply represented by a sum of sinusoidal
functions.
In other words, the Fourier series of a time-periodic function generates a
frequency spectrum of harmonically-related sine and cosine waveforms.
“Every signal has a spectrum and is also determined by its spectrum. You
can analyze the signal either in the time domain or in the frequency
domain.”
-- Professor Brad Osgood (Stanford University)
This leads to time duration versus frequency bandwidth duality in signals.
ES 442 Fourier Transform
4
Sinusoidal Waveforms are the Building Blocks in the Fourier Series
Harmonic Motion Produces Sinusoidal Waveforms
Past
Future
Mechanical
Oscillation
Time t
Sheet of paper unrolls in this direction
LC Tank Circuit
ES 442 Fourier Transform
Electrical
LC Circuit
Oscillation
5
Visualizing Time Domain & Frequency Domain of a Signal
Amplitude
Source: Agilent Technologies Application Note 150, Spectrum Analyzer Basics
ES 442 Fourier Transform
6
Example: Periodic Square Wave
Odd function
f (t ) 
4
sin( t )  13 sin(3 t )  15 sin(5 t )  71 sin(7 t ) 


Fundamental only
Five terms
Eleven terms
Forty-nine terms
t
http://ceng.gazi.edu.tr/dsp/fourier_series/description.aspx
ES 442 Fourier Transform
7
Example: Periodic Square Wave (continued)
ES 442 Fourier Transform
8
Square Wave From Fundamental + 3rd + 5th & 7th Harmonics
f (t ) 
4
sin( t )  13 sin(3 t )  15 sin(5 t )  71 sin(7 t ) 


  t
https://en.wikipedia.org/wiki/Fourier_series
ES 442 Fourier Transform
9
Another Example: Both Sine & Cosine Functions Required
Period T0
Note phase shift in the fundamental frequency sine waveform.
http://www.peterstone.name/Maplepgs/fourier.html#anchor2315207
ES 442 Fourier Transform
10
Gibb’s Phenomena at Discontinuities in Waveforms
Amplitude
The Gibbs phenomenon is an overshoot (or "ringing") of Fourier series occurring
at simple discontinuities.
time t
About 9% overshoot at discontinuities of waveform.
This is an artifact of the Fourier series representation.
ES 442 Fourier Transform
11
Fourier Series versus Fourier Transform
Continuous time
Discrete time
Periodic
Fourier
Series
Discrete
Fourier
Transform
Aperiodic
Continuous
Fourier
Transform
Discrete
Fourier
Transform
Fourier series for continuous-time periodic signals → discrete spectra
Fourier transform for continuous aperiodic signals → continuous spectra
ES 442 Fourier Transform
12
Definition of Fourier Transform



g(t )e
 j 2 ft
Figure 3.15
Lathi & Ding
dt
g(t )
G( f )



G( f )e
 j 2 ft
df
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.
ES 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 + .
G( f )  G( f ) exp  j ( f )
where |G(f)| is the continuous amplitude spectrum of g(t)
and
(f) is the continuous phase spectrum of g(t).
Supplemental:
Lathi & Ding
pp. 97-99
ES 442 Fourier Transform
14
Example: Rectangular Pulse
g(t )  rect(t )  (t / ) 
g(t )  rect(t )  (t / )

1
for 
0
for all t 
2
t


2
2
G(f)
Lathi & Ding
pp. 101-102
Example 3.2

 sinc(f )
1


2
0

2
2
time t

3


2


Remember  = 2f
ES 442 Fourier Transform
1

0
1

2
3


f
15
Two Definitions of the Sinc Function
sin( x )
sinc ( x ) 
(1)
x
and
sin( x )
sinc ( x ) 
(2)
x
Sometimes sinc (x)
is written as Sa (x).
x
Lathi and Ding, page 100, use the definition sinc (x) =
ES 442 Fourier Transform
sin( x )
x
16
Definition of the Sinc Function
Unfortunately, there are two definitions of the sinc function in use.
Format 1 (used in Lathi and Ding, fourth edition – pp. 100 – 102)
sinc( x ) 
sin( x )
x
Format 2 (used in many other text books)
sinc( x ) 
sin( x )
x
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 sinc(x) oscillates and monotonically
decreases as 1/x decrease with increasing |x|
5. sinc(x) is the Fourier transform of a single rectangular pulse
ES 442 Fourier Transform
17
Sinc Function in Two-Dimensions (e.g., Optics)
“Airy Disc”
Pardon
the digression!
ES 442 Fourier Transform
18
Sinc Function Tradeoff: Pulse Duration versus Bandwidth
G1 ( f )
T1

1
T1
G2 ( f )
1
T1
T2
f

g1 (t )
T1
2
T1 > T2 > T3
T1
2
t
G3 ( f )
T3
1
1

T3 g3 (t ) T3
T3 T3
2
2
ES 442 Fourier Transform
1
1
T2 g2 (t ) T2
T2
2
T2
2
f

f

t
Lathi & Ding
pp. 110-111
t
19
Sinc Function Appears in Both Pulse Train & a Single Pulse
Pulse Train – Fourier Series
Single pulse – Fourier Transform
f (t )  1 u(t  2 )  u(t  2 )
Discrete
Spectrum
Continuous
Spectrum
F f  
sin( ft )
( ft )
What does this imply for a digital binary communication signal?
ES 442 Fourier Transform
20
Some Insight into the Sinc Function
sin( x )
sinc( x ) 
x
1
x
sinc( x)
sin( x )
x

1
x
Lathi & Ding
pp. 100-101
ES 442 Fourier Transform
21
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
ES 442 Fourier Transform
22
Linearity (Superposition) Property
Given g1(t)  G1(f) and g2(t)  G2(f) ;
Then g1(t) + g2(t)  G1(f) + G2(f)
also kg1(t)  kG1(f) and kg2(t)  kG2(f)
(additivity)
(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 course.
ES 442 Fourier Transform
23
Time-Frequency Duality Property
Given g(t )  G( f ), then
g(t - t0 )  G( f ) e
and
g(t)e

j 2 ft0


g(t )e
 j 2 ft0
 G( f  f 0 )
 j 2 ft
dt
g(t )
G( f )



G( f )e
 j 2 ft
df
Lathi & Ding
pp. 106-109
This leads directly to the transform Duality Property
ES 442 Fourier Transform
24
Transform Duality Property
Given g(t )  G( f ), then
g(t )  G( f )
and
G(t )  g(  f )
Lathi & Ding
pp. 108-109
See illustration on next page for example!
ES 442 Fourier Transform
25
Illustration of Fourier Transform Duality
G1 ( f )
T1
Lathi & Ding
Page 109
g1 (t )

T1
2
T1
2
t

1
T1
1
T1
f
2
T1
2
T1
f1 
g 2 (t )
G2 ( f )
1
T1
T1

1
T1
1
T1
t
ES 442 Fourier Transform
T1
2
T1
2
f
26
Time-Scaling Property
Given g(t )  G( f ),
then for a real constant a,
f
1
g( at )  G( )
a
a
Lathi & Ding
pp. 110-112
Time compression of a signal results in spectral expansion and
time expansion of a signal results in spectral compression.
ES 442 Fourier Transform
27
Time-Shifting Property
Given g(t )  G( f ),
then g(t  t0 )  G( f )e
 j 2 ft0
Lathi & Ding
pp. 112-113
Delaying a signal by t0 seconds does not change its amplitude
spectrum, but the phase spectrum is changed by -2ft0.
Note that the phase spectrum shift changes linearly with frequency f.
=
Frequency f
 = 2
Frequency 2f
ES 442 Fourier Transform
28
Frequency-Shifting Property
aka Modulation Property
Given g(t )  G( f ),
then g(t )e
j 2 ft0
 G( f  f 0 )
Multiplication of a signal g(t) by the factor cos(2 fC )
places G(f) centered at f =  fC.
g(t )
2A
B
g(t )cos(2 fC )
t
g(t )
G( f )
g(t )  G( f )
t
g(t )
Lathi & Ding
pp. 114-119
B
f
A
 fC
2B
ES 442 Fourier Transform
fC
f
29
Frequency-Shifting Property (continued)
g(t )
G( f )
 a
g(t )cos(2 f 0 ) c
 
g(t )sin(2 f 0 ) e
 
G( f )
b
g ( f )
Lathi & Ding
Figure 3.21
Page 115
d
Note phase shift
f
 2
 2
Multiplication of a signal g(t) by the factor cos(2 f 0 )
places G(f) centered at f =  fC.
ES 442 Fourier Transform
30
Time Differentiation & Time Integration Property
Given g(t )  G( f )
For time differentiation:
dg(t )
 j 2 f G( f )
dt
d n g(t )
n
and

j
2

f
G( f )


n
dt
For time integration:
t


g( )d 
Lathi & Ding
pp. 121-123
G( f ) 1
 2 G(0) ( f )
j 2 f
ES 442 Fourier Transform
31
Area Under g(t) & Area Under G(f) Properties
Given g(t )  G( f ),

Then
 g(t )dt  G(0)

That is, the area under a function g(t) is equal to the value of its
Fourier transform G(f) at f = 0.
Given g(t )  G( f ),

Then
g(0) 
 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).
ES 442 Fourier Transform
32
Conjugate Functions Property
Given g(t )  G( f ),
Then for a complex-valued time function g(t),
we have
g * (t )  G * (  f )
where the star symbol (*) denotes the complex
conjugate operation.
The corollary to this property is
g * ( t )  G * ( f )
End of Fourier Transform Property Slides
ES 442 Fourier Transform
33
Some Important Fourier Transform Pairs
Impulse Function
Impulse train
Unit Rectangle Pulse
Unit Triangle Pulse
Exponential Pulse
Signum Function
ES 442 Fourier Transform
34
The Impulse Function (t)
 (t )  0 for all t  0
ES 442 Fourier Transform
35
Fourier Transform of the Impulse Function (t)
F  (t ) 

  (t )e
 j 2 ft
dt  e
 j 2 f (0)
1

because the exponential function to zero power is unity.
Therefor, we can write  (t)  1
g(t) =(t)
t
1
G(f) =1
f
(t) is often called
the Dirac delta function
ES 442 Fourier Transform
36
Inverse Fourier Transform of the Impulse Function (t)
F 1  ( f ) 

  ( f )e
j 2 ft
df  e
 j(0)t
1

because the exponential function to zero power is unity.
Therefor, we can write 1   (t)
1
G(f) = (f)
g(t)
t
This is just a DC signal.
0
f
A DC signal is
zero frequency.
ES 442 Fourier Transform
37
Fourier Transform of Impulse Train (t)
aka “Dirac Comb Function,” Shah Function & “Sampling Function”
Shah function (Ш(t) ):

Ш(t) =

  (t  nT )    (t  nT )
0
n 
n 1

n 1
Ш(t)
2T0 T0 0
2
n 
0
Ш (t ) dt  1
2
Period 
Period  T0
T0 2T0
t
2 f 0  f 0 0
ES 442 Fourier Transform
f0 2 f0
1
T0
f
38
Shah Function (Impulse Train) Applications
The sampling property is given by
Ш(t) f (t ) 


n 
f (n) (t  nT0 )
The “replicating property” is given by the convolution operation:
Ш(t) f (t ) 


n 
f (t  nT0 )
Convolution
Convolution theorem:
g1 (t ) g2 (t )  G1 ( f )G2 ( f ) and
g1 (t )g2 (t )  G1 ( f )G2 ( f )
ES 442 Fourier Transform
39
Sampling Function in Operation
Ш(t) f (t ) 


n 
f (n) (t  nT0 )
f (t )
t
 (t  nT0 )
T0
T0
f (0)
t
f (T0 )  f (1)
f (2T0 )  f (2)
t
ES 442 Fourier Transform
40
Fourier Transform of Complex Exponentials
F 1  ( f  f c ) 

 j 2 f t

(
f

f
)
e
df
C


Evaluate for f  f c
F 1  ( f  f c ) 

e j 2 f ct df  e j 2 f ct
f  fc
  ( f  f c )  e  j 2 f c t
F 1  ( f  f c ) 
and

Lathi & Ding
pp. 104
 j 2 f t

(
f

f
)
e
df
C


Evaluate for f   f c
F 1  ( f  f c ) 

e j 2 fct df  e j 2 f ct
f  f c
  ( f  f c )  e j 2 f c t
ES 442 Fourier Transform
41
Fourier Transform of Sinusoidal Functions
 j 2 f t
j 2 f t
c
c
Taking  ( f  f c )  e
and  ( f  f c )  e
We use these results to find FT of cos(2 ft ) and sin(2 ft )
Using the identities for cos(2 ft ) and sin(2 ft ),
*
j 2 f c t
 j 2 f c t 
j 2 f c t
 j 2 f c t 
cos(2 ft )  21  e
e
& cos(2 ft )  21j  e
e




Therefore,
cos(2 ft )  21  ( f  f c )   ( f  f c ) ,
sin(2 ft ) 
1
2j
and
 ( f  fc )   ( f  fc )
cos(2fct)
Lathi & Ding
pp. 105
sin(2fct)
-fc
-fc
fc
f
ES 442 Fourier Transform
fc
f
42
Fourier Transform of Signum (Sign) Function
sgn(t ) 
 1,
0,
 1,
for t  0
for t  0
for t  0
Sgn(t)
+1
-1
t
We approximate the signum function using
g(t ) 
exp(  at ), for t  0
0,
for t  0
 exp( at ), for t  0
+1
-1
 j 4 f
1
G( f )  2

, because
2
a  (2 f )
j f
  j 4 f   j
1
lim  2


2 
a

(2

f
)
a 0

  f j f
ES 442 Fourier Transform
t
Lathi & Ding
pp. 105-106
43
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
ES 442 Fourier Transform
44
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
ES 442 Fourier Transform
45
Fourier Transform of Cosine Signal
A cos(2 f c t ) 
A
 ( f  f c )   ( f  f c )

2
A
2
Re
A cos(2 f t )
-fc
Not in
Lathi & Ding
A
2
3D View
fc
f
Blue arrows indicate
positive phase direction
ES 442 Fourier Transform
46
Fourier Transform of Sine Signal
B sin(2 f c t )  j
B
 ( f  f c )   ( f  f c )

2
Re
B
2
B sin(2 f t )
-fc
B
2
fc
f
We must subtract 90
from cos(···) to get sin(···)
ES 442 Fourier Transform
47
Fourier Transform of Sine Signal (as usually shown in books)
 ( f0 )
sin(t )
t
FT
Imaginary axis
A
j
2
f0
 f0
j
ES 442 Fourier Transform
f
A
2
48
Fourier Transform of Phase Shifted Sinusoidal Signal
j j 2 ft
Re e
Re
A
2
 j 2 ft
-
B
2
 Re
 j  j 2 ft
e
Re
-fc
Re
A
2

j 2 ft
B
2
fc
f
2
 A  B
R     
 2  2
2
 A
and   tan 1   
 B
ES 442 Fourier Transform
49
Even, Odd Relationships for Fourier Transform (1)
g(t)
G(f)
Re
Im
t
Im
Real; Even
g(t)
G(f)
t
Real; Odd
f
Real; Even
Re
Im
Re
Re
Im
f
Imaginary; Odd
From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.
ES 442 Fourier Transform
50
Even, Odd Relationships for Fourier Transform (2)
g(t)
Re
G(f)
t
Im
Imaginary; Even
g(t)
f
Im
Imaginary; Even
Re
Im
Re
G(f)
t
Re
f
Im
Imaginary; Odd
Real; Odd
From: Frederic J. Harris, Trignometric Transforms, Spectral Dynamics Corporation, 1976.
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Effect of the Position of a Pulse on Fourier Transform
G( f ) 
Re(G( f ))  Im(G( f ))
2
2
t
Even function.
f
Both
must be
identical.
f
t
This time shifted pulse
is both even and odd.
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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/59528898E.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 of
all books covering the Fourier Transform (Bracewell gives many
insightful views and discussions on the FT).
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