Chapter 4
Fourier Series & Transforms
Basic Idea
notes
Taylor Series
• Complex signals are often broken into simple pieces
• Signal requirements
– Can be expressed into simpler problems
– The first few terms can approximate the signal
• Example: The Taylor series of a real or complex function
ƒ(x) is the power series
•
http://upload.wikimedia.org/wikipedia/commons/6/62/Exp_series.gif
Square Wave
S(t)=sin(2pft)
S(t)=1/3[sin(2p(3f)t)]
S(t)= 4/p{sin(2pft) +1/3[sin(2p(3f)t)]}
4  sin( 2pkft)
s (t )  A  
p K 1/ odd
k
Fourier Expansion
Square Wave
K=1,3,5
K=1,3,5, 7
Frequency Components of Square Wave
4  sin( 2pkft)
s (t )  A  
p K 1/ odd
k
K=1,3,5, 7, 9, …..
Fourier Expansion
Periodic Signals
• A Periodic signal/function can be
approximated by a sum (possibly
infinite) sinusoidal signals.
• Consider a periodic signal with
period T
• A periodic signal can be Real or
Complex
• The fundamental frequency: wo
• Example:
Periodic  x(t  nT )  x(t )
Re al  x(t )  cos(wot   )
Complex  x(t )  Ae jwot
wo  2p / To
Fourier Series
• We can represent all periodic signals
as harmonic series of the form
– Ck are the Fourier Series Coefficients; k is
real
– k=0 gives the DC signal
– k=+/-1 indicates the fundamental frequency
or the first harmonic w0
– |k|>=2 harmonics
Fourier Series Coefficients
• Fourier Series Pair
Series of complex
numbers
Defined over a period
of x(t)
• We have
Ck*  Ck
Ck*  e  jkwot & Ck  e  jkwot
Ck  Ck e j k
• For k=0, we can obtain the DC value which is the average
value of x(t) over one period
Euler’s Relationship
– Review Euler formulas
notes
Examples
• Find Fourier Series Coefficients for
x(t )  cos(wot )
C1=1/2; C-1=1/2; No DC
• Find Fourier Series Coefficients for
x(t )  sin( wot )
C1=1/2j; C-1=-1/2j; No DC
• Find Fourier Series Coefficients for
x (t )  cos( 2t  p / 4)
• Find Fourier Series Coefficients for
x (t )  sin 2 (wot )
notes
Different Forms of Fourier Series
• Fourier Series Representation has three different forms
Also:
Complex
Exp.
Also:
Harmonic
Which one is this?
What is the DC component?
What is the expression for Fourier Series Coefficients
4  sin( 2pkft)
s (t )  A  
p K 1/ odd
k
Examples
• Find Fourier Series
Coefficients for
• Find Fourier Series
Coefficients for
Remember:
Examples
notes
Find the Complex Exponential Fourier Series Coefficients
x(t )  10  3 cos wot  5 cos( 2wot  30o )  4 sin 3wot
textbook
Example
• Find the average power of x(t)
using Complex Exponential
Fourier Series – assuming x(t) is
periodic
 x(t )
P
2
P
 x(t )
2
dt
To
dt
To

x(t ) 
C e w
j
k  

x(t ) 
*
o tk
k
 Ck e  jwotk
*
k  
P   x(t ) x(t ) dt 
*
To

k  
 Ck Ck
*
k  
2

C

k
This is called the Parseval’s Identity
Example
• Consider the following
periodic square wave
• Express x(t) as a piecewise
function
• Find the Exponential
Fourier Series of
representations of x(t)
• Find the Combined
Trigonometric Fourier
Series of representations of
x(t)
• Plot Ck as a function of k
X(t)
To
-V
x(t ) 


2|Ck|
notes
To/2
V

2V  jp / 2 jwotk
e
e

k
p
k   / odd

4V
cos( kwot  90)

k 1 / odd kp

4V
sin( kwot )

k 1 / odd kp
|4V/p|
|4V/3p|
|4V/5p|
w0
3w0
5w0
Use a
Low Pass Filter to
pick any tone
you want!!
Practical Application
• Using a XTL oscillator which produces positive 1Vp-p how
can you generate a sinusoidal waveforms with different
frequencies?
Practical Application
• Using a XTL oscillator which produces positive 1Vp-p how
can you generate a sinusoidal waveforms with different
frequencies?
Square Signal
@ wo
Level Shifter
Filter @ [kwo]
Sinusoidal waveform
X(t)
1
To/2
@ [kwo]
To
X(t)
To/2
0.5
To
-0.5
x(t ) 


2V  jp / 2 jwotk
e
e

k   / odd kp
4V
 
sin( kwo t )
k 1 / odd kp
kwo
y(t )  B sin( kwot )
B changes depending on k value
Demo
Ck corresponds to frequency components
In the signal.
Example
•
Given the following periodic square wave, find the Fourier Series representations and
plot Ck as a function of k.
1
Note: sinc (infinity)  1 &
Max value of sinc(x)1/x
Sinc Function
Only a
function
of freq.
Note: First zero
occurs at Sinc (+/-pi)
Use the Fourier Series Table (Table 4.3)
• Consider the following
periodic square wave
• Find the Exponential
Fourier Series of
representations of x(t)
X(t)
To/2
V
To
-V

2V  jp / 2 jwotk
x(t )  
e
e
k   / odd kp
• X0V

2V jwotk
x(t )  0    j e
kp
k   / odd


2|Ck|

4V
cos( kwot  90)

k 1 / odd kp

4V
sin( kwot )

k 1 / odd kp
|4V/p|
|4V/3p|
|4V/5p|
w0
3w0
5w0
Fourier Series - Applet

2V jwotk
x(t )  0    j e
kp
k   / odd
http://www.falstad.com/fourier/
Using Fourier Series Table
• Given the following periodic square wave, find the Fourier Series
representations and plot Ck as a function of k. (Rectangular wave)
X01
C0=T/To
T/2=T1T=2T1
Ck=T/T0 sinc (Tkw0/2)
Tkwo 2T1
T
Ck  sin c

sin c(T1kwo )
To
2
To

2T1
 
sin c(T1kwo )e jwotk
k   To
Note: sinc (infinity)  1 &
Max value of sinc(x)1/x
Same as
before
Using Fourier Series Table
• Express the Fourier Series for a
triangular waveform?
Xo
To
•
Express the Fourier Series for a
triangular waveform that is
amplitude shifted down by –X0/2 ?
Plot the signal.
Fourier Series Transformation
• Express the Fourier Series for a
triangular waveform?
Xo
To
From the table:
Xo/2
•
Express the Fourier Series for a
triangular waveform that is
amplitude shifted down by –X0/2 ?
Plot the signal.
-Xo/2
To
Fourier Series Transformation
• Express the Fourier Series for a
triangular waveform?
Xo
To
From the table:

Xo
 2 X o jwotk
x(t ) 
 
e
2
2 k  / odd (kp 
Xo/2
•
Express the Fourier Series for a
triangular waveform that is
amplitude shifted down by –X0/2 ?
Plot the signal.
Only DC value changed!
-Xo/2
To
Xo
 x(t )
2

Xo Xo
 2 X o jwotk


 
e
2
2
2 k   / odd (kp 
y (t )  

 2 X o jwotk
e

2
k   / odd (kp 

Fourier Series Transformation
• Express the Fourier Series for a
sawtooth waveform?
Xo
To
From the table:
• Express the Fourier Series for
this sawtooth waveform?
Xo
1
To
-3
Fourier Series Transformation
• Express the Fourier Series for a
sawtooth waveform?
Xo
To
From the table:
• Express the Fourier Series for
this sawtooth waveform?
– We are using amplitude transfer
– Remember Ax(t) + B
• Amplitude reversal A<0
• Amplitude scaling |A|=4/Xo
• Amplitude shifting B=1

Xo
X o jp / 2 jwotk
x(t ) 
 
e e
2 k   /  0 (2kp 
Xo
1
To
-3
4
y (t ) 
x(t )  1
Xo
Example
Example
Fourier Series and Frequency Spectra
• We can plot the frequency spectrum or line spectrum of
a signal
– In Fourier Series k represent harmonics
– Frequency spectrum is a graph that shows the amplitudes and/or
phases of the Fourier Series coefficients Ck.
• Amplitude spectrum |Ck|
• Phase spectrum fk
• The lines |Ck| are called line spectra because we indicate the
values by lines
Schaum’s Outline Problems
• Schaum’s Outline Chapter 5 Problems:
– 4,5 6, 7, 8, 9, 10
• Do all the problems in chapter 4 of the textbook
• Skip the following Sections in the text:
– 4.5
• Read the following Sections in the textbook on your own
– 4.4