fnCh4.ppt ENGINEERING MATHEMATICS

Chapter 4
The Fourier Series and
Fourier Transform

• Let x(t) be a CT periodic signal with period
T, i.e.,
• Example: the rectangular pulse train
rectangular pulse train
Fourier Series Representation of
Periodic Signals
( ) ( ),
x t T x t t R
   

• Then, x(t) can be expressed as
where is the fundamental
fundamental
frequency
frequency (rad/sec) of the signal and
The Fourier Series
0
( ) ,
jk t
k
k
x t c e t


 
 
 
/ 2
/ 2
1
( ) , 0, 1, 2,
o
T
jk t
k
T
c x t e dt k
T



   
 
0 2 /T
 

is called the constant or dc component of x(t)
0
c

• A periodic signal x(t), has a Fourier series
if it satisfies the following conditions:
1. x(t) is absolutely integrable
absolutely integrable over any
period, namely
2. x(t) has only a finite number of maxima
finite number of maxima
and minima
and minima over any period
3. x(t) has only a finite number of
finite number of
discontinuities
discontinuities over any period
Dirichlet Conditions
| ( ) | ,
a T
a
x t dt a

   
 

• From figure , so
• Clearly x(t) satisfies the Dirichlet conditions and
thus has a Fourier series representation
Example: The Rectangular Pulse Train
2
T  0 2 / 2
  
 

Example: The Rectangular Pulse
Train – Cont’d
x(t) =
1
2
+
1
kπ
k=−
∞
k≠0
∞
∑ s
in
kπ
2
⎛
⎝
⎜
⎞
⎠
⎟ej
kπt
, t∈°

      
• By using Euler’s formula, we can rewrite
as
as long as x(t) is real
• This expression is called the trigonometric
trigonometric
Fourier series
Fourier series of x(t)
Trigonometric Fourier Series
0
( ) ,
jk t
k
k
x t c e t


 
 
 
0 0
1
( ) 2 | |cos( ),
k k
k
x t c c k t c t



   
 
dc component
dc component k
k-th harmonic
-th harmonic

• The expression
can be rewritten as
Example: Trigonometric Fourier
Series of the Rectangular Pulse Train
x(t) =
1
2
+
1
kπ
k=−
∞
k≠0
∞
∑ s
in
kπ
2
⎛
⎝
⎜
⎞
⎠
⎟ej
kπt
, t∈°
( 1)/ 2
1
1 2
( ) cos ( 1) 1 ,
2 2
k
k
k odd
x t k t t
k






 
 
     
 
 
 
 

• Given an odd positive integer N, define the
N-th partial sum of the previous series
• According to Fourier’s theorem
Fourier’s theorem, it should be
Gibbs Phenomenon
( 1)/ 2
1
1 2
( ) cos ( 1) 1 ,
2 2
N
k
N
k
k odd
x t k t t
k





 
 
     
 
 
 
 
lim | ( ) ( ) | 0
N
N
x t x t
 
 

Gibbs Phenomenon – Cont’d
3 ( )
x t 9 ( )
x t

Gibbs Phenomenon – Cont’d
21( )
x t 45 ( )
x t
overshoot
overshoot: about 9 % of the signal magnitude
(present even if )
N  

• Let x(t) be a periodic signal with period T
• The average power
average power P of the signal is
defined as
• Expressing the signal as
it is also
Parseval’s Theorem
/ 2
2
/ 2
1
( )
T
T
P x t dt
T 
 
0
( ) ,
jk t
k
k
x t c e t


 
 
 
2
| |
k
k
P c

 
 

• We have seen that periodic signals can be
represented with the Fourier series
• Can aperiodic signals
aperiodic signals be analyzed in terms of
frequency components?
• Yes, and the Fourier transform provides the
tool for this analysis
• The major difference w.r.t. the line spectra of
periodic signals is that the spectra of
spectra of
aperiodic signals
aperiodic signals are defined for all real
values of the frequency variable not just
for a discrete set of values
Fourier Transform


Frequency Content of the
Rectangular Pulse
( )
T
x t
( )
x t
( ) lim ( )
T
T
x t x t
 


• Since is periodic with period T, we
can write
Rectangular Pulse – Cont’d
( )
T
x t
0
( ) ,
jk t
T k
k
x t c e t


 
 
 
where
/ 2
/ 2
1
( ) , 0, 1, 2,
o
T
jk t
k
T
c x t e dt k
T



   
 

• What happens to the frequency components
of as ?
• For
• For
( )
T
x t T  
0:
k 
0 1/
c T

1, 2, :
k   
0 0
0
2 1
sin sin
2 2
k
k k
c
k T k
 
 
   
 
   
   
0 2 /T
 


plots of
vs.
for
| |
k
T c
0
k
 

2,5,10
T 

• It can be easily shown that
where
lim sinc ,
2
k
T
Tc



 
 
 
 
 

sin( )
sinc( )





• The Fourier transform of the rectangular
pulse x(t) is defined to be the limit of
as , i.e.,
Fourier Transform of the
Rectangular Pulse
( ) lim sinc ,
2
k
T
X Tc

 

 
 
  
 
 

k
Tc
T  
| ( ) |
X  arg( ( ))
X 

• Given a signal x(t), its Fourier transform
Fourier transform
is defined as
• A signal x(t) is said to have a Fourier
transform in the ordinary sense if the above
integral converges
The Fourier Transform in the
General Case
( )
X 
( ) ( ) ,
j t
X x t e dt

 


 
 
 

• The integral does converge if
1. the signal x(t) is “well-behaved
well-behaved”
2. and x(t) is absolutely integrable
absolutely integrable, namely,
1. Note: well behaved
well behaved means that the signal
has a finite number of discontinuities,
maxima, and minima within any finite time
interval
The Fourier Transform in the
General Case – Cont’d
| ( ) |
x t dt

 
 


• Consider the signal
• Clearly x(t) does not satisfy the first
requirement since
• Therefore, the constant signal does not have
a Fourier transform in the ordinary sense
Fourier transform in the ordinary sense
• Later on, we’ll see that it has however a
Fourier transform in a generalized sense
Fourier transform in a generalized sense
Example: The DC or Constant Signal
( ) 1,
x t t
 
| ( ) |
x t dt dt
 
   
 

 

• Consider the signal
• Its Fourier transform is given by
Example: The Exponential Signal
( ) ( ),
bt
x t e u t b

 
( ) ( )
0 0
( ) ( )
1
bt j t
t
b j t b j t
t
X e u t e dt
e dt e
b j

 



 
 



   


 
   




• If , does not exist
• If , and does not
exist either in the ordinary sense
• If , it is
Example: The Exponential Signal –
Cont’d
0
b  ( )
X 
0
b  ( ) ( )
x t u t
 ( )
X 
0
b 
1
( )
X
b j




2 2
1
| ( ) |
X
b




amplitude spectrum
amplitude spectrum
arg( ( )) arctan
X
b


 
  
 
phase spectrum
phase spectrum

Example: Amplitude and Phase
Spectra of the Exponential Signal
10
( ) ( )
t
x t e u t



• Consider
• Since in general is a complex
function, by using Euler’s formula
Rectangular Form of the Fourier
Transform
( ) ( ) ,
j t
X x t e dt

 


 
 
 
( )
X 
( ) ( )
( ) ( )cos( ) ( )sin( )
R I
X x t t dt j x t t dt
 
  
 
   
 
  
 
 
 
              
( ) ( ) ( )
X R jI
  
 

• can be expressed in
a polar form as
where
Polar Form of the Fourier Transform
( ) | ( ) | exp( arg( ( )))
X X j X
  

( ) ( ) ( )
X R jI
  
 
2 2
| ( ) | ( ) ( )
X R I
  
 
( )
arg( ( )) arctan
( )
I
X
R



 
  
 

• If x(t) is real-valued, it is
• Moreover
whence
Fourier Transform of
Real-Valued Signals
( ) ( )
X X
 

 
( ) | ( ) | exp( arg( ( )))
X X j X
  

 
| ( ) | | ( ) | and
arg( ( )) arg( ( ))
X X
X X
 
 
 
 
Hermitian
Hermitian
symmetry
symmetry

• Consider the even signal
• It is
Example: Fourier Transform of the
Rectangular Pulse
 
/ 2
/ 2
0
0
2 2
( ) 2 (1)cos( ) sin( ) sin
2
sinc
2
t
t
X t dt t

 
  
 





 
    
 
 
  
 


( ) sinc
2
X

 

 
  
 

amplitude
amplitude
spectrum
spectrum
phase
phase
spectrum
spectrum

• A signal x(t) is said to be bandlimited
bandlimited if its
Fourier transform is zero for all
where B
B is some positive number, called
the bandwidth of the signal
bandwidth of the signal
• It turns out that any bandlimited signal must
have an infinite duration in time, i.e.,
bandlimited signals cannot be time limited
Bandlimited Signals
( )
X  B
 

• If a signal x(t) is not bandlimited, it is said
to have infinite bandwidth
infinite bandwidth or an infinite
infinite
spectrum
spectrum
• Time-limited signals cannot be
bandlimited and thus all time-limited
signals have infinite bandwidth
• However, for any well-behaved signal x(t)
it can be proven that
whence it can be assumed that
Bandlimited Signals – Cont’d
lim ( ) 0
X


 

| ( ) | 0
X B
 
  
B being a convenient large number

• Given a signal x(t) with Fourier transform
, x(t) can be recomputed from
by applying the inverse Fourier transform
inverse Fourier transform
given by
• Transform pair
Transform pair
Inverse Fourier Transform
( )
X  ( )
X 
1
( ) ( ) ,
2
j t
x t X e d t

 


 
 
 
( ) ( )
x t X 


Properties of the Fourier Transform
• Linearity:
Linearity:
• Left or Right Shift in Time:
Left or Right Shift in Time:
• Time Scaling:
Time Scaling:
( ) ( )
x t X 
 ( ) ( )
y t Y 

( ) ( ) ( ) ( )
x t y t X Y
     
  
0
0
( ) ( ) j t
x t t X e 
 
 
1
( )
x at X
a a

 
  
 

• Time Reversal:
Time Reversal:
• Multiplication by a Power of t:
Multiplication by a Power of t:
• Multiplication by a Complex Exponential:
Multiplication by a Complex Exponential:
( ) ( )
x t X 
  
( ) ( ) ( )
n
n n
n
d
t x t j X
d



0
0
( ) ( )
j t
x t e X

 
 

• Multiplication by a Sinusoid (Modulation):
Multiplication by a Sinusoid (Modulation):
• Differentiation in the Time Domain:
Differentiation in the Time Domain:
 
0 0 0
( )sin( ) ( ) ( )
2
j
x t t X X
    
   
 
0 0 0
1
( )cos( ) ( ) ( )
2
x t t X X
    
   
( ) ( ) ( )
n
n
n
d
x t j X
dt
 


• Integration in the Time Domain:
Integration in the Time Domain:
• Convolution in the Time Domain:
Convolution in the Time Domain:
• Multiplication in the Time Domain:
Multiplication in the Time Domain:
1
( ) ( ) (0) ( )
t
x d X X
j
     

 
 

( ) ( ) ( ) ( )
x t y t X Y
 
 
( ) ( ) ( ) ( )
x t y t X Y
 
 

• Parseval’s Theorem:
Parseval’s Theorem:
• Duality:
Duality:
1
( ) ( ) ( ) ( )
2
x t y t dt X Y d
  



 
 
2 2
1
| ( ) | | ( ) |
2
x t dt X d
 


 
 
( ) ( )
y t x t

if
( ) 2 ( )
X t x
 
 

Properties of the Fourier Transform -
Summary

Example: Linearity
4 2
( ) ( ) ( )
x t p t p t
 
2
( ) 4sinc 2sinc
X
 

 
   
 
   
   

Example: Time Shift
2
( ) ( 1)
x t p t
 
( ) 2sinc j
X e 




 
  
 

Example: Time Scaling
2 ( )
p t
2 (2 )
p t
2sinc


 
 
 
sinc
2


 
 
 
time compression frequency expansion

time expansion frequency compression

1
a 
0 1
a
 

Example: Multiplication in Time
2
( ) ( )
x t tp t

2
sin cos sin
( ) 2sinc 2 2
d d
X j j j
d d
    

    
  
   
  
   
 
   
 

Example: Multiplication in Time –
Cont’d
2
cos sin
( ) 2
X j
  





Example: Multiplication by a Sinusoid
0
( ) ( )cos( )
x t p t t
 
 sinusoidal
burst
0 0
1 ( ) ( )
( ) sinc sinc
2 2 2
X
     
  
 
 
 
   
 
   
 
   
 

Example: Multiplication by a
Sinusoid – Cont’d
0 0
1 ( ) ( )
( ) sinc sinc
2 2 2
X
     
  
 
 
 
   
 
   
 
   
 
0 60 /sec
0.5
rad








Example: Integration in the Time
Domain
2 | |
( ) 1 ( )
t
v t p t


 
 
 
 
( )
( )
dv t
x t
dt


Domain – Cont’d
• The Fourier transform of x(t) can be easily
found to be
• Now, by using the integration property, it is
( ) sinc 2sin
4 4
X j
 


  
   
    
  
   
  
2
1
( ) ( ) (0) ( ) sinc
2 4
V X X
j
 
    
 
 
    
 

Domain – Cont’d
2
( ) sinc
2 4
V
 


 
  
 

• Fourier transform of
• Applying the duality property
Generalized Fourier Transform
( )
t

( ) 1
j t
t e dt

 



( ) 1
t
 

( ) 1, 2 ( )
x t t  
  

  
generalized Fourier transform
generalized Fourier transform
of the constant signal ( ) 1,
x t t
 

Generalized Fourier Transform of
Sinusoidal Signals
 
0 0 0
cos( ) ( ) ( )
t
       
   
 
0 0 0
sin( ) ( ) ( )
t j
       
   

Fourier Transform of Periodic Signals
• Let x(t) be a periodic signal with period T;
as such, it can be represented with its
Fourier transform
• Since , it is
0
0
2 ( )
j t
e 
  
 
0
( ) jk t
k
k
x t c e 

 
  0 2 /T
 

0
( ) 2 ( )
k
k
X c k
    

 
 


• Since
using the integration property, it is
Fourier Transform of
the Unit-Step Function
( ) ( )
t
u t d
  
 
 
1
( ) ( ) ( )
t
u t d
j
    

 
  


Common Fourier Transform Pairs

fnCh4.ppt ENGINEERING MATHEMATICS

More Related Content

Similar to fnCh4.ppt ENGINEERING MATHEMATICS (20)

More from Rethabile37 (16)

Recently uploaded (20)

fnCh4.ppt ENGINEERING MATHEMATICS