Transcript Document

Discrete-Time Fourier Transform
Quote of the Day
The profound study of nature is the most fertile
source of mathematical discoveries.
Joseph Fourier
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Frequency Response
• The frequency response defines a systems output
– for complex exponential at all frequencies
• If input signals can be represented as a sum of complex
exponentials
xn 
jkn

e
 k
k
– we can determine the output of the system
yn 
 
jk
jkn

H
e
e
 k
k
• Different from continuous-time frequency response
– Discrete-time frequency response is periodic with 2

He
j  2 r 
   hke

k  
 j  2 r k



 hke
k  

 j2 rk
e
 jk
 


 jk


h
k
e

k  
H ej2r   H ej
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
2
Discrete-Time Fourier Transform
• Many sequences can be expressed as a weighted sum of
complex exponentials as
1 
j
jn
xn 
X
e
e
d
(inversetransform)



2
 
• Where the weighting is determined as
    xne
Xe

j
 jn
(forwardtransform)
n  
• X(ej) is the Fourier spectrum of the sequence x[n]
– It specifies the magnitude and phase of the sequence
– The phase wraps at 2 hence is not uniquely specified
• The frequency response of a LTI system is the DTFT of the
impulse response
    hke
He
j

k  
Copyright (C) 2005 Güner Arslan
 jk
and
 
1 
j
jn
hn 
H
e
e
d



2
351M Digital Signal Processing
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Discrete-Time Fourier Transform Pair
    xne
Xe
j

 jn
and
n  
 
1 
j
jn
xn 
X
e
e
d



2
• Let’s show that they constitute a transform pair
– Substitute first equation into second to get
1  
 jm  jn
ˆ


xn 

x
m
e
e d 





2  m 

 1  jnm 
xm
e
d




 2

m 

– Evaluate the integral
 sinn  m
1  jn m
 0 for n  m

e
d


 n  m
2  

1
for n  m

 n  m
– Substitute the integral with this result to get
ˆ
xn 

 xmn  m  xn
m  
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
4
Existence of DTFT
• For a given sequence the DTFT exist if the infinite sum
convergence
    xne
Xe

j
 jn
n  
• Or
 
X e j  
    xne
Xe

j
n  
 jn

for all 

 xn e
n  
 jn


 xn  
n  
• So the DTFT exists if a given sequence is absolute summable
• All stable systems are absolute summable and have DTFTs
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
5
DTFT Demo
Square Wave
Triangular Wave
From Fundamentals of Signals and Systems Using the Web and Matlab
by Edward W. Kamen and Bonnie S. Heck
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
6