Transcript Document

Discrete-Time Fourier Transform Properties
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.
Absolute and Square Summability
• Absolute summability is sufficient condition for DTFT
• Some sequences may not be absolute summable but only
square summable

 xn  
2
n  
• To represent square summable sequences with DTFT
– We can relax the uniform convergence condition
– Convergence is in mean-squared sense
    xne
Xe

j
    xne
 jn
Xe
lim
M 
 jn
n  
n  


j
 Xe   X e 
j
M
j
2
0

– Error does not converge to zero for every value of 
– The mean-squared value of the error over all  does
Copyright (C) 2005 Güner Arslan
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Example: Ideal Lowpass Filter
• The periodic DTFT of the ideal lowpass filter is
 
Hlp e
j

  c
1


0 c    
• The inverse can be written as
 
1 
1 c jn
j
jn
hlp n 
Hlp e e d 
e d






c
2
2
sin cn
1
1
jn c

e  c 
e jcn  e  jcn 
2jn
2jn
n
 
•
•
•
•


Not causal
Not absolute summable but it has a DTFT?
The DTFT converges in the mean-squared sense
Role of Gibbs phenomenon
Copyright (C) 2005 Güner Arslan
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Example: Generalized DTFT
•
•
•
•
DTFT of xn  1
Not absolute summable
Not even square summable
But we define its DTFT as a pulse train
    2  2r 
Xe
j

r  
• Let’s place into inverse DTFT equation
 
1 
j
jn
xn 
X
e
e
d



2
 jn
1   



2



2

r
e d






2 r  


Copyright (C) 2005 Güner Arslan



e jnd  e j0n  1
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Symmetric Sequence and Functions
Conjugateantisymmetric
Conjugate-symmetric
xn  xe n  xo n
 
x e n 

 
x o n 
 


1
xn  x*  n
2
 
 
Xe ej  X*e e j
Xo ej  X*o e j
 
 X e   12 Xe   X e 
 
X ej  Xo ej  Xe ej X e e j 
Copyright (C) 2005 Güner Arslan

1
xn  x*  n
2
 
Function
 
xo n  x*o  n
xe n  x*e  n
Sequence

1
X e j  X* e  j
2
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j
j
*
 j
5
Symmetry Properties of DTFT
Sequence x[n]
Discrete-Time Fourier Transform X(ej)
x*[n]
X*(e-j)
x*[-n]
X*(ej)
Re{x[n]}
Xe(ej) (conjugate-symmetric part)
jIm{x[n]}
Xo(ej) (conjugate-antisymmetric part)
xe[n]
XR(ej)= Re{X(ej)}
xo[n]
jXI(ej)= jIm{X(ej)}
Any real x[n]
X(ej)=X*(e-j) (conjugate symmetric)
Any real x[n]
XR(ej)=XR(e-j) (real part is even)
Any real x[n]
XI(ej)=-XI(e-j) (imaginary part is odd)
Any real x[n]
|X(ej)|=|X(e-j)| (magnitude is even)
Any real x[n]
X(ej)=-X(e-j) (phase is odd)
xe[n]
XR(ej)
xo[n]
jXI(ej)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Example: Symmetry Properties
• DTFT of the real sequence x[n]=anu[n]
1
X e j 
1  ae j
• Some properties are
 
 
X e j
 
XR e j
 
XI e j
 
X e j
 
X e j
Copyright (C) 2005 Güner Arslan

if a  1

1
*
 j

X
e
1  ae j
1  a cos 
 j


X
e
R
1  a2  2a cos 
 a sin 
 j



X
e
I
1  a2  2a cos 
1

 X e  j
1  a2  2a cos 
  a sin  
 tan1 
  X e  j
 1  a cos  








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Fourier Transform Theorems
Sequence
DTFT
x[n]
y[n]
X(ej)
Y(ej)
ax[n]+by[n]
aX(ej)+bY(ej)
x[n-nd]
e jnd X ej
x[-n]
X(e-j)
nx[n]
dX e j
j
d
X(ej)Y(ej)
 
 
x[n]y[n]

 

1
j
j   
X
e
Y
e
d

2  
x[n]y[n]

 
2
1
j
Parseval's Theorem:  xn 
X
e
d

2  
n  


1
*
j
*
j
Parseval's Theorem:  xny n 
X
e
Y
e
d

2  
n  

2
   
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Fourier Transform Pairs
Sequence
DTFT
[n-no]
e jno

 2  2k 
1
k  
1
1  ae j
anu[n] |a|<1

1
    2k 
 j
1e
k  
1
  c
X e j  
0 c    
sinM  1 / 2  jM / 2
e
sin / 2
u[n]
sincn
n
1
xn  
0
 
0nM
otherwise

 2  
e jon
Copyright (C) 2005 Güner Arslan
k  
 e   

cos(on+)
o
k  
j
o
 2k 

 2k   e j  o  2k 
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