06/10/2016
DIGITAL SIGNAL PROCESSING
Frequency-Domain Representation
of
Discrete-Time Signals and Systems
1
Fourier Transform (FT)
Company
LOGO
• Analysis
Analysis
F
x(n) 

X (e j )
    xn e
X e

j
 jn
Fourier Transform
(FT)
X (e j )  F [ x(n)]
n  
Synthesis
x ( n) 
1
2


 X e e d
j
2
j
Inverse Fourier Transform
(IFT)
x(n)  F -1[ X (e j )]
www.company.com
1
06/10/2016
Fourier Transform (FT)
Company
LOGO
3
j
X (e ) 
n 
 x[n]e
 j n
is a complex-valued function
n  
X (e j )  X R (e j )  jX I (e j )
magnitude
X (e j )
X I ( e j )
phase
| X ( e j ) |
X ( e j )
X (e j ) | X (e j ) | e jX (e
j
)
X R ( e j )
j
X (ewww.company.com
)  X ( e j ) e j (  )
Company
LOGO
4
• Example:Find fourier transform of x(n):
x ( n)   ( n)
x ( n)   ( n  no )
x ( n)  u ( n)
n
1
x ( n)    u ( n)
2
n
x ( n)  2  u ( n)
x(n)  rect N (n)
www.company.com
2
06/10/2016
Key Issue
Company
LOGO
We need that |X(ej)|
<  for all 
Analysis
j
X (e ) 

 x ( n )e
 j n
n  
Does X(ej) exist
for all ?
Synthesis
1 
X (e j )e jn d

2  
x ( n) 
www.company.com
Fourier Transform (FT)
Sufficient Condition for Convergence
Company
LOGO
Sufficient Condition for Convergence

 | x(n) | 
| X (e j ) |  for all 
n  
| X ( e j ) | 

 x ( n ) e  j n 
n  


 | x(n) || e
 j n

 | x ( n )e
 j n
|
n  
|
n  


 | x ( n) |  
www.company.comn  
3
06/10/2016
Company
LOGO
Conjugate-Symmetric and
Conjugate-Antiymmetric
Sequences
• Conjugate-Symmetric Sequence
xe (n)  xe* (n)
Called an even
sequence if it is real.
• Conjugate-Antisymmetric Sequence
xo (n)   xo* (n)
Called an odd
sequence if it is real.
www.company.com
Company
LOGO
•
Fourier Transform (FT)
Sequence Decomposition
Any sequence can be expressed as the sum of a
conjugate-symmetric one and a conjugate-antisymmetric
one, i.e.,
x(n)  xe (n)  xo (n)
Conjugate
Symmetric
Conjugate
Antiymmetric
xe (n)  12 [ x(n)  x * (n)] xo (n)  12 [ x(n)  x * (n)]
www.company.com
4
06/10/2016
Company
LOGO
•
Fourier Transform (FT)
Sequence Decomposition
Any function can be expressed as the sum of a conjugatesymmetric one and a conjugate-antisymmetric one, i.e.,
X (e j )  X e (e j )  X o (e j )
Conjugate
Symmetric
Conjugate
Antiymmetric
X e (e j )  12 [ X (e j )  X * (e  j )]
X o (e j )  12 [ X (e j )  X * (e  j )]
www.company.com
Company
LOGO
Conjugate-Symmetric and
Conjugate-Antiymmetric Functions
• Conjugate-Symmetric Function
X e (e j )  X e* (e  j )
Called an even function if
it is real.
• Conjugate-Antisymmetric Function
X o (e j )   X o* (e  j )
Called an odd function if
it is real.
www.company.com
5
06/10/2016
Fourier Transform (FT)
Symmetric Properties
Company
LOGO
• Symmetric Properties
F
x(n) 

X (e j )
F
x(n) 

X (e j )

 x(n)e
 jn
n  
magnitude

 x ( n )e

jn
 X ( e  j )
n  
magnitude




phase



phase

www.company.com
Fourier Transform (FT)
Symmetric Properties
Company
LOGO
• Symmetric Properties
F
x * (n) 

X * (e j )
F
x(n) 

X (e j )

 x * (n)e
 jn

n  
 x(n)e 

jn *
n  
magnitude



phase

www.company.com
*
 

   x(n)e jn   X * (e j )
 n  

magnitude



phase

6
06/10/2016
Fourier Transform (FT)
Symmetric Properties
Company
LOGO
• Symmetric Properties
F
x(n) 

X (e j )
F
x * (n) 

X * (e j )
magnitude
magnitude





phase


www.company.com
Company
LOGO

phase
Fourier Transform (FT)
Symmetric Properties
• Symmetric Properties
F
x(n) 

X (e j )
F
Re{x(n)} 

X e (e j )
Re{x(n)}  12 [ x(n)  x * (n)]
1
2
F
j
*
 j
1
[ x(n)  x * (n)] 

)]
2 [ X (e )  X (e
F
j Im{x(n)} 

X o (e j )
j Im{x(n)}  12 [ x(n)  x * (n)]
www.company.com
2 [ x ( n) 
1
F
j
*
 j
1
x * (n)] 
)]
2 [ X (e )  X (e
7
06/10/2016
Company
LOGO
Fourier Transform (FT)
Symmetric Properties
• Symmetric Properties
F
x(n) 

X (e j )
F
xe (n) 

X R (e j )
xe (n)  12 [ x(n)  x * (n)]
F
j
*
j
1
1

2 [ x(n)  x * (n)] 
2 [ X (e )  X (e )]
F
xo (n) 

jX I (e j )
xo (n)  12 [ x(n)  x * (n)]
www.company.com
2 [ x ( n) 
1
Company
LOGO
F
j
1
x * ( n)] 
)  X * (e j )]
2 [ X (e
Fourier Transform (FT)
Symmetric Properties
• Symmetric Properties
F
x(n) 

X (e j )

F
x * (n) 

X * (e j )
magnitude
Facts:



phase
1. real part is even
X R (e j )  X R (e  j )
2. Img. part is odd
X I (e j )   X I (e j )
j
 j
3. Magnitude is even | X I (e ) || X I (e ) |
4. Phase is odd
X (e j )  X (e j )
www.company.com
8
06/10/2016
Fourier Transform Theorems
Company
LOGO
• Linearity
FT
FT
x1 (n) 
X 1 (e j ), x2 (n) 
X 2 (e j )
F
ax1 (n)  bx2 (n) 
aX 1 (e j )  bX 2 (e j )
• Example:Find fourier transform of
n
x( n)  2 x1 (n)  3 x2 (n)
n
1
x1 (n)    u (n)
2
1
x2 ( n )    u ( n )
 3
www.company.com
Company
LOGO
Fourier Transform Theorems
F
• Time Shifting x(n  nd ) 
e  jn X (e j )
d
F [ x(n  nd )] 

 x(n  n
d
n  


 x(n)e
)e  jn
 j ( n  n d )
n  
 e  jnd

 x(n)e
 jn
n  
 e jnd X (e j )
 Phase Change
www.company.com
9
06/10/2016
Fourier Transform Theorems
Company
LOGO
FT
• Frequency Shifting x (n) 
X (e j )
F
e j0n x(n) 

X (e j (0 ) )
F [e
j 0 n
x(n)] 


e
j0 n
n  

x(n)e  jn
 x ( n )e
 j (   0 ) n
n  
 X (e j (0 ) )
www.company.com
Company
LOGO
Fourier Transform Theorems
• Time Reversal
F
x (n) 
X ( e j )
F
x(n) 

X (e j )

F [ x(n)] 
 x (  n )e
n  


 x ( n )e
 jn
 j (  ) n
n  
 X (e  j )
www.company.com
10
06/10/2016
Fourier Transform Theorems
Company
LOGO
FT
X (e j )
• Differentiation in Frequency x(n) 
F
nx(n) 
j
F [nx(n)] 

 nx(n)e
d
X (e j )
d
 jn
n  
1 
de  jn
x
(
n
)

 j n  
d

d
d
 j
x(n)e  jn  j
X (e jn )

d n  
d

www.company.com
Fourier Transform Theorems
Company
LOGO
• The Convolution Theorem
y ( n) 

 x(k )h(n  k )  Y (e
F
j
)  X ( e j ) H ( e j )
k  
F [ y (n)] 

 y ( n)e
 jn

n  
k  

 

  x(k )h(n  k ) e  jn

n    k  


 x(k )  h(n  k )e


k  


n  

 jn






 x(k )  h(n)e
n  
 j ( n  k )





 j k 
x
(
k
)
e
h ( n ) e  j n 



k  
 n  


 X (e j ) H (e j )
www.company.com
11
06/10/2016
Fourier Transform Theorems
Company
LOGO
• The Modulation or Window Theorem
F
y (n)  x(n) w(n) 
Y (e j ) 

Y ( e j ) 
 w(n) x(n)e
 jn
n  
www.company.com

1 
X (e j )W (e j ( ) )d



2

1 
w(n)  X (e j )e jn d e  jn

 

2 n  

1  

  w(n) X (e j )e  j ( ) n d



2  n  


1
2

1
2


j 
X
(
e
)
w(n)e  j (  ) n d



 n  



  X (e
j

)W (e j (  ) )d
Fourier Transform Theorems
Company
LOGO
• Parseval’s Theorem

1
 x ( n) y * ( n)  2 

n  
Facts:

X (e j )Y * (e j )d
F
x(n) 
X ( e j )
F
y * (n) 
Y * ( e  j )
F
y (n)  x(n) w(n) 

Y (e j ) 


x(n) y * (n)e  j 
n  
www.company.com
1 
X (e j )W (e j ( ) )d
2 
1  
X (e j )Y (e  j ( ) )d




2 n 
Letting =0, then proven.
12
06/10/2016
Fourier Transform Theorems
Company
LOGO
• Parseval’s Theorem- Energy Preserving

1
 | x ( n) |  2 
2
n  

 | x ( n) |

2
n  


| X (e j ) |2 d

 x ( n) x * ( n)
n  

1 
X (e j )X * (e j )d
2  
Energy spectral density
www.company.com

1 
| X (e j ) |2 d
2 
j
s xx  X (e )
2
Fourier Transform
Sinusoidal and Complex Exponential
Sequences

Play an important role in DSP
Company
LOGO
 h( k ) x ( n  k )
y ( n)   h( k ) x ( n  k )
y ( n) 
x(n)  e jn
LTI
k  
k  
h(n)


 h( k )e
j ( nk )
k  
 

   h(k )e  jk e jn
 k  

 H (e j )e jn
www.company.com
13
06/10/2016
Company
LOGO
x ( n )  e j o n
LTI
y ( n) 

 h( k ) x ( n  k )
k  
h(n)
 H (e j0 )e jn0
y ( n )  H ( e j 0 ) x ( n )
www.company.com
Fourier Transform
Frequency Response
Company
LOGO
Frequency Response
X ( e j )
Y (e j )  X (e j ).H (e j )
H ( e j )
Y ( e j )
H (e ) 
X ( e j )
j
www.company.com
14
06/10/2016
Fourier Transform
Frequency Response
Company
LOGO
Frequency Response
j
H (e ) 

 h ( n )e
 jn
n  
H (e j )  H R (e j )  jH (e j )
H (e j ) | H (e j ) | eH ( e
j
)
phase
magnitude
www.company.com
Fourier Transform
Frequency Response
Company
LOGO
Example: Ideal Lowpass Filter
H ( e j )

 c
c

1 |  | c
H ( e j )  
0 c    
www.company.com
1 c
H (e j )e jn d



c
2
1 c jn

e d
2 c
1 c jn

e d ( jn)
2 jn c
h( n) 

1

e jn
2 jn

c
 c
sin c n
n
15
06/10/2016
Fourier Transform
Frequency Response
Company
LOGO
Example: Ideal Lowpass Filter
h( n) 
sin c n
n
n  0,1,2,
The ideal lowpass fileter
Is noncausal.
0.6
0.4
0.2
0
-0.2 www.company.com
-60
-40
-20
0
20
40
60
Fourier Transform
Frequency Response
Company
LOGO
Example: Ideal Lowpass Filter
h( n) 
sin c n
n
n  0,1,2,
TheTo
ideal
lowpass fileter
approximate
the
Is noncausal.
ideal lowpass filter
using a window.
0.6
j
sin c n  jn
e
0.2
n
n M
0.4 M
H (e ) 

0
-0.2
-60
-40
www.company.com
-20
0
20
40
60
16
06/10/2016
Fourier Transform Theorems
Company
LOGO
Example: Ideal Lowpass Filter
2
M=3
1
0
-1
-4
-3
-2
-1
0
1
2
3
4
3
4
3
4
2
M=5
H ( e j ) 
sin c n  jn
e

n
n M
1
M

0
-1
-4
-3
-2
-1
0
1
2
2
M=19
1
0
www.company.com
-1
-4
-3
-2
-1
0
1
2
17