Hossein Sameti
Department of Computer Engineering
Sharif University of Technology
• Eigen vector of matrix A:


AX  X
• In other words, once matrix A is multiplied by vector X,
the direction of X is preserved.
• Eigen function of a system:
Ф(n)
System
αФ(n)
Hossein Sameti, CE, SUT, Fall 1992
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cosn  j sin n
x(n)
LTI System
e jn
y(n)
H ( )e jn


k  
k  
y ( n)   x ( k ) h( n  k )   h( k ) x ( n  k )

H ( )   h(k )e  jk
k  

y ( n )   h ( k )e
k  
j ( n  k )

(  h(k )e  jk )e jn
k  
Frequency
response
y (n)  H ( )e jn
jn
H ( ) magnifies the input e
based on freq ω.
Clarification: Some textbooks use H (e j ) instead of
H(w ).
Hossein Sameti, CE, SUT, Fall 1992
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
H ( )   h(k )e  jk
1
k  


H (  2 )   h(k )e  j (  2 ) k   h(k )e  jk e  j 2k  H ( )
k  
k  
• Frequency response is periodic with the period of 2π. Implication?
Hossein Sameti, CE, SUT, Fall 1992
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cos 0 n :
Hossein Sameti, CE, SUT, Fall 1992
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
H ( )   h(k )e
 jk
k  
H ( ) 


  h ( k ) e  j k   h ( k )
k  
k  

 h( k )
k  
The same condition as the
stability condition
Hossein Sameti, CE, SUT, Fall 1992
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

DTFT{x(n)}  X ( )   x(n)e
Same mathematical representation
as the freq. response
 jn
n  
• Existence of DTFT:

 x ( n)  
n  
• Inverse DTFT:
x(n) is absolutely summable.
x(n) 
1
2

jn
X
(

)
e
d



Fourier analysis considers signals to be constructed from a sum of complex
exponentials with appropriate frequencies, amplitudes and phase.

Frequency components are the complex exponentials which, when added
together, make up the signal.
Hossein Sameti, CE, SUT, Fall 1992
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
IDTFT of the ideal low-pass filter:

  c
1
X ( )  

0 c    

c

c
x(n)  1  X ( )e jn d  1  e jn d
2
2
1
e jc n  e jc n
jn c
x(n) 
[e ]c 
j 2n
j 2n
x ( n) 
sin c n
n
Hossein Sameti, CE, SUT, Fall 1992
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sin(𝜔𝑐 𝑛)
1,
⟺𝑋 𝜔 =
0,
𝑛
|𝜔| ≤ 𝜔𝑐
𝜔𝑐 < 𝜔 ≤ 𝜋
Hossein Sameti, CE, SUT, Fall 1992
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x ( n)  a n u ( n )

n  jn
X ( )   a e
n 0
DTFT?

  (ae j ) n 
X ( ) 
n 0
1
1  ae  j
a 1
What happens if a>1?
1  a cos 
a sin 

j
1  a 2  2a cos 
1  a 2  2a cos 
X R ()
X I ()
Hossein Sameti, CE, SUT, Fall 1992
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x ( n)  a n u ( n )
X ( ) 
1
1  ae j
X ()  X () e jX ( )
2
X ( )  X ( ) X * ( )

1
1  a 2  2a cos 
X ( )  arctan(
a sin 
)
1  a cos 
Hossein Sameti, CE, SUT, Fall 1992
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Hossein Sameti, CE, SUT, Fall 1992
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Hossein Sameti, CE, SUT, Fall 1992
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x1(n)  X1()
x2 (n)  X 2 ()
• Linearity: a1x1(n)  a2 x2 (n)  a1X1()  a2 X 2 ()
• Time-shifting:
• Time-reversal:
x(n  k )  e jk X ()
x(n)  X ( )
• Convolution :
x(n)
LTI System
h(n)
y ( n)  x ( n ) * h( n)
y(n)
Y ( )  X ( ) H ( )
Hossein Sameti, CE, SUT, Fall 1992
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• Cross-correlation:
y(n)  x1(n) * x2 (n)
Y ()  X1() X 2 ()
•Frequency Shifting:
e j0 n x(n)
X (  0 )
•Parseval’s Theorem:

*
*
1
x
(
n
)
y
(
n
)

X
(

)
Y
( )d


2
2


2
y ( n)  x ( n)
 x(n)  21  X ( )

Hossein Sameti, CE, SUT, Fall 1992
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• Modulation:
x(n) cos 0n
1
2
X (  0 )  12 X (  0 )
• Multiplication:
x1(n) x2 (n)
1
2

 X1 ( )X 2 (   )d

• Differentiation in the freq. domain:
nx(n)
j
dX ( )
d
• Conjugation:
x* ( n )
X * ( )
Hossein Sameti, CE, SUT, Fall 1992
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• Conjugate Symmetric:
x(n)  x* (n)
• Conjugate Anti-Symmetric:
x(n)   x* (n)
• Why are these properties important?
Conjugate Symmetric
xe (n)  12 [ x(n)  x* (n)]
x(n)  xe (n)  xo (n)
Conjugate Anti-symmetric
xo (n)  12 [ x(n)  x* (n)]
X e ()  12 [ X ()  X * ()]
X ()  X e ()  X o ()
X o ()  12 [ X ()  X * ()]
Hossein Sameti, CE, SUT, Fall 1992
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x (n )
X ( )
x* ( n )
X * ( )
x* (n)
X * ( )
Re{ x(n)}
X e ()
j Im{x(n)}
X o ()
xe (n)
xo (n)
X R ()  Re{ X ()}
jX I ()  j Im{X ()}
Hossein Sameti, CE, SUT, Fall 1992
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x (n ) : real
x(n)  Re{ x(n)}
X ()  12 [ X ()  X * ()]
X ()  X e ()
X ( )  X * ( )
• If a sequence is real, then its DTFT is conjugate
symmetric.
Hossein Sameti, CE, SUT, Fall 1992
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x (n )
X ( )
x (n ) : real
X ( )  X * ( )
x (n ) : real
X R ()  X R ()
x (n ) : real
X I ()   X I ()
x (n ) : real
X ( )  X ( )
x (n ) : real
X ( )  X ( )
Hossein Sameti, CE, SUT, Fall 1992
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Proakis, et.al
Hossein Sameti, CE, SUT, Fall 1992
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Determining an inverse fourier transform
X (e
X (e
j
j
1
)
(1  ae  j  )(1  be  j  )
a /(a  b ) b /(a  b )
)

 j
1  ae
1  be  j 
b
 a

n
x [n ]  (
)a  (
)b n  u [n ]
a b
 a b

Hossein Sameti, CE, SUT, Fall 1992
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Determining the Impulse response from the frequency
response
Hossein Sameti, CE, SUT, Fall 1992
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Determining the Impulse response for a Difference
1
1
Equation
y [n ]  y [n  1]  x [n ]  x [n  1]
2
4
To find the impulse response h[n], we set x [n ]   [n ]
1
1
h [n ]  h [n  1]   [n ]   [n  1]
2
4
Applying the DTFT to both sides of equation. We obtain
1
1
H (e j  )  e  j  H (e j  )  1  e  j 
2
4
1
1 e  j
4
H (e j  ) 
1
1 e  j
2
Hossein Sameti, CE, SUT, Fall 1992
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Example:
x[n]  a nu[n  5]
1
 a u[n] 
1  ae j
 j 5
e
 a n5u[n  5] 
1  ae j
a 5e  j 5
n
 a u[n  5] 
1  ae j
n
Hossein Sameti, CE, SUT, Fall 1992
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

Reviewed Discrete-time Fourier Transform, some of its
properties and FT pairs
Next: the Z-transform
Hossein Sameti, CE, SUT, Fall 1992
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