17.10.2014
MEH329
DIGITAL SIGNAL PROCESSING
-3Discrete Time Systems
Instructor: Assoc.Prof.Dr. M. Kemal GÜLLÜ
Dept. Of Electronics & Telecomm. Eng.
Kocaeli University
http://akademikpersonel.kocaeli.edu.tr/kemalg/
Discrete-Time Systems
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Discrete-Time Systems
Example: Ideal Delay
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Discrete-Time Systems
Example: Moving Average
• For M1  1 and M 2  1 , the input sequence:
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Discrete-Time Systems
Example: Accumulator
y  n 
n
 x k 
k 
y  n  x  n 
n 1
 x k 
k 
 x  n   y  n  1
y  n 
or
initial condition
1
n
k 
k 0
 x k    x k 
n
 y  1   x  k 
k 0
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Discrete-Time Systems
Memoryless Systems
• A system memoryless if the output y[n]
depends only on x[n] at the same n.
y  n  x 2  n
(Memoryless)
y  n  x  n  nd  , nd  0
(Not Memoryless)
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Discrete-Time Systems
Linear Systems
• A system is linear if and only if superposition
principle is guaranteed.
– Superposition = Additivity + Homogenity (Scaling)
T x1  n  y1  n
T  x2  n  y2  n
T ax1  n  bx2  n  ay1  n  by2  n
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Discrete-Time Systems
Linear Systems
x1  n 
x2  n
x1  n 
x2  n
y1  n
a
system
b
system
y2  n
if w n  y  n 
system LINEAR!
a
x  n
b
w n
system
y  n
SUPERPOSITION = ADDITIVITY + HOMOGENEITY
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Discrete-Time Systems
Linearity Example: Ideal Delay System
y[n]  x[n  no ]
y1  n   x1  n  n0 
x  n   ax1  n   bx2  n 
y2  n   x2  n  n0 
y  n   x  n  n0 
w  n   ay1  n   by2  n 
 ax1  n  n0   bx2  n  n0 
 ax1  n  n0   bx2  n  n0 
w n  y  n 
the system is LINEAR!
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Discrete-Time Systems
Linearity Example
y[n]  x[n]  1
y1  n   x1  n   1
x  n   ax1  n   bx2  n 
y2  n   x2  n   1
y  n  x  n  1
 ax1  n   bx2  n   1
w  n   ay1  n   by2  n 
 ax1  n   a  bx2  n   b
w n   y  n 
the system is NOT LINEAR!
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Discrete-Time Systems
Time Invariant Systems
• A system is time invariant if a time shift or
delay of the input sequence causes a
corresponding shift in the output sequence.
T  x  n  y  n
T x  n  n0   y  n  n0 
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Discrete-Time Systems
Time Invariant Systems
x  n
x  n  nd 
system
delay
system
y  n
delay
w n
y  n  nd 
if w n  y  n  nd 
the system TIME INVARIANT
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Discrete-Time Systems
Time Invariance Example: Ideal Delay System
y[n]  x[n  no ]
w n  x  n  nd   n0 
y  n  x  n  n0 
y  n  nd   x  n  n0  nd 
w n   y  n 
the system is TIME INVARIANT!
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Discrete-Time Systems
Time Invariance: Example
y[n]  a n x[n]
w n  a x  n  nd 
n
y  n  a n x  n
y  n  nd   a n nd x  n  nd 
w n   y  n 
the system is NOT TIME INVARIANT!
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Discrete-Time Systems
Time Invariance: Example
y[n]  x[2n]
w n  x  2n  nd 
y  n   x  2n 
y  n  nd   x  2  n  nd  
w n   y  n 
the system is TIME VARIANT!
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Discrete-Time Systems
Causal Systems
• A system is causal if the output at n depends
only on the input at n and earlier inputs.
• Backward difference system:
y  n  x  n  x  n  1
causal
• Forward difference system:
y  n  x  n  x  n  1
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not causal
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Discrete-Time Systems
Stable Systems
• A system is stable if every bounded input
sequence produces a bounded output
sequence.
• Bounded input:
x  n  Bx  
• Bounded output:
y  n   By  
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Discrete-Time Systems
Stability: Example
y  n 
y  n 
n
n
 x k 
k 
0
,n0

,n0
 u  k   n  1
k 
Output has no finite upper bound. Therefore,
the system gives unbounded output for
bounded signal
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Discrete-Time Systems
Invertible Systems
• A system is invertible if the input sequence is
reconstituted using a system that takes y[n]
the as input.
x  n
D
y  n
D-1
y1[n]=x[n-1]
y1  n
y2[n]=x[n+1]
x  n
Example:
x  n
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y2  n  x  n
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Discrete-Time Systems
LTI Systems
• Linear Time-Invariant (LTI) Systems:
If the linearity property is combined with the
representation of a general sequence as a
linear combination of delayed impulses, then
it follows that a LTI system can be completely
characterized by its impulse response.
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Discrete-Time Systems
LTI Systems
x  n 

 x  k   n  k 
k 
 

y  n  T  x  n  y  n  T   x  k   n  k 
k 


y  n   x  k T   n  k 
k 
D
x  n  
 y  n
Convolution sum:
D
  n  
 h  n

D
  n  n0  
 h  n  n0 
y  n 
 x k  h n  k 
k 
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Discrete-Time Systems
LTI Systems Example: Bank Account
• Bank rate: 10% (yearly)
• Initial money: +1 TL (x[0]=1)
• Find the money at the end of the nth year.
y 0  x 0  1
y 1  x 1  y 0 1.1  0  11.1  1.1
y  2  x  2  y 1 1.1  0  1.11.1  1.21
y  n  x  n  y  n  1 1.1  0  y  n  1 1.1  1.1
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Discrete-Time Systems
LTI Systems Example: Bank Account
• If we consider 1 TL as unit impulse signal:
y  n 

 x k  h n  k 
k 

  x  k  1.1
nk
k 0
For example:
x  n  10  n  3  n  2  5  n  5
y 10  x 0 1.1
10  0
 x  2 1.1
10  2
 x 5 1.1
10 5
 10  2.594   3  2.144  5  1.611  27.563 TL
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Discrete-Time Systems
LTI Systems: Convolution
y  n 

 x k  h n  k 
k 
y  n  x  n  h  n
h[n]: impulse response of LTI system
y  n  n0  

 x k  h n  n
k 
0
 k
 x  n   h  n  n0 
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Discrete-Time Systems
Convolution: Analytical Example
x1  n   0.1 u  n ,
x2  n   0.2 u  n
n
n
x3  n  x1  n  x2  n  ?
x3  n 


k 
x1  k  x2  n  k  

  0.1 u k  0.2
k
k 
nk
u n  k 
What are the limits of this summation?
0k n
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Discrete-Time Systems
Convolution: Example
n
x3  n    0.1  0.2 
k
nk
k 0
x3  n   0.2 
n
n
  0.1  0.2
k
k
  0.2 
k 0
x3  n    0.2 
 0.5
n 1
n
n
  0.5
k
k 0
  0.5 
0
u n
0.5  1
n
n
  2  0.2    0.1  u  n 


n
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Discrete-Time Systems
Convolution: Example
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Discrete-Time Systems
Convolution: Example
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Discrete-Time Systems
Convolution
• Calculate the x[k]h[n-k] for each n to obtain
output signal y[n].
• For example:
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Discrete-Time Systems
Convolution: Analytical Example
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Discrete-Time Systems
Convolution: Analytical Example
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Discrete-Time Systems
Convolution: Analytical Example
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Discrete-Time Systems
Properties of LTI Systems
• Commutative:
• Distributive over addition:
• Associative:
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Discrete-Time Systems
Properties of LTI Systems
• Cascade Connection:
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Discrete-Time Systems
Properties of LTI Systems
• Parallel Connection:
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Discrete-Time Systems
Properties of LTI Systems
• Stability: All LTI systems are stable if and only
if:
• Proof:
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Discrete-Time Systems
Properties of LTI Systems
• Hence, y[n] is bounded if the eq. above is
ensured.
• For example: the ideal delay system is stable
since:
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Discrete-Time Systems
Properties of LTI Systems
• Moving average filter is stable since S is the
sum of a finite number of finite valued
samples:
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Discrete-Time Systems
Properties of LTI Systems
• The accumulator system:
is unstable since
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Discrete-Time Systems
Properties of LTI Systems
• Causality: A LTI system is causal if an only if
• The ideal delay system is causal if
nd  0
• The moving average system is causal if
 M1  0 & M 2  0
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Discrete-Time Systems
Properties of LTI Systems
• Fınıte Impulse Response (FIR) Systems:
– Systems with only a finite of nonzero values in
h[n] are called FIR systems.
• Infınıte Impulse Response (IIR) Systems:
– Systems with infinite length of nonzero values in
h[n] are called IIR systems.
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Discrete-Time Systems
Properties of LTI Systems
• FIR Examples:
– Ideal delay, moving average filter, forward and
backward systems…
– STABLE
• IIR Examples:
– Accumulator, filters …
– STABLE/UNSTABLE
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Discrete-Time Systems
Properties of LTI Systems
• Stability of an IIR system:
• The system is stable since
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