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MEH329DIGITAL SIGNAL PROCESSING
Dept. Of Electronics & Telecomm. Eng.Kocaeli University
-3-Discrete Time Systems
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Discrete-Time Systems
2MEH329 Digital Signal Processing
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Discrete-Time SystemsExample: Ideal Delay
3MEH329 Digital Signal Processing
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• For and , the input sequence:
4MEH329 Digital Signal Processing
Discrete-Time SystemsExample: Moving Average
1 1M 2 1M
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5MEH329 Digital Signal Processing
Discrete-Time SystemsExample: Accumulator
n
k
y n x k
1
1
n
k
y n x n x k
x n y n
1
0
0
1
n
k k
n
k
y n x k x k
y x k
or
initial condition
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Discrete-Time SystemsMemoryless Systems
• A system memoryless if the output y[n] depends only on x[n] at the same n.
6MEH329 Digital Signal Processing
2y n x n , 0d dy n x n n n
(Memoryless) (Not Memoryless)
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MEH329 Digital Signal Processing 7
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Discrete-Time SystemsLinear Systems
8MEH329 Digital Signal Processing
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Discrete-Time SystemsLinear Systems
9MEH329 Digital Signal Processing
system
system
1x n
2x n
1y n
2y n
a
b w n
SUPERPOSITION = ADDITIVITY + HOMOGENEITY
if
system LINEAR!
w n y n
a
b
1x n
2x n
system y n x n
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Discrete-Time SystemsLinearity Example: Ideal Delay System
10MEH329 Digital Signal Processing
[ ] [ ]oy n x n n
1 1 0
2 2 0
1 2
1 0 2 0
y n x n n
y n x n n
w n ay n by n
ax n n bx n n
1 2
0
1 0 2 0
x n ax n bx n
y n x n n
ax n n bx n n
the system is LINEAR!
w n y n
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Discrete-Time SystemsLinearity Example
11MEH329 Digital Signal Processing
[ ] [ ] 1y n x n
1 1
2 2
1 2
1 2
1
1
y n x n
y n x n
w n ay n by n
ax n a bx n b
1 2
1 2
1
1
x n ax n bx n
y n x n
ax n bx n
the system is NOT LINEAR!
w n y n
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Discrete-Time SystemsLinearity Example
MEH329 Digital Signal Processing 12
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Discrete-Time SystemsTime Invariant Systems
• A system is time invariant if a time shift ordelay of the input sequence causes acorresponding shift in the output sequence.
13MEH329 Digital Signal Processing
T x n y n
0 0T x n n y n n
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Discrete-Time SystemsTime Invariant Systems
14MEH329 Digital Signal Processing
delay
system
x n w nsystem
delay dy n n
dx n n
y n
if
the system TIME INVARIANTdw n y n n
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Discrete-Time SystemsTime Invariance Example: Ideal Delay System
15MEH329 Digital Signal Processing
[ ] [ ]oy n x n n
0dw n x n n n
0
0d d
y n x n n
y n n x n n n
the system is TIME INVARIANT!
w n y n
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Discrete-Time SystemsTime Invariance: Example
16MEH329 Digital Signal Processing
[ ] [ ]ny n a x n
ndw n a x n n
d
n
n nd d
y n a x n
y n n a x n n
the system is NOT TIME INVARIANT!
w n y n
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Discrete-Time SystemsTime Invariance: Example
17MEH329 Digital Signal Processing
[ ] [2 ]y n x n
2 dw n x n n
2
2d d
y n x n
y n n x n n
the system is TIME VARIANT!
w n y n
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Discrete-Time SystemsCausal Systems
• A system is causal if the output at n dependsonly on the input at n and earlier inputs.
• Backward difference system:
• Forward difference system:
18MEH329 Digital Signal Processing
1y n x n x n
1y n x n x n
causal
not causal
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Discrete-Time SystemsCausal Systems
MEH329 Digital Signal Processing 19
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Nedensel Sistemler
Nedensel olmayan bir sistem çıkışın uygun miktarda geciktirilmesiyle nedensel bir sistem haline getirilebilir.
Örneğin nedensel olmayan 2 ile aradeğerleme denklemini ele alalım.
Yukarıdaki sistemin nedensel hali
ile verilir. Nedensel denklem, nedensel olmayan denklemde n yerine n-1 yazılarak (veya eşdeğer olarak çıkış bir birim geciktirilerek) elde edilmiştir.
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Discrete-Time SystemsStable Systems
• A system is stable if every bounded inputsequence produces a bounded outputsequence.
• Bounded input:
• Bounded output:
21MEH329 Digital Signal Processing
xx n B
yy n B
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MEH329 Digital Signal Processing 22
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Discrete-Time SystemsStability: Example
23MEH329 Digital Signal Processing
n
k
y n x k
0 , 0
1 , 0
n
k
ny n u k
n n
Output has no finite upper bound. Therefore, the system gives unbounded output for
bounded signal
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Discrete-Time SystemsStability: Example
MEH329 Digital Signal Processing 24
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Discrete-Time SystemsInvertible Systems
• A system is invertible if the input sequence isreconstituted using a system that takes y[n]the as input.
25MEH329 Digital Signal Processing
D D-1 x n y n x n
y1[n]=x[n-1] y2[n]=x[n+1] x n 1y n 2y n x n
Example:
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• Example
MEH329 Digital Signal Processing 26
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Discrete-Time SystemsLTI Systems
• Linear Time-Invariant (LTI) Systems:If the linearity property is combined with therepresentation of a general sequence as alinear combination of delayed impulses, thenit follows that a LTI system can be completelycharacterized by its impulse response.
27MEH329 Digital Signal Processing
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Discrete-Time SystemsLTI Systems
MEH329 Digital Signal Processing 28
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Discrete-Time SystemsLTI Systems
29MEH329 Digital Signal Processing
k
x n x k n k
y n T x n k
y n T x k n k
k
y n x k T n k
0 0
D
D
D
x n y n
n h n
n n h n n
k
y n x k h n k
Convolution sum:
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MEH329 Digital Signal Processing 30
The relationship of an LTI system’s response with the input signal and the impulse response of the system is named as ‘‘convolution’
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Discrete-Time SystemsLTI 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.
31MEH329 Digital Signal Processing
0 0 1y x
1 1 0 1.1 0 1 1.1 1.1y x y
2 2 1 1.1 0 1.1 1.1 1.21y x y
1 1.1 0 1 1.1 1.1n
y n x n y n y n
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Discrete-Time SystemsLTI Systems Example: Bank Account
• If we consider 1 TL as unit impulse signal:
32MEH329 Digital Signal Processing
0
1.1
k
n k
k
y n x k h n k
x k
10 3 2 5 5x n n n n
10 0 10 2 10 510 0 1.1 2 1.1 5 1.1
10 2.594 3 2.144 5 1.611 27.563 TL
y x x x
For example:
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Discrete-Time SystemsConvolution: Analytical Example
33MEH329 Digital Signal Processing
1 20.1 , 0.2n n
x n u n x n u n
3 1 2 ?x n x n x n
3 1 2 0.1 0.2k n k
k k
x n x k x n k u k u n k
What are the limits of this summation?
0 k n
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Discrete-Time SystemsConvolution: Example
34MEH329 Digital Signal Processing
30
0.1 0.2n
k n k
k
x n
30 0
0.2 0.1 0.2 0.2 0.5n n
n k k n k
k k
x n
1 0
3
0.5 0.50.2
0.5 1
2 0.2 0.1
nn
n n
x n u n
u n
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Discrete-Time SystemsConvolution: Example
35MEH329 Digital Signal Processing
The output of an LTI system can be obtained as the superposition of responses to individual samples of the input. This approach is shown to estimate y[n] in the case of x[n] and h[n] given in the following:
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Discrete-Time SystemsConvolution: Example
36MEH329 Digital Signal Processing
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Discrete-Time SystemsConvolution: Example
37MEH329 Digital Signal Processing
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MEH329 Digital Signal Processing 38
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MEH329 Digital Signal Processing 39
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Discrete-Time SystemsConvolution
40MEH329 Digital Signal Processing
• Calculate the x[k]h[n-k] for each n to obtainoutput signal y[n].
• For example:
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Discrete-Time SystemsConvolution: Analytical Example
41MEH329 Digital Signal Processing
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MEH329 Digital Signal Processing 42
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MEH329 Digital Signal Processing 43
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MEH329 Digital Signal Processing 44
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45MEH329 Digital Signal Processing
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MEH329 Digital Signal Processing 46
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Discrete-Time SystemsProperties of LTI Systems
47MEH329 Digital Signal Processing
• Commutative:
• Distributive over addition:
• Associative:
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MEH329 Digital Signal Processing 48
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Discrete-Time SystemsProperties of LTI Systems
49MEH329 Digital Signal Processing
• Cascade Connection:
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Discrete-Time SystemsProperties of LTI Systems
50MEH329 Digital Signal Processing
• Parallel Connection:
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Basit Bağlama Biçimleri
Aşağıda verilen ayrık-zaman sisteminin eşdeğer impuls yanıtını bulalım.
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Basit Bağlama Biçimleri
Seri ve paralel bağlamanın özelliklerinden yararlanarak sistemi aşağıda gösterildiği gibi basitleştirebiliriz.
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Basit Bağlama Biçimleri
Eşdeğer impuls yanıtı h[n]
ile verilir. Yukarıdaki iki konvolüsyon terimini hesaplayalım.
O halde,
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Discrete-Time SystemsProperties of LTI Systems- Stability
54MEH329 Digital Signal Processing
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Discrete-Time SystemsProperties of LTI Systems
55MEH329 Digital Signal Processing
• For example: the ideal delay system is stablesince:
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Discrete-Time SystemsProperties of LTI Systems
56MEH329 Digital Signal Processing
• Moving average filter is stable since S is thesum of a finite number of finite valuedsamples:
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Discrete-Time SystemsProperties of LTI Systems
57MEH329 Digital Signal Processing
• The accumulator system:
is unstable since
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MEH329 Digital Signal Processing 58
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Discrete-Time SystemsProperties of LTI Systems
59MEH329 Digital Signal Processing
• Causality: A LTI system is causal if an only if
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MEH329 Digital Signal Processing 60
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MEH329 Digital Signal Processing 61
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MEH329 Digital Signal Processing 62
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Discrete-Time SystemsProperties of LTI Systems
63MEH329 Digital Signal Processing
• 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 SystemsProperties of LTI Systems
64MEH329 Digital Signal Processing
• FIR Examples:– Ideal delay, moving average filter, forward and
backward systems…– STABLE
• IIR Examples:– Accumulator, filters …– STABLE/UNSTABLE
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Discrete-Time SystemsProperties of LTI Systems
65MEH329 Digital Signal Processing
• Stability of an IIR system:
• The system is stable since