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MEH329DIGITAL SIGNAL PROCESSING
Dept. Of Electronics & Telecomm. Eng.Kocaeli University
-3-Discrete Time Systems
Discrete-Time Systems
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Discrete-Time SystemsExample: Ideal Delay
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• For and , the input sequence:
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Discrete-Time SystemsExample: Moving Average
1 1M 2 1M
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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
Discrete-Time SystemsMemoryless Systems
• A system memoryless if the output y[n] depends only on x[n] at the same n.
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2y n x n , 0d dy n x n n n
(Memoryless) (Not Memoryless)
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Discrete-Time SystemsLinear Systems
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Discrete-Time SystemsLinear Systems
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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
Discrete-Time SystemsLinearity Example: Ideal Delay System
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[ ] [ ]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
Discrete-Time SystemsLinearity Example
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[ ] [ ] 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
Discrete-Time SystemsLinearity Example
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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.
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T x n y n
0 0T x n n y n n
Discrete-Time SystemsTime Invariant Systems
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delay
system
x n w nsystem
delay dy n n
dx n n
y n
if
the system TIME INVARIANTdw n y n n
Discrete-Time SystemsTime Invariance Example: Ideal Delay System
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[ ] [ ]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
Discrete-Time SystemsTime Invariance: Example
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[ ] [ ]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
Discrete-Time SystemsTime Invariance: Example
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[ ] [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
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:
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1y n x n x n
1y n x n x n
causal
not causal
Discrete-Time SystemsCausal Systems
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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.
Discrete-Time SystemsStable Systems
• A system is stable if every bounded inputsequence produces a bounded outputsequence.
• Bounded input:
• Bounded output:
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xx n B
yy n B
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Discrete-Time SystemsStability: Example
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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
Discrete-Time SystemsStability: Example
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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.
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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:
• Example
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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.
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Discrete-Time SystemsLTI Systems
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Discrete-Time SystemsLTI Systems
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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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The relationship of an LTI system’s response with the input signal and the impulse response of the system is named as ‘‘convolution’
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.
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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
Discrete-Time SystemsLTI Systems Example: Bank Account
• If we consider 1 TL as unit impulse signal:
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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:
Discrete-Time SystemsConvolution: Analytical Example
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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
Discrete-Time SystemsConvolution: Example
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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
Discrete-Time SystemsConvolution: Example
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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:
Discrete-Time SystemsConvolution: Example
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Discrete-Time SystemsConvolution: Example
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Discrete-Time SystemsConvolution
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• Calculate the x[k]h[n-k] for each n to obtainoutput signal y[n].
• For example:
Discrete-Time SystemsConvolution: Analytical Example
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Discrete-Time SystemsProperties of LTI Systems
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• Commutative:
• Distributive over addition:
• Associative:
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Discrete-Time SystemsProperties of LTI Systems
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• Cascade Connection:
Discrete-Time SystemsProperties of LTI Systems
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• Parallel Connection:
Basit Bağlama Biçimleri
Aşağıda verilen ayrık-zaman sisteminin eşdeğer impuls yanıtını bulalım.
Basit Bağlama Biçimleri
Seri ve paralel bağlamanın özelliklerinden yararlanarak sistemi aşağıda gösterildiği gibi basitleştirebiliriz.
Basit Bağlama Biçimleri
Eşdeğer impuls yanıtı h[n]
ile verilir. Yukarıdaki iki konvolüsyon terimini hesaplayalım.
O halde,
Discrete-Time SystemsProperties of LTI Systems- Stability
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Discrete-Time SystemsProperties of LTI Systems
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• For example: the ideal delay system is stablesince:
Discrete-Time SystemsProperties of LTI Systems
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• Moving average filter is stable since S is thesum of a finite number of finite valuedsamples:
Discrete-Time SystemsProperties of LTI Systems
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• The accumulator system:
is unstable since
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Discrete-Time SystemsProperties of LTI Systems
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• Causality: A LTI system is causal if an only if
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Discrete-Time SystemsProperties of LTI Systems
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• 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.
Discrete-Time SystemsProperties of LTI Systems
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• FIR Examples:– Ideal delay, moving average filter, forward and
backward systems…– STABLE
• IIR Examples:– Accumulator, filters …– STABLE/UNSTABLE
Discrete-Time SystemsProperties of LTI Systems
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• Stability of an IIR system:
• The system is stable since