presentation-3 discrete time systemsehm.kocaeli.edu.tr/upload/duyurular/0910181256091b76e.pdf · ]...

MEH329DIGITAL SIGNAL PROCESSING

Dept. Of Electronics & Telecomm. Eng.Kocaeli University

-3-Discrete Time Systems

Discrete-Time Systems

2MEH329 Digital Signal Processing

Discrete-Time SystemsExample: Ideal Delay


• For and , the input sequence:


Discrete-Time SystemsExample: Moving Average

1 1M 2 1M


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.


2y n x n , 0d dy n x n n n

(Memoryless) (Not Memoryless)

MEH329 Digital Signal Processing 7

Discrete-Time SystemsLinear Systems


Discrete-Time SystemsLinear Systems


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


[ ] [ ]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


[ ] [ ] 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


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.


T x n y n

0 0T x n n y n n

Discrete-Time SystemsTime Invariant Systems


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


[ ] [ ]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


[ ] [ ]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


[ ] [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:


1y n x n x n

1y n x n x n

causal

not causal

Discrete-Time SystemsCausal Systems


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:


xx n B

yy n B

Discrete-Time SystemsStability: Example


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


Discrete-Time SystemsInvertible Systems

• A system is invertible if the input sequence isreconstituted using a system that takes y[n]the as input.


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


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.




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:


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.


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:


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


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


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



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


• Calculate the x[k]h[n-k] for each n to obtainoutput signal y[n].

• For example:

Discrete-Time SystemsConvolution: Analytical Example


Discrete-Time SystemsProperties of LTI Systems


• Commutative:

• Distributive over addition:

• Associative:



• Cascade Connection:



• Parallel Connection:

Basit Bağlama Biçimleri

Aşağıda verilen ayrık-zaman sisteminin eşdeğer impuls yanıtını bulalım.


Seri ve paralel bağlamanın özelliklerinden yararlanarak sistemi aşağıda gösterildiği gibi basitleştirebiliriz.


Eşdeğer impuls yanıtı h[n]

ile verilir. Yukarıdaki iki konvolüsyon terimini hesaplayalım.

O halde,

Discrete-Time SystemsProperties of LTI Systems- Stability




• For example: the ideal delay system is stablesince:



• Moving average filter is stable since S is thesum of a finite number of finite valuedsamples:



• The accumulator system:

is unstable since



• Causality: A LTI system is causal if an only if



• 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.



• FIR Examples:– Ideal delay, moving average filter, forward and

backward systems…– STABLE

• IIR Examples:– Accumulator, filters …– STABLE/UNSTABLE



• Stability of an IIR system:

• The system is stable since

presentation-3 discrete time systemsehm.kocaeli.edu.tr/upload/duyurular/0910181256091b76e.pdf · ]...

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