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Discrete-time Systems

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Discrete-time Systems. Prof. Siripong Potisuk. Input-output Description. A DT system transforms DT inputs into DT outputs. System Interconnection. - Build more complex systems - Modify response of a system. Response of an LTI System. (Also referred to as Impulse response). - PowerPoint PPT Presentation
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Discrete-time Systems Prof. Siripong Potisuk
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Page 1: Discrete-time Systems

Discrete-time Systems

Prof. Siripong Potisuk

Page 2: Discrete-time Systems

Input-output Description

A DT system transforms DT inputs into DT outputs

Page 3: Discrete-time Systems

System Interconnection

- Build more complex systems- Modify response of a system

Page 4: Discrete-time Systems

Response of an LTI System

Page 5: Discrete-time Systems
Page 6: Discrete-time Systems
Page 7: Discrete-time Systems

(Also referred to as Impulse response)

Page 8: Discrete-time Systems
Page 9: Discrete-time Systems
Page 10: Discrete-time Systems

Properties of Convolution Sum A discrete-time LTI system is completely characterized by its impulse response, i.e., completely determines its input-output behavior. There is only one LTI system with a given h[n]

The role of h [n] and x [n] can be interchanged

][][

,][][][][][][

nxnh

knrrhrnxknhkxnhnxrk

Commutative Property

Page 11: Discrete-time Systems

The Distributive Property

is equivalent to

Page 12: Discrete-time Systems

The Associative Property

is equivalent to

Page 13: Discrete-time Systems

- The impulse response of a causal LTI system must be zero before the impulse occurs.- Causality for a linear system is equivalent to the condition of initial rest.

0

][][][][][k

n

k

knxkhknhkxny

Stability for LTI Systems:

A necessary and sufficient condition for an LTI system tobe BIBO stable is that the impulse response is absolutelysummable.

Causality for LTI Systems:

Page 14: Discrete-time Systems

Time-domain Description of DT LTI Systems

A general Nth-order linear constant-coefficient differenceequation

M

k

N

kkk knyaknxb

any

0 10

][][1][

Recursive equation, i.e., expresses the output at time n interms of previous values of the input and output

Page 15: Discrete-time Systems

Solutions of LCCDE’s

- The complete solution depends on both the causal input x[n] and the initial conditions, y[-1], y[-2],……, y[-N ].

- The solution can be decomposed into a sum of two parts:

response state-zero therest) (initialonly input the todue is ][

responseinput -zero theinput) (noonly conditions initial the todue is ][

re whe][][][

1

0

10

ny

nynynyny

Page 16: Discrete-time Systems

Finite Impulse Response (FIR) System

][)(][ 0,NFor 0 0

knxabny

M

k

k

The equation is nonrecursive, i.e., previously computedvalues of the output are not used to recursively computethe present value of the output.

The impulse response is seen to have finite duration andgiven by

otherwise0

,0][

,0

Mnnh a

nb

Page 17: Discrete-time Systems

Infinite Impulse Response (IIR) System

recursive isequation difference the1,NFor

If the system is initially at rest, the impulse responsewill have infinite duration.

M

k

N

kkk knyaknxb

any

0 10

][][1][

Page 18: Discrete-time Systems

Example Consider the difference equation

rest. initial assume and ][][ where][]1[21][ nKnxnxnyny

.)21(]1[

21][][

,)21(]1[

21]2[]2[

,)21(]0[

21]1[]1[

,]1[21]0[]0[

2

Knynxny

Kyxy

Kyxy

Kyxy

n

][)21( ][ 1, Setting nunhK n


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