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)

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

The Distributive Property

is equivalent to

The Associative Property

is equivalent to

- 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:

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

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 the

rest) (initialonly input the todue is ][

responseinput -zero the

input) (noonly conditions initial the todue is ][

re whe][][][

1

0

10

ny

ny

nynyny

Finite Impulse Response (FIR) System

][)(][ 0,NFor 0 0

knxa

bny

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

otherwise

0

,0][

,0

Mnnh a

nb

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

][

Example Consider the difference equation

rest. initial assume and ][][ where][]1[2

1][ nKnxnxnyny

.)2

1(]1[

2

1][][

,)2

1(]1[

2

1]2[]2[

,)2

1(]0[

2

1]1[]1[

,]1[2

1]0[]0[

2

Knynxny

Kyxy

Kyxy

Kyxy

n

][)2

1( ][ 1, Setting nunhK n