Chapter 2.pdf

Prepared by Dr Nidal Kamel 1

Chapter 2

Linear Time-Invariant Systems


Introduction

Linearity and time invariance, play a fundamental role in

signal and systems analysis for two reasons:

Many physical processes can be modeled as linear time-invariant

(LTI) systems.

LTI systems can be analyzed intensively providing both insight into

their properties and a set of powerful tools form the core of signal

and system analysis.


Introduction

LTI systems are amenable to analysis because it possess the

superposition property.

This fact will allow us to develop complete characterization

of any LTI system in term of its response to unit impulse.

Such representation is referred to as convolution sum in

DT systems and convolution integral in CT systems.


DT Systems: The Convolution Sum Representation of DT Signals in term of impulses

1n 0,

1n ],1[]1[]1[

0n 0,

0n ],0[][]0[

-1n 0,

-1n ],1[]1[]1[

xnx

xnx

xnx

-k

knkx

nxnxnxnx

nxnxnxnx

-δ

...]3[]3[]2[]2[]1[]1[][]0[

]1[]1[]2[]2[]3[]3[...][


DT Systems: The Convolution Sum

Unit Impulse Response and Convolution Sum of LTI

If h[n] is the output of the system to δ[n] and hk [n] is the output

to δ[n-k], then the output of linear system to x[n] is:

[ ] δ

... [ 3] [ 3] [ 2] [ 2] [ 1] [ 1]

[0] [ ] [1] [ 1] [2] [ 2] [3] [ 3] ...

k -

x n x k n-k

x n x n x n

x n x n x n x n

3 2 1

1 2 3

[ ] ... [ 3] [ ] [ 2] [ ] [ 1] [ ]

[0] [ ] [1] [ ] [2] [ ] [3] [ ] ..

k

k -

y n x h n x h n x h n

x h n x h n x h n x h n

x k h n

Prepared by Dr Nidal Kamel

DT Systems: The Convolution Sum Unit Impulse Response and Convolution Sum of LTI

1 1

[ ] [ 1] [ 1] [0] [ ] [1] [ 1]

[ ] [ 1] [ ] [0] [ ] [1] [ ]

x n x n x n x n

y n x h n x h n x h n



The input-output relationship of a linear systems is given as:

If the linear system is also time-invariant, then,

Thus the output of LTI system is given

][][ knhnhk

-k

knhkxny ][

-k

knkxnx -δ][

[ ] [ ] [ ]k

y n x k h n k x n h n

8


Consider an LTI system with h[n] and x[n]

as shown in the figure. Find the system

output, y[n].

[ ] [ ] [ ]

[0] [ 0] [1] [ 1]

0.5 [ ] 2 [ 1]

k

y n x k h n k

x h n x h n

h n h n

10


[ ] [ ] [ ]k

y n x k h n k



1, 0 4[ ]

0,

, 0 6[ ]

0,

n

nx n

otherwise

nh n

otherwise



1, 0 4[ ]

0,

, 0 6[ ]

0,

k

kx k

otherwise

kh k

otherwise

0 0

n nn k

k k

y n h n k[ ] [ ]

4 4

0 0

n k

k k

y n h n k[ ] [ ]

4 4

6 6

n k

k n k n

y n h n k[ ] [ ]


CT Systems: The Convolution Integral Representation of CT Signals in Terms of Impulses

If we define

Since ΔδΔ(t) has unit amplitude, we have

otherwise 0,

t0 ,1

)(t

ˆ( ) ( )k

x t x k t k

14

CT Systems: The Convolution Integral

Convolution Integral Representation of LTI systems

The approximate representation of

x(t), is given as

Consequently, the response of

linear system is given as

k

ktkxtx )()(ˆ

)(ˆ)(ˆ thkxty

k

k


CT Systems: The Convolution Integral Convolution Integral Representation of LTI systems

If the system is linear -time invariant (LTI), then

As Δ →0 the summation becomes integral

)()(

)()()(

thtx

dthxty

ˆˆ( ) ( )k

y t x k h t k

16


Let x(t) be the input to LTI system with unit impulse response

h(t), where

)()(

0 ),()(

tuth

atuetx at

)(

)()()(

tuea

de

dthxty

at

ta

11

0

17


Consider the convolution of the

following two signals:

otherwise 0,

20 ,)(

,0

0 ,1)(

Tttth

otherwise

Tttx

18


There are three intervals:

2

2

2 2

( ) ( ) ( )

0, 0

1, 0

2

1( ) , 2

2

1 3, 2 3

2 2

0, 3

y t x h t d

t

t t T

y t Tt T T t T

t Tt T T t T

T

t


Properties of Linear-Time Invariant System

The LTI systems are represented in terms of their unit impulse

responses.

[ ] [ ] [ ] [ ] [ ]

( )

k

y n x k h n k x n h n

y t x h t d x t h t


Properties of Linear-Time Invariant System Commutative Property

Convolution is a commutative operation.

k

x n h n h n x n h k x n k

x t h t h t x t h x t d


Properties of Linear-Time Invariant System Distributive Property

Convolution is distributive over addition:

Parallel interconnected LTI systems

1 2 1 2

1 2 1 2

x n h n h n x n h n x n h n

x t h t h t x t h t x t h t

)()()(

)()()()(

)()()(

ththtx

thtxthtx

tytyty

21

21

21



Find the convolution of the following two sequences:

We may use the distributive property

1

[ ] 22

[ ] [ ]

n

nx n u n u n

h n u n

1 2

1 2

1 2

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]

y n x n x n h n

x n h n x n h n

= y n y n

23


11 2

0 0

n nk

k k

y n x k

[ ] [ ]

2

0 0

2 0

2 0

= 2

[ ] [ ]

[ ]

n nk

k k

k

k k

y n x k for n

x k for n

1 2[ ] [ ] [ ]y n y n y n


Properties of Linear-Time Invariant System Associative Property

Convolution is associative:

1 2 1 2

1 2 1 2

x n h n h n x n h n h n

x t h t h t x t h t h t


Properties of Linear-Time Invariant System Systems with and without Memory

System is memoryless if

This is true if

The convolution sum reduces to

[ ] [ ]y n bx n

y t bx t

[ ] [ ] [ ] [ ]y n x n b n bx n

y t x t b t bx t

[ ] [ ]h n b n

h t b t


Properties of Linear-Time Invariant System Invertibility of LTI Systems

The h(t) system is invertible if an

inverse system h1(t) exists.

The overall impulse response is

The same applies to DT system

1( ) ( ) ( )h t h t t

1[ ] [ ] [ ]h n h n n



Consider an LTI with impulse response h[n] = u[n], find the

output of the system.

This system is invertible and its inverse has impulse response

n

k

k

kx

knukxny

][

][][][

1[ ] [ ] [ 1]h n n n



The impulse response of inverse system is

We can verify this result by direct calculation:

1[ ] [ ] [ 1]h n n n

1[ ] [ ] [ ] [ ] [ 1

[ ]* [ ] - [ ]* [ -1]

[ ] - [ -1]

[ ]

h n h n u n n n

u n n u n n

u n u n

n


Properties of Linear-Time Invariant System Causality of LTI Systems

LTI system is causal if

For causal LTI system, the convolution become

( ) 0

[ ] 0 0

h t for t < 0

h n for n .

t

n

k

thxty

knhkxny

)()()(

][][][


Properties of Linear-Time Invariant System Causality of LTI Systems

Both the accumulator and its inverse are causal.

Causality for linear system is equivalent to the condition of

initial rest.

Initial rest: if the input to causal system is zero up to some point

in time, then the output must also be zero up to the same time.

Initial rest = no input no output

][][ nunh

[ ] [ ] [ 1]h n n n


Properties of Linear-Time Invariant System Stability of LTI Systems

A system is stable if BIBO

Consider input to LTI that is bounded in magnitude:

The magnitude of the output is:

[ ]x n B for all n.

[ ]k

y n h k x n k

[ ]

[ ]

k

k

y n h k x n k

B h k for all n



Thus, DT-LTI system is stable if

CT-LTI, the system is stable if

[ ]k

h k

( )h d



Find whether pure time shift system is stable or not.

thus the system is stable.

Find whether accumulator system, h[n] = u[n] is stable or not.

thus system is unstable.

0

0

[ ] 1

( ) ( 1

n n

h n n n

h d t d

0

0

[ ] [ ]

( )

n n

u n u n

u d d


Properties of Linear-Time Invariant System Unit step Response of an LTI

step response is obtained when u[n] is applied at the input

of the system

Thus h[n] can be recovered from s[n] using the relation

[ ] [ ] [ 1] [ ] [ ] [ 1]n u n u n h n s n s n

k

n

k

khknukh

nunhnhnuns

][][][

][][][][][


Properties of Linear-Time Invariant System Unit step Response of an LTI

For the CT system, the unit step response is given as:

In analogy to the DT part,

t

dhdtuh

tuththtuts

)()()(

)()()()()(

dt

tdsth

dt

tdut

)()(

)()(


Causal LTI Systems Described by Differential

and Difference Equations

Important class of CT systems has input-output relationships in

form of linear constant-coefficient differential equations.

A general Nth-order equation is given by

In case of N = 0, we have

0 0

( ) ( )k kN M

k kk kk k

d y t d x ta b

dt dt

00

1 ( )( )

kM

k kk

d x ty t b

a dt




auxiliary conditions are required for determination of input-

output relationship.

Different auxiliary conditions result in different input-output

relationships.

In practical systems we use auxiliary condition of initial rest.

Under condition of initial rest, the system is causal and LTI.

1

0 0

0 1

( ) ( )( ) ... 0

N

N

dy t d y ty t

dt dt




DT systems have input-output relationships in form of linear

constant-coefficients difference equations:

This form is called the recursive form.

0 0

0 10

[ ] [ ]

1[ ] [ ] [ ]

N M

k k

k k

M N

k k

k k

a y n k b x n k

y n b x n k a y n ka




When N = 0, we have the following form for input-output:

This often called nonrecursive equation.

By direct computation, this system has a finite impulse

response (FIR) of the form:

0 0

[ ] [ ]M

k

k

by n x n k

a

0

, 0[ ]

0,

nbn M

ah n

otherwise

0

[ ] [ ] [ ]M

k

y n h k x n k




Consider the difference equation with x[n] = bδ[n]. Find y[n].

Suppose we impose condition of initial rest .

Condition of initial rest implies x[n] = 0 for n < 0, and y[n] =

0 for n < 0.

1[ ] [ 1] [ ]

2

1[ ] [ ] [ 1]

2

y n y n x n

y n x n y n

41



Starting with this initial condition, we may solve for

successive values of y[n] for n ≥ 0 as follows:

System impulse response is

2

1[0] [0] [ 1] ,

2

1 1[1] [1] [0] ,

2 2

1 1[2] [2] [1]

2 2

1 1[ ] [ ] [ 1]

2 2

n

y x y b

y x y b

y x y b

y n x n y n b

1

[ ] [ ]2

n

h n u n


Summary

In this chapter we developed important representations

for LTI systems.

In discrete time we derived the representation of signals

as weighted sums of shifted impulse.

Next, we used this representation to derive the

convolution sum representation for LTI discrete systems.

In continuous time we derived representation of signals as

weighted integral of shifted unit impulse.

Later, this representation is used to derive the convolution

integral for LTI continuous systems.


Summary

Moreover, the convolution sum and integral provided us

with a means of analyzing the properties of LTI systems,

including causality and stability.

Important class of continuous time systems described by

linear constant-coefficients differential equations, is

discussed.

Discrete time systems represented by linear constant-

coefficients difference equations, are also discussed.

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