11

Linear Time-Invariant Systems

The convolution Intregral (Theorem)--System Impulse Response:

The system impulse response can be defined as the response h(t) as the response y(t) when

h(t)( ) ( )x t t= ( ) ( )y t h t=

13

The convolution Intregral (Theorem) Cont.

( ) ( ) ( ) ( )* ( )y t x h t dt x t h t

= =

Three different time scales are involvedExcitation time Response time tSystem-memory time t-This relationship is based on time domain analysis of linear time invariant systems. It states that the present value of the response of a linear time invariant system is weighted intregral over the past history of the input signal, weighted according to the impulse response of the system. Thus, the impulse response acts as a memory function for the system.

15

Important Properties of Convolution

Continuous time convolution satisfies the following important properties--Commutativity:

x(t)*h(t)=h(t)*x(t)

--Associativity:x(t)*h1(t)*h2(t)=[x(t)*h1(t)]*h2(t)=x(t)*[h1(t)*h2(t)]

h(t)x(t) y(t)

x(t)h(t) y(t)

h1(t)*h2(t)x(t) y(t)

h1(t) h2(t)x(t) y(t)

16

Important Properties of Convolution Cont.

-- Distributivity:x(t)*[h1(t)+h2(t)]=[x(t)*h1(t)]+[x(t)*h2(t)]

h1(t)

h2(t)

+x(t)y(t)

h1(t)+h2(t)x(t) y(t)

17

Important Properties of Convolution Cont.

( )* ( ) ( ) ( ) ( )x t t x t d x t

= =

( )* ( ) ( ) ( ) ( )t

x t u t x u t d x d

= =

( )* ( ) ( ) ( ) ( )x t t x t d x t

= =

( ) ( ) ( ) ( )x t t a x a t a =

( ) ( ) ( )x t t a dt x a

=

( )* ( ) ( )x t t a x t a =

AdministratorHighlight

AdministratorHighlight

18

Total Impulse Response

h1(t) h2(t)

h3(t) h4(t)

+x(t)y(t)

1 2 3 4( ) ( ) * ( ) ( ) * ( )h t h t h t h t h t= +

h(t)X(t) y(t)

21

Graphical Interpretation of Convolution Theorem

y(t)=rect(t/2a)*rect(t/2a)= x(t)*h(t)

( ) ( ) ( )

= Calculation of area of the product of the two signals

y t x h t d

=

-a a

1

-a a

1x(t) h(t)

t t

22

-a a

1

-a a

1x(t) h(t)

t t

-a a

1

( )x

-a a

1

t a+t-a+t t

( )h t

-a a

1

t

Area=0

Area={(a+t)-(-a)}*(1*1)=base*height

=t+2a

Area=0t=-2at

23

a+t-a+t t

( )h t

-a a

1

( )x

-a

24

-a a

1

( )x

a+t-a+t t

( )h t

0

25

-a a

1

( )x

a+t-a+t t

( )h t

Area=0

t

y(t)

2a-2a 0

Increasing area Decreasing area

y(t)=0 t -2a

=t+2a -2a

26

Properties of linear time-invariant systems

The impulse response of an LTI system represents a complete description of

the characteristics of the system.Memoryless LTI systems

Input-output relationshipy(t)=kx(t)

Impulse response by definition

( ) ( )h t k t=

27

Causal LTI systems

Causal LTI systemsFor a continuous-time system to be causal, y(t) must not depend on for

From equation h(t)=0 for t

( ) ( ) ( )y t x h t d

=

AdministratorHighlight

AdministratorHighlight

AdministratorHighlight

AdministratorHighlight

AdministratorHighlight

AdministratorHighlight

28

Invertible LTI systems

Invertible LTI systems Y(t)=h1(t)*h(t)*x(t)=x(t) so h1(t) must satisfy

h1(t)*h(t)=(t) Stable LTI systems Total h(t) must be finite i.e.

( )h t d t

<