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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=
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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.
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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)
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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)
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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 =
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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)
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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
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-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
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a+t-a+t t
( )h t
-a a
1
( )x
-a
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-a a
1
( )x
a+t-a+t t
( )h t
0
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-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
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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=
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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
=
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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
<