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LTI SYSTEM ANALYSIS:

CONVOLUTION INTEGRAL

Prof. Siripong Potisuk

CT Unit Impulse

Continuous-time impulse function

Properties:

−

== 1)( and 0 ,0)( dtttt

)( )()( )4

)()()()()3

)()()2 )(1

)()1

0

-

0

000

txdttttx

tttxtttx

ttta

at

=−

−=−

=−=

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Impulse Response

The output signal of an analog system at rest at t = 0 due

to a unit impulse

If h(t) is known for an LTI system, we can compute the

response to any arbitrary input using convolution

Analog LTI system is completely characterized in the time

domain by its impulse response since any arbitrary input

signal can be decomposed into a linear weighted sum of

scaled and time-shifted unit impulses

)}({ )( tHth =

Causality

- Causality for an LTI system is equivalent to the

condition of initial rest.

- The impulse response of a causal LTI system

must be zero before the impulse occurs

→ h(t) must be a causal signal

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A necessary and sufficient condition for a CT

LTI system to be BIBO stable is that the impulse

response is absolutely integrable.

BIBO stability

Example: if , the system is BIBO stable.||)( teth −=

Step Response

The output signal of an analog system at rest at t = 0 due

to a unit-step function

If known for an LTI system, we can apply the superposition

principle to compute the response to any arbitrary input

signal that can be decomposed into a linear weighted sum

of scaled and time-shifted unit-step functions

Used extensively in control-related applications in which

we are interested in how well the system tracks (follows) a

step input

)}({ )(step tuHty =

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Impulse and Ramp response from step response

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Representation of CT Signals Using Unit Impulses

Approximate x(t) as a sum of time-shifted and

scaled pulses

(Impulse Decomposition)

Contiguous-pulse approximation

to an arbitrary CT signal

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Response of a CT-LTI System

Impulse Response

Convolution Integral

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Application of linearity and time Invariance to find

the approximate system response

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Graphical Method for Computing

Convolution Integral

)(Flip

)( −⎯⎯ →⎯ hh

)(slide

)( −⎯⎯ →⎯− thh

)()(Multiply

)( −⎯⎯⎯⎯ →⎯− thxth

)(Integrate

)()( tythx ⎯⎯⎯⎯ →⎯−

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Example:

.)5()({4.0)( and )3()2()(

where),(*)( )( Evaluate

−−=−−+=

=

tututthtututx

thtxty

Positions of the time-shifted signal h(t – ) and x(t) for different

ranges of t: (a) t ≤ –2, (b) –2 < t ≤ 3

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Positions of the time-shifted signal h(t – ) and x(t) for different

ranges of t: (c) 3 < t ≤ 8, and (d) t > 8.

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2

(0.2)( 2) , 2 3

( ) (0.2)(16 6 ), 3 8

0 otherwise

t t

y t t t t

+ −

= + −

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Commutative & Distributive Properties

Associative Property

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Example: Determine the overall impulse response

of the LTI system shown, which is composed of

cascade and parallel connections of four simple LTI

subsystems.

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=−=

−

)()()( )()( txdtxttx

=−=

−

)()()( )()( tdttt

=−=

−

)()()( )()( tudtuttu

−=−−=−

−

)()()( )()( 000 ttxdttxtttx

Convolution with the delta function

(sifting property)

Convolution Integral of Two Causal Signals

−=t

tdthxty0

0,)()( )(

0,0 )( = tty

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Example:

0. ),()( and

)()( where),(*)( )( Evaluate

=

== −

tutx

tuethtxthty t

Note: If = 0, y(t) = u(t) u(t) = r(t) = t u(t)

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Time-Shifting property

x(t −T1) h(t −T2) = y(t −T1 −T2)

Given that y(t) = x(t) h(t),

Examples:

u(t) u(t −1) = r(t −1)

u(t) u(t −2) = r(t −2)

u(t −1) u(t −2) = r(t −3)

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Example Compute the response of an initially

uncharged RC circuit with = 1 by computing

the convolution integral with a triangular pulse

and the impulse response derived earlier.

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