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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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(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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