Time-Domain System Analysis M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1

Time-Domain System Analysis

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

1

Continuous Time

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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Impulse Response


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The Convolution Integral

ExactExcitation

ApproximateExcitation


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A Graphical Illustration of the Convolution Integral


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The process of convolving to find y(t) is illustrated below.


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Convolution Example


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Convolution Example


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Convolution Integral Properties


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The Unit Triangle Function

The unit triangle, is the convolution of a unit rectangle with Itself.


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System Interconnections

If the output signal from a system is the input signal to a secondsystem the systems are said to be cascade connected. It follows from the associative property of convolution that the impulse response of a cascade connection of LTI systems is theconvolution of the individual impulse responses of those systems.

CascadeConnection


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If two systems are excited by the same signal and their responsesare added they are said to be parallel connected.

It follows from the distributive property of convolution that theimpulse response of a parallel connection of LTI systems is the sum of the individual impulse responses.

ParallelConnection


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Unit Impulse Response and Unit Step Response


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Stability and Impulse Response


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Systems Described by Differential Equations


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MATLAB System Objects


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MATLAB System Objects


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Discrete Time


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Impulse Response


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Impulse Response


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Impulse Response Example


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System Response

• Once the response to a unit impulse is known, the response of any LTI system to any arbitrary excitation can be found

• Any arbitrary excitation is simply a sequence of amplitude-scaled and time-shifted impulses

• Therefore the response is simply a sequence of amplitude-scaled and time-shifted impulse responses


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Simple System Response Example


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More Complicated System Response Example

SystemExcitation

SystemImpulseResponse

SystemResponse


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The Convolution Sum


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A Convolution Sum Example


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Convolution Sum Properties


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Convolution Sum Properties (continued)


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Numerical Convolution


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Stability and Impulse Response


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The cascade connection of two systems can be viewed asa single system whose impulse response is the convolutionof the two individual system impulse responses. This is a direct consequence of the associativity property ofconvolution.


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The parallel connection of two systems can be viewed asa single system whose impulse response is the sumof the two individual system impulse responses. This is a direct consequence of the distributivity property ofconvolution.


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Unit Impulse Response and Unit Sequence Response


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Systems Described by Difference Equations


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Systems Described by Difference Equations


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Frequency Response


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Frequency Response


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Frequency Response Example


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Frequency Response Example


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Documents

Time-Domain System Analysis M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1