Time-Domain System Analysis

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Continuous Time

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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

Exact Excitation

Approximate Excitation

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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The Convolution Integral

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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

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A Graphical Illustration of the Convolution Integral

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A Graphical Illustration of the Convolution Integral

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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 second system 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 the convolution of the individual impulse responses of those systems.

Cascade Connection

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System Interconnections If two systems are excited by the same signal and their responses are added they are said to be parallel connected. It follows from the distributive property of convolution that the impulse response of a parallel connection of LTI systems is the sum of the individual impulse responses.

Parallel Connection

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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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Systems Described by Differential Equations

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Systems Described by Differential Equations

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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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Impulse Response Example

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Impulse Response Example

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Impulse Response Example

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Impulse Response Example

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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

System Excitation

System Impulse Response

System Response

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The Convolution Sum

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A Convolution Sum Example

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A Convolution Sum Example

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A Convolution Sum Example

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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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Numerical Convolution

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Numerical Convolution

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Numerical Convolution

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Numerical Convolution

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Stability and Impulse Response

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System Interconnections

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

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System Interconnections

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

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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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