Time-Domain Analysis of
Systems
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
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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.
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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.
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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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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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