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8/14/2019 signal and system Lecture 4
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Signals and SystemsFall 2003
Lecture #4
16 September 2003
1. Representation of CT Signals in terms of shifted unit impulses
2. Convolution integral representation of CT LTI systems3. Properties and Examples
4. The unit impulse as an idealized pulse that is
short enough: The operational definition of(t)
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Representation of CT Signals
Approximate any input x(t) as a sum of shifted, scaled
pulses
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has unit area
The Sifting Property of the Unit Impulse
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Response of a CT LTI System
LTI
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Example: CT convolution
Operation of CT Convolution
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-1
-1 0
0 1
1 2
2
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PROPERTIES AND EXAMPLES
1) Commutativity:
2)
4) Step response:
3) An integrator:
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DISTRIBUTIVITY
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ASSOCIATIVITY
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Consider response from initial rest to pulses of different shapes and
durations, but with unit area. As the duration decreases, the responsesbecome similar for different pulse shapes.
The impulse as an idealized short pulse
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The Operational Definition of the Unit Impulse (t)
(t) idealization of a unit-area pulse that is so short that, for
any physical systems of interest to us, the system responds
only to the area of the pulse and is insensitive to its duration
Operationally: The unit impulse is the signal which whenapplied to any LTI system results in an output equal to the
impulse response of the system. That is,
(t) is defined by what it does under convolution.
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The Unit Doublet Differentiator
Impulse response = unit doublet
The operational definition of the unit doublet:
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Triplets and beyond!
n is number of
differentiations
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Integrators
-1 derivatives" = integral I.R. = unit step
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Integrators (continued)
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Notation
Define
Then
E.g.
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Sometimes Useful Tricks
Differentiate first, then convolve, then integrate
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1 2
Example
1 2
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Example (continued)
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