signal and system lecture 4

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)

signal and system lecture 4

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