EEL 3135:
Dr. Fred J. Taylor, Professor
Lesson Title: ContinuousTime Signal and
Systems
Lesson Number: 26 (Section 95 to 910)
Background:
Continuoustime signals where introduced in Chapter 9.
The concept of linear convolution was
also defined and claimed to have the fundamental importance to continuous time signal
processing as it held in the study of discretetime signals and systems.
Along with the linear
convolution, several other important continuoustime signals were introduced including unit
impulses (
δ
(t)) and unit step (u(t)).
Convolution
Discretetime convolution was a fundamentally important to the study of discretetime linear
systems.
A completely complementary operation, called continuoustime convolution exists for
continuoustime signals and systems. Specifically, continuoustime convolution is defined by:
(
29
(
29
(
29
(
29
(
29
(
29
(
29
τ
d
h
t
x
d
t
h
x
t
x
t
h
t
y
∫
∫
∞
∞

∞
∞


=

=
=
1.
where h(t) is the linear system’s impulse response, x(t) the system input, and y(t) is the system’s
output.
Continuoustime convolution is associative, commutative, and distributive.
Example
Simulate the convolution of two 1 second pulses p(t)=u(t)u(t1).
The outcome is shown in
Figure 1.
>> t=0:.01:2.01;
{200 clock ‘ticks’ over 2 seconds}
>> p=[ones(100,1);zeros(100,1)];
{1 second pulse}
>> subplot(1,2,1);
>> plot(t,p)
>> y=conv(p,p);
{convolution}
>> t=0:.01:4.02;
{400 clock ‘ticks’ over 4 seconds}
>> subplot(1,2,2);
>> plot(t,0.01*y);
{note scale factor of 0.01}
1
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Dr. Fred J. Taylor, Professor
Figure 1:
Convolution of 2 pulses.
One second pulse shown over 2 second interval on left,
convolution of 2 pulse shown over 4 second interval on right.
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 Spring '08
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 LTI system theory, Dr. Fred J. Taylor, Dr. Fred J., Fred J. Taylor

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