EGR 277

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Lecture 11 EGR 261 Signals and Systems
Read Ch. 2, Sect. 1-8 in Linear Signals
Systems, 2nd Ed. by Lathi Ch. 13, Sect. 6 in
Electric Circuits, 9th Ed. by Nilsson

• Convolution
• In this class we will cover the following topics
• Convolution graphical approach
• Convolution direct integration approach
• Convolution using Laplace transforms to bypass
  the convolution integral

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Lecture 11 EGR 261 Signals and Systems
Procedure for evaluating y(t) x(t)h(t)
graphically (Note that x and h can be reversed in
the steps below (commutative property), depending
on which form of the convolution integral is
used.)

• Graph x(?) (with ? on the horizontal-axis).
• Invert h(?) to form h(-?). Then shift h(-?)
  along the ? axis t seconds for form h(t - ?).
  Note that as the time shift t varies, the
  waveform h(t - ?) will slide across x(?) and in
  some cases the waveforms will overlap.
• Determine the different ranges of t which result
  in unique overlapping portions of x(?) and h(t -
  ?). For each range determine the area under the
  product of x(?) and h(to - ?). This area is y(t)
  for the range.
• 4. Compile the results of y(t) for each range and
  graph y(t).

Which signal to invert and shift? Since x(t)h(t)
h(t)x(t), we can shift and delay either signal
(commutative property).
Hint Shift and delay the simplest waveform
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Lecture 11 EGR 261 Signals and Systems
Length of the response y(t) x(t)h(t) In
general if the length of the input x(t) is T1 and
the length of the impulse response h(t) is T2,
then the length of y(t) x(t)h(t) will be T1
T2.
Example Recall the example worked in the last
presentation where we determined y(t) x(t)h(t)
as shown below.
Length 1
Length 2
Length 12 3
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Lecture 11 EGR 261 Signals and Systems
Example 1 Find y(t) x(t)h(t) using the
graphical method. What should be the length of
y(t)?
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Lecture 11 EGR 261 Signals and Systems
Example 1 (continued)
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Lecture 11 EGR 261 Signals and Systems
Example 2 Find y(t) x(t)h(t) using the
graphical method. Which waveform is easiest to
shift and delay? What should be the length of
y(t)?
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Lecture 11 EGR 261 Signals and Systems
Example 2 (continued)
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Lecture 11 EGR 261 Signals and Systems
Example 3 (problem 2.4-18d in Lathi text) Find
y(t) x(t)h(t) using the graphical method.
Which waveform is easiest to shift and delay?
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Lecture 11 EGR 261 Signals and Systems
Example 3 (continued)
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Lecture 11 EGR 261 Signals and Systems

• Example 4
• Sketch x(t).
• Find H(s) for the circuit shown below. Also find
  h(t) from H(s).
• Find y(t) using Laplace transform techniques.
• Look familiar? This is the same problem used in
  Example 3. Show that y(t) found using
  convolution is exactly the same as y(t) found
  using Laplace transforms.

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Lecture 11 EGR 261 Signals and Systems
Example 4 (continued)
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Lecture 11 EGR 261 Signals and Systems
Example 5 Find y(t) x(t)h(t) using the
graphical method.
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Lecture 11 EGR 261 Signals and Systems
Example 5 (continued)
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Lecture 11 EGR 261 Signals and Systems
Convolution by direct integration Convolution can
be performed by direct evaluation of the
convolution integral (without sketching graphs)
for fairly straightforward functions. This
becomes difficult for piecewise-continuous
functions, such as were used in many of our
graphical examples.
Example 6 Evaluate y(t) x(t)h(t) for x(t)
u(t), h(t) u(t) by directly evaluating the
convolution integral rather than through a
graphical approach.
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Lecture 11 EGR 261 Signals and Systems
Example 7 Evaluate y(t) x(t)h(t) for x(t)
tu(t), h(t) u(t) by directly evaluating the
convolution integral rather than through a
graphical approach.
Example 8 Evaluate y(t) x(t)h(t) for x(t)
e-tu(t), h(t) u(t) by directly evaluating
the convolution integral rather than through a
graphical approach.
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Lecture 11 EGR 261 Signals and Systems
Table 2-1 from Signals Systems, 2nd Edition, by
Lathi