EGR 277

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Lecture 10 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

• Representation and Analysis of Systems
• The analysis of systems this course is limited to
  linear, time-invariant,
• continuous-time (LTIC) systems.
• LTIC systems can be characterized (described) in
  three ways
• 1) Differential equations a differential
  equation (DE) for y(t) in terms of x(t) found
  using standard D.E. methods (review of Ch. 7-8 in
  Nilsson text). The DE can be used to find the
  response to any input, x(t). Special cases
  include the unit step response (USR) and the
  impulse response, h(t).
• 2) Impulse response the system output, y(t),
  can be determined using the impulse response,
  h(t), through evaluation of the convolution
  integral
• Transfer function The transfer function, H(s)
  can be determined through Laplace transform
  analysis. H(s) can be used to find the USR,
  h(t), or the output y(t) for any input x(t).

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Lecture 10 EGR 261 Signals and Systems
Differential Equations System Representation As
we discussed the classification of systems, it
was stated that a linear system can be
represented by a linear differential equation of
the form
Example Determine the differential equation for
the output y(t) for any general input forcing
function x(t).
Note that this differential equation completely
characterizes the system.
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Lecture 10 EGR 261 Signals and Systems
Unit Step Response (USR) and Impulse Response,
h(t) Recall that we earlier found the unit step
response, USR, and the impulse response, h(t),
using s-domain techniques involving the transfer
function, H(s). To summarize, we found that
We can also find h(t) and USR from the
differential equation that characterizes a
system. Determining the USR and h(t) from a
Differential Equation We have just reviewed
writing and solving differential
equations. Recall that if the input, x(t), to the
differential equation is a unit step function,
then y(t) is called the unit step response
(USR). So, USR y(t) when x(t) u(t) The
impulse response can then be found using the
following relationship
Impulse response h(t) d/dtUSR(t)
?(t) d/dtu(t)
Also recall that
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Lecture 10 EGR 261 Signals and Systems
Example The USR for a system is y(t) USR 2
3e-4tu(t). Find the impulse response, h(t).
Hint Use the product rule.
Answer h(t) 5?(t) 12e-4tu(t)
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Lecture 10 EGR 261 Signals and Systems

• Example
• Write the differential equation for y(t) that
  characterizes the system (circuit).
• Find the unit step response, USR, by solving the
  differential equation for x(t) u(t).
• Find the impulse response, h(t), using the
  relationship h(t) d/dtUSR(t)

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Lecture 10 EGR 261 Signals and Systems
Example (continued)
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Lecture 10 EGR 261 Signals and Systems

• Determining a transfer function from a
  differential equation
• Taking the Laplace Transform of a differential
  equation will yield the transfer function if
• The differential equation uses a general input,
  x(t)
• The initial conditions are zero

Example Find H(s) for the D.E. shown below.
Invalid. Initial conditions are not zero.
Invalid. Input is not general.
Example Find H(s) for the D.E. shown below.
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Lecture 10 EGR 261 Signals and Systems
The Impulse Response and the Convolution
Integral Recall from our earlier study of systems
using Laplace transforms that the transfer
function, H(s), completely characterizes a system
and that the output y(t) can be determined for
any input x(t) using H(s).
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Lecture 10 EGR 261 Signals and Systems
Just as H(s) completely characterizes a system
and can be used to determine the output of the
system for any input, the impulse response h(t)
also completely characterizes the system.

• s-domain approach
• The output of a system can be determined with
  H(s) and Laplace transform techniques.
• time-domain approaches
• Differential equations (just covered)
• Convolution. The output of a system can be
  determined with h(t) and the convolution integral.

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Lecture 10 EGR 261 Signals and Systems

• Why use the convolution integral?
• It allows us to work directly in the time-domain.
  This is particularly useful with experimental
  data where using Laplace transforms might be
  difficult or impossible.
• Insight can be gained as to how closely the
  output waveform replicates the input waveform.

Development of the convolution integral The
following development is presented in section
13.6 of Electric Circuits, by Nilsson. A LTIC
system can be described by its impulse response,
h(t), as shown below.
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Lecture 10 EGR 261 Signals and Systems
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Lecture 10 EGR 261 Signals and Systems
This integral is referred to as the convolution
integral and is generally expressed as
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Lecture 10 EGR 261 Signals and Systems
Properties of the convolution integral Commutative
property Distributive property Associative
property Shift property
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Lecture 10 EGR 261 Signals and Systems
Why use graphical evaluation of the convolution
integral? The convolution integral can be
evaluated directly, but this can be very
difficult (especially for piecewise-continuous
functions), so a graphical approach is more
commonly used. However, even the graphical
approach is quite challenging. A detailed
example will follow shortly to illustrate the
procedure.

• Convolution can be understood by examining the
  graphical interpretation of the convolution
  integral. The graphical approach is helpful for
  the following reasons
• It is helpful in evaluating the convolution of
  complicated signals
• It allows us to grasp visually the convolution
  integrals result
• It allows us to perform convolution with
  signals that can only be described graphically.

Convolution Integral
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Lecture 10 EGR 261 Signals and Systems

• Procedure for evaluating the convolution integral
  graphically
• Graph x(?) (with ? on the horizontal-axis). (see
  note below)
• 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.
• Compile the results of y(t) for each range and
  graph y(t).

Note x(t) and h(t) can be reversed since
x(t)h(t) h(t)x(t), i.e., commutative
property. In general, it is easiest to invert
and delay the simplest function.
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Lecture 10 EGR 261 Signals and Systems
Example Find y(t) x(t)h(t) for x(t) and h(t)
shown below using the graphical method. Follow
the procedure listed on the previous slide.
Solution 1. Form x(?)
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Lecture 10 EGR 261 Signals and Systems
2. Form h(t - ?)
Note Remember that ? is the independent
variable, not t. t is simply a constant. h(t -
?) h(- ? t), so t is the amount of time shift
(to the left).
3. Consider different ranges of t
(time-shift) Each range of t selected should
result in different portions of the waveforms
overlapping. The convolution y(t) x(t)h(t) is
the area under the product of the overlapping
portions.
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Lecture 10 EGR 261 Signals and Systems
3A. First range selected (-? ltt lt 1) This
range results in no overlap for all values of ?
(i.e., -? lt ? lt ?).
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Lecture 10 EGR 261 Signals and Systems
3B. Second range selected (1 ltt lt 2) This
range results in an overlap from ? 0 to ? t-1.
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Lecture 10 EGR 261 Signals and Systems
3C. Third range selected (2 ltt lt 3) This
range results in an overlap from ? 0 to ? 1.
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Lecture 10 EGR 261 Signals and Systems
3D. Fourth range selected (3 ltt lt 4) This
range results in an overlap from ? t-3 to ? 1.
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Lecture 10 EGR 261 Signals and Systems
3E. Fifth range selected (4 ltt) This range
results in no overlap for all values of ? (i.e.,
-? lt ? lt ?).
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Lecture 10 EGR 261 Signals and Systems
5. Compile the results of y(t) over the five
ranges. Also graph y(t).
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Lecture 10 EGR 261 Signals and Systems
Example Find y(t) x(t)h(t) for x(t) and h(t)
shown below using the graphical method.
h(t)
x(t)
20
10
t
t
-1
3
6
0
0
4