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DORDT UNIVERSITY ENGINEERING DEPARTMENT
INTRODUCTION TO COMMUNICATION SYSTEMS—EGR 363
(Spring 2023)
Story version of PS#4
In this story version of the problem statements I shall repeat
the problem statements using only words. Although there will be
almost no mathematical symbolism to speed your way, perhaps you
will find hints of what the equations in the problem statements
mean—hints that may be hard for students to pick up on
until they gain a deeper understanding of the symbolism of the
equations.
Problem two-point-thirty-one.
By replacing the general input signal x with the unit-step input
signal in the convolution integral, prove that the response of a
linear-time-invariant system to the unit step function as the
input to the system is found by integrating the system's
impulse response from minus infinity to the time-instant at
which you would like to know what the output is.
Problem two-point-thirty-four-point-one
Use the convolution integral to compute the convolution of
an exponential-times-a-unit-step with an identical
exponential-times-a-unit-step. Let one exponential times a unit step play
the role of the input x and the other play the role of the
impulse response h in the convolution integral. Evaluate the
integral in a piecewise fashion according to the various overlap
conditions found by the mirror-and-slide technique. Hint:
The overlap conditions allow you to replace the infinite
limits of integration in the convolution integral with finite
limits that are valid only in a defined range of the amount
of slide, which is t. Also remember that t is not
the index of integration. That will hopefully simplify
the integral considerably for you.
Problem two-point-thirty-four-point-two
Use the convolution integral to compute the convolution of
a unit rectangle function with a unit triangle function.
Let the unit rectangle play the role of the input x and
let the unit triangle play the role of the impulse response
h in the convolution integral. Evaluate the
integral in a piecewise fashion according to the various overlap
conditions found by the mirror-and-slide technique. Hint:
The overlap conditions allow you to replace the infinite
limits of integration in the convolution integral with finite
limits that are valid only in a defined range of the amount
of slide, which is t. Also remember that t is not
the index of integration. That will hopefully simplify
the integral considerably for you.
Problem two-point-thirty-five
Prove that for a causal (what does "causal" mean?)
linear-time-invariant system, the convolution integral's
bounds can be modified by changing either the lower bound
(from minus infinity) to zero or the upper bound (from
infinity) to t.
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