Signals and Systems Need d e f
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Signals and Systems
Need d e f
![Q.1 Consider the signal given as:
h(t) = e-lu(t).
(1)
a) Find the time constant t of h(t) and find h(t), h(2t), h(3T), h(4r), h(5t).
b) We know that the output of a continuous-time LTI system, whose input x(t) and impulse
response h(t) are nonzero for -o < t <o, is given by the convolution integral
y(t) = x(t) + h(t) = S, x(t – t)h(t)dr = x(t)h(t – t)dr.
(2)
c) Now let x(t) = 8(t), the unit – impulse signal. Show that, using (2) and one of the
properties of the impulse signal 8(t), the output y(t) for this input is found as
y(t) = h(t).
d) Now let h(t) be given as in (1). Giving the mathematical justification, make the required
change in the integral limits of the convolution integral given in (2) according to the
impulse response signal h(t) given in (1) and rewrite (2).
e) Now let
x(t) = u(t),
(3)
the unit-step signal, which implies another change in the integral limits in (2). Giving the
mathematical justification, make the change in the integral limits of (2) according to (3).
f) Now substitute (1) and (3) into (2) and find y(t) by taking the integral given in (2) with
the modified limits and show that
У (t) %3D (1 — е-")u(t).
Hint: Use u(t- T) =1 for 0<T<t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F517118c9-6e6f-431b-a372-c9dae99b2fce%2Fc66bc145-0eed-4c3b-a64b-d8d99f3b53dd%2Fqrjk5j_processed.png&w=3840&q=75)
Transcribed Image Text:Q.1 Consider the signal given as:
h(t) = e-lu(t).
(1)
a) Find the time constant t of h(t) and find h(t), h(2t), h(3T), h(4r), h(5t).
b) We know that the output of a continuous-time LTI system, whose input x(t) and impulse
response h(t) are nonzero for -o < t <o, is given by the convolution integral
y(t) = x(t) + h(t) = S, x(t – t)h(t)dr = x(t)h(t – t)dr.
(2)
c) Now let x(t) = 8(t), the unit – impulse signal. Show that, using (2) and one of the
properties of the impulse signal 8(t), the output y(t) for this input is found as
y(t) = h(t).
d) Now let h(t) be given as in (1). Giving the mathematical justification, make the required
change in the integral limits of the convolution integral given in (2) according to the
impulse response signal h(t) given in (1) and rewrite (2).
e) Now let
x(t) = u(t),
(3)
the unit-step signal, which implies another change in the integral limits in (2). Giving the
mathematical justification, make the change in the integral limits of (2) according to (3).
f) Now substitute (1) and (3) into (2) and find y(t) by taking the integral given in (2) with
the modified limits and show that
У (t) %3D (1 — е-")u(t).
Hint: Use u(t- T) =1 for 0<T<t.
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