George Mason University
Signals and Systems I
Spring 2016
Problem Set #8
Assigned: March 22, 2016
Due Date: March 29, 2016
Assignment: Given below are two sets of problems. The first, Practice Problems, are optional
and for those of you who would like extra practice in solving problems using some of the concepts
developed in class. Only answers will be provided in the solution. The second set, Regular Problems,
which are starred, are to be turned in for grading at your normally scheduled recitation section.
Detailed solutions will be provided after the due date.
Practice Problems
Problem 8.1
Find the inverse Fourier transform of X(jω) = 2πδ(ω) + πδ(ω − 4π) + πδ(ω + 4π)
Answer: x(t) = 1 + cos(4πt).
Problem 8.2
Consider an LTI system with frequency response H(jω) = |H(jω)|ejφh (ω) . If the input is x(t) =
A cos(ω0 t + φ0 ) then it can be shown that the output has the form
y(t) = Ax(t − t0 )
where A is a non-negative real number representing an amplitude scaling factor and t0 is a time
delay.
(a) Express A in terms of |H(jω)|.
(b) Express t0 in terms of φh (ω).
Answer: (a) A = |H(jω0 )|,
(b) t0 = φh (ω0 )/ω0 .
Problem 8.3
A stable LTI filter has a frequency response
H(jω) =
a − jω
a + jω
where a > 0.
(a) Find the magnitude of the frequency response, |H(jω)|.
(b) The group delay is defined to be
dφh (ω)
dω
where φh (ω) is the phase of the frequency response. What is the group delay of this system?
τ (ω) = −
Answer: (a) 1,
(b)
a2 − ω 2
a2 + ω 2
2
Problem 8.4
If h(t) is an ideal low-pass filter with frequency response
1
; |ω| < 2π(2000)
H(jω) =
0
; otherwise
Find the frequency response of the filter that has an impulse response given by
g(t) = h(t) cos(12000πt)
Problem 8.5
Find the Fourier transform of x(t) = e−2|t| .
Answer: X(jω) =
4
.
4 + ω2
Problem 8.6
If X(jω) is the Fourier transform of x(t), what function has a Fourier transform X(jω) cos(2ω)?
Answer:
1
2 x(t
− 2) + 12 x(t + 2)
Problem 8.7
Find the generalized Fourier transforms of the following signals,
(a) x1 (t) = ejπ(t−1) .
(b) x2 (t) = t.
(c) x3 (t) = cos2 (4t).
Answers: (a) −2πδ(ω − π),
(b) 2πjδ 0 (ω),
(c) πδ(ω) + 0.5πδ(ω − 8) + 0.5πδ(ω + 8).
Problem 8.8
The Fourier transform of x(t) and v(t) are defined below,
(
2
; |ω| < π
X(jω) =
0
; otherwise
V (jω) = X(j(ω − 4π)) + X(j(ω + 4π))
(a) Find a closed-form expression for x(t).
Answer: (a) 2
sin(πt)
πt
(b) 4
(b) Find a closed-form expression for v(t).
sin(πt)
cos(4πt)
πt
Problem 8.9
Evaluate the integral
Z∞
t
−∞
Answer: 1/2π 3 .
2
sin t
πt
4
dt
Regular Problems
Problem 8.1F
A continuous-time signal x(t) has a Fourier transform
X(jω) =
1
b + jω
where b is a constant. Determine the Fourier transform V (jω) of the following signals,
(a) v(t) = x(5t − 4)
(b) v(t) = t2 x(t)
(c) v(t) = x(t)ej2t
(d) v(t) = x(t) cos(4t)
(e) v(t) =
d2 x(t)
dt2
(f) v(t) = x(t) ∗ x(t)
(g) v(t) = x2 (t)
(h) v(t) =
1
jt−b
Problem 8.2F
Find the inverse Fourier transform of the function
X(jω) =
sin(ω/2) −j2ω
e
2 + jω
Problem 8.3F
The periodic square wave shown in the figure below
x(t)
1
t
−1
1
2
−1
has a Fourier series expansion is given by
x(t) =
X
nodd
2 j2πnt
e
jπn
(a) Find the Fourier transform of x(t).
(b) Make a plot of X(f ), the Fourier transform expressed in Hz, versus f for |f | < 10.
Problem 8.4F
Consider an LTI system whose response to the input
x(t) = e−t + e−3t u(t)
is
y(t) = 2e−t + e−4t u(t)
(a) Find the frequency response of the system.
(b) Find the differential equation relating the input and the output of this system.
Problem 8.5F
Shown in the figure below is the frequency response of a low-pass differentiator.
|H(jω)|
φh (ω)
π/2
1
−3π
ω
−3π
ω
3π
3π
−π/2
For each of the following input signals, determine the filtered output y(t).
(a) x(t) = cos(2πt + θ).
(b) x(t) = cos(4πt + θ).
(c) x(t) is a half-wave rectified sine wave with a period of one,
(
sin(2πt)
; m ≤ t ≤ (m + 21 )
x(t) =
0
; (m + 21 ) ≤ t ≤ m, for any integer m
x(t)
1
t
−1
−0.5
1/2
1
1.5
George Mason University
Signals and Systems
Spring 2016
Problem Set #8
Homework Cover Sheet
Name:
Lecture Section: (1, Hayes) or (2, Griffiths)
Names of other students I discussed this problem set with:
Estimated percentage of how much of each problem has been completed:
1)
%
2)
%
3)
%
4)
%
5)
%
Total amount of time spent on this problem set:
(Hours)