MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 4
Issued,September 25,2003 Due,October 3,2003
REMINDER,Computer Lab #1 is due on October 8,
REMINDER,Quiz 1 will be held from 7:30 - 9:30 p.m,Tuesday,October 14 in Walker
Memorial,The quiz will cover material in Chapters 1 -3 of O & W,Lectures and Recitations
through September 26,Problem Sets # 1-3,and that part of Problem Set #4 involving
problems from Chapter 3,
Reading Assignments,
Lectures #7-8 & PS#4,Chapter 3 of O&W
Lectures #9-11 & PS#5,Chapters 4&5 of O&W,plus begin Chapter 6 (through
Section 6.2)
Exercise for home study (not to be turned in,although we will provide solutions),
(E1) O&W 3.63
Problems to be turned in:
Problem 1 Consider the LTI system with impulse response given in O&W 3.34,Find the
Fourier series representation of the output y(t) for the following input.
x(t)
4?3 2 3 5 6
· · ·
5?2
1
2
1
1 4
· · ·
t
1
Problem 2 The periodic triangular wave shown below has Fourier series coe?cients a
k
,
x(t)?
2
sin(k?/2)
e
jk?/2
,k?
= 0
1
a
k
=
j(k?)
2
1
,k = 0,
2
··· ···
4?2 0 2 4
t
Consider the LTI system with frequency response H(j?) depicted below,
H(j?)
x(t) H( )j? y(t)

3

2

1
1
2
3
A
1
A
2
A
3
Determine values of A
1
,A
2
,A
3
,?
1
,?
2
,and?
3
of the LTI?lter H(j?) such that

3?
y(t) = 1? cos t,
2
Problem 3 Consider a causal discrete-time LTI system whose input x[n] and output y[n]
are related by the following di?erence equation,
1
y[n]? y[n? 1] = x[n] + 2x[n? 4]
4
Find the Fourier series representation of the output y[n] when the input is
x[n] = 2 + sin(?n/4)? 2 cos(?n/2),
2
Problem 4 Specify the frequency response of a discrete-time LTI system so that if the
input is
x[n] = 2 + cos(?n)? sin(?n/2) + 2 cos(?n/4 +?/4)
then the output is
y[n] = 4? 2 sin(?n) + 2 cos(?n/4),
Problem 5 Compute the Fourier transform of each of the following signals,
(a) x(t) = e
|t|
cos 2t
(b) The signal x(t) depicted below,
x(t)
· · · · · ·
t
3?2 0 1 3 4
5
4?1
1
2 5
1
Problem 6 Determine the continuous-time signal corresponding to each of the following
transforms,
(a) X(j?) = j [?(? + 1)( 1)]? 3 [?() +?(? +?)]
(b) X(j?) = 2 sin(2/2)
3
Problem 7 Answer the questions asked in O&W 4.24 (a) for each of the following signals:
x
1
(t)
1
1
0 1
t
1
x
2
(t)
t
1
3?2?1 0 1 2 3
x
3
(t)
t
3?2?1
1 2 3 4
5 6
1
1
Problem 8 O&W 4.25,Do parts (b) - (f) plus the following new part (a)
(a) X(j?) can be written as A(j?)e
j?(j?)
where A(j?) and ω(j?) are real,Find ω(j?),
Reminder,The?rst 20 problems in each chapter of O&W have answers included at the
end of the text,Consider using these for additional practice,either now or as you study for
tests,
4