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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 8
Issued,November 4,2003 Due,November 19,2003
REMINDER,Quiz #2 will be held from 7:30 - 9:30 p.m,Thursday,November 13 in
Walker Memorial,The quiz will cover materials in Chapters 4 -7 (through Section 7.4) of
O&W,Lectures and Recitations through October 29,Problem Sets # 4-6,and that part of
Problem Set # 7 involving problems from Chapter 7,
Reading Assignments,
Lectures #16-18 & PS#8,Section 7.5,Chapters 8 & 9 (through Section 9.6) of
O&W
Lectures #18-20 & PS#9,Chapters 9 & 11 (through Subsection 11.3.4) of O&W
Exercise for home study (not to be turned in,although we will provide solutions),
Problems to be turned in,
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O&W 8.35
Problem 1 O&W 7.34
Problem 2 Consider the following system,
x
c
[n]

y
cH(e
j?
)? m
[n]
H(e
j?
)
cos[?
0
n]
1
2
3
[n]
0
=
sin[?
0
n]

M 0
M
H(e
j?
)
x
s
[n]
m
y
s
[n]
m denotes downsampling by m,and? m denotes upsampling by m as illustrated below,
1
x
?








?
x
in
[n]
m
x
out
[n] = x
in
[mn]
n
x
in
[n]
m
x
out
[n] =
x
in
m
n is an integer multiple of m
0 otherwise
x[n] is a real-valued DT signal whose DTFT for <? <? is given by
X(e
j?
)
0
=
2
,?
M
=
43
1

0 0
0?

0

M

0
+?
M
0

M
0
+?
M
(a) Sketch the DTFT for x
c
[n] and x
s
[n] for?2 2?,
(b) How much can one downsample without aliasing, i.e.,what is the maximum integer
value of m?
(c) Design a system which recovers the signal x[n] from y
c
[n] and y
s
[n],
Problem 3 Determine the Laplace transform and the associated region of convergence and
pole-zero plot for each of the following functions of time:
(a) x(t) = e
t
u(?t) + 2e
2t
u(t)
(b) x(t) = (e
t
cos t)u(?t) + u(?t)
Problem 4 Determine the function of time,x(t),for each of the following Laplace trans-
forms and associated region of convergence:
s? 25
(a) X(s) =,?3 < ↑e{s} < 4
s
2
s? 12
2s
2
+ 7s + 9
(b) X(s) =,↑e{s} >?2
(s + 2)
2
2
Problem 5 O&W 9.24 (f)
Problem 6 O&W 9.26,Also,determine if this system is stable and justify your answer,
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,
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