MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 7
Issued,October 28,2003 Due,November 5,2003
REMINDER,Computer Lab 2 is also due on November 7,
Reading Assignments,
Lectures #14-15 & PS#7,Chapter 7 (through Section 7.4) and Chapter 8 (through Section
8.4) of O&W
Lectures #16-18 & PS#8,Section 7.5 and Chapters 8 and 9 (through Section 9.6) of O&W
Exercise for home study (not to be turned in,although we will provide solutions),
O&W 7.28
O&W 8.23
Problems to be turned in,
Problem 1 A sinusoidal input signal,x(t) = cos(10t) is sampled and?ltered as shown
below,
x(t) ×
z(t)
H( ) y(t)j?
s(t)
1
The frequency response of the?lter is
|H(j?)| =
1,
0,
90 < |?| < 180
otherwise
H(j?) =?
200
,
as shown in the?gure below,
|H(j?)|?H(j?)
1
2
2
100
180?90 0 90 180
200?100 200
2?
(a) Suppose s(t) =?(t? kT) and T =, Provide a labeled sketch of Z(j?),the
90
k=
Fourier transform of z(t),
(b) Find y(t),assuming the s(t) and the value of T given in part (a),
2?
(c) The sampling function s(t) is changed to the form below,with T =,
90
(1) (1) (1) (1) (1)
t
T
2
T
3T
2
2T 0
T
2
T
3T
2
2T
· · · · · ·
(?1) (?1) (?1) (?1)
Find y(t),
2
Problem 2 O&W 7.30 except let the input be x
c
(t) =?(t? T/2),
Problem 3 Answer the same questions in O&W 7.31 except let h[n] be the unit sample
response of an LTI,causal system with di?erence equation
3 1
y[n] = y[n? 2] + x[n] + x[n? 1],
4 4
Problem 4 O&W 8.22
Problem 5 O&W 8.49
Problem 6 The transmission system depicted below is intended to allow a signal x(t) to
be transmitted through a communication channel that also carries other signals represented
by z(t),
Channel
q(t) r(t)
x(t)
p(t)
+
H
LP
(j?)
y(t)
H
LP
(jw)
× ×
C
z(t)
f
0?
f
cos(?
c
t)
Both x(t) and z(t) are bandlimited,and their Fourier transforms X(j?) and Z(j?) are real,
as sketched below,Notice that the bandwidth?
z
of Z(j?) is much greater than the band-
width?
x
of X(j?),
X(j?) Z(j?)
A B
x
x
z
z
3
(a) We wish to determine parameters for the transmission system so that the output y(t)
is equal to the input x(t),Determine the range of?
c
for which y(t) can be made equal
to x(t),Explain,
(b) Given a value of?
c
in the range speci?ed in part (a),determine the range of values of
f
and the value of C for which y(t) = x(t),Your expression may contain?
c
and/or
parameters of the Fourier transforms X(j?) and Z(j?),Brie?y explain your reasoning,
(c) Consider next what would happen if the channel also had appreciable delay,as depicted
below,Assuming that the parameters are chosen as in parts (a) and (b),?nd an
expression for the system’s frequency response,de?ned as
Y (j?)
H(j?) =
X(j?)
Channel
T
Delay
by
q(t) r(t)
x(t)
p(t)
+
H
LP
(j?)
y(t)
H
LP
(jw)
× ×
C
z(t)
f
0?
f
cos(?
c
t)
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
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 7
Issued,October 28,2003 Due,November 5,2003
REMINDER,Computer Lab 2 is also due on November 7,
Reading Assignments,
Lectures #14-15 & PS#7,Chapter 7 (through Section 7.4) and Chapter 8 (through Section
8.4) of O&W
Lectures #16-18 & PS#8,Section 7.5 and Chapters 8 and 9 (through Section 9.6) of O&W
Exercise for home study (not to be turned in,although we will provide solutions),
O&W 7.28
O&W 8.23
Problems to be turned in,
Problem 1 A sinusoidal input signal,x(t) = cos(10t) is sampled and?ltered as shown
below,
x(t) ×
z(t)
H( ) y(t)j?
s(t)
1
The frequency response of the?lter is
|H(j?)| =
1,
0,
90 < |?| < 180
otherwise
H(j?) =?
200
,
as shown in the?gure below,
|H(j?)|?H(j?)
1
2
2
100
180?90 0 90 180
200?100 200
2?
(a) Suppose s(t) =?(t? kT) and T =, Provide a labeled sketch of Z(j?),the
90
k=
Fourier transform of z(t),
(b) Find y(t),assuming the s(t) and the value of T given in part (a),
2?
(c) The sampling function s(t) is changed to the form below,with T =,
90
(1) (1) (1) (1) (1)
t
T
2
T
3T
2
2T 0
T
2
T
3T
2
2T
· · · · · ·
(?1) (?1) (?1) (?1)
Find y(t),
2
Problem 2 O&W 7.30 except let the input be x
c
(t) =?(t? T/2),
Problem 3 Answer the same questions in O&W 7.31 except let h[n] be the unit sample
response of an LTI,causal system with di?erence equation
3 1
y[n] = y[n? 2] + x[n] + x[n? 1],
4 4
Problem 4 O&W 8.22
Problem 5 O&W 8.49
Problem 6 The transmission system depicted below is intended to allow a signal x(t) to
be transmitted through a communication channel that also carries other signals represented
by z(t),
Channel
q(t) r(t)
x(t)
p(t)
+
H
LP
(j?)
y(t)
H
LP
(jw)
× ×
C
z(t)
f
0?
f
cos(?
c
t)
Both x(t) and z(t) are bandlimited,and their Fourier transforms X(j?) and Z(j?) are real,
as sketched below,Notice that the bandwidth?
z
of Z(j?) is much greater than the band-
width?
x
of X(j?),
X(j?) Z(j?)
A B
x
x
z
z
3
(a) We wish to determine parameters for the transmission system so that the output y(t)
is equal to the input x(t),Determine the range of?
c
for which y(t) can be made equal
to x(t),Explain,
(b) Given a value of?
c
in the range speci?ed in part (a),determine the range of values of
f
and the value of C for which y(t) = x(t),Your expression may contain?
c
and/or
parameters of the Fourier transforms X(j?) and Z(j?),Brie?y explain your reasoning,
(c) Consider next what would happen if the channel also had appreciable delay,as depicted
below,Assuming that the parameters are chosen as in parts (a) and (b),?nd an
expression for the system’s frequency response,de?ned as
Y (j?)
H(j?) =
X(j?)
Channel
T
Delay
by
q(t) r(t)
x(t)
p(t)
+
H
LP
(j?)
y(t)
H
LP
(jw)
× ×
C
z(t)
f
0?
f
cos(?
c
t)
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
