MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems Fall 2003
Quiz 2
Thursday,November 13,2003
Directions: The exam consists of 6 problems on pages 2 to 19 and work space on pages
20 and 21,Please make sure you have all the pages,Tables of Fourier series
properties as well as CT Fourier transform and DT Fourier transform properties
and tables are supplied to you at the end of this booklet,Enter all your work and
your answers directly in the spaces provided on the printed pages of this book-
let,Please make sure your name is on all sheets,You may use bluebooks for
scratch work,but we will not grade them at all,All sketches must be adequately
labeled,Unless indicated otherwise,answers must be derived or explained,not
just simply written down,This examination is closed book,but students may use
two 8 1/2 × 11 sheets of paper for reference,Calculators may not be used,
NAME,
Check your section Section Time Rec,Instr,
1 10-11 Prof,Zue
2 11-12 Prof,Zue
3 1- 2 Prof,Gray
4 11-12 Dr,Rohrs
5 12- 1 Prof,Voldman
6 12- 1 Prof,Gray
7 10-11 Dr,Rohrs
8 11-12 Prof,Voldman
Please leave the rest of this page blank for use by the graders:
Grader
1 15
2 20
3 25
4 25
5 15
100
Problem No,of points Score
Total
PROBLEM 1 (15%)
Consider the following system depicted below,
x(t)
z(t)
y(t)
SYSTEM A SYSTEM B
Overall System
The input-output relation for SYSTEM A is characterized by the following causal LCCDE,
dz(t) dx(t)
+ 6z(t) = + 5x(t),
dt dt
and the impulse response h
b
(t) for SYSTEM B is de?ned as,
h
b
(t) = e
10t
u(t),
Part a,What is the frequency response of the complete system? That is,given
F F
x(t) X(j?) and y(t) Y (j?),determine H(j?) =
Y (j?)
,
X(j?)
H(j?) =
2
Fall 2003,Quiz 2 NAME:
Work Page for Problem 1
3 Problem 1 continues on the following page.
Part b,What is the impulse response,h(t) of the complete system?
h(t) =
Part c,What is the differential equation that relates x(t) and y(t)?
4
Fall 2003,Quiz 2 NAME:
Work Page for Problem 1
5
PROBLEM 2 (20%)
Part a,Match the step response s(t) below to the correct frequency response and give a
brief justi?cation to your answer in the space provided in the next page,
s(t)
A
t
7? 7?
900 300
H
1
(j?)
1
2
c
c?
10
2
+?
c
(10
2
+?
c
)
(10
2
c
) 10
2
c
1
H
3
(j?) H
4
(j?)
1
3
1
2
3 4
c 0
c
c 0
c
0
H
2
( )
2
2
j?
6
Fall 2003,Quiz 2 NAME,
SYSTEM
Brief justi?cation (You can show why your answer is correct or show why the other three
systems are not correct),
7 Problem 2 continues on the following page.
Part b,Find?
c
and A,
c
= A =
8
Fall 2003,Quiz 2 NAME:
Work Page for Problem 2
9
PROBLEM 3 (25%)
Part a,Determine the Fourier transform R(e
j?
) of the following sequence,
1,0? n? M,M is a positive even integer
r[n] =
0,otherwise,
R(e
j?
) =
Part b,Consider the sequence
1 2?n
1? cos,0? n? M
w[n] = 2 M
0,otherwise,
where M is as de?ned in Part a,Express W (e
j?
),the Fourier transform of w[n] in terms of
R(e
j?
),the Fourier transform of r[n] above,
W (e
j?
) =
10
Fall 2003,Quiz 2 NAME:
Work Page for Problem 3
11 Problem 3 continues on the following page.
Part c,Is there a positive even integer M that will make W (e
j?
) real? If so,?nd the values
of M that satisfy this constraint,If not,explain why,
YES NO
Values of M Explanation,
12
Fall 2003,Quiz 2 NAME:
Work Page for Problem 3
13
PROBLEM 4 (25%)
Part a is independent of the other parts in this problem,
Part a,Consider the following system,
x(t) ×
H ( )
c 0
c
T
lp
j?
y(t)
2?
p(t) =?(t? kT ),?
s
=
T
k=
For this part,suppose
sin(4?t) sin(2?t)
x(t) = (?1)
t
,
t?t
and p(t) is an impulse train of frequency?
s
,H
lp
(j?) is a lowpass?lter whose gain is T
and cutoff frequency is?
c
,Determine the cutoff frequency?
c
and a frequency?
0
such that
y(t) = x(t) for any?
s
>?
0
,
0
=,?
c
=
14
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
15 Problem 4 continues on the following page.
For the rest of this problem,let’s consider the following system:
x
p
(t) y
p
(t)x[n]
x(t) ×
Impulse
to
Sequence
Sequence
to
Impulse
H ( )
2? 0 2?
T
1
lp
j?
y(t)
p
1
(t) =?(t? nT
1
) p
2
(t) =?(t? nT
2
)
n= n=
p
1
(t) is an impulse train whose fundamental period is T
1
and p
2
(t) is another impulse train
whose fundamental period is T
2
,H
lp
(j?) is a lowpass?lter whose gain is T
1
and cutoff
frequency is at?
c
,Note that x[n] = x(nT
1
) and y
p
(t) =
x[n]?(t? nT
2
),
n=
The input x(t) is a band limited real signal whose Fourier transform is shown below,
X(j?)
2 2?
1
Part b,Let’s de?ne
x
1
[n] = x(nT
1
),where T
1
= 1,
x
2
[n] = x(nT
1
),where T
1
=
1
3
,
In the given axes below and on the top of the next page,provide the labeled sketches of
X
1
(e
j?
) and X
2
(e
j?
),Fourier transforms of x
1
[n] and x
2
[n] respectively,
e
j?
) e
j?
)X
1
( X
2
(
2 22 2?
16
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
17 Problem 4 continues on the following page.
1
Part c,Suppose T
1
=
1
and T
2
=
2
,Provide a labeled sketch of Y (j?),Fourier transform
3
of the overall output y(t),
Y (j?)
18
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
19
PROBLEM 5 (15%)
We have a cascade of two stable CT LTI systems as shown below,
x(t)
1
z(t)
y(t)
1 + j?
SYSTEM G
Overall System
The straight line approximation of Bod·e plots of the overall system,H(j?) is shown in the
next page,
Find the frequency response,G(j?),of SYSTEM G,
G(j?) =
20
Fall 2003,Quiz 2 NAME:
20
0
20 log
10
|H(j?)|- dB
20
40
10
2
10
1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency?- rad/s
45
0
45
90
H(j?)- deg ?135
180
225
270
10
2
10
1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency?- rad/s
21
Fall 2003,Quiz 2 NAME,
Work Page
22
Fall 2003,Quiz 2 NAME,
Work Page
23
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems Fall 2003
Quiz 2
Thursday,November 13,2003
Directions: The exam consists of 6 problems on pages 2 to 19 and work space on pages
20 and 21,Please make sure you have all the pages,Tables of Fourier series
properties as well as CT Fourier transform and DT Fourier transform properties
and tables are supplied to you at the end of this booklet,Enter all your work and
your answers directly in the spaces provided on the printed pages of this book-
let,Please make sure your name is on all sheets,You may use bluebooks for
scratch work,but we will not grade them at all,All sketches must be adequately
labeled,Unless indicated otherwise,answers must be derived or explained,not
just simply written down,This examination is closed book,but students may use
two 8 1/2 × 11 sheets of paper for reference,Calculators may not be used,
NAME,
Check your section Section Time Rec,Instr,
1 10-11 Prof,Zue
2 11-12 Prof,Zue
3 1- 2 Prof,Gray
4 11-12 Dr,Rohrs
5 12- 1 Prof,Voldman
6 12- 1 Prof,Gray
7 10-11 Dr,Rohrs
8 11-12 Prof,Voldman
Please leave the rest of this page blank for use by the graders:
Grader
1 15
2 20
3 25
4 25
5 15
100
Problem No,of points Score
Total
PROBLEM 1 (15%)
Consider the following system depicted below,
x(t)
z(t)
y(t)
SYSTEM A SYSTEM B
Overall System
The input-output relation for SYSTEM A is characterized by the following causal LCCDE,
dz(t) dx(t)
+ 6z(t) = + 5x(t),
dt dt
and the impulse response h
b
(t) for SYSTEM B is de?ned as,
h
b
(t) = e
10t
u(t),
Part a,What is the frequency response of the complete system? That is,given
F F
x(t) X(j?) and y(t) Y (j?),determine H(j?) =
Y (j?)
,
X(j?)
H(j?) =
2
Fall 2003,Quiz 2 NAME:
Work Page for Problem 1
3 Problem 1 continues on the following page.
Part b,What is the impulse response,h(t) of the complete system?
h(t) =
Part c,What is the differential equation that relates x(t) and y(t)?
4
Fall 2003,Quiz 2 NAME:
Work Page for Problem 1
5
PROBLEM 2 (20%)
Part a,Match the step response s(t) below to the correct frequency response and give a
brief justi?cation to your answer in the space provided in the next page,
s(t)
A
t
7? 7?
900 300
H
1
(j?)
1
2
c
c?
10
2
+?
c
(10
2
+?
c
)
(10
2
c
) 10
2
c
1
H
3
(j?) H
4
(j?)
1
3
1
2
3 4
c 0
c
c 0
c
0
H
2
( )
2
2
j?
6
Fall 2003,Quiz 2 NAME,
SYSTEM
Brief justi?cation (You can show why your answer is correct or show why the other three
systems are not correct),
7 Problem 2 continues on the following page.
Part b,Find?
c
and A,
c
= A =
8
Fall 2003,Quiz 2 NAME:
Work Page for Problem 2
9
PROBLEM 3 (25%)
Part a,Determine the Fourier transform R(e
j?
) of the following sequence,
1,0? n? M,M is a positive even integer
r[n] =
0,otherwise,
R(e
j?
) =
Part b,Consider the sequence
1 2?n
1? cos,0? n? M
w[n] = 2 M
0,otherwise,
where M is as de?ned in Part a,Express W (e
j?
),the Fourier transform of w[n] in terms of
R(e
j?
),the Fourier transform of r[n] above,
W (e
j?
) =
10
Fall 2003,Quiz 2 NAME:
Work Page for Problem 3
11 Problem 3 continues on the following page.
Part c,Is there a positive even integer M that will make W (e
j?
) real? If so,?nd the values
of M that satisfy this constraint,If not,explain why,
YES NO
Values of M Explanation,
12
Fall 2003,Quiz 2 NAME:
Work Page for Problem 3
13
PROBLEM 4 (25%)
Part a is independent of the other parts in this problem,
Part a,Consider the following system,
x(t) ×
H ( )
c 0
c
T
lp
j?
y(t)
2?
p(t) =?(t? kT ),?
s
=
T
k=
For this part,suppose
sin(4?t) sin(2?t)
x(t) = (?1)
t
,
t?t
and p(t) is an impulse train of frequency?
s
,H
lp
(j?) is a lowpass?lter whose gain is T
and cutoff frequency is?
c
,Determine the cutoff frequency?
c
and a frequency?
0
such that
y(t) = x(t) for any?
s
>?
0
,
0
=,?
c
=
14
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
15 Problem 4 continues on the following page.
For the rest of this problem,let’s consider the following system:
x
p
(t) y
p
(t)x[n]
x(t) ×
Impulse
to
Sequence
Sequence
to
Impulse
H ( )
2? 0 2?
T
1
lp
j?
y(t)
p
1
(t) =?(t? nT
1
) p
2
(t) =?(t? nT
2
)
n= n=
p
1
(t) is an impulse train whose fundamental period is T
1
and p
2
(t) is another impulse train
whose fundamental period is T
2
,H
lp
(j?) is a lowpass?lter whose gain is T
1
and cutoff
frequency is at?
c
,Note that x[n] = x(nT
1
) and y
p
(t) =
x[n]?(t? nT
2
),
n=
The input x(t) is a band limited real signal whose Fourier transform is shown below,
X(j?)
2 2?
1
Part b,Let’s de?ne
x
1
[n] = x(nT
1
),where T
1
= 1,
x
2
[n] = x(nT
1
),where T
1
=
1
3
,
In the given axes below and on the top of the next page,provide the labeled sketches of
X
1
(e
j?
) and X
2
(e
j?
),Fourier transforms of x
1
[n] and x
2
[n] respectively,
e
j?
) e
j?
)X
1
( X
2
(
2 22 2?
16
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
17 Problem 4 continues on the following page.
1
Part c,Suppose T
1
=
1
and T
2
=
2
,Provide a labeled sketch of Y (j?),Fourier transform
3
of the overall output y(t),
Y (j?)
18
Fall 2003,Quiz 2 NAME:
Work Page for Problem 4
19
PROBLEM 5 (15%)
We have a cascade of two stable CT LTI systems as shown below,
x(t)
1
z(t)
y(t)
1 + j?
SYSTEM G
Overall System
The straight line approximation of Bod·e plots of the overall system,H(j?) is shown in the
next page,
Find the frequency response,G(j?),of SYSTEM G,
G(j?) =
20
Fall 2003,Quiz 2 NAME:
20
0
20 log
10
|H(j?)|- dB
20
40
10
2
10
1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency?- rad/s
45
0
45
90
H(j?)- deg ?135
180
225
270
10
2
10
1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency?- rad/s
21
Fall 2003,Quiz 2 NAME,
Work Page
22
Fall 2003,Quiz 2 NAME,
Work Page
23