MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems Fall 2003
Final Exam
Tuesday,December 16,2003
Directions: The exam consists of 7 problems on pages 2 to 33 and additional work space on
pages 34 to 37,Please make sure you have all the pages,Tables of Fourier series
properties,CT and DT Fourier transform properties and pairs,Laplace transform
and z-transform properties and pairs are supplied to you as a separate set of pages,
Enter all your work and your answers directly in the spaces provided on the
printed pages of this booklet,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 three 8 1/2 × 11 sheets of paper for reference,Calcula-
tors 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 30
2 15
3 35
4 30
5 30
6 25
7 35
200
Problem No,of points Score
Total
PROBLEM 1 (30 pts)
Let h(t) be a right sided impulse response of a system and its Laplace transform is given by
10(?s + 1)
H(s) =,
(s + 10)(s + 1)
Part a,Find the differential equation describing the system,
Part b,Is the system causal?
YES or NO
Brief explanation:
2
Fall 2003,Final Exam NAME:
Work Page for Problem 1
3 Problem 1 continues on the following page.
Part c,The response of this system to a positive step starts off in a negative direction before
turning around,Show this by?nding lim
t?0
+
ds(t)
,Justify your method,
dt
ds(t)
lim =
t?0
+
dt
Part d,Let H
I
(s) be the transfer function of a stable but noncausal inverse system of H(s),
i.e.,H
I
(s)H(s) = 1,Find H
I
(s) and its region of convergence,
H
I
(s) = ROC,
4
Fall 2003,Final Exam NAME:
Work Page for Problem 1
5
PROBLEM 2 (15 pts)
Consider the DT LTI system shown below,
x[n] H(e
j?
) y[n]
The input sequence is

5
x[n] = cos
2
n?
4
as sketched below,
2
x[n]
2
2?1
0 1
2 3
4 5
6
n
2
2
Determine and sketch y[n] if the magnitude and the phase of H(e
j?
) are given below,
|H(e
j?
)|?H(e
j?
)

1

2

2
6
Fall 2003,Final Exam NAME:
y[n] =
y[n]
6?5?4?3?2?1 1 2 3 4 5 6
n
7
Work Space for Problem 2
8
Fall 2003,Final Exam NAME,
PROBLEM 3 (35pts)
Consider the following system,
cos?
b
t cos?
c
t
x(t)
× H( )
x
c
(t)
×
y
c
(t)

b
b
1
H( )
× H( )
x
s
(t)
×
y
s
(t)
+ y(t)
j?
j?
j?
sin?
b
t sin?
c
t
The Fourier transform of x(t),X(j?) has real and imaginary parts given below,

b
b
1
e{X( )}

b
b
1
1
m{X( )}j? j?
For your convenience,the identical?gures above are attached along with the transform
tables,
9
Part a,Provide labeled sketches of the real and imaginary parts of X
s
(j?),
e{X
s
(j?)}
m{X
s
(j?)}
10
Fall 2003,Final Exam NAME:
Work Page for Problem 3
11 Problem 3 continues on the following page.
Part b,Provide labeled sketches of the real and imaginary parts of Y
s
(j?),
e{Y
s
(j?)}
m{Y
s
(j?)}
12
Fall 2003,Final Exam NAME:
Work Page for Problem 3
13 Problem 3 continues on the following page.
Part c,Y
c
(j?) has real imaginary parts as shown below

c
c

b
c
c
+?
b
1
4
e{Y
c
( )}

c
c

b
c
c
+?
b
1
4
1
4
m{Y
c
( )}j? j?
Provide labeled sketches of the real and imaginary parts of Y (j?),
e{Y (j?)}
m{Y (j?)}
14
Fall 2003,Final Exam NAME:
Work Page for Problem 3
15 Problem 3 continues on the following page.
Part d,What small change would you make in this system to create a lower sideband mod-
ulation?
16
Fall 2003,Final Exam NAME:
Work Space for Problem 3
17
PROBLEM 4 (30 pts)
Consider the following system,
x(t) ×
x
p
(t) Impulse
to
Sequence
x[n]
p(t) =
(t? kT)
k=
The Fourier transforms of x(t),x
p
(t),and x[n] are denoted respectively by X(j?),X
p
(j?)
and X(e
j?
),
Part a,If X(j?) is as shown below and T = 0.5 × 10
3
sec,provide labeled sketches of
X
p
(j?) and X(e
j?
),
X(j?)
2? × 10
3
2? × 10
3
1
X
p
(j?)
X(e
j?
)
32 2? 3
18
Fall 2003,Final Exam NAME:
Work Page for Problem 4
19 Problem 4 continues on the following page.

Part b,Using the same X(j?) and T as in Part a,determine
(i)
x(t)dt

(ii)
x[n],
n=
x(t)dt =,
x[n] =
n=
Part c,Now,assume only that x(t) is bandlimited,i.e.,X(j?) = 0 for |?|? W and is
otherwise arbitrary,
It has been claimed that,for suitable values of T,i.e.,T < A for some value A,the total area
under the continuous time input signal x(t) is T times the sum of the x[n],Do you believe
the claim,i.e.,is there any constraint between T and W which will guarantee that
T x[n] = x(t)dt?
n=

If your answer is yes,specify in terms of W,the smallest value of A for which the claim is
true,If your answer is no,explain clearly,
YES NO
A = Explanation,
20
Fall 2003,Final Exam NAME:
Work Space for Problem 4
21
PROBLEM 5 (30 pts)
Consider the following feedback system,
x(t) +
e(t)
G(s) H(s) y(t)
1
where H(s) =
s
is the plant,x(t) is the reference input,e(t) = x(t)? y(t) is the error
signal,and y(t) is the output of the plant H(s).
Part a,Is H(s) stable?
2
YES or NO
Brief explanation,
Part b,Find system functions
Y (s)
and
E(s)
X(s)
,Express your answers in terms of powers of s
X(s)
and G(s),
Y (s)
=
X(s)
E(s)
=
X(s)
22
Fall 2003,Final Exam NAME:
Work Page for Problem 5
23 Problem 5 continues on the following page.
Part c,Suppose G(s) = K
d
s + K
p
where K
d
and K
p
are real numbers,Find the values
of K
d
and K
p
such that the closed loop system is critically damped with undamped natural
frequency of 10 rad/s,
K
d
=,K
p
=
24
Fall 2003,Final Exam NAME:
Work Page for Problem 5
25
PROBLEM 6 (25 pts)
Consider the DT LTI system whose unit sample response,h[n] is shown below,
k? 1
a
k
b
k
c
n
h[n]
+ 1
where k is an unknown integer and a,b,and c are unknown real numbers,
It is known that h[n] satis?es the following conditions,
e
j?
)(i) Let H(e
j?
) be the Fourier transform of h[n],H( e
j?
is real and even,
(ii) If x[n] = (?1)
n
for all n,then y[n] = 0,
(iii) If x[n] =
1
2
n
u
[n] for all n,then y[2] =
9
2
.
Provide a labeled sketch of the output y[n] when the input x[n] is shown below,Your answer
should not include a,b,c,nor k,
x[n]
2
1 0 1
1
2
3 4 5
n
2
y[n]
n
6?5?4?3?2?1 1 2 3 4 5 6 7 8
26
9
Fall 2003,Final Exam NAME:
Work Space for Problem 6
27
PROBLEM 7 (35 pts)
Consider the?ve pole-zero plots below,Each plot corresponds to a DT LTI system function
whose unit sample response is real,Each plot is drawn to scale,Note that you have all the
information to solve the questions in this problem although some of the poles and zeros are
not labeled,For your convenience,the identical pole-zero plots to the ones on this page
are attached along with the transform tables,
1
2
1
2
×
×
m
e
1
A
×
×
1
2
1
2
m
e
1
B
1
2
1
2
×
×
0.6
0.6
m
e
1
C
×
×
×
×
0.6
0.6
0.6
0.6
m
e
1
D
×
×
×
×
0.6
0.6
0.6
0.6
0
m
e
1
E
28
Fall 2003,Final Exam NAME,
Part a,Which plot(s) can have an ROC so that it corresponds to a causal and stable system?
Which plot(s)?
Brief explanation,
Part b,Consider the following block diagram
x[n] H(z)
y[n]
G(z) w[n] = x[n]
H(z) is described by one or more of the pole-zero plots A-E,G(z),which does not corre-
spond to any of the pole-zero plots A-E,is a system such that w[n] = x[n],Which plot(s)
corresponds to H(z) such that both H(z) and G(z) are causal and stable?
Which plot(s)?
Brief explanation,
29 Problem 7 continues on the following page,
Work Page for Problem 7
30
Fall 2003,Final Exam NAME:
Work Space for Problem 7
31 Problem 7 continues on the following page.
1
Part c,Consider the following system with T = sec.
480
x
c
(t) C( ) ×
Impulse
to
Sequence
x[n]
H
d
(z)
y[n]
Sequence
to
Impulse
y
p
(t)
H ( )
T
T
T
y(t)j?
lp
j?
p(t) =?(t? nT)
T
n=
(i) Plot the frequency response C(j?) such that the entire system is LTI with the largest
possible bandwidth,
C(j?)
0
(ii) Assume that C(j?) is 1 for all? and x
c
(t) is suf?ciently band-limited so that the
Nyquist criteria is met,x
c
(t) consists of the superposition of s(t) which is the signal
you are interested in and a 60Hz sinusoidal interference,i.e.,
x
c
(t) = s(t) + cos(2? · 60t),
Which pole-zero plot corresponds to the best choice for H
d
(z) such that |Y (j?)|,the
magnitude of the Fourier transform of the overall output y(t) is approximately equal to
|S(j?)|,the magnitude of the Fourier transform of s(t)?
Which plot?
Brief explanation,
32
Fall 2003,Final Exam NAME:
Work Page for Problem 7
33
Fall 2003,Final Exam NAME,
Additional Work Page
There are no additional problems from this page on,Pages 34 to 37 are provided solely
as additional work pages,
34
Fall 2003,Final Exam NAME,
Additional Work Page
35
Fall 2003,Final Exam NAME,
Additional Work Page
36
Fall 2003,Final Exam NAME,
Additional Work Page
37