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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems Fall 2003
Quiz 1
Tuesday,October 14,2003
Directions,The exam consists of 5 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
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 booklet,
Please make sure your name is on all sheets,DO IT NOW!,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 one 8 1/2 × 11 sheet 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
18
20
20
21
21
100
Problem No,of points Score
Total
PROBLEM 1 (18%)
For the questions in this problem,no explanation is necessary,
Consider the following three systems,
SYSTEM A,y(t) = x(t + 2) sin(?t + 2),where= 0
1
n
SYSTEM B,y[n] =? (x[n] + 1)
2
n
2
SYSTEM C,y[n] = x [k + 1]? x[k]
k=1
where x and y are the input and output of each system.
Circle YES or NO for each of the following questions for each of these three systems.
SYSTEM A SYSTEM B SYSTEM C
Is the system linear?
Is the system time invariant?
Is the system causal?
Is the system stable?
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
YES NO
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 1
3
PROBLEM 2 (20%)
Consider a DT LTI system,H
2
with a unit sample response h
2
[n] = h[n]? h[n + 1],as shown
below,where h[n] =?[n][n? 1],You may remember from one of the lectures that h[n]
can be viewed as the unit sample response of a DT LTI system that acts as an edge detector,
The purpose of this problem is to develop an edge detector that is robust against additive
noise,
h
2
[n]
1 1
System H
2
x[n]
+
0
h
2
[n] np[n] y[n]
2?1 1 2
d[n]?2
Part a,Assume that the input to the system,p[n] is as shown below,and there is no noise,
i.e.,d[n] = 0 and p[n] = x[n],Provide a labeled sketch of y[n],the output of the system,
2 2 2 2
p[n]
2?1 0 1 2 3?3?4
n
y[n]
n
7?6?5?4?3?2?1 1 2 3 4 5 6 7
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 2
5 Problem 2 continues on the following page.
Part b,For the same input signal as Part a.,now assume that the noise signal is
d[n] =[n + 1],
Provide a labeled sketch of the output y[n],i.e.,the response to x[n] = p[n] + d[n],
y[n]
n
7?6?5?4?3?2?1 1 2 3 4 5 6 7
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 2
7 Problem 2 continues on the following page.
Part c,In order to use system H
2
as a part of an edge detector,we would like to add an
LTI system H
s
whose unit sample response,h
s
[n] is shown below,System H
s
smoothes out
effect of noise on x[n],The overall system can be represented as below,
2
h
s
[n]
2?1 0 1
n
1 1
System H
s
System H
2
p[n] +
x[n]
h
s
[n] h
2
[n] y
s
[n]
d[n]
Provide a labeled sketch of the overall output y
s
[n],when p[n] and d[n] are exactly the same
as in Part b,
y
s
[n]
n
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 2
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PROBLEM 3 (20%)
Consider the CT LTI system whose impulse response is given as,
h(t)
h(t)
1
x(t) y(t)
t
1 0 1
The following two parts can be done independently,
Part a,The input x(t),an impulse train starting at t = 2,is depicted below,
x(t)
0 1 2 3 4 5?1
a0a1a0a1a0a2a0a1a0
(1) (1) (1) (1)
t
Provide a labeled sketch of the corresponding output y(t),
y(t)
t
7?6?5?4?3?2?1 1 2 3 4 5 6 7
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Fall 2003,Quiz 1 NAME:
Work Space for Problem 3
11 Problem 3 continues on the following page.
Part b,For this part,the output y(t) is periodic and is depicted below:
3?3?6
6 9
2?2
4 8
y(t)
t
· · ·· · ·
2
2
Provide a labeled sketch of the input x(t) that produces this y(t),
x(t)
t
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 3
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PROBLEM 4 (21%)
Consider the following periodic triangular wave shown below,
x(t)
1?1
1
1
t
1/4
3/4 1/4
3/4
· · · · · ·
Part a,Determine the Fourier series coef?cients,a
k
for x(t),
a
k
=
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 4
15 Problem 4 continues on the following page.
Part b,Consider a causal LTI system,S,whose input-output relation is characterized by the
following stable linear constant coef?cient differential equation,
d
2
y dy
+ 4? + 4?
2
y(t) = 4?
2
x(t),
dt
2
dt
where x(t) is the input and y(t) is the output of the system,Suppose x(t) shown on the
previous page is applied to the system S as an input,Let b
k
be the Fourier coef?cients of the
corresponding output y(t),Find b
3
and b
3
,
b
3
= b
3
=,
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 4
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PROBLEM 5 (21%)
You are given the following facts about a discrete time sequence x[n],
(a) x[n] is real and odd,
(b) x[n] is periodic with period N = 6,
1
(c) |x[n]|
2
= 10,
N
n=<N >
(d) (?1)
n/3
x[n] = 6j,
n=<N >
(e) x[1] > 0,
Find an expression of x[n] in the form of sines and cosines,
x[n] =
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Fall 2003,Quiz 1 NAME:
Work Page for Problem 5
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Fall 2003,Quiz 1 NAME,
Work Page
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Fall 2003,Quiz 1 NAME,
Work Page
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