ENEE 241 02
∗
HOMEWORK ASSIGNMENT 2
Due Tue 02/15
Problem 2A
Use your calculator for algebraic calculations only. Solutions based on trial and error, inspection
of numerical plots, etc., are not acceptable.
Consider the sinusoid
x
(
t
) =
A
cos(
Ω
t
+
φ
), where
A >
0 and
φ
∈
(
−
π
,
π
]. Time
t
is in seconds.
It is known that
•
x
(
t
)
≥
2
.
4 for exactly 18% of each period;
•
it takes 0.123 seconds for the value of the sinusoid to drop from 2
.
4 to the next minimum
(“valley”);
•
the first zero of the sinusoid in positive time occurs at
t
= 0
.
040 (seconds).
(i) (5 pts.)
Determine the amplitude
A
.
(ii) (5 pts.)
Determine the period and angular frequency
Ω
of
x
(
t
).
(iii) (5 pts.)
Determine the initial phase
φ
of
x
(
t
) as a fraction of
π
.
(iv) (5 pts.)
Write simple MATLAB code which computes and plots two periods of
x
(
t
) starting
at
t
= 0, using 150 uniformly spaced samples per period (i.e., a total of 300 samples). Attach a
printout of the code and a plot of the result.
Problem 2B
Consider the discretetime sinusoids
x
[
n
] = cos
5
π
n
8
+
π
4
and
y
[
n
] = cos
10
π
n
7
−
2
π
3
(i) (3 pts.)
What is the fundamental period of each sinusoid?
(ii) (4 pts.)
Use MATLAB to generate separate plots of
x
[
n
] and
y
[
n
] for
n
= 0
, . . . ,
111.
(iii) (3 pts.)
If
N
x
and
N
y
are the two fundamental periods found in part (i), show that
u
[
n
] =
x
[
n
] +
y
[
n
] is also periodic with period
N
u
=
N
x
N
y
. Is
N
u
the fundamental period of
u
[
n
]?
(iv) (3 pts.)
An equivalent form for
y
[
·
] is
y
[
n
] = cos (
ω
n
+
φ
)
where
ω
is between 0 and
π
. What are the values of
ω
and
φ
?
(v) (2 pts.)
The sequence
v
[
·
] is formed by taking every other sample in
x
[
·
], i.e.,
v
[
n
] =
x
[2
n
]
Write an equation for
v
[
n
]. What is the period of
v
[
·
]?
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 Spring '08
 staff
 Cos, Period, Fundamental frequency, pts

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