Digital Signal Processing Reference
In-Depth Information
x(t) = e
j2t
(c)
(d)
(e)
(f)
x(t) = cos t + sin2t
x(t) = e
j(5t +p)
x(t) = e
-j10t
+ e
j15t
2.13.
For each signal, if it is periodic, find the fundamental period
T
0
and the fundamental
frequency
v
0
.
Otherwise, prove that the signal is not periodic.
(a)
(b)
(c)
(d)
x(t) = cos 3t + sin 5t.
x(t) = cos 6t + sin 8t + e
j2t
.
x(t) = cos t + sin pt.
x
1
(t) = sin(
p
6
)
x
2
(t) = sin(
p
9
).
x(t) = x
1
(t) + x
2
(3t)
where
and
2.14.
(a)
Consider the signal
x(t) = 4 cos(12t + 40°) + sin16t.
If this signal is periodic, find its fundamental period and its fundamental
frequency Otherwise, prove that the signal is not periodic.
(b)
Repeat Part (a) for the signal
T
0
v
0
.
x(t) = cos 4t + 3e
- j12t
(c)
Repeat Part (a) for the signal
x(t) = cos 2pt + sin 6t.
(d)
Repeat Part (a) for the signal
x
4
(t) = x
1
(t) + x
2
(t) + x
3
(t),
where
q
n=-
q
t
+
n
0.2
a
5p
6
p
4
¢
≤
x
1
(t) = cos(pt), x
2
(t) =
rect
,
and
x
3
(t) = 4 sin
t +
b
.
2.15.
Suppose that
x
1
(t)
is periodic with period
T
1
and that
x
2
(t)
is periodic with period
T
2
.
(a)
Show that the sum
x(t) = x
1
(t) + x
2
(t)
is periodic only if the ratio
T
1
/T
2
is equal to a ratio of two integers
k
2
/k
1
.
(b)
Find the fundamental period
T
0
of
x
(
t
), for
T
1
/T
2
= k
2
/k
1
.
2.16.
Find
q
d(at - b)sin
2
(t - 4)dt,
L
-
q
where
a 7 0.
(
Hint
: Use a change of variables.)
2.17.
Express the following in terms of
x
(
t
):
q
1
2
L
y(t) =
x(t)[d(t - 2) + d(t + 2)]dt.
-
q
