Digital Signal Processing Reference
In-Depth Information
(e)
The product of two odd functions is odd.
(f)
The product of an even function and an odd function is odd.
9.13.
Suppose that the signals
x
1
[n],
x
2
[n]
and
x
3
[n]
are given by
a
2pn
10
a
2pn
25
, and x
3
[n] = e
j2pn/20
.
x
1
[n] = cos
b
,
x
2
[n] = sin
b
(a)
Determine whether
x
1
[n]
is periodic. If so, determine the number of samples per
fundamental period.
(b)
Determine whether
x
2
[n]
is periodic. If so, determine the number of samples per
fundamental period.
(c)
Determine whether
x
3
[n]
is periodic. If so, determine the number of samples per
fundamental period.
(d)
Determine whether the sum of
x
1
[n], x
2
[n],
and
x
3
[n]
is periodic. If so, determine
the number of samples per fundamental period.
9.14.
Consider the discrete-time signals that follow. For each signal, determine the fundamental
period
N
0
if the signal is periodic; otherwise, prove that the signal is not periodic.
x[n] = e
j5pn/7
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
x[n] = e
j5n
x[n] = e
j2pn
x[n] = e
j0.3n/p
x[n] = cos(3pn/7)
x[n] = e
j0.3n
x[n] = e
j5pn/7
+ e
j2pn
x[n] = e
j5pn/7
+ e
j2pn
+ cos(3pn/7)
x[n] = e
j.3n
+ e
j2pn
9.15.
(a)
A continuous-time signal
x(t) = cos
2pt
is sampled every
T
seconds, resulting in
the discrete-time signal
x[n] = x(nT).
Determine whether the sampled signal is
periodic for
(i) (ii)
(iii) (iv)
(v) (vi)
(b)
For those sampled signals in part (a) that are periodic, find the number of periods
of in one period of
x
[
n
].
(c)
For those sampled signals in part (a) that are periodic, find the number of samples
in one period of
x
[
n
].
T = 1
s
T = 0.1
s
T = 0.125
s
T = 0.130
s
4
3
s
T = 5
s
T =
x(t)
9.16.
A continuous-time signal is sampled at a 10-Hz rate, with the resulting discrete-time
signals as given. Find the time constant
x(t)
t
for each signal, and the frequency
v
of the si-
nusoidal signals.
x[n] = (.3)
n
(a)
(b)
(c)
(d)
x[n] = (.3)
n
sin(n + 1)
x[n] = (.3)
n
cos(n)
x[n] = (-.3)
n
