Digital Signal Processing Reference
In-Depth Information
(b)
Sketch the signals and verify the results of part (a).
(c)
Find the even part and the odd part of each of the signals.
9.10.
(a)
Given in Figure P9.10 are the
parts
of a signal
x
[
n
] and its even part
x
e
[n],
for
only
n G 0.
Note that
x
e
[n] = 2, n G 0.
Complete the plots of
x
[
n
] and
x
e
[n],
and give a
plot of the odd part,
x
o
[n],
of the signal. Give the equations used for plotting each
part of the signals.
(b)
In Figure P9.10, let with all other values unchanged. Give the changes in
this case for the results of part (a).
x[0] = 0,
x
[
n
]
6
4
2
• • •
5
4
3
2
1
0
1
2
3
4
5
n
x
e
[
n
]
2
• • •
5
4
3
2
1
0
1
2
3
4
5
n
Figure P9.10
9.11.
Let
x
e
[n] and x
o
[n]
be the even and odd parts, respectively, of
x
[
n
].
(a)
Show that
x
o
[0] = 0
and that
x
e
[0] = x[0].
(b)
Show that
q
n=-
q
x
o
[n] = 0.
(c)
Show that
q
n=-
q
q
n=-
q
x[n] =
x
e
[n].
(d)
Do the results of part (c) imply that
n
2
n
2
x[n] =
a
x
e
[n],
a
n=n
1
n=n
1
where
n
1
and
n
2
are any integers? Why?
9.12.
Give proofs of the following statements:
(a)
The sum of two even functions is even.
(b)
The sum of two odd functions is odd.
(c)
The sum of an even function and an odd function is neither even nor odd.
(d)
The product of two even functions is even.
