Digital Signal Processing Reference
In-Depth Information
2.9.
The average value
A
x
of a signal
x
(
t
) is given by
T
1
2T
L
A
x
=
lim
T:
q
x(t)dt.
-T
Let
x
e
(t)
be the even part and
x
o
(t)
be the odd part of
x
(
t
).
(a)
Show that
T
1
2T
L
lim
T:
q
x
o
(t)dt = 0.
-T
(b)
Show that
T
1
2T
L
lim
T:
q
x
e
(t)dt = A
x
.
-T
(c)
Show that
x
o
(0) = 0
and
x
e
(0) = x(0).
2.10.
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.
(e)
The product of two odd functions is even.
(f)
The product of an even function and an odd function is odd.
2.11.
Given in Figure P2.11 are the parts of a signal
x
(
t
) and its odd part
x
o
(t),
for
t Ú 0
only;
that is,
x
(
t
) and
x
o
(t)
for
t 6 0
are not given. Complete the plots of
x
(
t
) and
x
e
(t),
and
give a plot of the even part,
x
e
(t),
of
x
(
t
). Give the equations used for plotting each part
of the signals.
2.12.
Prove mathematically that the signals given are periodic. For each signal, find the fun-
damental period
T
0
and the fundamental frequency
v
0
.
(a)
(b)
x(t) = sin(8t + 30°)
x(t) = 7 sin3t
x
(
t
)
2
2
1
0
1
2
t
x
e
(
t
)
2
Figure P2.11
2
1
0
1
2
t
