Digital Signal Processing Reference
In-Depth Information
and
P
z
(
t
+
8)
=
P
z
(
t
);
4
2
=
1
8
=
1
8
=
100
8
z
(
t
)
2
d
t
average power
P
z
25 d
t
=
12
.
5;
−
4
−
2
∞
z
(
t
)
2
d
t
energy
E
z
=
=∞.
−∞
Because the signal has finite power (0
<
P
z
=
12
.
5
< ∞
)
,
z
(
t
) is a power
signal.
Example 1.7
Consider the following DT sequence:
−
0
.
5
k
e
k
≥
0
f
[
k
]
=
0
k
<
0
.
Determine if the signal is a power or an energy signal.
Solution
The total energy of the DT sequence is calculated as follows:
∞
∞
∞
1
1
−
e
f
[
k
]
2
−
0
.
5
k
2
−
1
)
k
E
f
=
=
e
=
(e
=
≈
1
.
582
.
−
1
k
=−∞
k
=
0
k
=
0
Because
E
f
is finite, the DT sequence
f
[
k
] is an energy signal.
In computing
E
f
, we make use of the geometric progression (GP) series to
calculate the summation. The formulas for the GP series are considered in
Appendix A.3.
Example 1.8
Determine if the DT sequence
g
[
k
]
=
3 cos(
π
k
/
10) is a power or an energy
signal.
Solution
The DT sequence
g
[
k
]
=
3 cos(
π
k
/
10) is a periodic signal with a fundamental
period of 20. All periodic signals are power signals. Hence, the DT sequence
g
[
k
] is a power signal.
Using Eq. (1.15), the average power of
g
[
k
]isgivenby
19
19
1
20
π
k
10
9
20
1
2
2
π
k
10
9 cos
2
P
g
=
=
1
+
cos
k
=
0
k
=
0
19
19
9
40
+
9
40
2
π
k
10
=
1
cos
.
k
=
0
k
=
0
term I
term II
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