Digital Signal Processing Reference
In-Depth Information
Clearly, the summation represented by term I equals 9(20)
/
40
=
4
.
5. To com-
pute the summation in term II, we express the cosine as follows:
19
19
19
9
40
1
2
[e
j
π
k
/
5
+
e
9
80
(e
j
π/
5
)
k
+
9
80
−
j
π
k
/
5
]
=
−
j
π/
5
)
k
.
term II
=
(e
k
=
0
k
=
0
k
=
0
Using the formulas for the GP series yields
19
=
1
−
(e
j
π/
5
)
20
1
−
(e
j
π/
5
)
1
−
e
j
π
4
1
−
(e
j
π/
5
)
1
−
1
1
−
(e
j
π/
5
)
(e
j
π/
5
)
k
=
=
=
0
k
=
0
and
19
−
j
π/
5
)
20
1
−
(e
j
π/
5
)
−
j
π
4
=
1
−
(e
1
−
e
1
−
1
1
−
(e
j
π/
5
)
−
j
π/
5
)
k
(e
=
=
=
0
.
1
−
(e
j
π/
5
)
k
=
0
Term II, therefore, equals zero. The average power of
g
[
k
] is therefore given
by
P
g
=
4
.
5
+
0
=
4
.
5
.
In
general,
a
periodic
DT
sinusoidal
signal
of
the
form
x
[
k
]
−
A
cos
A
2
/
2.
(
ω
0
k
+ θ
) has an average power
P
x
=
1.1.5 Deterministic and random signals
If the value of a signal can be predicted for all time (
t
or
k
) in advance without
any error, it is referred to as a
deterministic signal
. Conversely, signals whose
values cannot be predicted with complete accuracy for all time are known as
random signals
.
Deterministic signals can generally be expressed in a mathematical, or graph-
ical, form. Some examples of deterministic signals are as follows.
(1) CT sinusoidal signal:
x
1
(
t
)
=
5 sin(20
π
t
+
6);
(2) CT exponentially decaying sinusoidal signal:
x
2
(
t
)
=
2e
−
t
sin(7
t
);
e
j4
π
t
t
<
5
(3) CT finite duration complex exponential signal:
x
3
(
t
)
=
0
elsewhere;
−
2
k
;
(4) DT real-valued exponential sequence:
x
4
[
k
]
=
4e
−
2
k
(5) DT
exponentially
decaying
sinusoidal
sequence:
x
5
[
k
]
=
3e
16
π
k
5
sin
.
Unlike deterministic signals, random signals cannot be modeled precisely.
Random signals are generally characterized by statistical measures such as
means, standard deviations, and mean squared values. In electrical engineering,
most meaningful information-bearing signals are random signals. In a digital
communication system, for example, data are generally transmitted using a
sequence of zeros and ones. The binary signal is corrupted with interference
from other channels and additive noise from the transmission media, resulting
in a received signal that is random in nature. Another example of a random
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