1
Copyright
(c) Andreas Spanias
Ch 8 - 1
DSP EEE 407/591
by
Andreas Spanias, Ph.D.
Chapter 8
[email protected]
http://www.eas.asu.edu/~spanias
Copyright
(c) Andreas Spanias
Ch 8 - 2
Introduction to Random Signal Processing
Deterministic Signals:
are characterized explicitly by a mathematical
equation and their value at any time can be predicted exactly.
Random Signals:
are not modeled explicitly by a mathematical
equation, however, they are usually characterized by their statistics.
Most “real-life” signals, such as those encountered in speech,
communications,,
and radar are random
Our discussion concentrates on signals whose D.C. content and
power spectra do not change with time are constant.
Also we will
be concerned with signals.
whose statistics can be measured by time
averaging.
We call such signals wide-sense
stationary
and
ergodic
.

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2
Copyright
(c) Andreas Spanias
Ch 8 - 3
Random Signal Processing - Some Definitions
µ
x
N
n
N
N
E
x n
N
x n
=
=
+
→ ∞
= −
∑
[
(
)]
(
)
lim
1
2
1
The mean
is defined as:
Statistical expectation; for ergodic
signals it is computed as a time average
The variance
σ
µ
µ
µ
x
x
x
x
E
x n
x n
E x
n
2
2
2
=
−
−
=
−
[( ( )
)( ( )
)]
[
( )]
The variance is a measure of
dispersion from the mean
Copyright
(c) Andreas Spanias
Ch 8 - 4
The mean,
, is the DC component.
The mean squared
is the DC power.
The mean square
is the total average power.
The variance
is the AC power.
The standard deviation
is the rms value.
µ
x
µ
x
2
E x
n
[
( )]
2
σ
x
2
σ
x
Examples, Interpretations, and Analogies to electrical signals
0
10
20
30
40
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80
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-4
-3
-2
-1
0
1
2
3
σ
2
1
>
σ
2
2
σ
2
1
σ
2
2
µ
1
=
µ
2
=0