Digital Signal Processing Reference
In-Depth Information
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Chapter 8
Stochastic Signal Processing
In the previous Chapters, the signals we considered were all
deterministic
signals in the sense that they could either be expressed in analytic form
(such as
x
n
) or they could be explicitly described in terms of
their samples, such as in the case of finite-length signals. When designing
a signal processing system, however, it is very rare that we know exactly the
expression for the set of all the possible input signals (in some sense, if we
did, we would not need a processing system at all.) Fortunately, very often
this set can be characterized in terms of the
statistical
properties of its mem-
ber signals; this entails leaving the domain of deterministic quantities and
entering the world of stochastic processes. A detailed and rigorous treat-
ment of statistical signal processing is beyond the scope of this topic; here,
we only consider elementary concepts and restrict ourselves to the discrete-
time case. We will be able to derive that, fundamentally, in the case of sta-
tionary random signals, the standard signal processing machinery that we
have seen so far (and especially the usual filter design techniques) is still ap-
plicable with very intuitive results. To establish a coherent notation, we start
by briefly reviewing some elementary concepts of probability theory.
[
n
]=(
1
−
λ
)
λ
8.1 Random Variables
Probability Distribution.
Consider a real-valued random variable
X
tak-
ing values over
.Therandomvariable
(1)
is characterized by its
cumulative
distribution function F
X
(cdf ) which is defined as
F
X
(
α
)=
P
[
X
≤
α
]
,
α
∈
(1)
Note that in this Chapter, random quantities will be indicated by
uppercase
variables.
