Digital Signal Processing Reference
In-Depth Information
little information, and therefore statistical averages such as mean, autocor-
relation, and power spectral density are commonly used to specify stochastic
signals. We start by defining the statistical mean and autocorrelation commonly
used to define a stochastic signal. If
x
[
k
]
,
x
[
k
1
]
,
x
[
k
2
] are discrete random vari-
ables taking on values from the set
x
m
, −∞ ≤
m
≤∞
at times
k
,
k
1
, and
k
2
,
respectively, the mean and autocorrelation functions are defined as follows:
∞
mean
E
x
[
k
]
=
x
m
P
[
x
[
k
]
=
x
m
];
(17.5)
m
=−∞
autocorrelation
R
xx
[
k
1
,
k
2
]
=
E
x
[
k
1
]
x
[
k
2
]
∞
∞
=
x
m
x
n
P
[
x
[
k
1
]
=
x
m
;
x
[
k
2
]
=
x
n
]
.
m
=−∞
n
=−∞
(17.6)
In Eqs. (17.5) and (17.6), the operator
E
denotes the expectation and
P
[
x
[
k
]
=
x
m
] is the probability that
x
[
k
] takes on the value
x
m
. Likewise,
P
[
x
[
k
1
]
=
x
m
;
x
[
k
2
]
=
x
n
] refers to the joint probability for random signals
x
[
k
1
] and
x
[
k
2
]
observed at time instants
k
1
and
k
2
. Estimating the mean and autocorrelation of
a stochastic signal is difficult in general. In many applications, random signals
satisfy the following two properties.
(1) The mean
E
{
x
[
k
]
}
is constant and independent of time.
(2) The autocorrelation
E
{
x
[
k
1
]
x
[
k
2
]
}
depends upon the duration between the
observation instants
k
1
and
k
2
. In other words, the autocorrelation is inde-
pendent of the observation instants and is only determined by the duration
between the two observations.
Such signals are referred to as wide-sense stationary (WSS) random signals.
Sometimes, these are referred to as weak-sense stationary or second-order sta-
tionary random signals. Mathematically, the aforementioned two properties of
the WSS signals can be expressed as follows:
E
x
[
k
]
= µ
x
;
mean
(17.7)
R
xx
[
k
1
,
k
2
]
=
R
xx
[
k
1
−
k
2
]
=
R
xx
[
m
]
.
autocorrelation
(17.8)
The DTFT of the autocorrelation
R
xx
[
m
] of a WSS signal is referred to as the
power spectral density, which is defined as follows:
∞
R
xx
[
m
]e
−
j
Ω
m
.
power spectral density
S
xx
(
Ω
)
=
(17.9)
m
=−∞
Equations (17.8) and (17.9) are widely used to estimate the spectral content
of WSS signals, and the equations require the probability density functions
to estimate the spectral content, which is generally not known in most signal
processing applications. In the following, we present a method, based on the
periodogram, to estimate the spectral content of stochastic signals from a finite
number of observations.
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