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Random Processes and Spectral Analysis Chap. 6
THEOREM.
variables.
A random process may be described by an indexed set of random
Referring to Fig. 6–1, define a set of random variables v 1 = v(t 1 ), v 2 = v(t 2 ),. . . , where v(t)
is the random process. Here, the random variable v j = v(t j ) takes on the values described by
the set of constants {v(t j , E i ), for all i}.
For example, suppose that the noise source has a Gaussian distribution. Then any of the
random variables will be described by
1
f vj (v j ) = e -(v j-m j ) 2 /(2sj 2 )
(6–1)
12ps j
where v j ! v(t j ). We realize that, in general, the probability density function (PDF) depends
implicitly on time, since m j and s j respectively correspond to the mean value and standard
deviation measured at the time t = t j . The N = 2 joint distribution for the Gaussian source
for t = t 1 and t = t 2 is the bivariate Gaussian PDF f v (v 1 , v 2 ), as given by Eq. (B–97), where
v 1 = v(t 1 ) and v 2 = v(t 2 ).
To describe a general random process x(t) completely, an N-dimensional PDF, f x (x),
is required, where x = (x 1 , x 2 ,.. .,
x j ,... , x N ), x j ! x(t j ), and N → q. Furthermore, the
N-dimensional PDF is an implicit function of N time constants t 1 , t 2 , . .. , t N , since
f x (x) = f x (x(t 1 ), x(t 2 ), .. ., x(t N ))
(6–2)
Random processes may be classified as continuous or discrete. A continuous random
process consists of a random process with associated continuously distributed random variables
v j = v(t j ). The Gaussian random process (described previously) is an example of a continuous
random process. Noise in linear communication circuits is usually of the continuous type.
(In many cases, noise in nonlinear circuits is also of the continuous type.) A discrete random
process consists of random variables with discrete distributions. For example, the output of
an ideal (hard) limiter is a binary (discrete with two levels) random process. Some sample
functions of a binary random process are illustrated in Fig. 6–2.
Example 6–1 GAUSSIAN PDF
Plot the one-dimensional PDF for a Gaussian random variable where the mean is -1 and the
standard deviation is 2. See Example6_01.m for the solution.
Stationarity and Ergodicity
DEFINITION.
t 1 , t 2 ,...,
t N ,
A random process x(t) is said to be stationary to the order N if, for any
f x (x(t 1 ), x(t 2 ), . . . , x(t N )) = f x (x(t 1 + t 0 ), x(t 2 + t 0 ), . . . , x(t N + t 0 ))
(6–3)
where t 0 is any arbitrary real constant. Furthermore, the process is said to be strictly
stationary if it is stationary to the order N → q.