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Sec. 6–9 Summary 475
6–9 SUMMARY
A random process is the extension of the concept of a random variable to random waveforms.
A random process x(t) is described by an N-dimensional PDF, where the random variables are
x 1 = x(t 1 ), x 2 = x(t 2 ),..., x N = x(t N ). If this PDF is invariant for a shift in the time origin, as
N → q, the process is said to be strict-sense stationary.
The autocorrelation function of a random process x(t) is
In general, a two-dimensional PDF of x(t) is required to evaluate R x (t 1 , t 2 ). If the process is
stationary,
where t = t 2 - t 1 .
If x(t) is a constant and if R x (t 1 , t 2 ) = R x (t), the process is said to be wide-sense stationary.
If a process is strict-sense stationary, it is wide-sense stationary, but the converse
is not generally true.
A process is ergodic when the time averages are equal to the corresponding ensemble
averages. If a process is ergodic, it is also stationary, but the converse is not generally true.
For ergodic processes, the DC value (a time average) is also X dc = x(t) and the RMS value
(a time average) is also X rms = 2 x2 (t).
The power spectral density (PSD), x( f ), is the Fourier transform of the autocorrelation
function R x (t) (Wiener–Khintchine theorem):
The PSD is a nonnegative real function and is even about f = 0 for real processes. The PSD
can also be evaluated by the ensemble average of a function of the Fourier transforms of the
truncated sample functions.
The autocorrelation function of a wide-sense stationary real random process is a real
function, and it is even about t = 0. Furthermore, R x (0) gives the total normalized average
power, and this is the maximum value of R x (t).
For white noise, the PSD is a constant and the autocorrelation function is a Dirac delta
function located at t = 0. White noise is not physically realizable, because it has infinite
power, but it is a very useful approximation for many problems.
The cross-correlation function of two jointly stationary real processes x(t) and
y(t) is
and the cross-PSD is
R x (t 1 , t 2 ) = x(t 1 )x(t 2 ) =
L
q
R x (t 1 , t 2 ) = x(t 1 )x(t 1 + t) = R x (t)
q
-q L-q
x (f) = [R x (t)]
R xy (t) = x(t)y(t + t)
x 1 x 2 f x (x 1 , x 2 ) dx 1 dx 2
xy (f) = [R xy (t)]