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Random Processes and Spectral Analysis Chap. 6
The definitions of the correlation functions can be generalized to cover complex random
processes.
DEFINITION.
The autocorrelation function for a complex random process is
R g (t 1 , t 2 ) = g*(t 1 )g(t 2 )
(6–33)
where the asterisk denotes the complex conjugate.
Furthermore, the complex random process is stationary in the wide sense if g(t) is a
complex constant and R g (t 1 , t 2 ) = R g (t), where t = t 2 - t 1 . The autocorrelation of a wide-sense
stationary complex process has the Hermitian symmetry property
R g (-t) = R g
*(t)
(6–34)
DEFINITION.
and g 2 (t) is
The cross-correlation function for two complex random processes g 1 (t)
R g1 g 2
(t 1 , t 2 ) = g 1
*(t 1 )g 2 (t 2 )
(6–35)
When the complex random processes are jointly wide-sense stationary, the crosscorrelation
function becomes
R g1 g 2
(t 1 , t 2 ) = R g1 g 2
(t)
where t = t 2 - t 1 .
In Sec. 6–7, we use these definitions in the statistical description of bandpass random
signals and noise.
6–2 POWER SPECTRAL DENSITY
Definition
A definition of the PSD x ( f ) was given in Chapter 2 for the case of deterministic waveforms
by Eq. (2–66). Here we develop a more general definition that is applicable to the spectral
analysis of random processes.
Suppose that x(t, E i ) represents a sample function of a random process x(t). A truncated
version of this sample function can be defined according to the formula
x T (t, E i ) = b x(t, E i), ƒtƒ 6 1 2 T
(6–36)
0, t elsewhere
where the subscript T denotes the truncated version. The corresponding Fourier transform is
q
X T (f, E i ) = x T (t, E i ) e -j2pft dt
L
-q
T/2
= x(t, E i )e -j2pft dt
L
-T/2
(6–37)