B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

468 RANDOM PROCESSES AND SPECTRAL ANALYSIS
provided foo I r !Rx ( r )e-i 211 fr d r is bounded. Thus, the PSD of a wide-sense stationary
random process is the Fourier transform of its autocorrelation function,*
(9. 16)
This is the well-known Wiener-Khintchine theorem, first presented in Chapter 3.
From the discussion thus far, the autocorrelation function emerges as one of the most
significant entities in the spectral analysis of a random process. Earlier we showed heuristically
how the autocorrelation function is connected with the frequency content of a random process.
The autocorrelation function Rx ( r) for real processes is an even function of r . This can
be proved in two ways. First, because IXT (f)l 2 = IXT (f)Xr if) i = IXT (f)XT (-f)I is an even
function off, Sx if) is also an even function off, and Rx (r), its inverse transform, is also an
even function of r (see Prob. 3.1-1). Alternately, we may argue that
Letting t - r = CJ, we have
Rx (r) = x(t)x(t + r) and Rx (-r) = x(t)x(t - r)
Rx (-r) = x(CJ)x(CJ + r) = Rx (r) (9.17)
The PSD Sx (f) is also a real and even function off.
The mean square value x 2 (t) of the random process x(t) is R x (O),
Rx (O) = x(t)x(t) = x 2 (t) = x 2 (9.18)
The mean square value x 2 is not the time mean square of a sample function but the ensemble
average of the squares of all sample function amplitudes at any instant t.
The Power of a Random Process
The power Px (average power) of a wide-sense random process x(t) is its mean square value
x 2 • From Eq. (9. 16),
Hence, from Eq. (9 .18),
P x = x 2 = Rx (O) = L: Sx if) df
(9.19a)
Because Sx (f) is an even function off, we have
(9. 19b)
where f is the frequency in hertz. This is the same relationship as that derived for deterministic
signals in Chapter 3 [Eq. (3.81)]. The power Px is the area under the PSD. Also, P x = x 2 is
the ensemble mean of the square amplitudes of the sample functions at any instant.
* It can be shown that Eq. (9.15) holds also for complex random processes, for which we define
R x (T) = x*(t)x(t + ,).