Page 218 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES 21 1

Property 3 [Eq. (6.15)J is easily obtained by setting z = 0 in Eq. (6.12). If we assume that X(t) is a
voltage waveform across a 1-Q resistor, then E[X2(t)] is the average value of power delivered to the
1-Q resistor by X(t). Thus, E[x~(~)] often called the average power of X(t). Properties 1 and 2 are
is
verified in Prob. 6.13.
In case of a discrete-time random process X(n), the autocorrelation function of X(n) is defined by
Rx(k) = E[X(n)X(n + k)] (6.1 6)
Various properties of Rx(k) similar to those of RX(z) can be obtained by replacing z by k in Eqs. (6.13)
to (6.15).

B. Cross-Correlation Functions
The cross-correlation function of two continuous-time jointly WSS random processes X(t) and
Y(t) is defined by

Properties of RAT) :

These properties are verified in Prob. 6.14. Two processes X(t) and Y(t) are called (mutually) orthog-
onal if
RXy(z) = 0 for all z (6.21)

Similarly, the cross-correlation function of two discrete-time jointly WSS random processes X(n) and
Y(n) is defined by
Rxy(k) = E[X(n) Y(n + k)] (6.22)
and various properties of Rxy(k) similar to those of RXy(z) can be obtained by replacing z by k in Eqs.
(6.18) to (6.20).

C. Power Spectral Density:
The power spectral density (or power spectrum) Sx(o) of a continuous-time random process X(t) is
defined as the Fourier transform of RX(z):

Thus, taking the inverse Fourier transform of Sx(o), we obtain

Equations (6.23) and (6.24) are known as the Wiener-Khinchin relations.

Properties of SAo) :
1. SAo) is real and Sx(o) 2 0.