Page 221 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Properties of SAC&)
:
Unlike S,(SZ), which is a real-valued function of w, Sxy(Q), in general, is a complex-valued func-
tion.

6.4 WHITE NOISE
A continuous-time white noise process, W(t), is a WSS zero-mean continuous-time random
process whose autocorrelation function is given by

where 6(2) is a unit impulse function (or Dirac 6 function) defined by

where @(r) is any function continuous at z = 0. Taking the Fourier transform of Eq. (6.43), we obtain

which indicates that X(t) has a constant power spectral density (hence the name white noise). Note
that the average power of W(t) is not finite.
Similarly, a WSS zero-mean discrete-time random process W(n) is called a discrete-time white noise
if its autocorrelation function is given by

where S(k) is a unit impulse sequence (or unit sample sequence) defined by

Taking the Fourier transform of Eq. (6.46), we obtain

Again the power spectral density of W(n) is a constant. Note that Sw(R + 2n) = Sw(Q) and the
average power of W(n) is o2 = Var[W(n)], which is finite.

6.5 RESPONSE OF LINEAR SYSTEMS TO RANDOM INPUTS
A. Linear Systems:

A system is a mathematical model of a physical process that relates the input (or excitation)
signal x to the output (or response) signal y. Then the system is viewed as a transformation (or
mapping) of x into y. This transformation is represented by the operator T as (Fig. 6-1)

If x and y are continuous-time signals, then the system is called a continuous-time system, and if x
and y are discrete-time signals, then the system is called a discrete-time system. If the operator T is a