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

460 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Fi g ure 9.4
Autocorrelation
functions for a
slowly varying
and a rapidly
varying random
process.
x(t)
y(t)
(a) Y1 Y2 (b)
Rx(r)
(c)
random process. The spectral content of a process depends on the rapidity of the amplitude
change with time. This can be measured by correlating amplitudes at t1 and t1 + r. On average,
the random process x(t) in Fig. 9.4a is a slowly varying process in comparison to the process
y(t) in Fig. 9.4b. For x(t), the amplitudes at t1 and t1 + r are similar (Fig. 9.4a), that is, have
stronger correlation. On the other hand, for y(t), the amplitudes at t1 and t1 + T have little
resemblance (Fig. 9 .4b ), that is, have weaker correlation. Recall that correlation is a measure of
the similarity of two RVs. Hence, we can use correlation to measure the similarity of amplitudes
at t1 and t2 = t1 + T . If the RVs x(t1) and x(t2) are denoted by x1 and x2, respectively, then
for a real random process,* the autocorrelation function R x (t1 , t2) is defined as
(9.3a)
This is the correlation of RVs x(t 1 ) and x(t2), indicating the similarity between RVs x(t1) and
x(t2). It is computed by multiplying amplitudes at t 1 and t2 of a sample function and then
averaging this product over the ensemble. It can be seen that for a small r, the product x1 x2
will be positive for most sample functions of x(t), but the product YIY2 is equally likely to
be positive or negative. Hence, x1x2 will be larger than YI Y2· Moreover, XJ and x2 will show
correlation for considerably larger values of T, whereas YI and y2 will lose correlation quickly,
even for small r, as shown in Fig. 9.4c. Thus, R x (t 1 , t2), the autocorrelation function of x(t),
provides valuable information about the frequency content of the process. In fact, we shall
show that the PSD of x(t) is the Fourier transform of its autocorrelation function, given by
(for real processes)
R x (tl , t2) = X J X2
= 1_:1_: x1x2px(x1, x2; t1 , t2)dx1 dx2 (9.3b)
* For a complex random process x(t), the autocorrelation function is defined as