and
g
xy
n, k Efxn À m
x
nyk À m
y
kg r
xy
n, k À m
x
nm
y
k:
8:1:5
Correlation is a veryuseful DSP tool for detecting signals that are corrupted
byadditive random noise, measuring the time delaybetween two signals, determining
the impulse response of a system (such as obtain the room impulse response used in
Section 4.5.2), and manyothers. Signal correlation is often used in radar, sonar, digital
communications, and other engineering areas. For example, in CDMA digital commu-
nications, data symbols are represented with a set of unique key sequences. If one of
these sequences is transmitted, the receiver compares the received signal with every
possible sequence from the set to determine which sequence has been received. In radar
and sonar applications, the received signal reflected from the target is the delayed
version of the transmitted signal. Bymeasuring the round-trip delay, one can determine
the location of the target.
Both correlation functions and covariance functions are extensivelyused in analyzing
random processes. In general, the statistical properties of a random signal such as the
mean, variance, and autocorrelation and autocovariance functions are time-varying
functions. A random process is said to be stationaryif its statistics do not change
with time. The most useful and relaxed form of stationaryis the wide-sense stationary
(WSS) process. A random process is called WSS if the following two conditions are
satisfied:
1. The mean of the process is independent of time. That is,
Exn m
x
,
8:1:6
1.
where m
x
is a constant.
2. The autocorrelation function depends onlyon the time difference. That is,
r
xx
k Exn kxn:
8:1:7
Equation (8.1.7) indicates that the autocorrelation function of a WSS process is inde-
pendent of the time shift and r
xx
k denotes the autocorrelation function of a time lag of
k samples.
The autocorrelation function r
xx
k of a WSS process has the following important
properties:
1. The autocorrelation function is an even function of the time lag k. That is,
r
xx
Àk r
xx
k:
8:1:8
2. The autocorrelation function is bounded bythe mean squared value of the process
expressed as
jr
xx
kj r
xx
0,
8:1:9
INTRODUCTION TO RANDOM PROCESSES
353