Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 43

1.4 Applications of Analog Signal Analysis

1.4.1 Signal Detection in Noise

In Sect. 1.2.4 the autocorrelation function was defined for deterministic periodic
and non-periodic signals. The autocorrelation function of random signals was
defined in Sect. 1.3.3.5 ; that definition, however, requires ensemble averaging, and
in many practical situations, an ensemble of experimental realizations is not
available. For this reason it is common in practice to compute the autocorrelation
function of random signals using the definition in Sect. 1.2.4 —that definition uses
a time average in its formulation rather than an ensemble average. This practice is
strictly valid only when the random signals are ergodic; i.e., for signals whose
ensemble average equals the time average. There are many signals for which this
represents a reasonable approximation to reality. With this approach, the definition
for the autocorrelation function of random signals becomes:

Z

T = 2

R x ð s Þ¼ 1
T

x ð k Þ x ð s þ k Þ dk ;

T = 2

while the cross-correlation between two random signals is defined as:

Z

T = 2

R xy ð s Þ¼ 1
T

x ð k Þ y ð s þ k Þ dk :

T = 2

Naturally occurring noise is often un-correlated with deterministic signals. It is
also often very nearly uncorrelated with itself (i.e., almost white), so that its
autocorrelation function is almost a delta function (see Sect. 1.3.3.8 ). These
properties are exploited in many practical schemes for detecting signals in noise.
Consider, for example, the scenario in which a deterministic signal x(t) is trans-
mitted across a communication channel, and a corrupted signal y(t) =
x(t) ? n(t) is received, One often has to decide whether or not there is a message
present inside the received signal or not. One simple way to inform this decision is
to correlate y(t) with itself. The autocorrelation of the signal with itself is:

Z

T = 2

R y ð s Þ¼ 1
T

½ x ð k Þþ n ð k Þ½ x ð k þ s Þþ n ð k þ s Þ dk :

ð 1 : 29 Þ

T = 2

¼ R x ð s Þþ R n ð s Þþ 2R xn ð s Þ

If the noise is white then R n (s) is a spike and R xn (s) is approximately zero. Now for
most communication signals, R y (s) exhibits a shape which is more than simply a
spike and/or a negligibly small background noise. If such a shape is present in the
received signal, then, one can infer that a true message is present.