Page 44 - Introduction to Information Optics

1.5. Signal Analysis 29

1.5.2. STATISTICAL SIGNAL DETECTION

In the preceding section we illustrated that signal detection can be achieved
by improving the output signal-to-noise ratio. In fact, the increase in signal-to-
noise ratio is purposely to minimize the probability of error in signal detection.
However, in certain signal detections, an increase in the signal-to-noise ratio
does not necessarily guarantee minimizing the detection error. Nevertheless,
minimizing the detection error can always be obtained by using a decision
process.
Let us consider the detection of binary signals. We establish a Bayes's
decision rule, in which the conditional probabilities are given by

(1.98)
r\p)

One can write

P(a = Q/b) = P(a = Q)P(b/a = 0)
P(0 = l/b) P(a = l)p(b/a = 1)' " J

where P(0) is the a priori probability of 0; that is, a = I corresponds to the
signal presented, and 0 = 0 corresponds to no signal.
A logical decision rule is that, if P(0 = Q/b) > P(a = l/b), we decide that
there is no signal (0 = 0) for a given b. However, if P(0 = Q/b) < P(a = l/b),
then we decide that there is a signal (0 = 1) for a given b. Thus, the Bayes's
decision rule can be written as:

Accept a = Q if

w
« = l) (UOO)

Accept a — 1 if

W «=0) Pjo^J)
;
P(b/a = 1) P(a = 0)

Two possible errors can occur: if we accept that the received event b contains
no signal (a = 0), but the signal does in fact exist (a = 1); and vice versa. In
other words, the error of accepting a — 0 when a — 1 has actually occurred is
a miss, and the error of accepting 0= 1 when a = I has actually not occurred
is a false alarm.