Page 144 - Introduction to Statistical Pattern Recognition

126 Introduction to Statistical Pattern Recognition

0 bserved
at t=T

m,(t)
Fig. 4-1 Block diagram of a correlation classifier.

We can see that the classifier (4.3) compares the difference in the correlations
of X with M1 and M2 with a threshold to make a decision. Thus, we may call
it a correlation classifier. The structure of the correlation classifier is shown in
Fig. 4-1, and is written as
WI
MYX - MTX 5 c . (4.6)
cu,
If c is selected as (MTM, - M;M2)/2 - In P I/P2, (4.6) becomes identical to
(4.3). Thus, in order for the correlation classifier to be the Bayes classifier, the
distributions must be normal with the equal covariance matrix I for both o1
and 02.

Matched Filter

The correlation between Mi and X can also be considered as the output
of a linear filter. Suppose we construct functions g;(t) such that
gi(T - t) = mi(r) . (4.7)

The relation between gi(t) and mi(f) is illustrated in Fig. 4-2. Then, clearly,

Thus, the correlation is the output of a linear filter whose impulse response is
gi(t). This filter is called a matchedflter. The matched filter classifier, which