A full exposition of the statistical theory of
signal detection that is outlined here can be found in the
monographs
[87, 46, 83, 82, 56, 37
, 67]
. A general introduction to stochastic processes is given
in
[85]
. Advanced treatment of the subject can be found in
[54, 86]
.
The problem of detecting the signal in noise
can be posed as a statistical hypothesis testing problem. The
null hypothesis
is that the signal is absent from the data and the
alternative hypothesis
is that the signal is present. A
hypothesis test
(or
decision rule)
is a partition of the observation set into two sets,
and its complement
. If data are in
we accept the null hypothesis, otherwise we reject it. There are
two kinds of errors that we can make. A type I error is choosing
hypothesis
when
is true and a type II error is choosing
when
is true. In signal detection theory the probability of a type I
error is called the
false alarm probability, whereas the probability of a type II error is called the
false dismissal probability
.
is the
probability
of detection
of the signal. In hypothesis testing the probability of a type I
error is called the
significance of the test, whereas
is called the
power of the
test
.
The problem is to find a test that is in some way optimal. There are several approaches to find such a test. The subject is covered in detail in many books on statistics, for example see references [44, 34, 52] .
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