Page 444 - Fundamentals of Radar Signal Processing

integrand can be either positive or negative, depending on the values of λ and

the relative values of p (y|H ) and p (y|H1). The integral is therefore maximized
y
0
y
by including in all the points, and only the points, in the N-dimensional space
1
for which p (y|H ) + λp (y|H ) > 0, that is, is all points y for which p (y|H ) >
1
1
y
1
y
y
0
– λp (y|H ). This leads directly to the decision rule:
y
0
(6.6)

Equation (6.6) is known as the likelihood ratio test (LRT). Although
derived from the point of view of determining what values of y should be
assigned to decision region , it in fact allows one to skip over explicit
1
determination of and gives a rule for optimally guessing, under the Neyman-
1
Pearson criterion, whether a target is present or not based directly on the
observed data y and a threshold –λ (which must still be computed). This
equation states that the ratio of the two PDFs, each evaluated for the particular
observed data y, should be compared to a threshold. If that “likelihood ratio”
exceeds the threshold, choose hypothesis H , i.e., declare a target to be present.
1
If it does not exceed the threshold, choose H and declare that a target is not
0
present. Under the Neyman-Pearson optimization criterion, the probability of a
false alarm cannot exceed the original design value P . Note again that models
FA
of p (y|H ) and p (y|H1) are required in order to carry out the LRT. Finally,
y
y
0
realize that in computing the LRT, the data processing operations to be carried
out on the observed data y are being specified. What exactly the required
operations are depends on the particular PDFs.
The LRT is as ubiquitous in detection theory and statistical hypothesis
testing as is the Fourier transform in signal filtering and analysis. It arises as the
solution to the hypothesis testing problem under several different decision
criteria, such as the Bayes minimum cost criterion, or maximization of the
probability of a correct decision. Substantial additional detail is provided in

Johnson and Dudgeon (1993) and Kay (1998). As a convenient and common
shorthand, it is convenient to express the LRT in the following notation:

(6.7)

From Eq. (6.6), Λ(y) = p (y|H )/p (y|H ) and η = –λ.
y
y
0
1
Because the decision depends only on whether the LRT exceeds the
5
threshold or not, any monotone increasing operation can be performed on both
sides of Eq. (6.7) without affecting the values of observed data y that cause the