Page 443 - Fundamentals of Radar Signal Processing

(6.2)

Because probability density functions are nonnegative, Eq. (6.2) proves a

claim made earlier, namely that P and P must rise or fall together. As the
FA
D
region grows to include more of the possible observations y, either integral
1
encompasses more of the N-dimensional space and therefore integrates more of
the nonnegative PDF. The opposite is true if shrinks. That is, as grows or
1
1
3
shrinks, both P and P must decrease or increase. In order to increase
D
FA
detection probability, the false alarm probability must be allowed to increase as
well. Loosely speaking, to achieve a good balance of performance, the points
that contribute more probability mass to P than to P are assigned to . If the
1
FA
D
system can be designed so that p (y|H ) and p (y|H ) are as disjoint as possible,
0
1
y
y
this task becomes easier and more effective. This point will be revisited later.
6.1.2 The Likelihood Ratio Test
The Neyman-Pearson criterion is motivated by the goal of obtaining the best
possible detection performance while guaranteeing that the false alarm
probability does not exceed some tolerable value. Thus, the Neyman-Pearson
decision rule is to

(6.3)

where α is the maximum allowable false alarm probability. This optimization
4
problem is solved by the method of Lagrange multipliers. Construct the function

(6.4)

To find the optimum solution, maximize F and then choose λ to satisfy the
constraint criterion P = α. Substituting Eq. (6.2) into Eq. (6.4)
FA

(6.5)

Remember that the design variable here is the choice of the region . The first
1
term in the second line of Eq. (6.5) does not depend on , so F is maximized by
1
maximizing the value of the integral over . Since λ could be negative, the
1