Page 442 - Fundamentals of Radar Signal Processing

Probability of The probability that a target is declared (i.e., H is
1
False Alarm, P : chosen) when a target is in fact not present. 0
FA The probability that a target is not declared (i.e., H is
Probability of
Miss, P : chosen) when a target is in fact present.
M
Note that P = 1 – P . Thus, P and P suffice to specify all of the
D
M
FA
D
probabilities of interest. As the latter two definitions imply, it is important to
realize that, because the problem is statistical, there will be a finite probability
that the decisions will be wrong. 2

6.1.1 The Neyman-Pearson Detection Rule

The next step in making a decision is to decide what the rule will be for
deciding what constitutes an optimal choice between the two hypotheses. This is
a rich field. The Bayes optimization criterion assigns a cost or risk to each of
the four possible combinations of actual state (target present or not) and
decision (select H or H ). In radar, it is more common to use a special case of
1
0
the Bayes criterion called the Neyman-Pearson criterion. Under this criterion,
the decision process is designed to maximize the probability of detection P D
under the constraint that the probability of false alarm P does not exceed a set
FA
value. The achievable combinations of P and P are affected by the quality of
FA
D
the radar system and signal processor design. However, it will be seen that for a
fixed system design, increasing P implies increasing P as well. The radar
D
FA
system designer will generally decide what rate of false alarms can be tolerated
based on the implications of acting on a false alarm, which may include using

radar resources to start a track on a nonexistent target, or in extreme cases even
firing a weapon! Recalling that the radar may make tens or hundreds of
thousands, even millions of detection decisions per second, values of P must
FA
–8
–4
generally be quite low. Values in the range of 10 to 10 are common, and yet
may still lead to false alarms every few seconds. Higher-level logic
implemented in downstream data processing, beyond the scope of this book, is
often used to reduce the number or impact of false alarms.
Each vector of measured data values y can be considered to be a point in
N-dimensional space. To have a complete decision rule, each point in that space
(each possible combination of N measured data values) must be assigned to one
of the two allowed decisions, H or H . Then, when the radar measures a
0
1
particular set of data values (“observation”) y, the system declares either
“target absent” or “target present” based on the preexisting assignment of y to
either H or H . Denote the set of all observations y for which H will be chosen
1
1
0
as the region . Note that is not necessarily a connected region. General
1
1
expressions can now be written for the probabilities of detection and false
alarm as integrals of the joint PDFs over the region in a N-dimensional space:
1