344 Lecture 18 
faa) 
a 
Fig. 18.1. Receiver; operating 
characteristic curve. 
fe) =| 
p(FA) 
Let us imagine many independent successive trials using the detection 
likelihood-ratio processor in the f,,{X) and {,(X) acoustic fields. The (a priori) 
probability of occurrence of SN isp(SN), and p(W) is the a priori probability of 
occurrence of Non any trial. Athresholdconstant K is compared with the likeli- 
hood-ratio processor output at each trial. If the processor output exceeds K , the 
decision SV is made (5sy); otherwise, the decision N is made(6,). There are four 
possible combinations of event SW or W and decision 5, or dy: 
(SV, dsy) = detection (D) 
(SV, dy) = miss 
(WV, 6sy) = false alarm (FA) 
(VN, dy) = correct dismissal 
If p(D) =p(SN, dsy) is the probability of detection developed as the number 
of trials becomes infinite, and p(FA) is the probability of false alarm similarly 
developed, then it is found that both are functions of K. If K is large, both p(D) 
and p(FA) are small; if K is small, both p(D) and p(FA) are large. Generally, if K 
is varied, a receiver operating characteristic will be traced out for 
a given f,(X) and f(x), as shown in Fig. 18.1. 
If £,(X) or fy(X) is changed, a family of operating characteristics will be 
generated. Further, if some processor other than /(X) is used in the trials, a 
different p(D) and p(FA) will result, and will be to the right and below the operat- 
ing characteristic; i.e., the processor is not as efficient as I(X). From this brief 
discussion of operating characteristics, the mechanism for comparing different 
states of knowledge and/or different processors in terms of p(D) and p(FA) should 
be clear. A fuller discussion of the use of operating characteristic curves and 
receiver efficiency is given in [1]. 
18.3.3.1.Examples 
Three examples will be given: 
I. Signal is known exactly in time and space (waveform and wavefront); 
