at termination. Since Wald's approximation equations yield true values for tests 

 in which there can be no excess over either bound, exact results can be obtained 

 without resorting to unwieldy iterative computational methods. The necessary 

 equations can be put in terms of quotients of polynomials;* those for the C = 6 

 case, for example, are 



E(„|p)= l-2p^3p-4-3p- ^ pj^iarmlp 



l-4p + 7p'-6p' + 3p''' l-4p + 7p'-6p' + 3p'' 



Eo(n) = E(,(«1p(,) is assigned a value, and the computer solves for p^ = p„ and 

 a^ = Plalarm|poi and, given a value of S, for p(S), P/\(S) = l-Pialarm|p(S)!, and 

 and E(n\p(S)). 



Detector Optimization 



Since M^ = 1, the mean number of pulses per beam position in the noise- 

 only situation is N = 1 + a^Ng for a system with a single-pulse stage A, and 

 N - E^in) + a^iVg for a system with a sequential stage A. 



For each Ng there is a value of N for which Pq is maximized when the SNR 

 and FAR are fixed and optimum thresholds are used. The value of N at which this 

 maximum P^, occurs is close to 1 for small Ng but in typical cases reaches 

 N= 1 +Ng for Ng greater than about 6 or 8. As /V is increased while holding S, 

 a^Qg/A/ and Ng constant, a^ increases, and therefore (b^ decreases and (since Og 

 decreases for fixed a^a^/N) Pg increases. For the sequential case the increase 

 of a^ with N depends on the fact that a^ increases with £„(«) (see fig. 7) for tests 

 (fixed C) of the class considered. The effect of the opposite behaviors of the 

 per-stage miss probabilities on Pq = (1-p^) (1-Pg) accounts for the existence 

 of a maximum P^. Graphs of P^ versus N are given as figure 8 for a system 

 with fixed-sample stages and as figure 9 for a system with a sequential first- 

 stage. (Rice distribution; S = 8 dB/pulse; M^= 1; a^ag = N x 10'°; optimum 

 Kg for each Ng and N.) Since a larger iV results in fewer opportunities to detect 

 the target, the actual optimum values of N (those which maximize the cumulative 

 detection probability) are smaller than those for which Pq is maximized. One 

 conclusion from these results is that when /Vg is limited to a relatively small 

 value by certain factors in a particular application - possibly by the length of 

 time the target remains in the resolution cell or by high-target density, clutter, 

 or other conditions occasionally present which cause a large proportion of the 

 stage A's to alarm — then increasing N beyond a certain value will not improve 

 even the single-scan detection probability. In fact, N can often be set con- 

 siderably smaller than that value with little loss in P^ and perhaps no loss in 

 the cumulative detection probability. 



♦These are obtained from equations (5:20), (5:19) and (5:23) in reference 1 by letting 



Pi = l-Po>y2 and logll^ = (C-1) Iogi^= (C-1) log i— ^, where Pi is the value 



" Pi Po 



of p when p(S) = p^. They are also derivable by the method of difference equations 



("the classical ruin problem,")" or by using Markov chain theory. 



22 



