IS predetermined by equipment restraints or other considerations. The following 

 properties of the two-stage system are pertinent to the problem of selecting a 

 suitable value of N. 



1. The single-scan detection probability P^ increases with N. Figure 4 

 illustrates this and also indicates how the optimum value of N^ changes with N. 

 (Fixed-sample stages; Rice distribution; S= 5 dB/pulse; M^^ = 1; a^ag = N \ 10"'; 

 K^, Ng, Kg optimum; FAR = 3.6 x Mg x PRF x 10'% where Mg may vary.) 



2. The average number of scans across a target staying in the surveillance 

 zone a relatively long time is approximately inversely proportional to iv if the 

 number of beam positions is large and the target density is low. 



3. The cumulative probability of detection (probability of at least one detec- 

 tion before a given range) tends to increase slightly with N but in an irregular 

 manner. Property 3 is the resultant of properties 1 and 2. It has not been verified, 

 but is supported by calculations of the quantity i_(i_Pp)constant/,v^ y^heie P^ is 



a function of N and of the optimum test parameters for that N. This quantity is an 

 indicative measure of the cumulative detection probability; it is based on the 

 fact that the probability of detection is equal to I-(I-Pq)^ for a target scanned 

 L times while at the same range, with L approximately inversely proportional 

 to N by property 2. Curves of l-d-P^) ^/'^ versus N are given in figure 5 for 

 the values of Pq plotted in figure 4. 



4. The length of an optimized two-stage test in the case of a detection (or 

 at least a stage-A alarm) tends to increase with N. Figure 6 illustrates this 

 with a plot of N^ + Ng versus N, where Ng is optimized for each N^ and N. 

 Although N^ + A/g is an upper bound to N(S), it varies with N in a similar 

 manner and is approximately equal to N iS) for signals somewhat stronger than 

 that for which the /Vg's are optimized here. _ 



Properties 3 and 4 suggest that a small N is at least nearly as efficient 

 as a large N, so that choosing the most suitable average scanning rate is largely 

 a matter of determining (and this involves examining the corresponding single- 

 scan detection probability curve) the most desirable value of the number of scans 

 across a target (of particular velocity) while it travels through the surveillance zone. 



Consider now the case where the PRF is adjustable in that it may be set at 

 one of a wide range of values. The optimum choice of the combination of PRF, 

 average scanning rate, and N then depends on the particular application. For a 

 situation in which the radar is peak-power limited and the cost of the power can 

 be disregarded, the PRF might be best set at the highest rate allowing the de- 

 sired maximum unambiguous range. If, on the other hand, the major limitation is 

 that of average power, then the most efficient procedure would probably be to 

 maximize the transmitted energy per pulse by using a small N and, for that N, the 

 lowest PRF that permits the antenna to scan across the target a reasonable 

 number of times. 



It is fair to conclude that a small N is suitable in many applications. Also, 

 note in figure 4 that for N as large as 2. 6 the N^ = 1 case is still better (in the 

 particular situation considered and in other typical situations) than the Nj^ = 2 case. 

 The subsequent analysis is restricted to cases with N^ = 1 and £„(«) < 2 for these 

 two reasons and because computationally they are the most tractable. 



17 



