where a^iag is the per-bin false-alarm probability for all M^ Mg range bins and 

 N is the average number of pulses per beam position when only noise is present. 

 When comparisons are made where the number of range bins and the PRF are 

 fixed, the FAR is made the same for all cases by holding a^ag constant if N is 

 fixed or a^a^/N constant if N varies. 



Sample Sizes 



When applying two-stage detection to a particular detection situation, it 

 probably will not be immediately evident what values of N, N^ (or £„(«)), Ng 

 and a^ are best to use. A relation among these quantities is 



for fixed-sample stages, and 



N= E,{n) + a^/Vg 



when the first stage is sequential and involves a single range bin. (Sequential 

 tests are discussed later; calculations were not made for the multiple-resolution- 

 eleraent type.) 



The nature of binomial tests makes it necessary to use numerical rather 

 than analytical methods to find the optimum values of the unspecified quantities, 

 but the amount of computation time required on a high-speed computer is moderate. 



When values of iV, M^, /V^ (or £„(«)), S-S^,* S, and a^ag (or a^ag/iV) 

 are specified, the optimum Ng is the value yielding the maximum single-scan 

 detection probability 



P^iS) = |i-Pa(Sa)] [i-Pb<S)] 



For each value of Ng considered, the values of a^ and ag must be determined 

 before calculating Pq(S); the value of a^ is determined from the equation for N 

 (an additional relation between a^ and E„in) must be satisfied in the sequential 

 case), and Og is found from fixed ct^ag. It is important to keep in mind that the 

 optimum value of Ng varies with S and therefore varies with range for a target of 

 given cross section. For simplicity it is assumed that the same number Ng of 

 pulses are transmitted no matter which stage-A bin alarms. 



The optimization of N^, given N, and the optimization of N are 

 discussed later. 



Another sample size of interest is the mean number of pulses when a 

 signal is present. When M^=_l, this quantity is NiS) = N^ + [l-p(S)] Ng for 

 the fixed-sample case and is N(S) = Ein) + [l-|3(S)] Ng for the sequential case, 

 where Ein) is a function of S. 



*S^ and S are the per-pulse SNR's in stage A and stage B, respectively; their difference is 

 assumed constant with range and cross section, and is zero for all results presented. 



14 



