in the signal case, where 



S = 10 log.o aV2 



is the mean power SNR in dB/pulse. 



Let S^ be the SNR corresponding to a "design" target (a target with 

 specified radar cross section) at range R^^^^- The fourth-power relation between 

 range and received echo-power gives as the SNR of this design target at range r 



S= Sir)= Srf -40 log^or 



where r is the normalized range ^/^max- 



Binomial Fixed-Sample-Size Testing 



In a stage of the fixed-sample type, a binomial fixed-sample-size test is 

 performed for each range bin. An individual test is performed by comparing, for 

 each of N successive independent pulses, the sample voltage (the amplitude of 

 the envelope at the time delay corresponding to the given range bin) with the 

 quantization level q, and deciding "alarm" if at least K of the N voltages exceed 

 q and "noise only" otherwise. If p is the probability for a particular range bin 

 that the sample voltage for any given pulse will exceed the threshold q, the 

 probability for that range bin that K out of N independent samples exceed q is the 

 cumulative binomial probability 



D{N,K,p)= y ('Y)pMl-p/ 



(For convenience the subscripts designating stage are omitted here.) For fixed N 

 and a each possible value of the threshold K determines a value of Po, from 

 a = D(N,K,Po), and therefore determines a value of q. Then each K, from its q, 

 also determines a value of piS) for any given value of S. The optimum value of K 

 for the given value of S is that which minimizes the miss probability (3(K,S) = 

 1 -D{N,K,p(S)). Whenever convenient, (3(K,S) will be written (3(S) or (B, and 

 since the optimum values of the thresholds are used for each S in all calculations, 

 p(S) and p will be the minimum p(K,S) for that particular N, a, and S. Ideally, for 

 each range-bin test the threshold K should be chosen to optimize detection for 

 the SNR of the design target at that range, i.e., for Sir), where arbitrarily r could 

 be the center of the bin. Generally, however, the use of a different quantization 

 level q for each bin is impractical in implementation. It is fortunate, therefore, 

 that using the same value of K for all bins results in miss probabilities which do 

 not differ appreciably from those in the ideal case. In figure 2, curves of ^{K,S) 

 are given for a typical stage-B test (/V= 12; a=6 x 10"^ Rice distribution); note 

 that K= 7 is optimum over effectively the entire range of (3(S), as S varies, and 

 that the curves are relatively flat over several values of K. 



12 



