and no alarm. Computations will be for both binomial sequential testing and 

 binomial fixed-sample-size testing. 



3. Very few pulses. In a given beam position, a small number N^ of pulses 

 are transmitted for stage-A fixed-sample-size testing, or a variable number for 

 sequential testing. £„ in) denotes the mean number of pulses in a sequential 

 stage-A when only noise is present. 



STAGE B 



1. Stage B used infrequently. In a given beam position, a stage B is used 

 only if one or more alarms occur in stage A. 



2. Short pulses or pulse compression, many range bins. The M^ wide bins 

 of stage A are each subdivided into Mg radially narrow bins. (The equivalent 

 single-stage system must therefore examine M 4 Mg bins.) The term pulse will 

 refer to the entire waveform transmitted each pulse repetition period. 



3. Not all bins examined. Stage-B tests are performed for those stage-B 

 bins contained in a stage-A bin which alarmed.* Since the probability of a false 

 alarm in stage A is small, the probability of multiple false alarms in stage A is 

 very small; thus the number of stage-B bins examined when a stage B is used is 

 usually Mg in noise-only cases. 



4. Same pulse repetition frequency as in stage A. 



5. Many pulses. Stage-B testing is the performance of a number of binomial 

 fixed-sample-size tests, one for each bin being examined. The sample size iVg of 

 these tests is considerably larger than N ^ or £„(«). 



6. Same SNR as for stage A in results shown. However, the equations given 

 throughout are also valid when the SNR (signal-to-noise ratio per pulse) is dif- 

 ferent in stage B. 



Distribution of Signal and Noise 



The graphical results presented assume pulse-to-pulse independence and 

 are for the Rayleigh and Rice cases, which correspond respectively to Swerling's 

 rapidly fluctuating target type 2 and nonfluctuating target type. Computations for 

 the rapidly fluctuating target of type 4 were made but for brevity are not included, 

 since in every situation the type 4 curve fell almost entirely between the curves 

 for type 2 and the nonfluctuating type. 



*It would be better to examine, for each stage-A bin that alarms, the stage-B bins contained 

 in a somewhat larger range interval (covering the alarming stage-A bin). An analysis of this 

 more complicated case is unnecessary, since the higher threshold(s) required to provide the 

 same false-alarm rate generally result in only a very slight loss in detection probability. 



10 



