ANALYSIS OF SPECIAL CASES 



General Description 



Sequential Cases. STAGE A: Binomial Wald' sequential test, single- 

 range-bin type (M^= 1), with E„{n) less than about two. STAGE B: Mg range 

 bins (radially narrow). Returns from /Vg pulses used in Mg binomial fixed-sample- 

 size tests. 



Fixed-sample Cases. STAGE A: One pulse, one range bin (/V^ = 1, M^ = 1). 

 STAGE B: Mg range bins. Returns from Ng pulses used in Mg binomial fixed- 

 sample-size tests. 



The detection capabilities of these two-stage detectors will be compared 

 with that of a conventional single-stage detector, one that employs a binomial 

 fixed-sample-size test for each of m= Mg range bins. 



A Sequential Test For Stage A 



When there is only one resolution element to be tested and £„(«) is to be 

 less than about two, the "best" binomial sequential test is a very simple form 

 of Wald's test, one which may be performed in the following manner. A random 

 variable y^ is assigned the value 1 if the sample voltage x for the i^^ pulse exceeds 

 q, and the value (-1) otherwise. After each pulse, say the n"^, one of three 

 possible actions is taken. (C is an integer larger than two and is specified in 

 accordance with desired test characteristics.) 



If < 1 +yj +• • • +>'„ < C, testing is continued with another pulse. 



If C 11 +yj + • •■ + >'„, testing is terminated with an alarm. 



If l+yi + '' • + 3'„ lO, testing is terminated without an alarm. 

 A test of this form is "best" in the following sense. 



1. It is a probability-ratio test for (and only for) any Po and piS) such that 

 piS) - l-Po- (The probability that y^ equals 1 is Po when noise only is present 

 and p{S) when the SNR has value S.) Since the SNR varies widely with range and 

 cross section, the choice of the value S for which p{S) = 1-Po as the value for 

 which to optimize the test is often as fair as any. 



2. For any binomial sequential test having an E„{n) < 2, there is some C 

 and q such that a test of the above form has the same E„(w), approximately the 

 same a, and a (3(S) which is not appreciably larger for any S (assuming any 

 particular functional form of the noise and signal-plus-noise distributions and 

 assuming q adjustable for each test). This was apparent from an examination, 

 for a wide variety of binomial sequential tests, of the trade-offs in detection 

 characteristics that occur from test to test; the nature of binomial tests does 

 not permit direct comparisons. 



3. A test of this form is much simpler to analyze than any other binomial 

 sequential test. Note that the sum 1 +yi +• ••+y„ will equal one of the bounds 



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