For target types with only scan-to-scan independence (slowly fluctuating 

 types 1 and 3 for example) it would be necessary to calculate the single-scan 

 detection probability 



'Pd{S)= fPoiS) g{S;S) 



dS 



where giS;S) is the density function of the SNR with mean 5' and Pq (S) is the 

 single-scan detection probability for a given value of the SNR. 



The equations below are stated in the form for linear detection, but the 

 results based on these may be used for square-law detection as well. 



When the output voltage x of an envelope detector has the Rayleigh 

 probability density function 



l(x) = ^ exp {-xV2a') ix > 0, 2a' = E{x')) 



a 



where a' = 1 when only noise is present and a^ > 1 when a signal is present, 

 then the probability that x exceeds a quantization level q is 



p= Po= exp(-(?V2) 

 in the noise-only case and is 



p= p{S)^ exp(-qV2a') 

 in the signal case, where 



S= 10 log,o(2cr'-l) 



is the mean power SNR in dB/pulse. 



For a nonfluctuating target the normalized output voltage x (in units of 

 rms i-f voltage) after linear envelope detection is assumed to have the well-known 

 Rice density function 



where a is the ratio of peak i-f signal voltage to rms i-f noise voltage and /„(•) is 

 the modified Bessel function of first kind, zero order. The probability that x 

 exceeds q is 



p = Po = exp(-flV2) 

 in the noise-only case and is (ref. 7, page 159) 



p= p(S)= exp [=^^y (9/a)J Ijiaq) 



7=1 



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