NAFI TR-lUl+0 



where Zy = tlireshold value of Z, and 



/ 



The subscripts and 1 refer to W and S + N, respectively, and Zo and Zi 

 are the output means . 



Then 



where 



Define two new variables, 







/I 



■' / ? 



3o r " 



?^-i'r 







St^ 



If the PpA is specified as some constant value, then §ot is likewise a 

 constant, say Cq , so that 



- c - 



/^j - Hr 



'O T 



Thus 



and 





+ ?o 



OCJ 





Under the non-restrictive assumption Zi > Zo , i.e., the mean with S + N 

 is greater than the mean with W, the probability of detection is maxi- 

 mized "by maximizing the quantity 



Maximizing Pp while Pp * is held constant is just the Neyman-Pearson 

 criterion, which differs from the Bayes criterion only in the specifica- 

 tion of the threshold of the likelihood ratio ( 1, pp 69-70) • Consequent- 

 ly maximizing d, or equivalently, maximizing the output SNE, satisfies 

 the Bayes criterion for Gaussian output statistics. 



D-3 



