NAFI TR-li|UO 



This appendix expands and clarifies some of the statements 

 made in the text. The statements are of a general nature and rather 

 lengthy, and, as such were deemed inappropriate for inclusion in the 

 main body. The numbers of the headings refer to the footnote references 

 in the text . 



(l) That Bayes and maximum SWR lead to the same optimum con- 

 figuration. 



This statement is based upon a number of approximations 

 and assumptions which may, at first glance, appear un- 

 justifiable; however, experience indicates that the 

 statement, though perhaps not rigorous, is reasonably 

 correct. The logic leading to it is based on the 

 following: 



a. If the noise into the square -law device is bandlimited 

 white and Gaussian, then the equi-spaced Wyquist samples of a segment 



T seconds long are independent and obey the chi-squared distribution at 

 the output of the integrator (summer for sampled values). (l,pp 77-78) 



b. For TW > 50, the chi-square distribution can be satisfac- 

 torily approximated as Guassian (I8). This is simply an illustration 

 of the Central Limit Theorem, but note that the chi-square distribution 

 is sufficiently well behaved for the Gaussian approximation to be valid 

 at rather small values of TW. 



c. If the output of the processor is Gaussian, the statisti- 

 cal description of its performeiice__ip readily characterized by a single 

 parameter d, defined as d = — =* = "• 



where it is assumed that o^ = a = o^ (n) = a^ (S + N) . It will now be shown 

 that maximization of this parameter results in a Bayes optimization for 

 Guassian output statistics. 



The system false alarm and detection probabilities are 



Pd ^I f,(^)^^? 



^ 



D-2 



