NAFI TR-lij-UO 



It Is important to realize the context in which these statements 

 are made. In order for the maximization of d to have meaning, a process- 

 or structure must be specified; one cannot derive the structure from d, 

 but can optimally choose parameters of the structure by maximizing d. 

 For an unstructured problem in which one is attempting to determine the 

 best manner to process raw data, resort must be made to likelihood ratio 

 techniques (the Bayes criterion). 



The preceding logic is then applied to the problem at hand by 

 stating that although the output statistics may not be chi-square (the 

 noise is not necessarily white), they probably possess the characteris- 

 tic of the chi-sq.uare distribution that the output distribution approaches 

 Gaussian for reasonable values of TWe , say TWe greater than 100 or so. 

 Then, by assuming small input SKR's, so that Oj {'3 + N) ^<^z{n), one can 

 state that maximizing d (or the output SNR) satisfies the Bayes criterion. 



(2) That the best one can do where highly limited a priori sig- 

 nal information is available is to whiten the noise and bandlimit the input. 



The work contained in the body of this report can be considered an 

 an attempt to justify this statement. The discussion in Section 8 is 

 directly applicable; as stated there, it is likely that there exists a 

 compromise filter, one which does not necessarily whiten the noise and 

 would have a spectral shape somehow dependent upon the expected range of 

 signal variations -- analogous to the mismatched Echart filter for simple 

 spectra treated in Section 7 -- but that such a filter would be difficult 

 to find since it would have to be determined by trial and error techniques. 

 The quantitative results in this report indicate that the gains achievable 

 from such a filter are minimal, and that a bandlimited pre-whitening system 

 should prove satisfactory. 



In the practical situation, of course, one is never going to 

 build filters that actually whiten the noise or sharply bandlimit the in- 

 put. The spectral fluctuations of the input noise and considerations of 

 physical realizability theory make these idealistic concepts; the power 

 of the dollar over system realization makes their approximations doubtful. 

 The results in this report tend to substantiate the hope that these 

 limitations do not severely compromise performance. 



