H = c: + i££(C;VQ-,Uf.(N-i)ri][A!-eA,&(Wi)l„^ 



A.- 1 J-Jt+t 



0) 



To find A that minimizes H, consider 

 o 



(7.11) 



This requires that Aq be of the form Aq = S(£,,m,fQ) and a resulting 

 value of H of the form 



H = C>e|;t(cSQ.^)-[N(N-0.>][S(i>,,Oj 



(7.12) 



2 2 2 

 Since Cq, C^js, and Q^^- are all nonnegative, a minimum H results when 



S(£,m,f ) is a maximum. A choice of %^ and m^ that maximizes S(^,m,fQ) 



implies a 6^ = arctan -^.^1%^ which is optimum. Remember that we are 



assuming Kp = Ji^ + ^2 holds, along with the wave equation. The results 



then for each f^ is a two-sided energy spectrum estimate A^(fQ,eQ). 



Appendix B contains a listing of a FORTRAN II program for finding 



k^^t^,^^ , the least square wave fit, from a set of spectral matrices 



obtained from the task SWOC data collection and analysis system 



described in Bennett et al (June 1964) . 



Examples of least square single-wave train fit analysis from 

 Bennett (March 1968) are shown in Figure 7 . 



A more complete collection of the directional spectra calculated 

 from data collected off Panama City, Florida, is given in Appendix A, 

 and in Bennett (November 1967), and Bennett and Austin (September 1968), 

 an unpublished Laboratory Technical Note TN160. 



We would actually like a continuous estimate of S'(£.,m,f). Con- 

 sider then a third method. From Equation (3.8) we have for a pair of 

 detectors, as illustrated in Figure 8, that 



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32 



