7. DIRECTIONAL ANALYSIS FROM THE CROSS SPECTRAL MATRIX 



First, we consider a fundamental approach. We have for a pair of 

 detectors I and J located at (x^,y^) and (x^,y^) respectively, the 

 cross spectral matrix, P-^^ (f ) = C^j (f ) + IQ^-Cf) and more impor- 

 tantly <t>l.i (f ) the phase lead of I over J given by, 



<p.w = — [fgj 



(7.1) 



The actual phase lead may differ from this value since the true phase 

 lead 6 is some one of the values 



9:M) -^ftaiT 



where 



K = 0»il,i2.,- • • 



Consider a single wave of frequency f with corresponding wave 

 length Aq and wave number Kq = 1/Xq, and find the direction the wave 

 must travel; i.e., fit a single wave to the spectral matrix results for 

 the detectors I and J. 



Let D-j^j be the distance between I and J. The distance between 

 I and J in wave lengths is KqD^j . In radians this is ZirKoDi-j. From 

 this relation we get 



or that only values of h such that 



- ^'^ K.D^^$ (f./f) + ^i atr ^ £tt K.Djj (7.2) 



give physically acceptable candidates for the value of 4). If D^j < Aq/2 

 only one h value is valid. If Aq/2 < D-j^^ < A^ at most two values of h 

 are valid, etc. 



The problem is how to find the true direction, Bq, of a single 

 wave given the above possible value(s) of ((>. Consider a given value 

 of <\> in terms of wave length units and we obtain 



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