By definition it follows that 



C(X,Y,f) = Re[P(X,Y,f)] 



Q(X,Y,f) = Im[P(X,Y,f)] 

 where -co < f < «> . 



The function C(X,Y,f) is called the cospectrum and Q(X,Y,f) is called 

 the quadrature spectrum. Both are spectral density functions. 



Explicitly, we can think of (xq, y^) and (xj^, y^^) as being the 

 location of elements of a probe-array with space separation (X,Y). 



The first step to find (or estimate) S(&,m,f) (see Equation (3.9)) 

 is to find P(X,Y,f) related to a pair of array elements. This is a 

 problem of estimating the cospectrum and quadrature spectrum of a two- 

 dimensional (vector) stationary Gaussian process. Goodman (Mar 1967 - 

 Chapter 3) has an excellent treatment of this subject, which we will 

 discuss. Kinsman (1965 - Chapters 7-9) also discusses the subject. 

 The essence of the problem is that if P(X,Y,f) is continuous and negli- 

 gible for |f I > fji» then R(X,Y,T) can be obtained by a time average over 

 a particular realization N(X,Y,t) instead of having to average (find the 

 expected value) over the ensemble. This says that we can find R(X,Y,T) 

 by obtaining two time series (realizations) rQ(t) and T-^(t) measured 

 over time at only two points; e.g., (xq, yo) and (x^, yi) where 

 (xi - Xq) = X and (y^ - y^) = Y. The relationship between (r (t) , r^(t)) 

 and R(X,Y,T) -» < T < °<> is 





, (4.8) 



- 'A 



where ro(t) is a realization measured over time at (x , y ) and r-|^(t) 

 is measured at (x^, y^) • We can simplify the notation by an expression 

 for a time average given by 



R(X,Y,T) = Roi(T) = ro(t)ri(t+T). 

 It follows from Equation (3.7) that C(X,Y,f) = CQi(f) = 

 J Roi (T) cos(2TTfT)dt, 



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