where the single prime indicates observations with respect to the first array heading and the double 

 prime indicates observations with respect to the second array heading. For the time being we shall con- 

 sider only the Fourier components of order n, for if we can determine each Fourier coefficient separately 

 we can determine the function which they collectively represent, and vice versa. Let us call the meeis- 

 ured values c and d, where: 



d = d„+n2. (26) 



where c„ and d„ are true veilues and n.^ and n^ are noises or measurement errors. For convenience, we 

 shall assume that «, and n„ are mutually independent Gaussian variables with zero mean and veiriance 



The components of order n of the true field are characterized by the two coefficients, a„ and b„ . 

 If a priori estimates of the distributions of these coefficients are needed, we will assume that the nth 

 order component of the field has the form 



a„ cos nd + 6„ sin nd = A cos (n6 - 6), (27) 



where A has a Raleigh distribution with variance 2af and 5 is uniformly distributed from zero to 2it and 

 independent of A. This seemingly capricious choice of distributions has the quality of making a„ and b„ 

 independent Gaussian variables with zero mean and a common variance a^ relative to an angular coordi- 

 nate system with any reference direction whatsoever. 



It is easy to work out the deterministic relationships among these nth order coefficients, as follows: 



c„ = a„ cos n4> - 6„ sin n<}> (28) 



d„ = a„ cos n0 + 6„ sin n<t> (29) 



2 cos n0 



(30) 



" 2 sm n(p 



Noise- Free Measurements 



Suppose that noises m, cind n„ (Eq. (26)) cire identically zero, i.e., that the measurements of the 

 observed field are exact. Then Equations (30) and (31) can be used to generate the following estimates: 



a = 7 =a_ (32) 



and 



2 cos n<f> 



b-i^-K- (33) 



2 sm n0 



These estimates are exact and free from any error or bias. 



897 



