REFERENCE 

 DIRECTION 



FIRST 



ARRAY 



HEADING 



Fig. 5. Relations among array headings, assumed reference 

 direction, and an arbitrary direction vector 



and the quantity we can observe is the part, Igid), which is symmetric with respect to the array axis. 

 Under these circumstances expressions (22) and (23) are quite different, and the signal processor which 

 minimizes one cannot be expected to minimize the other. 



In many cases, the bias (Eq. 21) can be reduced to zero or to a provable irreduceable minimum 

 without exhausing all the degrees of freedom available in the estimation process. Under such a circum- 

 stance, we may have a constrained criterion of error, such as minimize the mean-square error (Eq. 22) 

 subject to the constraint that the bias (Eq. 21) is zero or at the irreduceable minimum. Another type of 

 constraint which is relevant to the measurement of the intrinsically positive noise intensity distribution, 

 1(6), is to require that estimator 1(6) be greater than or equal to zero for all values of 0, for as shown in 

 Ref. 1, the irreduceable-bias minimum-mean-square-error estimator based on real observations may have 

 negative values. 



Measurements Made with Two Arbitrary Array Headings 



Let us suppose, as in Fig. 5, that we measure an arbitrary intensity field, 1(9), twice with linear 

 arrays. For convenience, the reference array for angular measurement is chosen halfway between the 

 two array headings, the true azimuth is represented by 0, and the directions of the two array headings 

 are, respectively, + 4> and - 0. This results in the angular relationships shown in Fig. 5. If the true field 

 is given in Eq. (4), then the fields observed are those given in Eq. (25) 



I' = -f + I, c„ cos n (d + (/)) 



(24) 



and 



l"=Y+'^d„ cos (d - <t>). 



(25) 



896 



