3 Solution for the complex gains

Assuming that the antenna dependent complex gains are independent, with a gaussian probability density function (this implies that the real and imaginary parts are independently gaussian random processes), one can estimate gis by minimizing, with respect to gis, the function S given by

     ∑  |               |
S =     |ρOibjs-  gi g⋆j ρ∘ij|2 wij
     i,j
     i⁄=j
(4)

where wij = 1∕σij2, σ ij being the variance on the measurement of ρijObs

Dividing the above equation by ρij (the source model, which is presumed to be known - it is trivially known for an unresolved source), and writing ρijObs∕ρ ij = X ij, we get

     ∑  ||          ⋆||2
S =      Xij - gi gj  wij
     ii,j⁄=j
(5)

If ρij represents the structure of the source accurately, X ij will have no source dependent terms and is purely a product of the two antenna dependent complex gains.

Expanding Eq. 5, we get

    ∑   [                                     ]
S =       |Xij|2 - g ⋆igjXij - gig⋆jX ⋆ij + gig⋆igjg⋆j wij
     i,j
     i⁄=j
(6)

Evaluation ∂S _ ∂gi and equating it to zero 3, we get

∂S      ∑   [                     ]
--⋆- =       - gjXijwij + gigjg⋆jwij   =  0
∂gi       j
         j⁄=i
(7)

or

       ∑
          Xijgjwij
       jj⁄=i
gi =   ∑------2----
        j  |gj|wij
       j⁄=i
(8)

This can also be derived by equating the partial derivatives of S with respect to real and imaginary parts of gi as shown in the appendix.

Since the antenna dependent complex gains also appear on the right-hand side of Eq. 8, it has to be solved iteratively starting with some initial guess for gjs or initializing them all to (1,0).

Eq. 8 can be written in the iterative form as:

                ⌊ ∑       n-1          ⌋
                |  j Xijg j  wij       |
 n      n-1     | j⁄=i-------------   n-1|
gi  =  gi   + α |⌈ ∑   ||gn-1||2w   - gi  |⌉
                   j    j      ij
                  j⁄=i
(9)

where n is the iteration number and 0 < α < 1. Convergence would be defined by the constraint

|Sn - Sn-1| < δ
(10)

(the change in S from one iteration to another) where δ is the tolerance limit.