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 $g_i$s by minimizing, with respect to $g_i$s, the function $S$ given by

$$S = \sum_{{i,j} \atop {i \ne j}}{\left\vert\rho_{ij}^{Obs} - g_i g_j^\star \rho_{ij}^\circ\right\vert}^2 w_{ij}$$ (4)

where $w_{ij}=1/\sigma^2_{ij}$, $\sigma_{ij}$ being the variance on the measurement of $\rho^{Obs}_{ij}$

Dividing the above equation by $\rho_{ij}^\circ$ (the source model, which is presumed to be known - it is trivially known for an unresolved source), and writing $\rho_{ij}^{Obs}/\rho_{ij}^\circ = X_{ij}$, we get

$$S = \sum_{{i,j} \atop {i \ne j}}{\left\vert X_{ij} - g_i g_j^\star\right\vert}^2 w_{ij}$$ (5)

If $\rho_{ij}^\circ$ 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

$$S=\sum_{{i,j} \atop {i \ne j}}\left[ \left\vert X_{ij}\right\vert... ...X_{ij} - g_i g_j^\star X_{ij}^\star + g_i g_i^\star g_j g_j^\star\right] w_{ij}$$ (6)

Evaluation ${\partial S \over \partial g_i^\star}$ and equating it to zero ³, we get

$${\partial S \over \partial g_i^\star}  =  \sum_{j \atop {j \ne i}}\left[-g_j X_{ij} w_{ij} +g_i g_j g_j^\star w_{ij}\right]  =  0$$ (7)

or

$$g_i = {\sum\limits_{j \atop {j \ne i}} X_{ij} g_j w_{ij} \over \sum\limits_{j \atop {j \ne i}} \left\vert g_j\right\vert^2 w_{ij}}$$ (8)

This can also be derived by equating the partial derivatives of $S$ with respect to real and imaginary parts of $g_i$ 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 $g_j$s or initializing them all to (1,0).

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

$$g_i^n = g_i^{n-1} + \alpha\left[{\sum\limits_{j \atop {j \ne i}} ... ...j \atop {j \ne i}} \left\vert g_j^{n-1}\right\vert^2 w_{ij}} - g_i^{n-1}\right]$$ (9)

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

$$\left\vert S_n-S_{n-1}\right\vert < \delta$$ (10)

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