Computation of antenna based complex gains

The normalized cross-correlation function (the correlator output), measured by an interferometer using two antennas labeled by $i$ and $j$, in the limit $I \ll T^s_i/\eta_i$, can be written as:

$$\begin{split}\rho_{ij}^{Obs}=\rho^{Obs}(u_{ij},v_{ij},w_{ij})... ...} &{dl dm\over\sqrt{(1-l^2-m^2)}} + \epsilon_{ij}\end{split}$$ (15.1)

where $I(l,m)$ is the sky surface brightness, $\eta_i$ is the sensitivity and $T^s_i$ is the system temperature of the antenna $i$ in units of Kelvin/Jy and Kelvin respectively, $\epsilon_{ij}$ is the additive noise on the baseline $i$-$j$, and $\phi_i$ is the antenna based phase of the signal. The rest of the symbols have the usual meaning.

In practice, however, the antenna based amplitude ( $\sqrt{{\eta_i}/{T^s_i}}$) and phase ($\phi_i$) are potentially time varying quantities. This could be due to changes in the ionosphere, temperature variations, ground pick up, antenna blockage, noise pick up by various electronic components, background temperature, etc. Treating the quantities under the square root in the above equation as the antenna dependent amplitude gains, these can be written as complex gains $g_i= a_ie^{-\iota \phi_i}$ where $a_i=\sqrt{\eta_i/T^s_i}$. For an unresolved source at the phase tracking center, variations in this amplitude will be indistinguishable from a variations in the ratio of $\eta$ and $T^s$.

In terms of $g_i$s, we can write Equation D.1 as

$$\rho_{ij}^{Obs} = g_i g^\star_j \rho^\circ_{ij} + \epsilon_{ij}$$ (15.2)

where

$$\rho_{ij}^\circ=\int\limits_{-\infty}^{+\infty}\int\limits_{-\inf... ...pi\iota(u_{ij}l+v_{ij}m+w_{ij}\sqrt{1-l^2-m^2})} {dl dm\over\sqrt{(1-l^2-m^2)}}$$ (15.3)

The use of the word ``antenna based gains'' for $g_i$s result in confusion for many and needs some clarifications. $g_i$s are called complex ``gains'' since they multiply with the complex quantity $\rho_{ij}$. For an unresolved source, $\left\vert g_i\right\vert$ represents the fraction of correlated signal and $arg(g_i)$ represents the phase of the correlated part of the signal from the antenna with respect to the phase reference (usually the reference antenna). It is in this sense that it is referred to as ``antenna based'' gains. However, as defined here, they include $T^s$ which in turn includes the sky background temperature. They are therefore a function of direction in the sky. However, here we assume that the angular scale over which $g_i$s vary is larger than the antenna primary beam (isoplanatic case).

For an unresolved source at the phase tracking center, all terms in the exponent of $\rho_{ij}^\circ$ are exactly zero. $\rho_{ij}^\circ$ in this case would be proportional to the flux density of the source.

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

$$S = \sum_{{i,j} \atop {i \ne j}}{\left\vert\rho_{ij}^{Obs} - g_i g_j^\star \rho_{ij}^\circ\right\vert}^2 w_{ij}$$ (15.4)

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

$$S = \sum_{{i,j} \atop {i \ne j}}{\left\vert X_{ij} - g_i g_j^\star\right\vert}^2 w_{ij}$$ (15.5)

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 Equation D.5, we get

$$S=\sum_{{i,j} \atop {i \ne j}}\left[ \left\vert X_{ij}\right\vert... ...X_{ij} - g_i g_j^\star X_{ij}^\star + g_i g_i^\star g_j g_j^\star\right] w_{ij}$$ (15.6)

Evaluation ${\partial S \over \partial g_i^\star}$ and equating it to zero ^15.1, we get

$${\partial S \over \partial g_i^\star} =  \sum_{j \atop {j \ne i}}\left[-g_j X_{ij} w_{ij} +g_i g_j g_j^\star w_{ij}\right] =  0$$ (15.7)

or

$$g_i = {\sum\limits_{j \atop {j \ne i}} X_{ij} g_j w_{ij} \over \sum\limits_{j \atop {j \ne i}} \left\vert g_j\right\vert^2 w_{ij}}$$ (15.8)

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 Section D.3.

Since the antenna dependent complex gains also appear on the right-hand side of Equation D.8, it has to be solved iteratively starting with some initial guess for $g_j$s or initializing them all to 1.

Equation D.8 can be written in the iterative form as:

$$g_i^n = g_i^{n-1} + \alpha\left[g_i^{n-1}-{\sum\limits_{j \atop {... ...sum\limits_{j \atop {j \ne i}} \left\vert g_j^{n-1}\right\vert^2 w_{ij}}\right]$$ (15.9)

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

$$\left\vert S_n-S_{n-1}\right\vert < \delta$$ (15.10)

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