7.1.1 Baseline based vs antenna based gains

Since amplitude and phase distortions have different physical origins it is generally useful to write

$$\ensuremath{\mathcal{G}} _{ij}(t) = g_i(t) g_j^*(t) = a_i(t) a_j(t) e^{i(\phi_i(t)-\phi_j(t))}$$ (7.2)

Here we have split the gains into antenna based factors. This is generally legitimate since the gains represent properties of the data acquisition chains which are in the analogue part of the system. The correlator itself is a digital machine and we assume it is perfectly working (including the clipping correction). This assumption is certainly valid when considering a single frequency and a single instant. When we start averaging in time or frequency, the average of the product may not be the product of averages, and we may have some baseline-based effects.

The baseline-based gains can be determined by observing a point source. This is usually a strong quasar. In that case the true visibilities $\ensuremath{V} _{ij}(t)$ should all be equal to the quasar flux density S. Then

$$\ensuremath{\mathcal{G}} _{ij}(t) = \frac{\ensuremath{\widetilde{V}} _{ij}(t)}{S}$$ (7.3)

The antenna gains g_i(t) can also be deduced from the non-linear set of equations:

$$g_i(t) g_j^*(t) = \frac{\ensuremath{\widetilde{V}} _{ij}(t)}{S}$$ (7.4)

This is a system with N complex unknowns and N (N-1)/2 equations. In terms of real quantities there are N (N-1) measured values (amplitudes and phases; there are only 2 N-1 unknowns since one may add a phase factor to all complex gains without affecting the baseline-based complex gains. When N is larger than 2 the system is over determined and may be solved by a method of least squares.

If we note $\ensuremath{\widetilde{V}} _{ij} = \tilde{A}_{ij} e^{i \tilde{\varphi}_{ij}}$, the equations for phases are simply:

$$\phi_i - \phi_j = \tilde{\varphi}_{ij}$$ (7.5)

It can be shown that the least-squares solutions (when the same weight is given to all baselines, and if we impose the condition $\sum_{j=1,N} \phi_J = 0$), is given by:

$$\phi_i = \frac{1}{N} \sum_{j\neq i} \tilde{\varphi}_{ij}$$ (7.6)

For the amplitudes we can define in order to get a linear system:

$$\gamma_i = \log{g_i}, \tilde{\alpha}_{ij} = \log{\tilde{A}_{ij}}$$ (7.7)

$$\gamma_i + \gamma_j \tilde{\alpha}_{ij}$$ = (7.8)

This time the least square solution is, when the same weight is given to all baselines:

$$\gamma_i = \frac{1}{N-1}\sum_{j\neq i}\alpha_{ij} - \frac{1}{(N-1)(N-2)}\sum_{j\neq i}\sum_{k\neq i, >j}\alpha_{jk}$$ (7.9)

Obviously this antenna gain determination needs at least three antennas. For three antennas it reduces to the obvious result:

$$g_1 = \frac{\tilde{A}_{12}\tilde{A}_{13}}{\tilde{A}_{23}}$$ (7.10)

These formulas can be generalized to the cases where the baselines have different weights.

It can be seen in the above formulas that the precision to which the antenna phases and amplitudes is determined is improved by a factor $\sqrt{N}$ over the precision of the measurement of the baseline amplitudes and phases.