4 Interpretation of the equation

Eq. 8 offers itself for some intuitive understanding in the following way.

X_ij is a product of two complex numbers, namely g_i and g_j^⋆, which we wish to determine. X_ij is itself derived from the measured quantity V _ij^Obs. Numerically speaking, each term in the summation of the numerator of Eq. 8 will involve g_i (via X_ij) and the multiplication of X_ij with g_jw_ij would give g_i an effective weight of ²w _ij. Since the denominator is the sum of this effective weight, the right-hand side of Eq. 8 can be interpreted as the weighted average of g_i over all correlations with antenna i.

In the very first iteration, when g_j = (1, 0), the normalization would be incorrect since the numeric value of g_j as it appears inside X_ij would be different from that used in the denominator of Eq. 8. However, as the estimates of g_js improve with iterations, the equation would progressively approach a true weighted average equation. The speed of convergence will depend upon the value of α and the convergence would be defined by the constraint in Eq. 10. In the ideal case when the true value of all g_is is known, right hand side of Eq. 8 also reduces to g_i.

Estimating g_i for an antenna, by averaging over the measurements from all baselines in which it participates (for an unresolved source) makes sense since for an N element array, g_i would be present in N-1 measurements (all the X_ij|_j=1,N;j≠i) and the best estimate of g_i would be the weighted average of all these measurements. Proper weight for g_i, buried in each of the products X_ij, can be found heuristically as follows. g_i, estimated from the measurements of a given baseline, must obviously be weighted by the signal-to-noise ratio on that baseline. This is w_ij in the above equations. It must also be weighted by the amplitude gain of the other antenna making the baseline, namely g_j, to account for variation in antenna sensitivities and T_sys. The total weight for g_i would then be ²w _ij, the sum of which appears in the denominator of Eq. 8. Knowing that ideally X_ij = g_ig_j^⋆, each of the X _ij|_j=1,N must be multiplied by g_jw_ij (to apply the the above mentioned weights to g_i), before being summed for all values of j and normalized by the sum of weights to form the weighted average of g_i. One thus arrives at Eq. 8 using these heuristic arguments.