Computation of antenna based complex gains

| (D.1) |

where I(l,m) is the sky surface brightness, η_{i} is the sensitivity and T_{i}^{s} is the system temperature
of the antenna i in units of Kelvin/Jy and Kelvin respectively, ϵ_{ij} is the additive noise on the
baseline i-j, and ϕ_{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 () and phase (ϕ_{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_{i}e^{-ιϕi} where a_{i} = . For an unresolved source at the phase tracking center,
variations in this amplitude will be indistinguishable from a variations in the ratio of η and
T^{s}.

In terms of g_{i}s, we can write Equation D.1 as

| (D.2) |

where

| (D.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 ρ_{ij}.
For an unresolved source, 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
ρ_{ij}^{∘} are exactly zero. ρ_{ij}^{∘} 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

| (D.4) |

where w_{ij} = 1∕σ_{ij}^{2}, σ_{ij} being the variance on the measurement of ρ_{ij}^{Obs}

Dividing the above equation by ρ_{ij}^{∘} (the source model, which is presumed to be known
– it is trivially known for an unresolved source), and writing ρ_{ij}^{Obs}∕ρ_{ij}^{∘} = X_{ij}, we
get

| (D.5) |

If ρ_{ij}^{∘} 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

| (D.6) |

Evaluation ∂S _
∂g_{i}^{⋆} and equating it to zero ^{1} ,
we get

| (D.7) |

or

| (D.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:

| (D.9) |

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

| (D.10) |

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

Equation D.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 Equation D.8 will involve g_{i} (via X_{ij}) and the
multiplication of X_{ij} with g_{j}w_{ij} would give g_{i} an effective weight of ^{2}w_{ij}. Since the
denominator is the sum of this effective weight, the right-hand side of Equation D.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 Equation D.8. However, as the estimates of g_{j}s 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 Equation D.10. In
the ideal case when the true value of all g_{i}s is known, right hand side of Equation D.8 also
reduces of g_{i}.

Estimating g_{i} for an antenna, by averaging over the measurements from all baselines in which
it participates (for a unresolved source) makes sense since for an N element array, g_{i} would be
present in N-1 measurements (all the _{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^{s}. The total weight for g_{i} would then be ^{2}w_{ij}, the sum of which appears
in the denominator of Equation D.8. Knowing that ideally X_{ij} = g_{i}g_{j}^{⋆}, each of the
_{j=1,N} must be multiplied by g_{j}w_{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 Equation D.8 using these heuristic
arguments.

For an unresolved source of known brightness I, in the limit T^{a} ≪ T^{s}, ρ_{ij}^{∘} = I and Equation D.1
can be written as

| (D.11) |

where η_{i} = A_{e}∕2k_{b}, A_{e} is the effective area of the dish, k_{b} is the Boltzman’s constant
and

| (D.12) |

Hence, knowing η_{i}, T_{i}^{s} can be estimated from the amplitude of the antenna dependent complex
gains.

All contributions to ρ_{ij}^{Obs}, which cannot be factored into antenna dependent gains, will result
in the reduction of . η remaining constant, this will be indistinguishable from an increase in the
effective system temperature. Since majority of later processing of interferometry data for
mapping (primary calibration, bandpass calibration, SelfCal, etc.) is done by treating
the visibility as a product of two antenna based numbers, this is the effective system
temperature which will determine the noise in the final map (though, as a final step in
the mapping process, baseline based calibration can possibly improve the noise in the
map).

In the normal case of no significant baseline based terms (ϵ_{ij}) in X_{ij}, the system temperature
as measured by the above method will be equivalent to any other determination of
T_{i}^{s}.

T^{s} can also be determined by recording interferometric data for a strong point source with and
without an independent noise source of known temperature at each antenna. In this
case

| (D.13) |

where g_{i}^{ON} and g_{i}^{OFF } are the antenna dependent gains with and without the noise source of
temperature T^{n}. Note that η_{i} does not enter this equation. Also, T^{n} should be such that
≥ 0.1 to ensure that the correlated signal is measured with sufficient
signal-to-noise ratio. For example, for P-band, a calibrator with P-band flux density > 5 Jy must
be used.

g_{i}s are complex functions. One can therefore write S in terms of g_{i}^{I} and g_{i}^{p}, the real and
imaginary parts of g_{i} and minimize with respect to g_{i}^{I} and g_{i}^{p} separately. It is shown here that
the complex arithmetic achieves exactly this and the results are same as that given by complex
calculus. The superscripts I and R in the following are used to represent the real and imaginary
parts of complex quantities.

Expanding Equation D.5, ignoring w_{ij}s and writing it in terms of real and imaginary parts we
get

| (D.14) |

where

| (D.15) |

Taking partial derivative of S with respect to g_{i}^{p} and reintroducing w_{ij}, we get

| (D.16) |

Therefore,

| (D.17) |

Equating ∂S _
∂g_{i}^{p} to zero, we get

| (D.18) |

Similarly

| (D.19) |

Therefore the equivalent imaginary part of Equation D.18 is

| (D.20) |

writing g_{i} = g_{i}^{p} + ιg_{i}^{I} and substituting for g_{i}^{p} and g_{i}^{I} from Equation D.18 and D.20 respectively,
we get

| (D.21) |

This is same as Equation D.8, which was arrived at by evaluating a complex derivative of
Equation D.5 as ∂S∕∂g_{i}^{⋆}, treating g_{i} and g_{I}^{⋆} as independent variables. Evaluating ∂S
∂g_{i} = 0 would
give the complex conjugate of Equation D.21. Hence, ∂S∕∂g_{i} gives no independent information
not present in ∂S∕∂g_{i}^{⋆}.