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Standard Decomposition of Visibilities

For any given complex number $Z$, let us call $PZ$ its phase, $AZ$ its amplitude, and $Z^*$ its complex conjugate. The observed visibility $V_{ijk}(t)$ is a complex number representing the amplitude and phase of the signal detected on baseline $ij$, from spectral channel $k$, at time $t$,

\begin{displaymath}
V_{ijk}(t) = AV_{ijk}(t) . \exp(-i.PV_{ijk}(t))
\end{displaymath} (1)

This visibility is the Fourier transform of the product of the primary beam patterns of the antennas and the brightness distribution of the observed source, sampled at the point $(u(t),v(t))_{ij}$ corresponding to baseline $ij$ at time $t$,

\begin{displaymath}
V_{ijk}(t) = FT ( B_i(x,y,x_0+x_i,y_0+y_i) B^*_j(x,y,x_0+x_j,y_0+y_j)
I(x,y,k) ) (u,v)_{ij}
\end{displaymath} (2)

where If we further assume that all antennas are equal, their pointing errors are negligible, their beam shape does not depend on the pointing direction, and the fractional bandwidth is small ( $\delta\nu/\nu << 1$), this equation reduces to
\begin{displaymath}
V_{ijk}(t) = FT ( P(x,y) I(x,y,k) ) (u,v)_{ij}
\end{displaymath} (3)

where $P(x,y)$ is the power beam pattern of the antennas.

The antennas, receivers, cables, and correlators all introduce additional modifications to this visibility. These perturbations can be formally decomposed into

\begin{displaymath}
V_{ijk} = A_i A^*_j S_{ik} S^*_{jk} C_{ijk} R_{ijk} + O_{ijk} + N_{ijk}
\end{displaymath} (4)

where

Provided the design of the interferometer system is adequate, the terms appearing in this decomposition have the following properties:

Let $ W = V - O - N = V - N $ to first order. Here, $PN_{ijk}(t)$ is a random phase and $AN_{ijk}(t)$ has known variance, <$AN$>. Then

\begin{displaymath}
PW_{ijk}(t) = PA_i(t)+PS_{ik}(t)-PA_j(t)-PS_{jk}(t)+PC_{ijk}(t)+PR_{ijk}(t)
\end{displaymath} (5)


\begin{displaymath}
PW_{ijk}(t) = PV_{ijk}(t) + PD_{ijk}(t)
\end{displaymath} (6)

where

Then, to first order,

Similar relations can be expressed for the intensities:


\begin{displaymath}
AW_{ijk}(t) = AA_i(t).AS_{ik}(t).AA_j(t).AS_{jk}(t)).AC_{ijk}(t).AR_{ijk}(t)
\end{displaymath} (7)

and
\begin{displaymath}
AW_{ijk}(t) = AV_{ijk}(t) + AD_{ijk}(t)
\end{displaymath} (8)

where

The purpose of calibration is to determine as best as possible these various functions, taking advantage of the time independence of some parameters, and of the weak chromaticity of the atmosphere.


next up previous contents
Next: Baseline versus Antenna based Up: Appendix: Calibration Principles Previous: Appendix: Calibration Principles   Contents
Gildas manager 2018-09-25