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 $t$,

$$V_{ijk}(t) = AV_{ijk}(t) . \exp(-i.PV_{ijk}(t))$$ (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 $t$,

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

where

• $FT$ is the Fourier Transform operation, $x,y$ are integration parameters,
• $I(x,y,k)$ is the brightness distribution of the source at frequency $k$,
• $B_i(x,y,x',y')$ is the voltage pattern of antenna $i$ pointed in direction $(x',y')$,
• $(x_0,y_0)$ is the pointing direction of the antennas,
• $(x_i,y_i)$ is the pointing error of antenna $i$, and
• $(u,v)_{ij}$ are the projected coordinates of baseline $ij$, in wavelength units.

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

$$V_{ijk}(t) = FT ( P(x,y) I(x,y,k) ) (u,v)_{ij}$$ (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

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

where

• $A_i$ is the complex gain of antenna $i$ (amplitude and phase),
• $S_{ik}$ is the complex gain of channel $k$ for antenna $i$,
• $C_{ijk}$ is the complex gain of channel $k$ for correlator entry $ij$,
• $R_{ijk}$ is the theoretical visibility of the source,
• $O_{ijk}$ is a non random error on channel $k$ for correlator entry $ij$. These errors have various origins, such as finite bandpass, bandpass mismatch between antennas $i$ and $j$, etc...), and
• $N_{ijk}$ is a random error due to detection noise.

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

• $N_{ijk}(t)$ is normally distributed with known variance.
• $O_{ijk}$ is negligible with respect to $N_{ijk}$. We will assume $O_{ijk} = 0$ in the following discussion.
• $C_{ijk}(t)$ is only weakly time dependent. This factor is introduced essentially by analog filters in the correlator and residual (constant) delay offsets between subbands; it may depend strongly on $k$.
• $S_{ik}(t)$ is only weakly time dependent. This factor is introduced by receivers and IF cables. The frequency ($k$) dependence is weak.
• $A_i(t)$ depends on antenna pointing (amplitude only), focus (amplitude and phase), and on atmosphere (amplitude and phase).

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

$$PW_{ijk}(t) = PA_i(t)+PS_{ik}(t)-PA_j(t)-PS_{jk}(t)+PC_{ijk}(t)+PR_{ijk}(t)$$ (5)

$$PW_{ijk}(t) = PV_{ijk}(t) + PD_{ijk}(t)$$ (6)

where

• $PA_i$ is the instrumental phase on antenna $i$
• $PS_{ik}$ is the relative phase of channel $k$ for antenna $i$
• $PC_{ijk}$ is the relative phase of channel $k$ for correlator entry $ij$, independent of the $PS_{ik}$
• $PR_{ijk}$ is the source phase on baseline $ij$ for channel $k$. $R_{ijk}(t)$ only depends on $ij$ through the coordinates of antennas $i$ and $j$, via $(u(t),v(t))_{ij}$.
• $PD_{ijk}$ is a phase noise introduced by measurement noise $N_{ijk}$

Then, to first order,

• $PC_{ijk}$ can be measured independently, and is only weakly time dependent,
• $PS_{ik}$ is constant, providing the receiver is not retuned,
• $PA_i$ is time variable, on many different timescales, and
• $PD_{ijk}$ has known variance, depending on $W_{ijk}$/<$AN$ >.

Similar relations can be expressed for the intensities:

$$AW_{ijk}(t) = AA_i(t).AS_{ik}(t).AA_j(t).AS_{jk}(t)).AC_{ijk}(t).AR_{ijk}(t)$$ (7)

and

$$AW_{ijk}(t) = AV_{ijk}(t) + AD_{ijk}(t)$$ (8)

where

• $AA_i$ is the gain of antenna $i$ (including effects due to atmospheric absorption, focus, receiver gain, pointing, etc...),
• $AS_{ik}$ is the relative gain of channel $k$ for antenna $i$,
• $AC_{ijk}$ is the relative gain of channel $k$ for correlator entry $i,j$, measured independently from $AS_{ik}$,
• $AR_{ijk}$ is the Source intensity on baseline $ij$. $AR_{ijk}(t)$ depends on $ij$ only through antenna coordinates.
• $AD_{ijk}$ has known variance <$AN$>.

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.