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 obsrved 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,x₀+x_i,y₀+y_i) B^*_j(x,y,x₀+x_j,y₀+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')
• (x₀,y₀) is the pointing direction of the antennas
• (x_i,y_i) is the pointing error of antenna i.
• (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 iand 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 part of this term,
• 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.