# Gain Components

Gain components calculate the complex-valued gains and then perform the computation of the observed visibilities, $V^\text{OBS}$, from the RFI visibilities, $V^\text{RFI}$, and the astronomical visibilities, $V^\text{AST}$. This is typically done in a linear manner as

$$V^\text{OBS}_{pq} = G_p \left( V_{pq}^\text{RFI} + V_{pq}^\text{AST} \right) G^*_q$$

The indices $p$ and $q$ are the antennas forming a baseline $pq$. Both components produce a gain on the full $(t, \nu)$ grid and hold it constant over both axes; they differ in whether it is fitted. {class}`~tabascal.components.gains.ConstGains` fits one complex gain per antenna, and {class}`~tabascal.components.gains.UnitaryGains` fits nothing and leaves it at 1.

## Unitary Gains - {class}`~tabascal.components.gains.UnitaryGains`

Unitary gains are perfect gains, i.e $G(t, \nu) = 1$. Therefore,

$$V^\text{OBS}_{pq} = V_{pq}^\text{RFI} + V_{pq}^\text{AST}$$

## Constant Gains - {class}`~tabascal.components.gains.ConstGains`

One complex direction-independent gain per antenna, constant over time and frequency, and fitted:

$$V^\text{OBS}_{pq} = g_p g_q^* \left( V_{pq}^\text{RFI} + V_{pq}^\text{AST} \right)$$

This is the static gain of the array rather than a time-variable one, so it adds only $2 n_\text{ant} - 2$ parameters. It is written in the gauge the data can actually see: the log amplitudes are carried by $n_\text{ant} - 1$ parameters on an orthonormal basis of the zero-sum subspace, so the geometric mean of $|g_p|$ is exactly 1 and the gain carries no absolute flux scale, and the phase of the reference antenna `gains.ref_ant` is pinned to 0, so it carries no absolute phase either. Both directions the likelihood is flat along are removed from the parameters themselves, rather than left in as coordinates the data cannot see.

The gain is only identifiable against a model term it cannot deform, so `ConstGains` pairs with {class}`~tabascal.components.rfi_signal.ComplexRFIConstAnt` (whose RFI amplitude has no per-antenna freedom of its own) and with a rigid sky, {class}`~tabascal.components.ast_signal.FixedDiscreteSky` and {class}`~tabascal.components.ast_vis.DiscreteSkyVis`. Combining it with {class}`~tabascal.components.rfi_signal.ComplexRFIVarAnt` raises a warning naming the flat direction. See the [gains configuration section](../config.md#a-constant-gain-per-antenna) for the configuration keys, the identifiability rules and how to initialise the fit at a previously measured gain.