7.1 Definitions and formalism

As has been seen in the previous lectures, each interferometer baseline provides a measurement of the source visibility at a given point in the u,v plane of spatial frequencies; the source brightness distribution can then be reconstructed by an appropriate Fourier Transform.

In reality things are not so simple. Interferometers are designed with a lot of care; however many electronic components will have variable gains both in amplitude and in phase; these variations will affect the results and have to be taken out. It is generally sometimes more efficient to have a slightly varying instrument response, and a more sensitive instrument, than a very stable one with less sensitivity, provided the varying terms in the response are slow and may be easily calibrated out. At millimeter wavelengths the atmospheric absorption and path length fluctuations will dominate the instrument imperfections in most cases.

For a given observation, if we interpret the correlator response (amplitude and phase) as the source visibility, ignoring any imperfections, we have an observed (apparent) visibility $\ensuremath{\widetilde{V}} _{ij}(t)$, where i,j are antenna numbers, $\nu$ the frequency and t is time. If the true source visibility is $\ensuremath{V} _{ij}(t)$, we may define :

$$\ensuremath{\widetilde{V}} _{ij}(t) = \ensuremath{\mathcal{G}... ...j}(t) \ensuremath{V} _{ij}(t) + \epsilon_{ij}(t)+ \eta_{ij}(t)$$ (7.1)

where the $\ensuremath{\mathcal{G}} _{ij}(\nu,t)$ are the complex gains of each baseline. $\eta_{ij}(t)$ is a noise term resulting from thermal fluctuations in the receivers; $\epsilon_{ij}(t)$ is an offset term. This assumes that the system is linear. $\pi$ phase switching applied on the first local oscillators is a very efficient method of suppressing the offsets $\epsilon_{ij}(t)$; they are generally negligible and will not be considered any further.