An aperture synthesis array measures the visibilities at discrete
points in the -domain. The visibilities are Fourier
transformed to get the *Dirty Map* and the weighted -sampling
function is Fourier transformed to get the *Dirty Beam* using the
efficient FFT algorithm. This lecture describes the entire chain of
data processing required to inverted the visibilities recorded as a
function of (), and the resulting errors/distortions in the
final image. In this entire lecture, the `' operator
represents convolution operation, the `.' operator represents
point-by-point multiplication and the `
' operator represents the
Fourier transform operator.

As described earlier, the visibility measured by an aperture synthesis
telescope is related to the sky brightness distribution as

where is the

(11.1.3) |

This function essentially assigns a weight of unity to all measured
points and zero everywhere else in the -plane. Fourier
transform of is referred to as the *Dirty Beam*. As written
in Eq. 11.1.2, the *Dirty Beam* is the transfer function of
the instrument used as an imaging device. The shape of the *Dirty
Beam* is a function of the -coverage which in turns is a
function of the location of the antennas. *Dirty Beam* for a
fully covered -plane will be equal to
where is the
largest antenna spacing for which a measurement is available. The
width of the main lobe of this function is proportional to
. The resolution of such a telescope is therefore
roughly
and can be interpreted as the size
of an equivalent lens. For a real -coverage however, is
not flat till and has `holes' in between representing
un-sampled points. The effect of this missing data is to
increase the side-lobes and make the *Dirty Beam* noisy, but in a
deterministic manner. Typically, an elliptical gaussian can be fitted
to the main lobe of the *Dirty Beam* and is used as the resolution
element of the telescope. The fitted gaussian is referred to as the
*Synthesized Beam*.

The *Dirty Map* is a convolution of the true brightness
distribution and the *Dirty Beam*. is almost never a
satisfactory final product since the side-lobes of (which are due
to missing spacings in the -domain) from a strong source in the map
will contaminate the entire map at levels higher than the thermal
noise in the map. Without removing the effect of from the map,
the effective RMS noise in the map will be much higher than the
thermal noise of the telescope and will result into obscuration of
faint sources in the map. This will be then equivalent to reduction
in the dynamic range of the map. The process of De-convolving is
discussed in a later lecture, which effectively attempts to estimate
from such that is minimized consistent with the
estimated noise in the map.

To use the FFT algorithm for Fourier transforming, the irregularly
sampled visibility needs to be gridded onto a regular grid of
cells. This operation requires interpolation to the grid points and
then re-sampling the interpolated function. To get better control on
the shape of the *Dirty Beam* and on the signal-to-noise ratio in
the map, the visibility is first re-weighted before being gridded.
These operations are described below.