Introduction

An aperture synthesis array measures the visibilities at discrete points in the $uv$-domain. The visibilities are Fourier transformed to get the Dirty Map and the weighted $uv$-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 ($u,v,w$), and the resulting errors/distortions in the final image. In this entire lecture, the `$\star$' operator represents convolution operation, the `.' operator represents point-by-point multiplication and the ` $\rightleftharpoons$' operator represents the Fourier transform operator.

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

$$V \rightleftharpoons I,$$ (11.1.1)

where $\rightleftharpoons$ denotes the Fourier Transform. The above equation is true only for the case of continuous sampling of the $uv$-plane such that $V$ is measured for all values of ($u,v$). However since there are finite antennas in an array, $uv$-plane is sampled at descreet $uv$ points and Eq. 11.1.1 has to be written as

$$V.S \rightleftharpoons I*DB (=I^d),$$ (11.1.2)

where $I^d$ is the Dirty Map, $I$ is the true brightness distribution, $DB$ is the Dirty Beam and $S$ is the $uv$-sampling function given by

$$S(u,v)=\sum_{k} \delta(u-u_k,v-v_k),$$ (11.1.3)

where $u_k$ and $v_k$ are the actual ($u,v$) points measured by the telescope. The pattern of all the measured ($u,v$) points is referred to as the $uv$-coverage.

This function essentially assigns a weight of unity to all measured points and zero everywhere else in the $uv$-plane. Fourier transform of $S$ 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 $uv$-coverage which in turns is a function of the location of the antennas. Dirty Beam for a fully covered $uv$-plane will be equal to $sin(\pi l \lambda/u_{max})/(\pi l \lambda/u_{max})$ where $u_{max}$ is the largest antenna spacing for which a measurement is available. The width of the main lobe of this function is proportional to $\lambda/u_{max}$. The resolution of such a telescope is therefore roughly $\lambda/u_{max}$ and $u_{max}$ can be interpreted as the size of an equivalent lens. For a real $uv$-coverage however, $S$ is not flat till $u_{\rm max}$ and has `holes' in between representing un-sampled $(u,v)$ 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. $I^d$ is almost never a satisfactory final product since the side-lobes of $DB$ (which are due to missing spacings in the $uv$-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 $DB$ 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 $I$ from $I^d$ such that $(I-I^d)*DB$ is minimized consistent with the estimated noise in the map.

To use the FFT algorithm for Fourier transforming, the irregularly sampled visibility $V(u,v)$ 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.