15.2 Sampling & Aliasing

Sampling on a regular grid is equivalent to multiplying by a series of periodically spaced delta functions, i.e. the so-called shah function $\ensuremath{\mathrm{III}}$:

$$[\frac{1}{\Delta u}] \ensuremath{\mathrm{III}}(\frac{u}{\Delta u}) = \sum_{k=-\infty}^{\infty} \delta(u-k\Delta u)$$ (15.10)

The Fourier Transform of the shah function above is the shah function

$$\ensuremath{\mathrm{III}}(x \Delta u) = \frac{1}{\Delta u} \sum_{m=-\infty}^{\infty} \delta(x-\frac{m}{\Delta u})$$ (15.11)

Hence, sampling the visibilities $V$ results in convolving its Fourier Transform $\hat{V}$ by a periodic shah function. This convolution reproduces in a periodic way the Fourier Transform of the visibilities $\hat{V}$.

If the Fourier Transform of the visibilities $\hat{V}$, i.e. the brightness distribution $B I$, has finite support $\Delta X$, the replication poses no problem provided the support is smaller than the periodicity of the shah function, i.e.

$$(\Delta u)^{-1} \geq (\Delta X) {\hskip 1.0cm} \Delta u \leq (\Delta X)^{-1}$$ (15.12)

If not, data outside $(\Delta u)^{-1}$ are aliased in the imaged area $(\Delta u)^{-1}$.

In aperture synthesis, finite support is ensured to first order by the finite width of the antenna primary beam $B$. However, strong sources in the antenna sidelobes may be aliased if the imaged area is too small. Moreover, the noise does not have finite support. White noise in the $uv$ plane would result in white noise in the map plane. In practice, the noise in the $uv$ plane is not completely white. However, it is support limited (since only a finite region of the $uv$ plane is sampled in any experiment). Accordingly, its Fourier Transform in the map plane is not support limited. Noise aliasing thus occurs, and produces an increased noise level at the map edges.