\delta_{G}}{\delta^2_{A} + \delta^2_{G}} \right ) \label{eq:prodconv}\\
& = &  \frac{i}{w} \underbrace{\exp \left ( - \frac{t^2}{\delta_R} \right ) }_\textrm{real envelope}
\underbrace{ \exp \left (  - \frac{i t^2}{ \delta_I} \right )}_\textrm{complex chirp} \label{eq:conv} \\\end{aligned}$$

The width of the envelope is given by $$\delta_R = \frac{\delta_A^2 + \delta_G^2}{\delta_A}, \label{eq:deltar}$$

and width of the complex chirp is

$$\delta_I = \frac{\delta_A^2 + \delta_G^2}{\delta_G}.$$

The convolution function is, therefore, a complex Gaussian, which can be expressed as the product of real-valued Gaussian envelope (the $\delta_R$ term in equation \[eq:conv\]) with a complex chirp (the $\delta_I$ term). Clearly, as $|w|$ increases, the width of the Gaussian envelope $\delta_R$ increases.

For a given visibility indexed by $j$ the form of the convolution function can be calculated from its $w$ coordinate using: $$C(t, w_j) = \frac{i}{w_j} \exp{\left ( \frac{-t^2}{\delta_j} \right )} \label{eq:conv_ctw}$$

where $$\delta_j = \delta_A - i \frac{w_j}{\pi} \label{eq:conv_variance}$$

is the width of the relevant convolution function, and $w_j$ is the $w$ coordinate of the visibility.

Conversion from wavelengths to pixels
-------------------------------------

Up to this point, the $u,v,w$ co-ordinates and the Gaussian widths have been in units of wavelength. For the remainder of this paper, width will often be quoted in terms of pixels. We use the terms 