# W Projection

## 1 W-Projection

### 1.1 The W Problem

A baseline’s projection in the direction of phase center, ${\displaystyle w}$, introduces a phase error that increases away from phase center.

W-Projection is the name of an algorithm used to correct for a problem that arises in making wide-field images with baselines that have a non-zero projection toward the desired phase center (called the ${\displaystyle w}$ component of the baseline).

According to traditional synthesis imaging, each baseline, measured in wavelengths is projected toward the desired phase center, resulting in three coordinates, ${\displaystyle (u,v,w)}$ which enter into the full (idealized) measurement equation as:

${\displaystyle V(u,v,w)=\int {\!\!\int {I(l,m)e^{-2\pi i(ul+vm+w{\sqrt {1-l^{2}-m^{2}}}}dl~dm}}.\,\!}$

A typical next step is to assume that ${\displaystyle l}$ and ${\displaystyle m}$ are small, so that ${\displaystyle {\sqrt {1-l^{2}-m^{2}}}}$ is nearly ${\displaystyle 1}$. This is called the flat-sky approximation. Under this approximation, the ${\displaystyle e^{-2\pi iw}}$ term can be removed by applying the appropriate phasing to the measured visibility, and the resulting measurement equation describes a 2D Fourier transform, where ${\displaystyle u}$ and ${\displaystyle v}$ are taken to be samples of the ${\displaystyle uv}$-plane that is the Fourier complement of the image plane.

This works fine in the flat-sky approximation (which holds out to ${\displaystyle \pm 10^{\circ }}$ or so, where ${\displaystyle l\equiv \sin \theta \approx \theta }$), but it fails dramatically for wide fields of view. It fails because multiplying by a phasor to remove the ${\displaystyle e^{-2\pi iw}}$ phase term works fine at phase center, where ${\displaystyle {\sqrt {1-l^{2}-m^{2}}}=1}$, but farther from phase center, this is not the correct phasor. The ${\displaystyle w}$-term introduces a spatially-varying departure from the 2D Fourier transform that was measurement equation under the flat-sky approximation. It makes point sources away from phase center de-cohere and become very messy.

If we weren’t so attached to using Fourier transforms for synthesis imaging, this wouldn’t be a big deal. If you image by phasing all of your visibilities to each ${\displaystyle l,m}$ separately, using the proper phasing to remove the ${\displaystyle w}$-term at that particular location, you’ll do just fine. The problem is, that’s just computationally intractable. Synthesis imaging is really only practical because FFTs are extremely computationally efficient. So what to do? You could design an array where ${\displaystyle w=0}$ for all baselines relative to a chosen phase center (and this really isn’t a bad idea). This array would sample a “true” ${\displaystyle uv}$-plane at ${\displaystyle w=0}$ that can be directly Fourier transformed to a wide-field image without ${\displaystyle w}$-term phase errors. But if you want to use earth-rotation synthesis, or track a region of sky, you need a more flexible solution.

### 1.2 The Solution: W-Projection

The solution is reasonably straight-forward to understand, if we look at the ${\displaystyle w}$-term is a function that is multiplying ${\displaystyle I}$ in our 2D Fourier transform. If we define:

${\displaystyle W\equiv e^{-2\pi iw{\sqrt {1-l^{2}-m^{2}}}}\,\!}$

then the full curved-sky measurement equation is:

${\displaystyle V(u,v,w)=\int {\!\!\int {I(l,m)\cdot W(l,m)e^{-2\pi i(ul+vm)}dl~dm}}\,\!}$

By the convolution theorem, we know that a multiplication in image ${\displaystyle (l,m)}$ domain is a convolution in ${\displaystyle (u,v)}$ domain. That means that a visibility measured at height ${\displaystyle w}$ f is actually just sampling the “true” ${\displaystyle w=0}$ ${\displaystyle uv}$-plane, convolved with the Fourier transform of ${\displaystyle W}$:

${\displaystyle V(u,v,w)=V(u,v,0)*{\hat {W}}(u,v)\,\!}$

In practice, there’s really no hope of truly undoing this convolution, since any one baseline will only sample ${\displaystyle V(u,v,0)*{\hat {W}}(u,v)}$ at one ${\displaystyle (u,v)}$ coordinate. However, we can project our measured visibility ${\displaystyle V(u,v,w)}$ back into the ${\displaystyle V(u,v,0)}$ plane in a way that reflects the appropriate weights and phases of the nearby ${\displaystyle uv}$-pixels that got multiplied and summed together to produce our measurement.

So this is W-projection. You project ${\displaystyle V(u,v,w)}$ into the ${\displaystyle V(u,v,0)}$ plane by convolving with the inverse of ${\displaystyle W}$, which happens to be ${\displaystyle W^{*}}$. This corresponds to ${\displaystyle W}$ computed for ${\displaystyle -w}$ instead of ${\displaystyle w}$. In practice, this convolution can happen at the same time normal gridding does. And suddenly point sources are point sources, all the way down to the edge of your image. Cool.