Delay Imaging

Delay Imaging

Delay imaging is an interesting and useful application of radio interferometry that gets to the heart of what an interferometer does, and avoids some of the complexity associated with aperture synthesis. In a nutshell, delay imaging is a way of using the frequency information in a wide-bandwidth interferometer to make a one-dimensional projected image of the sky using only one baseline (i.e. two antennas paired together). Here’s how it works.

The geometric interpretation of the delay spectrum measured by an interferometer. The left plot shows how two sources with identical spectra can have differing geometric delays g) owing to their positions relative to the two antennas being correlated. The right plot shows how a strictly geometric interpretation of a delay spectrum is violated by the fact that the Fourier transform of the spectrum of each source also enters the delay spectrum centered at the appropriate τg.

In a previous lecture on interferometry, we showed that, for a baseline $\vec b$, the geometric time delay between when a wavefront hits one antenna and the other is given by:

$\tau _ g = \frac{\vec b\cdot \hat s}{c}, \,\!$

where $\hat s$ is a unit vector pointing in the direction of a source (see the figure above). Now we also know from a previous lecture on correlation that the correlator at the heart on an interferometer, when computing the correlation between two time-domain voltage streams (say, f(t) and g(t)) from two antennas, computes $[f\star g](\tau )$ by computing the frequency-dependent correlation coefficient between two antennas:

$\mathcal{C}_{ij}(\nu )=\mathcal{F}(f_ i)\cdot \mathcal{F}(f_ j) \,\!$

which, according to the correlation theorem, is equal to $\mathcal{F}(f\star g)$. We argued that since, for most radio astronomy, we’re interested in the frequency spectrum of the correlation function anyway, this was a fine quantity for the correlator to output.

Well, delay imaging is just taking the inverse Fourier transform of the correlation (or “visibility”) spectrum after all. For a source that arrives at delay τg, this gives us :

$\mathcal{F}^{-1}(\mathcal{C}_{ij}(\nu )) = |f\cdot g| \delta (\tau _ g) \,\!$

In fact, this helps us understand what $\mathcal{C}_{ij}$ was in the first place. It’s the delay-frequency Fourier transform of delta function above:

$\mathcal{C}_{ij}(\nu ) = \hat f(\nu ) \hat g(\nu ) e^{-2\pi i \tau _ g \nu } \,\!$

Now, if f and g are voltage streams from two antenas that are seeing the same sky, their product, in frequency space, will just be the spectrum of the source at τg. This means that, if we assume our antennas are well-calibrated, that when we look at the sky with this one baseline , we see the superposition of the sine waves in frequency from a bunch of sources at their respective delays. Since we’re talking sky now, I’m going to write V (the visibility) instead of $\mathcal{C}$ for the correlation coefficient:

$V(\nu )=\sum _ n{S_ n(\nu ) e^{-2\pi i \tau _{g,n} \nu }}. \,\!$

And now, if I take the inverse Fourier transform of V(ν) over some finite bandwidth, I get:

\begin{align} \hat V(\tau ) & = \int _{\nu _0}^{\nu _1}{\sum _ n{S_ n(\nu ) e^{-2\pi i \tau _{g,n}\nu }}e^{2\pi i \tau \nu }d\nu }\\ & =\sum _ n{\delta (\tau _ g) * \int _{\nu _0}^{\nu _1}{S_ n(\nu ) e^{2\pi i\tau \nu }d\nu }} \end{align}\,\!

And this final equation is telling us that, for sufficiently wide bandwidths such that the integration interval in the Fourier transform of the spectrum of the source isn’t overly bothersome (we can be more quantitative, but let’s leave it at that for now), the Fourier transform of frequency-dependent visibilities gives us a sum of delta functions centered at the delay of each source, and that each delta function is convolved by the Fourier transform of the frequency spectrum of the respective source. And because delay represents the projection of the sky along the axis of the baseline, this “delay transform” of a visibility provides a one-dimensional, projected image of the sky.

The delay spectra measured by baselines of four different lengths for a simulated sky consisting of several celestial sources, one of which has a patently unsmooth spectrum (top region of each panel). Vertical axis shows time as the earth rotates, horizontal axis shows delay.