Radio interferometry, Fourier transforms, and the guts of radio interferometry (Part 1)
Today’s post comes from Dr Enno Middelberg and is the first part of two explaining in more detail about radio interferometry and the techniques used in producing the radio images in Radio Galaxy Zoo.
I have written in an earlier post about the basic idea of how to increase the resolution of a radio telescope: use many telescopes, separated by kilometers, and observe the same object with all. Here is a little more information about how this works.
At the very heart of an interferometer is the van Cittert-Zernike theorem: it essentially states that the degree of similarity of the electric field at two locations is a measure of the Fourier transform of the sky brightness distribution. Now that’s a big bite to swallow, but let me explain it in less confusing words: the electric field is all we can measure – radio waves are electromagnetic waves, and radio telescopes are sensitive to the electric field. Now we can build a radio telescope in a way that it produces as its output a voltage which is proportional to the electric field which the antenna receives from, e.g., a galaxy. Much of the signal will be noise from our own Milky Way, the atmosphere and the electronics which amplify the feeble signals, but a tiny little bit of the signal will be caused by radio waves from space, and both antennas will receive a little bit of these. Now suppose we have two telescopes separated by 1 km or so, and both telescopes produce such voltages which contain a little bit of this signal. The voltages are digitised and the two data streams are fed into a correlator. The correlator is a computer which takes the two data streams and calculates their correlation coefficient, which is an indicator for their similarity. If the two data streams have nothing in common (for example, because an unexperienced PhD student pointed the two antennas in different directions 🙂 ) then the correlation coefficient will be zero, which is to say that they are not similar at all. However, if the two telescopes point at the same source, the data streams will have a few bits in common, and the correlator spits out a correlation coefficient which is not zero. This is our measurement!
Now that we’ve that out of the way, we need to talk about Fourier transforms. The van Cittert-Zernike theorem states that the correlation coefficient is a measure for the Fourier transform of the sky brightness. Now what is a Fourier transform? The Fourier transform is an ingenious way of representing a mathematical function with a sum of sine and cosine functions. That is, if I take a large number of sine and cosine functions with various (but carefully selected!) frequencies and amplitudes, then their sum will be an accurate representation of another function, for example a square wave or a sawtooth. Check out the Wikipedia page on Fourier series (which are related to Fourier transforms, but easier to understand), which has a number of nice animations to illustrate this, such as this one:
You can also play with Paul Falstad’s Java applet to see how to construct functions using sine and cosine waves interactively – very instructive! In part 2 of this post I will explain how astronomers use 2D Fourier transforms to assemble images of the radio sky.