Signals in Radio Astronomy

A fundamental property of the radio waves emitted by cosmic sources is that they are stochastic in nature, i.e. the electric field at Earth due to a distant cosmic source can be treated as a random process^2.2. Random processes can be simply understood as a generalization of random variables. Recall that a random variable $x$ can be defined as follows. For every outcome $o$ of some given experiment (say the tossing of a die) one assigns a given number to $x$. Given the probabilities of the different outcomes of the experiment one can then compute the mean value of $x$, the variance of $x$ etc. If for every outcome of the experiment instead of a number one assigns a given function to $x$, then the associated process $x(t)$ is called a random process. For a fixed value of $t$, $x(t)$ is simply a random variable and one can compute its mean, variance etc. as before.

A commonly used statistic for random processes is the auto-correlation function. The auto-correlation function is defined as

$$r_{xx}(t,\tau) = \bigl<x(t)x(t+\tau)\bigr>$$

where the angular brackets indicate taking the mean value. For a particularly important class of random processes, called wide sense stationary (WSS) processes the auto-correlation function is independent of changes of the origin of $t$ and is a function of $\tau$ alone, i.e.

$$r_{xx}(\tau) = \bigl<x(t)x(t+\tau)\bigr>$$

For $\tau = 0$, $r(\tau)$ is simply the variance $\sigma^2$ of $x(t)$ (which for a WSS process is independent of $t$).

The Fourier transform $S(\nu)$ of the auto-correlation function is called the power spectrum, i.e.

$$S(\nu) = \int_{-\infty}^\infty r_{xx}(\tau)e^{-i2\pi\tau\nu} d\tau$$

Equivalently, $S(\nu)$ is the inverse Fourier transform of $r(\tau)$ or

$$r_{xx}(\tau) = \int_{-\infty}^\infty S(\nu)e^{i2\pi\tau\nu} d\nu$$

Hence

$$r_{xx}(0) = \sigma^2 = \int_{-\infty}^\infty S(\nu)d\nu$$

i.e. since $\sigma^2$ is the ``power'' in the signal, $S(\nu)$ is a function describing how that power is distributed in frequency space, i.e. the ``power spectrum''.

A process whose auto-correlation function is a delta function has a power spectrum that is flat - such a process is called ``white noise''. As mentioned in Section 2.2, many radio astronomical signals have spectra that are relatively flat; these signals can hence be approximated as white noise. Radio astronomical receivers however have limited bandwidths, that means that even if the signal input to the receiver is white noise, the signal after passing through the receiver has power only in a finite frequency range. Its auto-correlation function is hence no longer a delta function, but is a sinc function (see Section 2.5) with a width $\sim 1/\Delta \nu$, where $\Delta \nu$ is the bandwidth of the receiver. The width of the auto-correlation function is also called the ``coherence time'' of the signal. The bandwidth $\Delta \nu$ is typically much smaller than the central frequency $\nu$ at which the radio receiver operates. Such signals are hence also often called ``quasi-monochromatic'' signals. Much like a monochromatic signal can be represented by a constant complex phasor, quasi-monochromatic signals can be represented by complex random processes.

Given two random processes $x(t)$ and $y(t)$, one can define a cross-correlation function

$$r_{xy}(\tau) = \bigl<x(t)y(t-\tau)\bigr>$$

where one has assumed that the signals are WSS so that the cross-correlation function is a function of $\tau$ alone. The cross-correlation function and its Fourier transform, the cross power spectrum, are also widely used in radio astronomy.

We have so far been dealing with random processes that are a function of time alone. The signal received from a distant cosmic source is in general a function both of the receivers location as well as of time. Much as we defined temporal correlation functions above, one can also define spatial correlation functions. If the signal at the observer's plane at any instant is E(r), then spatial correlation function is defined as:

$$V({\bf x}) = \bigl< E({\bf r}) E^{*}(\bf {r+x}) \bigr>$$

Note that strictly speaking the angular brackets imply ensemble averaging. In practice one averages over time^2.3 and assumes that the two averaging procedures are equivalent. The function $V$ is referred to as the ``visibility function'' (or just the ``visibility'') and as we shall see below, it is of fundamental interest in interferometry.

Footnotes

... process^2.2

see Chapter 1 for a more detailed discussion of topics discussed in this section.

... time^2.3

For typical radio receiver bandwidths of a few MHz, the coherence time is of the order of micro seconds, so in a few seconds time one gets several million independent samples to average over.