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 process2.2. Random processes can be simply understood as a generalization of random variables. Recall that a random variable can be defined as follows. For every outcome of some given experiment (say the tossing of a die) one assigns a given number to . Given the probabilities of the different outcomes of the experiment one can then compute the mean value of , the variance of etc. If for every outcome of the experiment instead of a number one assigns a given function to , then the associated process is called a random process. For a fixed value of , 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
The Fourier transform of the auto-correlation function
is called the power spectrum, i.e.
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 , where is the bandwidth of the receiver. The width of the auto-correlation function is also called the ``coherence time'' of the signal. The bandwidth is typically much smaller than the central frequency 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 and , one can define
a cross-correlation function
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