Introduction

The record of the electric field $E(t)$, received at a point on earth from a source of radio waves can be called a ``signal'', so long as we do not take this to imply intelligence at the transmitting end. Emanating as it does from a large object with many independently radiating parts, at different distances from our point, and containing many frequencies, this signal is naturally random in character. In fact, this randomness is of an extreme form. All measured statistical properties are consistent with a model in which different frequencies have completely unrelated phases, and each of these phases can vary randomly from $0$ to $2\pi$. A sketch of such a signal is given in Fig. 1.1. The strength (squared amplitude or power) of the different frequencies $\omega$ has a systematic variation which we call the ``power spectrum'' $S(\omega)$. This chapter covers the basic properties of such signals, which go by the name of ``time-stationary gaussian noise''. Both the signal from the source of interest, as well as the noise added to this cosmic signal by the radio telescope recievers can be described as time-stationary gaussian noise. The word noise of course refers to the random character. ``Noise'' also evokes unwanted disturbance, but this of course does not apply to the signal from the source (but does apply to what our receivers unavoidably add to it). The whole goal of radio astronomy is to receive, process, and interpret these cosmic signals, (which were, ironically enough, first discovered as a ``noise'' which affected trans-atlantic radio communication). ``Time-Stationary'' means that the signal in one time interval is statistically indistinguishable from that in another equal duration but time shifted interval. Like all probabilistic statements, this can never be precisely checked but its validity can be made more probable (circularity intended!) by repeated experiments. For example, we could look at the probability distribution of the signal amplitude. An experimenter could take a stretch of the signal say, from times $0$ to $T$, select $N$ equally spaced values $E(t_i), i$ going from 1 to $N$, and make a histogram of them. The property of time stationarity says that this histogram will turn out to be (statistically) the same -- with calculable errors decreasing as N increases! -- if one had chosen instead the stretch from $t$ to $t+T$, for any $t$. The second important characteristic property of our random phase superposition of many frequencies is that this histogram will tend to a gaussian, with zero mean as N tends to infinity.

Figure 1.1: A signal made by superposition of many frequencies with random phases