2.1 Random signals

Random signals are random variables which evolve, often with time (e.g. audio noise), but also with distance (e.g. intensityin an image of a random texture), or sometimes another parameter.

They can be described as usual by their cdf and either their pmf (if the amplitude is discrete, as in a digitized signal) ortheir pdf (if the amplitude is continuous, as in most analogue signals).

However a very important additional property is how rapidly a random signal fluctuates. Clearly a slowly varying signal suchas the waves in an ocean is very different from a rapidly varying signal such as vibrations in a vehicle. We will seelater in how to deal with these frequency dependent characteristics of randomness.

For the moment we shall assume that random signals are sampled at regular intervals and that each signal is equivalent to asequence of samples of a given random process, as in the following examples.

Example - detection of a binary signal in noise

We now consider the example of detecting a binary signal after it has passed through a channel which adds noise. Thetransmitted signal is typically as shown in (a) of .

In order to reduce the channel noise, the receiver will include a lowpass filter. The aim of the filter is to reducethe noise as much as possible without reducing the peak values of the signal significantly. A good filter for this has ahalf-sine impulse response of the form:

This filter will convert the rectangular data bits into sinusoidally shaped pulses as shown in (b) of and it will also convert wide bandwidth channel noise into the form shown in (c) of . Bandlimited noise of this form will usually have an approximately Gaussian pdf.

Because this filter has an impulse response limited to just one bit period and has unit gain at zero frequency (the areaunder $h(t)$ is unity), the signal values at the center of each bit period at the detector will still be $\pm (1)$ . If we choose to sample each bit at the detector at this optimal mid point, the pdfs of the signal plus noise atthe detector will be shown in .

Let the filtered data signal be $D(t)$ and the filtered noise be $U(t)$ , then the detector signal is

Similarly the probability of error when the data = $-1$ is then given by:

From we may obtain the probability of error in thebinary detector, which is often expressed as the bit error rate or BER . For example, if $(\mathrm{error})=210^3$ , this would often be expressed as a bit error rate of $210^3$ , or alternatively as 1 error in 500 bits (on average).

The argument ( $\frac{v_0}{\sigma }$ ) in is the signal-to-noise voltage ratio (SNR) atthe detector, and the BER rapidly diminishes with increasing SNR (see ).