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Random Signals

Module by: Nick Kingsbury

Summary: This module introduces random signals.

Random signals are random variables which evolve, often with time (e.g. audio noise), but also with distance (e.g. intensity in 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) or their 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 such as the waves in an ocean is very different from a rapidly varying signal such as vibrations in a vehicle. We will see later in (Reference) 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 a sequence of samples of a given random process, as in the following examples.
Figure 1: Detection of signals in noise: (a) the transmitted binary signal; (b) the binary signal after filtering with a half-sine receiver filter; (c) the channel noise after filtering with the same filter; (d) the filtered signal plus noise at the detector in the receiver.
Figure 2: The pdfs of the signal plus noise at the detector for the two ±1 ± 1 . The vertical dashed line is the detector threshold and the shaded area to the left of the origin represents the probability of error when data = 1.

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. The transmitted signal is typically as shown in (a) of Figure 1.
In order to reduce the channel noise, the receiver will include a lowpass filter. The aim of the filter is to reduce the noise as much as possible without reducing the peak values of the signal significantly. A good filter for this has a half-sine impulse response of the form:
ht=π2TbsinπtTbif0≤t≤Tb0otherwise h t 2 Tb t Tb 0 t Tb 0 (1)
Where Tb Tb = bit period.
This filter will convert the rectangular data bits into sinusoidally shaped pulses as shown in (b) of Figure 1 and it will also convert wide bandwidth channel noise into the form shown in (c) of Figure 1. 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 area under ht h t is unity), the signal values at the center of each bit period at the detector will still be ±1 ± 1 . If we choose to sample each bit at the detector at this optimal mid point, the pdfs of the signal plus noise at the detector will be shown in Figure 2.
Let the filtered data signal be Dt D t and the filtered noise be Ut U t , then the detector signal is
Rt=Dt+Ut R t D t U t (2)
If we assume that +1 + 1 and -1-1 bits are equiprobable and the noise is a symmetric zero-mean process, the optimum detector threshold is clearly midway between these two states, i.e. at zero. The probability of error when the data = +1 + 1 is then given by:
Prerror|D=+1=PrRt<0|D=+1=FU-1=∫-∞-1fUudu D + 1 error D + 1 R t 0 FU -1 u -1 fU u (3)
where FU FU and fU fU are the cdf and pdf of UU. This is the shaded area in Figure 2.
Similarly the probability of error when the data = -1-1 is then given by:
Prerror|D=-1=PrRt>0|D=-1=1-FU+1=∫1∞fUudu D -1 error D -1 R t 0 1 FU + 1 u 1 fU u (4)
Hence the overall probability of error is:
Prerror=Prerror|D=+1PrD=+1+Prerror|D=-1PrD=-1=∫-∞-1fUuduPrD=+1+∫1∞fUuduPrD=-1 error D + 1 error D + 1 D -1 error D -1 u -1 fU u D + 1 u 1 fU u D -1 (5)
since fU fU is symmetric about zero Prerror=∫1∞fUuduPrD=+1+PrD=-1=∫1∞fUudu error u 1 fU u D + 1 D -1 u 1 fU u To be a little more general and to account for signal attenuation over the channel, we shall assume that the signal values at the detector are ±v0 ± v0 (rather than ±1 ± 1 ) and that the filtered noise at the detector has a zero-mean Gaussian pdf with variance σ2 σ 2 :
fUu=12πσ2ⅇ-u22σ2 fU u 1 2 σ 2 u 2 2 σ 2 (6)
and so
Prerror=∫ v0 ∞fUudu=∫ v0 σ∞fUσuσdu=Q v0 σ error u v0 fU u u v0 σ fU σ u σ Q v0 σ (7)
where
Qx=12π∫x∞ⅇ-u22du Q x 1 2 u x u 2 2 (8)
This integral has no analytic solution, but a good approximation to it exists and is discussed in some detail in (Reference).
From Equation 7 we may obtain the probability of error in the binary detector, which is often expressed as the bit error rate or BER. For example, if Prerror=2×103 error 2 10 3 , this would often be expressed as a bit error rate of 2×103 2 10 3 , or alternatively as 1 error in 500 bits (on average).
The argument ( v0σ v0 σ ) in Equation 7 is the signal-to-noise voltage ratio (SNR) at the detector, and the BER rapidly diminishes with increasing SNR (see (Reference)).

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