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# White and Coloured Processes

Module by: Nick Kingsbury. E-mail the author

Summary: This module introducts white, near white and colored processes.

## White Noise

If we have a zero-mean Wide Sense Stationary process XX, it is a White Noise Process if its ACF is a delta function at τ=0 τ 0 , i.e. it is of the form:

r X X τ= PX δτ r X X τ PX δ τ
(1)
where PX PX is a constant.

The PSD of XX is then given by

SX ω= PX δτe(iωτ)dτ= PX e(iω0)= PX SX ω τ PX δ τ ω τ PX ω 0 PX
(2)
Hence XX is white, since it contains equal power at all frequencies, as in white light.

PX PX is the PSD of XX at all frequencies.

But:

Power  of  X=12π SX ωdω= Power  of  X 1 2 ω SX ω
(3)
so the White Noise Process is unrealizable in practice, because of its infinite bandwidth.

However, it is very useful as a conceptual entity and as an approximation to 'nearly white' processes which have finite bandwidth, but which are 'white' over all frequencies of practical interest. For 'nearly white' processes, r X X τ r X X τ is a narrow pulse of non-zero width, and SX ω SX ω is flat from zero up to some relatively high cutoff frequency and then decays to zero above that.

## Strict Whiteness and i.i.d. Processes

Usually the above concept of whiteness is sufficient, but a much stronger definition is as follows:

Pick a set of times t1 t2 tN t1 t2 tN to sample Xt X t .

If, for any choice of t1 t2 tN t1 t2 tN with NN finite, the random variables X t1 X t1 , X t2 X t2 , X tN X tN are jointly independent, i.e. their joint pdf is given by

f X ( t1 ) , X ( t2 ) ,     X ( tN ) x1 x2 xN =i=1N f X ( ti ) xi f X ( t1 ) , X ( t2 ) ,     X ( tN ) x1 x2 xN i 1 N f X ( ti ) xi
(4)
and the marginal pdfs are identical, i.e.
f X ( t1 ) =f X ( t2 ) ==f X ( tN ) = fX f X ( t1 ) f X ( t2 ) f X ( tN ) fX
(5)
then the process is termed Independent and Identically Distributed (i.i.d).

If, in addition, fX fX is a pdf with zero mean, we have a Strictly White Noise Process.

An i.i.d. process is 'white' because the variables X ti X ti and X tj X tj are jointly independent, even when separated by an infinitesimally small interval between ti ti and tj tj .

## Additive White Gaussian Noise (AWGN)

In many systems the concept of Additive White Gaussian Noise (AWGN) is used. This simply means a process which has a Gaussian pdf, a white PSD, and is linearly added to whatever signal we are analysing.

Note that although 'white' and Gaussian' often go together, this is not necessary (especially for 'nearly white' processes).

E.g. a very high speed random bit stream has an ACF which is approximately a delta function, and hence is a nearly white process, but its pdf is clearly not Gaussian - it is a pair of delta functions at +V + V and V V , the two voltage levels of the bit stream.

Conversely a nearly white Gaussian process which has been passed through a lowpass filter (see next section) will still have a Gaussian pdf (as it is a summation of Gaussians) but will no longer be white.

## Coloured Processes

A random process whose PSD is not white or nearly white, is often known as a coloured noise process.

We may obtain coloured noise Yt Y t with PSD SY ω SY ω simply by passing white (or nearly white) noise Xt X t with PSD PX PX through a filter with frequency response ω ω , such that from this equation from our discussion of Spectral Properties of Random Signals.

SY ω= SX ω|ω|2= PX |ω|2 SY ω SX ω ω 2 PX ω 2
(6)
Hence if we design the filter such that
|ω|= SY ω PX ω SY ω PX
(7)
then Yt Y t will have the required coloured PSD.

For this to work, SY ω SY ω need only be constant (white) over the passband of the filter, so a nearly white process which satisfies this criterion is quite satisfactory and realizable.

Using this equation from our discussion of Spectral Properties of Random Signals and Equation 1, the ACF of the coloured noise is given by

r Y Y τ= r X X τ*hτ*hτ= PX δτ*hτ*hτ= PX hτ*hτ r Y Y τ r X X τ h τ h τ PX δ τ h τ h τ PX h τ h τ
(8)
where hτ h τ is the impulse response of the filter.

This Figure from previous discussion shows two examples of coloured noise, although the upper waveform is more 'nearly white' than the lower one, as can be seen in part c of this figure from previous discussion in which the upper PSD is flatter than the lower PSD. In these cases, the coloured waveforms were produced by passing uncorrelated random noise samples (white up to half the sampling frequency) through half-sine filters (as in this equation from our discussion of Random Signals) of length Tb =10 Tb 10 and 5050 samples respectively.

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