4.5 White and coloured processes

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This module introducts white, near white and colored processes.

White noise

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

${r}_{XX}(\tau )={P}_{X}\delta (\tau )$
where ${P}_{X}$ is a constant.

The PSD of $X$ is then given by

${S}_{X}(\omega )=\int {P}_{X}\delta (\tau )e^{-(i\omega \tau )}\,d \tau ={P}_{X}e^{-(i\omega \times 0)}={P}_{X}$
Hence $X$ is white , since it contains equal power at all frequencies, as in white light .

${P}_{X}$ is the PSD of $X$ at all frequencies.

But:

S X ω
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 finitebandwidth, but which are 'white' over all frequencies of practical interest. For 'nearly white' processes, ${r}_{XX}(\tau )$ is a narrow pulse of non-zero width, and ${S}_{X}(\omega )$ 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 $\{{t}_{1}, {t}_{2}, \dots , {t}_{N}\}$ to sample $X(t)$ .

If, for any choice of $\{{t}_{1}, {t}_{2}, \dots , {t}_{N}\}$ with $N$ finite, the random variables $X({t}_{1})$ , $X({t}_{2})$ , $\dots$ $X({t}_{N})$ are jointly independent , i.e. their joint pdf is given by

and the marginal pdfs are identical, i.e.
${f}_{X\left({t}_{1}\right)}={f}_{X\left({t}_{2}\right)}=\dots ={f}_{X\left({t}_{N}\right)}={f}_{X}$
then the process is termed Independent and Identically Distributed (i.i.d) .

If, in addition, ${f}_{X}$ is a pdf with zero mean, we have a Strictly White Noise Process .

An i.i.d. process is 'white' because the variables $X({t}_{i})$ and $X({t}_{j})$ are jointly independent, even when separated by an infinitesimally small interval between ${t}_{i}$ and ${t}_{j}$ .

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 towhatever 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 whiteprocess, but its pdf is clearly not Gaussian - it is a pair of delta functions at $+(V)$ and $-V$ , 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 stillhave 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 $Y(t)$ with PSD ${S}_{Y}(\omega )$ simply by passing white (or nearly white) noise $X(t)$ with PSD ${P}_{X}$ through a filter with frequency response $ℋ(\omega )$ , such that from this equation from our discussion of Spectral Properties of Random Signals.

${S}_{Y}(\omega )={S}_{X}(\omega )\left|ℋ(\omega )\right|^{2}={P}_{X}\left|ℋ(\omega )\right|^{2}$
Hence if we design the filter such that
$\left|ℋ(\omega )\right|=\sqrt{\frac{{S}_{Y}(\omega )}{{P}_{X}}}$
then $Y(t)$ will have the required coloured PSD.

For this to work, ${S}_{Y}(\omega )$ need only be constant (white) over the passband of the filter, so a nearly white process which satisfies this criterion is quite satisfactory andrealizable.

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

${r}_{YY}(\tau )=({r}_{XX}(\tau ), h(-\tau ), h(\tau ))=({P}_{X}\delta (\tau ), h(-\tau ), h(\tau ))=({P}_{X}h(-\tau ), h(\tau ))$
where $h(\tau )$ is the impulse response of the filter.

This Figure from previous discussion shows two examples of coloured noise, although the upper waveform is more 'nearlywhite' 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 colouredwaveforms were produced by passing uncorrelated random noise samples (white up to half the sampling frequency) throughhalf-sine filters (as in this equation from our discussion of Random Signals) of length ${T}_{b}=10$ and $50$ samples respectively.

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