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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 τ 0 , i.e. it is of the form:

r X X τ P X δ τ
where P X is a constant.

The PSD of X is then given by

S X ω τ P X δ τ ω τ P X ω 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:

Power of X 1 2 ω 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 X X τ is a narrow pulse of non-zero width, and S X 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 t 1 t 2 t N to sample X t .

If, for any choice of t 1 t 2 t N with N finite, the random variables X t 1 , X t 2 , X t N are jointly independent , i.e. their joint pdf is given by

f X ( t 1 ) , X ( t 2 ) ,     X ( t N ) x 1 x 2 x N i 1 N f X ( t i ) x i
and the marginal pdfs are identical, i.e.
f X ( t 1 ) f X ( t 2 ) f X ( t N ) 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 ω simply by passing white (or nearly white) noise X t with PSD P X through a filter with frequency response ω , such that from this equation from our discussion of Spectral Properties of Random Signals.

S Y ω S X ω ω 2 P X ω 2
Hence if we design the filter such that
ω S Y ω P X
then Y t will have the required coloured PSD.

For this to work, S Y ω 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 Y Y τ r X X τ h τ h τ P X δ τ h τ h τ P X h τ h τ
where 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 '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.

Questions & Answers

what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
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LITNING
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Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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Tarell
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Source:  OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3
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