<< Chapter < Page
  Random processes   Page 1 / 1
Chapter >> Page >
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

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 ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
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
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Random processes' conversation and receive update notifications?

Ask