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Second-order description

Practical and incomplete statistics

The mean function of a random process X t is defined as the expected value of X t for all t 's.
μ X t X t x x f X t x continuous k x k p X t x k discrete
The autocorrelation function of the random process X t is defined as
R X t 2 t 1 X t 2 X t 1 x 2 x 1 x 2 x 1 f X t 2 X t 1 x 2 x 1 continuous k l x l x k p X t 2 X t 1 x l x k discrete


If X t is second-order stationary, then R X t 2 t 1 only depends on t 2 t 1 .

R X t 2 t 1 X t 2 X t 1 x 1 x 2 x 2 x 1 f X t 2 X t 1 x 2 x 1
R X t 2 t 1 x 1 x 2 x 2 x 1 f X t 2 - t 1 X 0 x 2 x 1 R X t 2 t 1 0

If R X t 2 t 1 depends on t 2 t 1 only, then we will represent the autocorrelation with only one variable τ t 2 t 1

R X τ R X t 2 t 1 R X t 2 t 1


  1. R X 0 0
  2. R X τ R X τ
  3. R X τ R X 0

X t 2 f 0 t Θ ω and Θ is uniformly distributed between 0 and 2 . The mean function

μ X t X t 2 f 0 t Θ θ 0 2 2 f 0 t θ 1 2 0

The autocorrelation function

R X t τ t X t + τ X t 2 f 0 t τ Θ 2 f 0 t Θ 1 2 2 f 0 τ 1 2 2 f 0 2 t τ 2 Θ 1 2 2 f 0 τ 1 2 θ 0 2 2 f 0 2 t τ 2 θ 1 2 1 2 2 f 0 τ
Not a function of t since the second term in the right hand side of the equality in [link] is zero.

Toss a fair coin every T seconds. Since X t is a discrete valued random process, the statistical characteristics can be captured by the pmf and the mean function is written as

μ X t X t 1 2 -1 1 2 1 0
R X t 2 t 1 k k l l x k x l p X t 2 X t 1 x k x l 1 1 1 2 -1 -1 1 2 1
when n T t 1 n 1 T and n T t 2 n 1 T
R X t 2 t 1 1 1 1 4 -1 -1 1 4 -1 1 1 4 1 -1 1 4 0
when n T t 1 n 1 T and m T t 2 m 1 T with n m
R X t 2 t 1 1 n T t 1 n 1 T n T t 2 n 1 T 0
A function of t 1 and t 2 .

Wide Sense Stationary
A process is said to be wide sense stationary if μ X is constant and R X t 2 t 1 is only a function of t 2 t 1 .

If X t is strictly stationary, then it is wide sense stationary. The converse is not necessarily true.

Autocovariance of a random process is defined as
C X t 2 t 1 X t 2 μ X t 2 X t 1 μ X t 1 R X t 2 t 1 μ X t 2 μ X t 1

The variance of X t is Var X t C X t t

Two processes defined on one experiment ( [link] ).

The crosscorrelation function of a pair of random processes is defined as
R X Y t 2 t 1 X t 2 Y t 1 y x x y f X t 2 Y t 1 x y
C X Y t 2 t 1 R X Y t 2 t 1 μ X t 2 μ Y t 1
Jointly Wide Sense Stationary
The random processes X t and Y t are said to be jointly wide sense stationary if R X Y t 2 t 1 is a function of t 2 t 1 only and μ X t and μ Y t are constant.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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