copyright robert j. marks ii ece 5345 random processes - example random processes

copyright Robert J. Marks II

ECE 5345Random Processes - Example Random Processes


Example RP’sExample Random Processes GaussianRecall Gaussian pdf

Let Xk=X(tk) , 1 k n. Then if, for all n, the corresponding pdf’s are Gaussian, then the RP is Gaussian.

The Gaussian RP is a useful model in signal processing.

)(K)(

2

1

2/12/

1

|K|2

1)(

mxmx

nX

T

exf


Flip TheoremLet A take on values of +1 and -1 with equal probability

Let X(t) have mean m(t) and autocorrelation RX

Let Y(t)=AX(t)

Then Y(t) has mean zero and autocorrelation RX

What about the autocovariances?


Multiple RP’sX(t) & Y(t)

Independence

(X(t1), X(t2), …, X(tk ))

is independent to

(Y(1), Y( 2), …, Y( j ))

…for All choices of k and j and

all sample locations


Multiple RP’sX(t) & Y(t)

Cross Correlation

RXY(t, )=E[X(t)Y()] Cross-Covariance

CXY(t, )= RXY(t, ) - E[X(t)] E[Y()]

Orthogonal: RXY(t, ) = 0

Uncorrelated: CXY(t, ) = 0 Note: Independent Uncorrelated, but not

the converse.


Example RP’sMultiple Random Process Examples Example

X(t) = cos(t+), Y(t) = sin(t+),

Both are zero mean.Cross Correlation=?

p.338


Example RP’sMultiple Random Process Examples Signal + Noise

X(t) = signal, N(t) = noise

Y(t) = X(t) + N(t)

If X & N are independent,RXY=? p.338

Note: also, var Y = var X + var N

Nvar

XvarSNR


Example RP’sMultiple Random Process Examples (cont) Discrete time RP’s

X[n]MeanVarianceAutocorrelationAutocovariance

Discrete time i.i.d. RP’s Bernoulli RP’s Binomial RP’s p.340

Binary vs. Bipolar Random Walk p.341-2


Autocovariance of Sum Processes

X[k]’s are iid.

Autocovariance=?

n

kn ]k[XS

1

Xn]S[E n

)Xvar(n]Svar[ n



When i=j, the answer is var(X). Otherwise, zero.How many cases are there where i = j?

k

jj

n

ii

kn

kknnS

XXXXE

XkSXnSE

SSSSEknC

11

)()(

))((

))((),(

)Xvar()k,nmin()k,n(CS )k,nmin(



For Bernoulli sum process,

For Bipolar case

pq)k,nmin()k,n(CS

pq)Xvar(

pq)Xvar( 4

pq)k,nmin()k,n(CS 4


Continuous Random Processes

Poisson Random Process Place n points randomly on line of length T

T

tp;qp

k

n]sintpokPr[ knk

T

t

Choose any subinterval of length t.

The probability of finding k points on the subinterval is



Poisson Random Process (cont) The Poisson approximation: For k big and p small…

!

)/(

!

)(]points Pr[

/

k

Tnte

k

npeqp

k

nk

kTnt

knpknk



The Poisson Approximation… For n big and p small (implies k << n since p k/n<<1)

!

)(

k

npeqp

k

n knpknk

!!

)1)...(2)(1(

)!(!

!

k

n

k

knnnn

knk

n

k

n k

npnknkn eppq )()1()1(

Here’s why…



Poisson Random Process (cont)

Let n such that =n/T = frequency of points remains constant.

!

)(] intervalon points Pr[

k

tetk

kt

!

)/(

!

)(]points Pr[

/

k

Tnte

k

npeqp

k

nk

kTnt

knpknk



Poisson Random Process (cont)

This is a Poisson process with parameter

occurrences per unit time Examples: Modeling

Popcorn Rain (Both in space and time) Passing cars Shot noise Packet arrival times

!k

)t(e]tkPr[

kt intervalon points

t



Poisson Counting Process

Poisson Points

!k

)t(e]k)t(XPr[

kt

)t(X



Recall for Poisson RV with parameter a

Poisson Counting Process Expected Value is thus

t)]t(X[E

!k

)a(e]kXPr[

ka a)Xvar(X



The Poisson Counting Process is independent increment process. Thus, for t and j i,

)!(

)(

!

])()(Pr[])(Pr[

])()(,)(Pr[

])(,)(Pr[

)(

ij

et

i

et

ijtXXitX

ijtXXitX

jXitX

tijti



Autocorrelation: If > t

tt

tt)t(t

)t(XE)t(X)(XE)t(XE

)t(XE)t(X)(X)t(XE

)t(X)t(X)(X)t(XE

)(X)t(XE),t(RX

2

2

2

2

2

),tmin(t),t(RX 2

t



Autocovariance of a Poisson sum process

),tmin(

t),tmin(t

)(XE)t(XE),t(R),t(C XX

2



Other RP’s related to the Poisson process Random telegraph signal

)t(X

Poisson Points


Poisson Random Processes Random telegraph signal

|t|X e),t(C 2

||2)]([ tetXE

PROOF…


Poisson Random Processes Random telegraph signal. For t>0,

odd is 0on points ofnumber Pr

even is 0on points ofnumber Pr

]1)(Pr[)1(1)(Pr1)]([

,t)(

,t)(

tXtXtXE


Poisson Random Processes Random telegraph signal. For t>0,

te

tte

,t)(

t

t

cosh

...!4

)(

!2

)(1

even is 0on points ofnumber Pr42


Poisson Random Processes Random telegraph signal. For t>0.

Similarly…

)sinh(

...!5

)(

!3

)(


53

te

ttte

,t)(

t

t


Poisson Random Processes Random telegraph signal. For t>0.

Thus

0;

)sinh()cosh(


even is 0on points ofnumber Pr

]1)(Pr[)1(1)(Pr1)]([

2

te

tte

,t)(

,t)(

tXtXtXE

t

t

||2)]([ tetXE For all t…


Poisson Random Processes Random telegraph signal. For t > ,

X(t)

X()

-1

1

1-1

1)(Pr1)(|1)(Pr

1)(Pr1)(|1)(Pr

1)(,1)(Pr

1)(,1)(Pr]1)()(Pr[

XXtX

XXtX

XtX

XtXXtX



)(cosh

even is ),(on points ofnumber Pr

1)(|1)(Pr

1)(|1)(Pr

)(

te

t

XtX

XtX

t

eet

XXtX

XtX

t )cosh()(cosh

1)(Pr1)(|1)(Pr

1)(,1)(Pr

)(

Thus…



)cosh()(cosh

1)(Pr1)(|1)(Pr

1)(,1)(Pr

tet

XXtX

XtX

And…

)sinh()(cosh

1)(Pr1)(|1)(Pr

1)(,1)(Pr

tet

XXtX

XtX

X(t)

X()

-1

1

1-1


Poisson Random Processes Random telegraph signal. For t > .

Onward…

)(sinh

odd is ),(on points ofnumber Pr

1)(|1)(Pr

1)(|1)(Pr

)(

te

t

XtX

XtX

t



)cosh()(sinh

1)(Pr1)(|1)(Pr

1)(,1)(Pr

tet

XXtX

XtX

And…

)sinh()(sinh

1)(Pr1)(|1)(Pr

1)(,1)(Pr

tet

XXtX

XtX

X(t)

X()

-1

1

1-1



)sinh()(cosh)cosh()(sinh

)sinh()(sinh)cosh()(cosh

1)()(Pr11)()(Pr1

)()(),(

tt

tt

X

etet

etet

XtXXtX

XtXEtR

X(t)

X()

-1

1

1-1

In general… ||2)()(),(),( tXX eXtXtRtC



Other RP’s related to the Poisson process Poisson point process, Z(t)

Let X(t) be a Poisson sum process. Then

pp.352

)St()t(Xdt

d)t(Z n

n

)t(Z

Poisson Points



Other RP’s related to the Poisson process Shot Noise, V(t)

Z(t) V(t)

pp.352

)St(h)t(V nn

Poisson Points

h(t)

)t(V



Wiener Process Assume bipolar Bernoulli sum process with jump

bilateral height h and time interval E[X(t)]=0; Var X(n) = 4npqh 2 = nh 2 Take limit as h 0 and 0 keeping = h 2 /

constant and t = n . Then Var X(t) t By the central limit theorem, X(t) is Gaussian with zero

mean and Var X(t) = t

We could use any zero mean process to generate the Wiener process.

p.355



Wiener Processes: =1



Wiener processes in financeS= Price of a Security. = inflationary force. If there is no risk…interest earned is proportional to investment.

Solution isWith “volatility” , we have the most commonly used model in finance for a security:

V(t) is a Wiener process.

Sdt

dSdt)t(S)t(dS

teS)t(S 0

)t(dV)t(Sdt)t(S)t(dS

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