Post on 19-Dec-2015
copyright Robert J. Marks II
ECE 5345Random Processes - Example Random Processes
copyright Robert J. Marks II
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
copyright Robert J. Marks II
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?
copyright Robert J. Marks II
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
copyright Robert J. Marks II
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.
copyright Robert J. Marks II
Example RP’sMultiple Random Process Examples Example
X(t) = cos(t+), Y(t) = sin(t+),
Both are zero mean.Cross Correlation=?
p.338
copyright Robert J. Marks II
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
copyright Robert J. Marks II
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
copyright Robert J. Marks II
Autocovariance of Sum Processes
X[k]’s are iid.
Autocovariance=?
n
kn ]k[XS
1
Xn]S[E n
)Xvar(n]Svar[ n
copyright Robert J. Marks II
Autocovariance of Sum Processes
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(
copyright Robert J. Marks II
Autocovariance of Sum Processes
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
copyright Robert J. Marks II
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
copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process (cont) The Poisson approximation: For k big and p small…
!
)/(
!
)(]points Pr[
/
k
Tnte
k
npeqp
k
nk
kTnt
knpknk
copyright Robert J. Marks II
Continuous Random Processes
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…
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
Poisson Counting Process
Poisson Points
!k
)t(e]k)t(XPr[
kt
)t(X
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
Autocovariance of a Poisson sum process
),tmin(
t),tmin(t
)(XE)t(XE),t(R),t(C XX
2
copyright Robert J. Marks II
Continuous Random Processes
Other RP’s related to the Poisson process Random telegraph signal
)t(X
Poisson Points
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal
|t|X e),t(C 2
||2)]([ tetXE
PROOF…
copyright Robert J. Marks II
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
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t>0,
te
tte
,t)(
t
t
cosh
...!4
)(
!2
)(1
even is 0on points ofnumber Pr42
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t>0.
Similarly…
)sinh(
...!5
)(
!3
)(
odd is 0on points ofnumber Pr
53
te
ttte
,t)(
t
t
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t>0.
Thus
0;
)sinh()cosh(
odd is 0on points ofnumber Pr
even is 0on points ofnumber Pr
]1)(Pr[)1(1)(Pr1)]([
2
te
tte
,t)(
,t)(
tXtXtXE
t
t
||2)]([ tetXE For all t…
copyright Robert J. Marks II
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
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t > ,
)(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…
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t > ,
)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
copyright Robert J. Marks II
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
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For 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
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For 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
copyright Robert J. Marks II
Poisson Random Processes Random telegraph signal. For t > .
)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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
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
copyright Robert J. Marks II
Continuous Random Processes
Wiener Processes: =1
copyright Robert J. Marks II
Continuous Random Processes
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