copyright robert j. marks ii ece 5345 random processes - example random processes
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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?
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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
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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.
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Example RP’sMultiple Random Process Examples Example
X(t) = cos(t+), Y(t) = sin(t+),
Both are zero mean.Cross Correlation=?
p.338
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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
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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
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Autocovariance of Sum Processes
X[k]’s are iid.
Autocovariance=?
n
kn ]k[XS
1
Xn]S[E n
)Xvar(n]Svar[ n
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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(
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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
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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
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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
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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…
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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
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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
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Continuous Random Processes
Poisson Counting Process
Poisson Points
!k
)t(e]k)t(XPr[
kt
)t(X
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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
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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
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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
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Continuous Random Processes
Autocovariance of a Poisson sum process
),tmin(
t),tmin(t
)(XE)t(XE),t(R),t(C XX
2
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Continuous Random Processes
Other RP’s related to the Poisson process Random telegraph signal
)t(X
Poisson Points
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Poisson Random Processes Random telegraph signal
|t|X e),t(C 2
||2)]([ tetXE
PROOF…
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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
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Poisson Random Processes Random telegraph signal. For t>0,
te
tte
,t)(
t
t
cosh
...!4
)(
!2
)(1
even is 0on points ofnumber Pr42
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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
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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…
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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
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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…
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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
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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
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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
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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
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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
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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
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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
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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
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Continuous Random Processes
Wiener Processes: =1
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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