Introduction to Stochastic Processes2.142001/08/032005/01/05 22:24:24.026 US/CentralBehnaamAazhangaaz@ece.rice.eduDineshRajandinesh@ece.rice.eduMohammadJaberBorranmohammad@ece.rice.eduRoyHarha@rice.eduShawnStewartmrshawn@alumni.rice.eduBehnaamAazhangaaz@ece.rice.edudistributionrandom processstationarystochastic processDescribes signals that cannot be precisely characterized.Definitions, distributions, and stationarityStochastic Process
Given a sample space, a stochastic process is an indexed collection
of random variables defined for each
ωΩ.
ttXtω
Received signal at an antenna as in .
For a given t,
Xtω
is a random variable with a distribution
First-order distributionFXtbXtbωΩXtωbFirst-order stationary process
If
FXtb
is not a function of time then
Xt
is called a first-order stationary process.
Second-order distributionFXt1,Xt2b1b2Xt1b1Xt2b2
for all
t1,
t2,
b1,
b2Nth-order distributionFXt1,Xt2,…,XtNb1b2…bNXt1b1…XtNbNNth-order stationary : A
random process is stationary of order
N if
FXt1,Xt2,…,XtNb1b2…bNFXt1+T,Xt2+T,…,XtN+Tb1b2…bN
Strictly stationary : A process is strictly stationary if it
is Nth order stationary for all
N.
Xt2f0tΘω
where
f0
is the deterministic carrier frequency and
Θω:→Ω
is a random variable defined over
and is assumed to be a uniform random variable;
i.e.,
fΘθ12θ0FXtbXtb2f0tΘbFXtb2f0tΘbb2f0tΘFXtbθ2f0tb2f0t12θb2f0t2f0t1222b12fXtxx11x11x2x10
This process is stationary of order 1.
The second order stationarity can be determined by first considering
conditional densities and the joint density. Recall that
Xt2f0tΘ
Then the relevant step is to find
Xt1x1Xt2b2
Note that
Xt1x12f0tΘΘx12f0tXt22f0t2x12f0t12f0t2t1x1FXt2,Xt1b2b1x1b1fXt1x1Xt1x1Xt2b2
Note that this is only a function of
t2t1.
Every T seconds, a fair coin
is tossed. If heads, then
Xt1
for
nTtn1T.
If tails, then
Xt-1
for
nTtn1T.
pXtx12x112x-1
for all
t.
Xt
is stationary of order 1.
Second order probability mass function
pXt1Xt2x1x2pXt2|Xt1x2|x1pXt1x1
The conditional pmf
pXt2|Xt1x2|x10x2x11x2x1
when
nTt1n1T
and
nTt2n1T
for some n.
pXt2|Xt1x2|x1pXt2x2
for all
x1
and for all
x2
when
nTt1n1T
and
mTt2m1T
with
nmpXt2Xt1x2x10x2x1for nTt1,t2n1TpXt1x1x2x1for nTt1,t2n1TpXt1x1pXt2x2nmfor nTt1n1TmTt2m1T