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Course by: Laurence Riddle. E-mail the author

# Introduction to Stochastic Processes

Module by: Behnaam Aazhang. E-mail the author

Summary: Describes signals that cannot be precisely characterized.

## Definitions, distributions, and stationarity

Definition 1: Stochastic Process
Given a sample space, a stochastic process is an indexed collection of random variables defined for each ωΩ ω Ω .
t,tR: X t ω t t X t ω
(1)

### Example 1

Received signal at an antenna as in Figure 1.

For a given tt, X t ω X t ω is a random variable with a distribution

### First-order distribution

F X t b=Pr X t b=Pr ωΩ X t ωb F X t b X t b ω Ω X t ω b
(2)

Definition 2: First-order stationary process
If F X t b F X t b is not a function of time then X t X t is called a first-order stationary process.

### Second-order distribution

F X t 1 , X t 2 b 1 b 2 =Pr X t 1 b 1 X t 2 b 2 F X t 1 , X t 2 b 1 b 2 X t 1 b 1 X t 2 b 2
(3)
for all t 1 R t 1 , t 2 R t 2 , b 1 R b 1 , b 2 R b 2

### Nth-order distribution

F X t 1 , X t 2 , , X t N b 1 b 2 b N =Pr X t 1 b 1 X t N b N F X t 1 , X t 2 , , X t N b 1 b 2 b N X t 1 b 1 X t N b N
(4)

NNth-order stationary : A random process is stationary of order NN if

F X t 1 , X t 2 , , X t N b 1 b 2 b N = F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N F X t 1 , X t 2 , , X t N b 1 b 2 b N F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N
(5)

Strictly stationary : A process is strictly stationary if it is NNth order stationary for all NN.

### Example 2

X t =cos2π f 0 t+Θω X t 2 f 0 t Θ ω where f 0 f 0 is the deterministic carrier frequency and Θω : ΩR Θ ω : Ω is a random variable defined over π π and is assumed to be a uniform random variable; i.e., f Θ θ={12π  if  θ π π 0  otherwise   f Θ θ 1 2 θ 0

F X t b=Pr X t b=Prcos2π f 0 t+Θb F X t b X t b 2 f 0 t Θ b
(6)
F X t b=Prπ2π f 0 t+Θarccosb+Prarccosb2π f 0 t+Θπ F X t b 2 f 0 t Θ b b 2 f 0 t Θ
(7)
F X t b=(π)2π f 0 t(arccosb)2π f 0 t12πd θ +arccosb2π f 0 tπ2π f 0 t12πd θ =(2π2arccosb)12π F X t b θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2
(8)
f X t x=dd x 11πarccosx={1π1x2  if  |x|10  otherwise   f X t x x 1 1 x 1 1 x 2 x 1 0
(9)
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

X t =cos2π f 0 t+Θ X t 2 f 0 t Θ
(10)
Then the relevant step is to find
Pr X t 2 b 2 | X t 1 = x 1 X t 1 x 1 X t 2 b 2
(11)
Note that
( X t 1 = x 1 =cos2π f 0 t+Θ)(Θ=arccos x 1 2π f 0 t) X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t
(12)
X t 2 =cos2π f 0 t 2 +arccos x 1 2π f 0 t 1 =cos2π f 0 ( t 2 t 1 )+arccos x 1 X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1
(13)

F X t 2 , X t 1 b 2 b 1 = b 1 f X t 1 x 1 Pr X t 2 b 2 | X t 1 = x 1 d x 1 F X t 2 , X t 1 b 2 b 1 x 1 b 1 f X t 1 x 1 X t 1 x 1 X t 2 b 2
(14)
Note that this is only a function of t 2 t 1 t 2 t 1 .

### Example 3

Every TT seconds, a fair coin is tossed. If heads, then X t =1 X t 1 for nTt<(n+1)T n T t n 1 T . If tails, then X t =-1 X t -1 for nTt<(n+1)T n T t n 1 T .

p X t x={12  if  x=112  if  x=-1 p X t x 1 2 x 1 1 2 x -1
(15)
for all tR t . X t X t is stationary of order 1.

Second order probability mass function

p X t 1 X t 2 x 1 x 2 = p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 p X t 1 X t 2 x 1 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1
(16)

The conditional pmf

p X t 2 | X t 1 x 2 | x 1 ={0  if   x 2 x 1 1  if   x 2 = x 1 p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1
(17)
when nT t 1 <(n+1)T n T t 1 n 1 T and nT t 2 <(n+1)T n T t 2 n 1 T for some nn.
p X t 2 | X t 1 x 2 | x 1 = p X t 2 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2
(18)
for all x 1 x 1 and for all x 2 x 2 when nT t 1 <(n+1)T n T t 1 n 1 T and mT t 2 <(m+1)T m T t 2 m 1 T with nm n m
p X t 2 X t 1 x 2 x 1 ={0  if   x 2 x 1 for  nT t 1 , t 2 <(n+1)T p X t 1 x 1   if   x 2 = x 1 for  nT t 1 , t 2 <(n+1)T p X t 1 x 1 p X t 2 x 2   if   nm for  (nT t 1 <(n+1)T)(mT t 2 <(m+1)T) p X t 2 X t 1 x 2 x 1 0 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 p X t 2 x 2 n m for  n T t 1 n 1 T m T t 2 m 1 T
(19)

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