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# Introduction to Stochastic Processes

Summary: Describes signals that cannot be precisely characterized.

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## 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 .
X t ω for all tRωΩ X t ω  for all  t ω Ω
(1)

### Example 1

Received signal at an antenna

For a given t, 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 Pr X t b Pr 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

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

### 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 Pr X t 1 b 1 X t N b N
(4)

Definition 3: Nth-order stationary
A random process is stationary of order N 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)
Definition 4: Strictly stationary
A process is strictly stationary if it is Nth order stationary for all N.

### Example 2

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

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

X t =cos2π f 0 t+Θ X t 2 π f 0 t Θ
(10)
Pr X t 2 b 2 | X t 1 = x 1 =???? Pr X t 2 b 2 | X t 1 = x 1 ????
(11)
( X t 1 = x 1 =cos2π f 0 t+Θ)(Θ=arccos x 1 2π f 0 t) X t 1 x 1 2 π f 0 t Θ Θ x 1 2 π 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 π f 0 t 2 x 1 2 π f 0 t 1 2 π f 0 t 2 t 1 x 1
(13)

F X t 2 , X t 1 b 2 b 1 = b 1 Pr( X t 2 b 2 )|( X t 1 = x 1 ) f X t 1 x 1 d x 1 F X t 2 , X t 1 b 2 b 1 x 1 b 1 Pr X t 2 b 2 | X t 1 x 1 f X t 1 x 1
(14)

### Example 3

Every T 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)
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 2 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 2 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 n.
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 )(nT t 1 , t 2 <(n+1)T) p X t 1 x 1   if  ( x 2 = x 1 )(nT t 1 , t 2 <(n+1)T) p X t 2 x 2 p X t 1 x 1   otherwise   p X t 2 X t 1 x 2 x 1 0 x 2 x 1 n T t 1 , t 2 n 1 T p X t 1 x 1 x 2 x 1 n T t 1 , t 2 n 1 T p X t 2 x 2 p X t 1 x 1
(19)

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