2.8 Introduction to stochastic processes

Principles of digital communications Page 1 / 1

Definitions, distributions, and stationarity

Stochastic Process

Given a sample space, a stochastic process is an indexed collection of random variables defined for each

ω Ω

t t X_{t} ω

Received signal at an antenna as in [link] .

For a given $t$ , $X_{t} ω$ is a random variable with a distribution

First-order distribution

F_{X_{t}} b X_{t} b ω Ω X_{t} ω b

First-order stationary process

F_{X_{t}} b

is not a function of time then

X_{t}

is called a first-order stationary process.

Second-order distribution

F_{X_{t_{1}}, X_{t_{2}}} b_{1} b_{2} X_{t_{1}} b_{1} X_{t_{2}} b_{2}

for all

t_{1}

t_{2}

b_{1}

b_{2}

Nth-order distribution

F_{X_{t_{1}}, X_{t_{2}}, \dots, X_{t_{N}}} b_{1} b_{2} … b_{N} X_{t_{1}} b_{1} … X_{t_{N}} b_{N}

$N$ th-order stationary : A random process is stationary of order $N$ if

F_{X_{t_{1}}, X_{t_{2}}, \dots, X_{t_{N}}} b_{1} b_{2} … b_{N} F_{X_{t_{1} + T}, X_{t_{2} + T}, \dots, X_{t_{N} + T}} b_{1} b_{2} … b_{N}

Strictly stationary : A process is strictly stationary if it is $N$ th order stationary for all $N$ .

$X_{t} 2 f_{0} t Θ ω$ where $f_{0}$ is the deterministic carrier frequency and $Θ ω : \to Ω$ is a random variable defined over and is assumed to be a uniform random variable; i.e. , $f_{Θ} θ 12 θ 0$

F_{X_{t}} b X_{t} b 2 f_{0} t Θ b

F_{X_{t}} b 2 f_{0} t Θ b b 2 f_{0} t Θ

F_{X_{t}} b θ 2 f_{0} t b 2 f_{0} t 12 θ b 2 f_{0} t 2 f_{0} t 12 2 2 b 12

f_{X_{t}} x x 1 1 x 1 1 x 2 x 1 0

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} 2 f_{0} t Θ

Then the relevant step is to find

X_{t_{1}} x_{1} X_{t_{2}} b_{2}

Note that

X_{t_{1}} x_{1} 2 f_{0} t Θ Θ x_{1} 2 f_{0} t

X_{t_{2}} 2 f_{0} t_{2} x_{1} 2 f_{0} t_{1} 2 f_{0} t_{2} t_{1} 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

Note that this is only a function of

t_{2} t_{1}

Every $T$ seconds, a fair coin is tossed. If heads, then $X_{t} 1$ for $n T t n 1 T$ .If tails, then $X_{t} -1$ for $n T t n 1 T$ .

p_{X_{t}} x 12 x 1 12 x -1

for all

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}

The conditional pmf

p_{X_{t_{2}} | X_{t_{1}}} x_{2} | x_{1} 0 x_{2} x_{1} 1 x_{2} x_{1}

when

n T t_{1} n 1 T

and

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}

for all

x_{1}

and for all

x_{2}

when

n T t_{1} n 1 T

and

m T t_{2} m 1 T

with

n m

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

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Source: OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1

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