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

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Principles of digital communications' conversation and receive update notifications?

Ask

	Precalculus By OpenStax Read Online Course
	Interventionism Mixed Economy By Robert Murphy Start Test
	27 AP Key Terms 27 The Reproductive System By OpenStax Start Key Terms
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz
	42 Biology 42 The Immune System MCQ By OpenStax Start Quiz
	Introduction to sociology 2e By OpenStax Read Online Course
	6 Microbiology Midterm Practice By Madison Christian Start Test
©flickr:	Biology 1 By Jill Zerressen Start Quiz
	18 Biology 18 Evolution the Origin of Species MCQ By OpenStax Start Quiz
	5 AP 05 Integumentary System MCQ By OpenStax Start Quiz