Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Digital Communication Systems » Introduction to Stochastic Processes

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

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 .
X t ω  ,   tR    t t X t ω
(1)

Example 1

Received signal at an antenna as in Figure 1.

Figure 1
Figure 1 (Figure3-1.png)

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.

Figure 2
Figure 2 (Figure3-3a.png)

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)

Figure 3
Figure 3 (Figure3-3b.png)

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 .

Figure 4
Figure 4 (Figure3-4.png)

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)

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks