topic 5_intro to random processes

8/10/2019 Topic 5_Intro to Random Processes

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Topic 5: Stochastic/Random Processes Introduction(Read Chapter 9 and some sections in Chapter 10] in Papoulis) -2 weeks

A general random, or stochastic, process can be described as: Collection of time functions (signals) corresponding to various outcomes

of random experiments (events).

Collection of random variables observed at different times.

Rather than consider fixed random variablesX, Y, etc. or even

sequences of i.i.d random variables, we consider sequencesX0

,X1

,

X2, . WhereXtrepresent some random quantity at time t.

In general, the valueXtmight depend on the quantityXt-1at time t-1, oreven the valueXsfor other timess < t.

Examples of random processes in communications:

Channel noise,

Information generated by a source,

Interference.

Examples in other fields: Daily stream flow, hourly rainfall of storm

events , stock index,

Note: we will use the terms stochastic and random process interchangeably 1


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A RANDOM VARIABLEX, is a rule for assigning to everyoutcome, !, of an experiment a numberX(!).

Note:Xdenotes a random variable andX(!)denotes a particular value.

A RANDOM PROCESSX(t) is a rule for assigning to every !,afunctionX(t,!).

Note: for notational simplicity we often omit the dependence on !.

Random Processes

An example of a stochastic processX(t) is shown below

time

a sample path

a random variable for each fixed t

t

X(t)

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Random

Event

0

Random Process

t

t

1+n(t)

0+n(t)

1

Random

Process

0=!

1=!

( )tX ,!

Ensemble average = 0.5

Time average = 1

Time average = 0

A random processes can be either discrete-time or continuous-time.

A random processes can be either discrete valued or continuous valued.

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Classification of Stochastic Processes

Four classes of stochastic processes:

Discrete-state process!chain

Discrete-time process!stochastic sequence {Xn| n!T} (e.g.,probing a system every 10 ms.)

Continuous time and discrete state!quantized signal

Continuous time and continuous state!white noise


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Types of Random Processes

5


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Random Process for a Continuous Sample Space

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Stochastic/Random Processes Are Important

Most of the real life signals (observations) are contaminated by random

noise.

Noise contamination may lead to unpredictablechanges in the parameters

(e.g., amplitude and phase) of the signal.

Many information-bearing signals are random (the data bits are random).

Signals (processes)

Deterministicsignal.

One possible value for any time

instance. Therefore, we canpredict the exact value of the

signal for a desired time instance.

Random (stochastic):

Many (infinitely) possible values for

any time instance. Therefore, we canpredict only the expected value of

the signal for a desired time

instance.

Stock market, speech, medical

data, communication signals,


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Chapter 4aStochastic Processes(a bit more formally)

A deterministic model predicts a single outcome from a giveset of circumstances;

A stochastic model predicts a set of possible outcomes

weighted by their likelihood or probabilities.

A stochastic process is a special stochastic model that dealwith a class of random variables.

A stochastic process{X(t), t"T} is a collection of random

variablesX(t) indexed by t. That is, for each t"T, X(t) is a

random variable, with toften interpreted as time and the values

ofX(t) are referred to as statesof the process (especially if thevalues ofX(t) are discrete).

e.g., X(t) = the received signal voltage

X(t) = number of customers in a supermarket at time t;

X(t) = the quantity of a commodity in inventory at time t.8


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Chapter 4a

Stochastic Processes-2

The set Tis called the index set of the process. When Tis

countable, the stochastic process is said to be a discrete-timeprocess; if Tis an interval of the real line, it is said to be a

continuous-time process.

The set of possible values that the random variableX(t) can assume

is called the state space of the stochastic process.

If the state space is countable,X(t) is a discrete-state process.

Otherwise it is a continuous-state process.

Thus, a stochastic process is a family of random variables that

describes the evolution through time of some (physical) process.

Stochastic processes are classified by the type of state space, by the

index set T, and the dependence relations among the random variables

X(t) and their distributions --- giving the names such as Gaussian,

Markov,and Poisson Process.9


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Random Processes Recall that a random variable,X, is

a rule for assigning to every

outcome, !, of an experiment anumberX(!).

Note:Xdenotes a randomvariable andX(!)denotes a

particular value.

A random processX(t)is a rule for

assigning to every !,a functionX(!,t).

Note: for notational simplicitywe often omit the dependenceon !.

Analytical descriptionX(t) =X(t,!)

where !is an outcome of a randomevent.

Statistical description: for any

integer n and any choice of (t1,

t2, . . ., tn) the joint PDF of (X(t1), X

(t2

), . . ., X(tn

) ) is known. 10


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Example: Analytical Description ---Random Phase Sine Wave

Let X(t) =A cos(2"f0t +")where "is a random variable uniformlydistributed on [0,2#).Ais a fixed amplitude.

Complete statistical description ofX(to)is:

Let Y= 2"f0t +" [which is a uniform RV]

Then, we need to transform fromy tox:

pX(x)dx = pY(y1)dy + pY(y2)dy

We need bothy1andy2because for a givenxthe equationx=A

cos yhas two solutions in [0,2#).

Since

SincepYis uniform in [2#f0t, 2#f0t + 2#], we get

2 2sin

dxA y A x

dy= = !

( ) 2 21

0 elsewhere

X

A x Ap x A x!

"# <


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Example: Statistical Description---Gaussian Random Process

Suppose a random processX(t)has the property that forany nand (t0,t1, . . .,tn) the joint density function of {x(ti)}is a jointly distributed Gaussian vector with zero meanand covariance

This gives complete statistical description of the randomprocessX(t).

The Gaussian Random Process is unique in that the firstand second-order statistics completely describe theprocess.

( )2 min ,ij i jt t! !=

12

(5-2)


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Random Processes Basic Concepts

The univariate probability density function describes the general distribution of

the magnitude of the random process, but it gives no information on the time or

frequency content of the process.

We will use the multi-variate PDF/PMF, as well as other parameters such ascorrelation function and power spectral density to describe the Random Process.

fX(x)

time, t

x(t)

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Probability Distribution of a Random Process

For any stochastic process with index setI, its probability

distribution function is uniquely determined by its finite dimensional

distributions.

The kdimensional distribution function of a process is defined by

for any and any real numbersx1, ,xk.

The distribution function tells us everything we need to know about

the process {Xt}.

However, in many cases the joint distribution function is not

available and so other metrics are used to describe a random

process.

( ) kttkXX

xXxXPxxFkktt

!!= ,...,,...,11,..., 11

Ittk!,...,1

14

(5-3)


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Moments of Stochastic Process

We can describe a stochastic process via its moments, i.e.,

We often use the first two moments.

The mean function of the process is

The variance function of the process is

The covariance function betweenXt,Xsis

The correlation function betweenXt,Xsis

These moments are often functions of time, but not always

(!stationarity)

( ) ( ) etc.,, 2sttt

XXEXEXE !

( ) .tt

XE =

( ) .2ttXVar !=

( ) ( )( )( )ssttst

XXEXX !!=,Cov

( ) ( )

22

,Cov,

st

st

st

XXXX

!!

" =

15

(5-4c)

(5-4d)

(5-4b)

(5-4a)


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Example: Consider the complex RV

Then the second moment ofZis given by

Z = X(t)dt!T

T

" .

E[| Z |2]=E[ZZ*]= E{X(t1)X

*(t2 )}dt1 dt2

!T

T

"!T

T

"

= RXX(t1, t2 )dt1 dt2

!T

T

"!T

T

"

16

(5-5a)

(5-5b)


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Stationary Random Processes

Means of collecting statistics:

Sample records which are individual representations of the underlyingprocess.

Ensemble averaging: properties of the process are obtained by

averaging over a collection or ensembleof sample records using

values at corresponding times

Time averaging: properties are obtained by averaging over a singlerecord in time.

Stationary Processes ----ensemble averages do not vary with time

Ergodic process: stationary process in which averages from a single

record are the same as those obtained from averaging over the

ensemble.

Strictly stationarity: the joint PDF is time invariant

(Second-order)Wide-sense stationary (WSS): the first and second

moments are time invariant.

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Ergodic Random Process

(ergodic stationary)

Definition: A random process is ergodicif all time averages of any samplefunction are equal to the corresponding ensemble averages

Example, for ergodic processes, can use ensemble statistics to compute:

DC values

RMS values

Ergodic processes are always stationary; Stationary processes are not necessarilyergodic.

Example: X(t) =A sin(2"f0t +")

Aand !0are constants; "0is a uniformly distributed RV from [-#,#);tis time.

Mean (Ensemble statistics)

( ) ( ) ( )01

sin 02

xm x x f d A t d !

"!

" " " # " " !

$

%$ %= = = + =& &

( )2

2 2 2

0

1sin

2 2x

AA t d

!

!

" # $ $ !%

= + =&

18

(5-6a)

(5-6b)


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Example: Ergodic Process

Mean (Time Average) T is large

Variance

The ensemble and time averages are thesame, at least for the the momentsconsidered, so the process is ergodic

( ) ( )00

1sin 0lim

T

T

x t A t dtT

! "

#$

= + =%

( ) ( )2

2 2 2

00

1sin

2lim

T

T

Ax t A t dt

T! "

#$

= + =%

19

(5-7a)

(5-7b)


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Strictly Stationary Processes

A process is said to be strictly stationary if has the samejoint distribution as . That is, if

If {Xt} is a strictly stationary process and then, the mean

function is a constant and the variance function is also a constant.

Moreover, for a strictly stationary process with first two momentsfinite, the covariance function, and the correlation function dependonly on the time differences.

This means that the process behaves similarly (follows the same

PDF) regardless of when you measure it. Is the random process from the coin tossing experiment stationary?

ktt XX ,...,1

!+!+ ktt XX ,...,

1

( ) ( )kXXkXX

xxFxxFkttktt

,...,,...,1,...,1,...,

11 !+!+=

!

topic 5_intro to random processes

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