Elements Of Stochastic Processes

Stochastic Processes

Elements of Stochastic ProcessesBy

Mahdi Malaki

Outline

Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

Basic Definitions Suppose a set of random variables

indexed by a parameter Tracking these variables with respect to

the parameter constructs a process that is called Stochastic Process.

i.e. The mapping of outcomes to the real (complex) numbers changes with respect to index.

,...,...,,, 21 nXXXtX

Basic Definitions (cont’d)

In a random process, we will have a family of functions called an ensemble of functionsensemble of functions

t

1,tX

t

2,tX

t

3,tX

Basic Definitions (cont’d)

With fixed “beta”, we will have a “time” “time” function called sample path.

Sometimes stochastic properties of a random process can be extracted just from a single sample path. (When?)

Basic Definitions (cont’d)

With fixed “t”, we will have a random variable.

With fixed “t” and “beta”, we will have a real (complex) number.

Basic Definitions (cont’d)

Example I Voltage of a generator with fixed frequency

Amplitude is a random variables

tAtV cos.,

Basic Definitions (cont’d)

Equality Ensembles should be equal for each “beta”

and “t”

Equality (Mean Square Sense) If the following equality holds

Sufficient in many applications

,, tYtX

0,, 2 tYtXE

Basic Definitions (cont’d)

First-Order CDF of a random process

First-Order PDF of a random process

xtXtxFX Pr,

txFx

txf XX ,,

Basic Definitions (cont’d)

Second-Order CDF of a random process

Second-Order PDF of a random process

22112121 Pr,;, xtXandxtXttxxFX

212121

2

2121 ,;,.

,;, ttxxFxx

ttxxf XX

Basic Definitions (cont’d)

nth order can be defined. (How?)

Relation between first-order and second-order can be presented as

Relation between different orders can be obtained easily. (How?)

ttxftxf XX ,;,,

Basic Definitions (cont’d)

Mean of a random process

Autocorrelation of a random process

Fact: (Why?)

dxtxfxtXEt X ,.

212121212121 ,;,..., dxdxttxxfxxtXtXEttR X

2)(

2)(, tXtXttR

Basic Definitions (cont’d)

Autocovariance of a random process

Correlation Coefficient

Example

2211

212121

.

.,,

ttXttXE

ttttRttC

2221112

21 ,,2, ttRttRttRtXtXE

2211

2121

,.,

,,

ttCttC

ttCttr

Basic Definitions (cont’d)

Example

Poisson Process

Mean

Autocorrelation

Autocovariance

kt

tk

etK .

!

.

tt .

21212

1

21212

221

...

...,

ttttt

tttttttR

2121 ,min., ttttC

Basic Definitions (cont’d)

Complex process Definition

Specified in terms of the joint statistics of two real processes and

Vector Process A family of some stochastic processes

tYitXtZ .

tX tY

Basic Definitions (cont’d)

Cross-Correlation

Orthogonal Processes

Cross-Covariance

Uncorrelated Processes

2121 ., tYtXEttRXY

212121 .,, ttttRttC YXXYXY

0, 21 ttRXY

0, 21 ttCXY

Basic Definitions (cont’d)

aa-dependent processes

White Noise

Variance of Stochastic Process

0, 2121 ttCatt

0, 2121 ttCtt

tttC X2,

Basic Definitions (cont’d)

Existence Theorem For an arbitrary mean function For an arbitrary covariance function

There exist a normal random process that its mean is and its covariance is

t

t

21, ttC

21, ttC

Outline

Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

Stationary/Ergodic Processes Strict Sense Stationary (SSS)

Statistical properties are invariant to shift of time origin

First order properties should be independent of “t” or

Second order properties should depends only on difference of times or

…

xftxf XX ,

;,,;, 21212112 xxfttxxftt XX

Stationary/Ergodic Processes (cont’d)

Wide Sense Stationary (WSS) Mean is constant

Autocorrelation depends on the difference of times

First and Second order statistics are usually enough in applications.

tXE

RtXtXE .

Stationary/Ergodic Processes (cont’d)

Autocovariance of a WSS process

Correlation Coefficient

2 RC

0CC

r

Stationary/Ergodic Processes (cont’d)

White Noise

If white noise is an stationary process, why do we call it “noise”? (maybe it is not stationary !?)

aa-dependent Process

aa is called “Correlation Time”

.qC

aC 0

Stationary/Ergodic Processes (cont’d)

Example SSS

Suppose aa and bb are normal random variables with zero mean.

WSS Suppose “ ” has a uniform distribution in

the interval

tbtatX sin.cos.

tatX cos.

,

Stationary/Ergodic Processes (cont’d)

Example Suppose for a WSS process X(8) and X(5) are random variables

eAR .

3200

8.525858 222

RRR

XXEXEXEXXE

Stationary/Ergodic Processes (cont’d)

Ergodic Process Equality of timetime properties and statisticstatistic properties.

First-Order Time average Defined as

Mean Ergodic ProcessMean Ergodic Process

Mean Ergodic Process in Mean Square SenseMean Ergodic Process in Mean Square Sense

T

TTt dttX

TtXtXE

2

1lim

tXEtXE

02

1.1

lim02

0

2

T

XTt dT

CT

tXE

Stationary/Ergodic Processes (cont’d)

Slutsky’s TheoremSlutsky’s Theorem A process X(t) is mean-ergodicmean-ergodic iff

Sufficient Conditions

a)

b)

01

lim0

T

T dCT

0

dC

0lim CT

Outline

Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

Stochastic Analysis of Systems Linear Systems

Time-Invariant Systems

Linear Time-Invariant Systems

Where h(t) is called impulse responseimpulse response of the system

babtxaSystembtyatxSystemty ,..

txSystemtytxSystemty

txthtytxSystemty *

Stochastic Analysis of Systems (cont’d)

Memoryless Systems

Causal Systems

Only causal systems can be realized. (Why?)

00 tttxSystemty

00 tttxSystemty

Stochastic Analysis of Systems (cont’d)

Linear time-invariant systems Mean

Autocorrelation

txEthtxthEtyE **

2*21121 *,*, thttRthttR xxyy

Stochastic Analysis of Systems (cont’d)

Example I System:

Impulse response:

Output Mean:

Output Autocovariance:

t

dxty0

).(

tUdtttht

0

).(

t

dttxEtyE0

.

0 0

2121 ..,, ddttRttR xxyy

Stochastic Analysis of Systems (cont’d)

Example II System:

Impulse response:

Output Mean:

Output Autocovariance:

txdt

dty

tdt

dth

txEdt

dtyE

2121

2

21 ,.

, ttRtt

ttR xxyy

Outline

Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

Power Spectrum Definition

WSS process Autocorrelation

Fourier Transform of autocorrelation

deRS j.

tX R

Power Spectrum (cont’d)

Inverse trnasform

For real processes

deSR j.2

1

0

0

.cos.1

.cos.2

dSR

dSS

Power Spectrum (cont’d)

For a linear time invariant system

Fact (Why?)

xx

xxyy

SH

HSHS

.

..

2

*

dHStyVar xx2

.2

1

Power Spectrum (cont’d)

Example I (Moving Average) System

Impulse Response

Power Spectrum

Autocorrelation

Tt

Tt

dxT

ty .2

1

.

.sin

T

TH

22

2

.

.sin

T

TSS xxyy

T

Txxyy dR

TTR

2

2

..2

12

1

Power Spectrum (cont’d)

Example II

System

Impulse Response

Power Spectrum

txdt

dty

.iH

xxyy SS .2

Question?Question?