Stochastic Processes - SHARIF UNIVERSITY OF...

Hamid R. Rabiee

Stochastic Processes

Elements of Stochastic Processes

Lecture II

Fall 2014

Overview

Reading Assignment

Chapter 9 of textbook

Further Resources

MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic

Processes, 2nd ed., Academic Press, New York, 1975.

Outline

Basic Definitions

Statistics of Stochastic Processes

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)

A stochastic process x(t) is a rule for assigning

to every a function x(t, ).

ensemble of functions: Family of all functions a

random process generates

Basic Definitions (cont’d)

With fixed (zeta), we will have a

“time” function called sample path.

Are sample paths sufficient for

estimating stochastic properties of a

random process?

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 “zeta”, we will have a real (complex)

number

Basic Definitions (cont’d)

Example I

Brownian Motion

Motion of all particles (ensemble)

Motion of a specific particle (sample path)

Basic Definitions (cont’d)

Example II

Voltage of a generator with fixed frequency

Amplitude is a random variable

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

Statistics of Stochastic Processes

First-Order CDF of a random process

First-Order PDF of a random process

xtXtxFX Pr,

txFx

txf XX ,,

Statistics of Stochastic Processes (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

Statistics of Stochastic Processes (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 ,;,,

Statistics of Stochastic Processes (cont’d)

Mean of a random process

Autocorrelation of a random process

Fact: (Why?)

dxtxfxtXEt X ,.

212121212121 ,;,..., dxdxttxxfxxtXtXEttR X

2)(

2)(, tXtXttR

Statistics of Stochastic Processes (cont’d)

Autocovariance of a random process

Correlation Coefficient

Example

2211

212121

.

.,,

ttXttXE

ttttRttC

2221112

21 ,,2, ttRttRttRtXtXE

2211

2121

,.,

,,

ttCttC

ttCttr

End of Section 1

Any Question?

Outline

Basic Definitions (cont’d)

Stationary/Ergodic Processes

Stochastic Analysis of Systems

Power Spectrum

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)

a-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

0C

Cr

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 !?)

a-dependent Process

a is called “Correlation Time”

.qC

aC 0

Stationary/Ergodic Processes (cont’d)

Example

SSS

Suppose a and b 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

8.525858222

RRR

XXEXEXEXXE

Stationary/Ergodic Processes (cont’d)

Ergodic Process

Equality of time properties and statistic properties.

First-Order Time average

Defined as

Mean Ergodic Process

Mean Ergodic Process in Mean Square Sense

T

T

Tt dttXT

tXtXE2

1lim

tXEtXE

02

1.1

lim02

0

2

T

XTt dT

CT

tXE

Stationary/Ergodic Processes (cont’d)

Slutsky’s Theorem

A process X(t) is mean-ergodic iff

Sufficient Conditions

a)

b)

01

lim

0

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 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

dxty

0

).(

tUdtttht

0

).(

t

dttxEtyE

0

.

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 xx

2.

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

T

xxyy dRTT

R2

2

..2

12

1

Power Spectrum (cont’d)

Example II

System

Impulse Response

Power Spectrum

txdt

dty

.iH

xxyy SS .2

Acknowledgement

Thank to A. Jalali and M. H. Rohban

Next Lecture

Markov Processes

&

Markov Chains