stochastic processesce.sharif.edu/courses/90-91/1/ce695-1/resources/root...s. karlin and h. m....

Hamid R. Rabiee

Stochastic Processes

Elements of Stochastic Processes

Lecture II

with thanks to Ali Jalali

1

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.

2 Stochastic Processes

Outline

Basic Definitions

Statistics of Stochastic Processes

Stationary Processes

Stochastic Analysis of Systems

Power Spectrum

Ergodic Processes


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.

X (t , β )= (X 1(β), X 2(β), .. . , X n(β ), . ..)


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



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?)



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

With fixed “t” and “zeta”, we will have a real (complex)

number



Example I

Brownian Motion

Motion of all particles (ensemble)

Motion of a specific particle (sample path)



Example II

Voltage of a generator with fixed frequency

Amplitude is a random variable

tAtV cos.,



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

txFx

txf XX ,,

xtXtxFX Pr,


Statistics of Stochastic Processes (cont’d)

Second-Order CDF of a random process

Second-Order PDF of a random process

212121

2

2121 ,;,.

,;, ttxxFxx

ttxxf XX

22112121 Pr,;, xtXandxtXttxxFX



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 ,;,,



Mean of a random process

Autocorrelation of a random process

Fact: (Why?)

dxtxfxtXEt X ,.

212121212121 ,;,..., dxdxttxxfxxtXtXEttR X

2)(

2)(, tXtXttR



Autocovariance of a random process

Correlation Coefficient

Example

2211

212121

.

.,,

ttXttXE

ttttRttC

2211

2121

,.,

,,

ttCttC

ttCttr

2221112

21 ,,2, ttRttRttRtXtXE


End of Section 1

Any Question?


Outline




Power Spectrum

Ergodic Processes



Example

Poisson Process

Mean

Autocorrelation

Autocovariance

21212

1

21212

221

...

...,

ttttt

tttttttR

2121 ,min., ttttC

tt .

kt

tk

etK .

!

.



Complex process

Definition

Specified in terms of the joint statistics of

two real processes and

Vector Process

A family of some stochastic processes

tX tY

tYitXtZ .



Cross-Correlation

Orthogonal Processes

Cross-Covariance

Uncorrelated Processes

2121 ., tYtXEttRXY

0, 21 ttRXY

212121 .,, ttttRttC YXXYXY

0, 21 ttCXY



a-dependent processes

White Noise

Variance of Stochastic Process

0, 2121 ttCatt

0, 2121 ttCtt

tttC X2,


22


Normal Process

A Process is called Normal, if the random

variables are jointly normal for any

and .

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

tx

)(,..., 21 txtx n

21,...,tt

Stochastic Processes

Outline

Basic Definitions



Power Spectrum

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

…

;,,;, 21212112 xxfttxxftt XX

xftxf XX ,


Stationary 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 .


Autocovariance of a WSS process

Correlation Coefficient


2 RC

0C

Cr


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


Example

SSS

Suppose a and b are normal random variables

with zero mean.

Why is it SSS?

WSS

Suppose “ ” has a uniform distribution in the

interval

Why is it WSS?


tbtatX sin.cos.

tatX cos.

,


Example

Suppose for a WSS process

X(8) and X(5) are random variables


3200

8.525858222

RRR

XXEXEXEXXE

eAR .


End of Section 2

Any Question?


Outline

Basic Definitions



Power Spectrum

Ergodic Processes



Linear Systems

Time-Invariant Systems

Linear Time-Invariant Systems

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

babtxagbtyatxgty ,..

txgtytxgty

txthtytxgty *


Stochastic Analysis of Systems (cont’d)

Memoryless Systems

Causal Systems

Only causal systems can be realized. (Why?)

00 txgty

00 tttxgty



Linear time-invariant systems

Mean

Autocorrelation

2

*

21121 *,*, thttRthttR xxyy

txEthtxthEtyE **



Example I

System:

Impulse response:

Output Mean:

Output Autocovariance:

tUdtttht

0

).(

t

dxty

0

).(

t

dttxEtyE

0

.

0 0

2121 ..,, ddttRttR xxyy



Example II

System:

Impulse response:

Output Mean:

Output Autocovariance:

txdt

dty

tdt

dth

txEdt

dtyE

2121

2

21 ,.

, ttRtt

ttR xxyy


Outline

Basic Definitions



Power Spectrum

Ergodic Processes


Power Spectrum

Definition

WSS process

Autocorrelation

Fourier Transform of autocorrelation

R

tX

deRS j.


Power Spectrum (cont’d)

Inverse trnasform

For real processes

deSR j.2

1

0

0

.cos.1

.cos.2

dSR

dSS



For a linear time invariant system

Fact (Why?)

xx

xxyy

SH

HSHS

.

..

2

*

dHStyVar xx

2.

2

1



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



Example II

System

Impulse Response

Power Spectrum

txdt

dty

.iH

xxyy SS .2


Outline

Basic Definitions



Power Spectrum

Ergodic Processes


Ergodic Processes

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


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


stochastic processesce.sharif.edu/courses/90-91/1/ce695-1/resources/root...s. karlin and h. m....

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