STOCHASTIC SIGNAL PROCESSING

PRESENTED BY ILA SHARMA

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OUTLINE

Introduction to probability Random variables Moments of random variables Stochastic or Random processes Basic types of Stochastic Processes

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PROBABILITY THEORY

Probability theory begins with the concept of a probability space, which is a collection of three items (Ω,F, P);

Ω = Sample space F = Event space or field F,

P = Probability measure. This (Ω,F, P) is collectively called a

probability space or an experiment.

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AXIOMATIC DEFINATION OF PROBABILITY

Given a sample space Ω, and a field F of events defined on Ω, we define probability Pr[.] as a measure on each event E belongs to F, such that:

Pr[E]>= 0, Pr[Ω] = 1, Pr[E U F] = Pr[E] + Pr[F], if EF = Ø.

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RANDOM VARIABLE

A Real Random Variable X(.) is a mapping from sample space(Ω) to the real line, which assigns a number X(ç) to every outcome ç belongs to sample space(Ω).

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MEAN AND VARIANCE

The expected value (or mean) of an RV is defined as:

The variance of an RV X is defined as:

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VARIANCE AND CORRELATION

The variance of an RV X is defined as:

We can define the covariance between two random variables as:

dxdyyxpyxyxEyx yxyx ),())(())((), cov(

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CONTINUED…………

For a discrete random variable representing the samples of a time series, we can estimate this directly from the signal as:

Two random variables are said to be uncorrelated if

n

x knxnxN

kR ][][1

0),cov( yx

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RANDOM PROCESS

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AUTO CORRELATION FUNCTION

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BASIC TYPES OF RANDOM PROCESS

GAUSSIAN PROCESS

MARKOV PROCESS

STATIONARY PROCESS

WHITE PROCESS

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GAUSSIAN PROCESS A random process X(t) is a Gaussian

process if for all n and for all , the random variables has a jointly Gaussian density function, which may expressed as

Where ->

1 2( , , , )nt t t

2 ( ), ( ), , ( )i nX t X t X t

1/ 2 1/ 2

1 1( ) exp[ ( ) ( )]

2(2 ) [det( )]T

nf x x m C x m

C

: n random variables: mean value vector: nxn covariance matrix

2[ ( ), ( ), , ( )]Ti nx X t X t X t

( )m E X

(( )( ))ij i i j jC c E x m x m

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MARKOV PROCESS

Markov process X(t) is a random process whose past has no influence on the future if its present is specified. If , then

Or if

1n nt t

1 1[ ( ) | ( ) ] [ ( ) | ( )]n n n n n nP X t x X t t t P X t x X t

2 1...nt t t

1 2 1 1[ ( ) | ( ), ( ),..., ( )] [ ( ) | ( )]n n n n n n nP X t x X t X t X t P X t x X t

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STATIONARY PROCESS

Definition of Autocorrelation

Where X(t1),X(t2) are random variables obtained at t1,t2

Definition of stationary A random process is said to stationary, if its

mean(m) and covariance(C) do not vary with a shift in the time origin

A process is stationary if

1 2 1 2( , ) [ ( ) ( )]XR t t E X t X t

( ( ) constantk XE X t m

1 2 1 2( , ) ( ) ( )X X XR t t R t t R

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WHITE PROCESS

A random process X(t) is called a white process if it has a flat power spectrum. If Sx(f) is constant for all f

It closely represent thermal noise

f

Sx(f)

The area is infinite(Infinite power !)

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REFERENCES

Stark & Woods : Probability and Random Processes with Applications to Signal Processing, Chapters 1-3 &7.

Edward R. Dougherty : Random process for image and signal processing, Chapters 1-2.

T. Chonavel : Stochastic signal processing.

Robert M. Gray & Lee D. Davisson: An Introduction to Statistical Signal Processing.

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THANK

YOU