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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