spectral representation

7/30/2019 spectral representation

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SPECTRAL

REPRESENTATION

By- Saurabh Shukla

R.No-ICE2012005M.Tech


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1. SERIES REPRESENTATION OF

STOCHASTIC PROCESSa) Fourier series

b) Mean Square Periodicity

c) Representation of ACF in Fourier series

d) Fourier series representation of periodic stochasticprocess

e) Karhunen-Loeve expansion


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2. FOURIER SERIES

a) What is Fourier Series?

b) How it decomposes the periodic signals?

c) What is Periodicity?

d) What is Orthogonality relationship?

e) What are the existence condition of FourierSeries?


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3. MEAN SQAURE PERIODICITY


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a) Let we have a stochastic processX(t) in

b) can this continuous information be represented in terms of acountable set of random variables whose relative importance

decrease under some arrangement?

c) To get to know this it is best to start with Mean-Square periodicprocess.

d) A stochastic processX(t) is said to be mean square (M.S) periodic,

if for some T > 0

i.e

e) Proof: supposeX(t) is M.S. periodic. Then

,0 Tt

.allfor0])()([2

ttXTtXE

( ) ( ) with 1 for all .X t X t T probability t

--(1)


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But from Schwarz inequality

Thus the left side equals

or

i.e.,

i.e.,X(t) is mean square periodic.

.periodwithperiodicis)( TR

2 2 2*

1 2 2 1 2 2

0

[ ( ){ ( ) ( )} ] [ ( ) ] [ ( ) ( ) ]E X t X t T X t E X t E X t T X t

* *

1 2 1 2 2 1 2 1[ ( ) ( )] [ ( ) ( )] ( ) ( )E X t X t T E X t X t R t t T R t t

Thenperiodic.is)(Suppose)( R

0)()()0(2]|)()([| *2 RRRtXtXE

( ) ( ) for anyR T R

.0])()([2

tXTtXE --(2)

--(3)

*

1 2 2[ ( ){ ( ) ( )} ] 0E X t X t T X t


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4. REPRESENTATION OF ACF IN FOURIER

SERIES


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Thus ifX(t) is mean square periodic, then is periodic and let

represent its Fourier series expansion. Here

In a similar manner define

Notice that are random variables, and

0

0

1( ) .

T jn

nR e d

T

0

0

1

( )

T jk t

kc X t e dt T

kck ,

--(5)

--(6)

)(R

0

0

2( ) ,

jn

nR eT

--(4)

0 1 0 2

0 1 0 2

0 2 1 0 1

* *

1 1 2 22 0 0

2 1 1 22 0 0

( ) ( )

2 1 2 1 10 0

1[ ] [ ( ) ( ) ]

1 ( )

1 1[ ( ) ( )]

m

T Tjk t jm t

k m

T T jk t jm t

T T jm t t j m k t

E c c E X t e dt X t e dtT

R t t e e dt dtT

R t t e d t t e dtT T


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i.e., form a sequence of uncorrelated random variables.

0 1

,

1 ( )*

10

0,[ ] { }

0 .m k

T mj m k t

k m m T

k mE c c e dt

k m

--(7)

nnnc }{


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5. REPRESENTATION OF M.S. PERIODIC

PROCESS IN FOURIER SERIES


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Now on considering the partial sum-

Theorem- We shall show that in the mean square

sense as

i.e.,

Proof:

0( ) .N

jk tN k

K N

X t c e

)()(~

tXtXN

--(8)

.N

.as0])(~

)([2

NtXtXE N --(9)

2 2 *

2

[ ( ) ( ) ] [ ( ) ] 2 Re[ ( ( ) ( )]

[ ( ) ].

N N

N

E X t X t E X t E X t X t

E X t

--(10)


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0

0

0

2

* *

( ) *

0

( )

0

[ ( ) ] (0) ,

[ ( ) ( )] [ ( )]

1[ ( ) ( ) ]

1[ ( ) ( )] .

k

kk

N

jk tN k

k N

NT jk t

k N

N NT jk t

kk N k N

E X t R

E X t X t E c e X t

E X e X t dT

R t e d tT

--(12)

Similarly

i.e.,

0 02 ( ) ( )* *

2

[ ( ) ] [ [ ] .

[ ( ) ( ) ] 2( ) 0 as

Nj k m t j k m t

N k m k m kk m k m k N

N

N k kk k N

E X t E c c e E c c e

E X t X t N

--(13)

0

( ) , .

jk t

kkX t c e t

--(14)

and

But,


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CONCLUSION DRAWN FROM

EQUATION -14


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a) Thus mean square periodic processes can be represented in the form

of a series as in (14).

b) The stochastic information is contained in the random variables

c) Further these random variables are uncorrelated

and their variances

i.e

d) Thus if the power P of the stochastic process is finite, then the

positive sequence converges, and hence

e) This implies that the random variables in (14) are of relatively less

importance as and a finite approximation of the series in

(14) is indeed meaningful.

*

,( { } )k m k k mE c c 0 as .k k

., kck

2(0) [ ( ) ] .kk

R E X t P

kk

0 as .k k

,k


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6-KARHUNEN-LOEVE EXPANSION


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a) QUES-What about a general stochastic process, that is not

mean square periodic? Can it be represented in a similar

series fashion as in (14),

b) if not in the whole interval then take in finite

support of

c) Suppose that it is indeed possible to do so for any arbitrary

process X(t) in terms of a certain sequence of orthonormal

functions

, t

0 ?t T


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i.e.,

where

and in the mean square sense

Further, as before, we would like the cks to be uncorrelated random

variables. If that should be the case, then we must have

Now

1

)()(~

nkk tctX --(15)

--(16)

--(17)

( ) ( ) in 0 .X t X t t T

*

,[ ] .k m m k mE c c --(18)

* * *

1 1 1 2 2 20 0

* *

1 1 2 2 2 10 0

*

1 1 2 2 2 10 0

[ ] [ ( ) ( ) ( ) ( ) ]

( ) { ( ) ( )} ( )

( ){ ( , ) ( ) }

T T

k m k m

T T

k m

T T

k XX m

E c c E X t t dt X t t dt

t E X t X t t dt dt

t R t t t dt dt

--(19)

*

0

*

,0

( ) ( )

( ) ( ) ,

T

k k

T

k n k n

c X t t dt

t t dt


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and

Substituting (19) and (20) into (18), we get

Since (21) should be true for every we must have

or

*

1 1 2 2 2 1 10 0( ){ ( , ) ( ) ( )} 0.

XX

T T

k m m mt R t t t dt t dt

1 2 2 2 10( , ) ( ) ( ) 0,

XX

T

m m mR t t t dt t

( ), 1 ,k t k

*

, 1 1 10 ( ) ( ) .

T

m k m m k mt t dt

--(20)

--(21)

1 2 2 2 1 10( , ) ( ) ( ), 0 , 1 .

XX

T

m m mR t t t dt t t T m --(22)


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a) This is the desired uncorrelated condition in (18) which is

translated into the integral equation in (22) and it is known

as the Karhunen-Loeve or K-L. integral equation

b) The functions are not arbitrary and they must be

obtained by solving the integral equation in (22).They areknown as the eigenvectors of the autocorrelation function

of

c) Similarly the set represent the eigenvalues of the

autocorrelation function.

1)}({ kk t

1 2( , ).XXR t t

1{ }k k


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END OF

SERIES REPRESNTATION

OF

STOCHASTIC PROCESS

THANK YOU

By- Saurabh Shukla

R.No-ICE2012005

M.Tech

spectral representation

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