a discussion on “wiener – khinchin theorem and its applications”

Indian Institute of Science Education and Research, Kolkata 14th December 2010

1 | Term Paper on ‘Wiener – Khinchin Theorem & its applications’ by Harsh Purwar (07MS – 76)

A Discussion on “Wiener – Khinchin Theorem and its applications”

Harsh Purwar (07MS – 76) Student, Non-Equilibrium Statistical Mechanics (ID – 419)

Indian Institute of Science Education and Research, Kolkata

Power Spectral Density

Power spectral density or power spectrum in short, of a stationary random process is an important quantitative characterizer of the random process. Consider a general stationary random process, ( ). In general, the plot of ( ) versus may be expected to be a highly irregular curve – the random process is, after all, a ‘noise’ in a sense. We may ask: perhaps a Fourier transform of the function ( ) would give us some insight into its time variation, by decomposing into its individual harmonics? But ( ), regarded as a

function of , need not be integrable, in general. To be precise ∫ | ( )|

may not be finite. Therefore

the Fourier transform may not exist. However, a much regular function of can be associated with the process ( ): namely its autocorrelation function. Basically power spectrum of a stationary process is a measure of how much of the intensity of the fluctuating signal ( ) lies in an infinitesimal frequency window centered at the frequency . Suppose we monitor the process (or signal) ( ) over a very long interval of time, say from up to . The power spectral density ( ) is then defined as,

( )

|∫

( )|

( )

Note: It is evident from the definition above that ( ) is real and positive.

Wiener – Khinchin Theorem

Observe; above definition do not have any average over all realizations of the random process. An averaging of this sort is not necessary because the process is supposed to be ergodic: given a sufficient amount of time (ensured by passing to the limit ), ( ) will take on all values in its sample space. Consequently, we may expect ( ) to be expressible in terms of a statistical average. Indeed, this is the

content of the Wiener – Khinchin theorem. Statement: The power spectral density ( ) of a stationary random process ( ) is equal to the Fourier

transform of its autocorrelation function. That is,

( )

∫

⟨ ( ) ( )⟩ ( )

Proof: We have,

|∫

( )|

∫

∫

( ) ( ) ( )



∫

∫

( ) ( ) ( )

As left hand side is a real quantity, the imaginary part, ( ) of ( ) must vanish. This actually is true because ( ) is an odd function of and vanishes when integrated. Now introducing function we have,

|∫

( )|

∫

∫

( ) ( ) ( ) ( ( ) ( ))

∫

∫

( ) ( ) ( ) ( ) ∫

∫

( ) ( ) ( ) ( )

∫

{

∫

( ) ( ) ( ) ( )⏟

∫

( ) ( ) ( ) ( )⏟

}

∫

{

∫

( ) ( ) ( ) ( )⏟

∫

( ) ( ) ( ) ( )⏟

}

The second integral above has as can be seen from the limits of integration but in this regime ( ) vanishes. Similarly the fourth term also vanishes and we are left with,

|∫

( )|

∫

∫

( ) ( ) ( ) ∫

∫

( ) ( ) ( )

function has been replaced by 1. Now interchanging the dummy variable and in the second term we get,

|∫

( )|

∫

∫

( ) ( ) ( ) ∫

∫

( ) ( ) ( )

∫

∫

( ) ( ) ( )

Changing variable from to . Implies and we get,

|∫

( )|

∫

∫

( ) ( )

∫

∫

( ) ( )

Interchanging the order of integration we get,

|∫

( )|

∫

∫

( ) ( )

Now again change variable to , implying . We get,

|∫

( )|

∫

∫

( ) ( )

∫

∫

( ) ( )

Substituting this back in the definition of ( ) we have,

( )

∫

∫

( ) ( )



∫

∫

( ) ( )

Due to the ergodicity of the process ( ) we can write,

∫

( ) ( ) ⟨ ( ) ( )⟩

Now using the fact that the autocorrelation function of a stationary random process is a function only of the difference of the two time arguments involved, and not of the two arguments separately, we have,

⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩ Hence,

( )

∫

⟨ ( ) ( )⟩ ( )

Considering ( ) being a classical variable i.e. to say, ⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩

And as , we have

( )

∫

⟨ ( ) ( )⟩

∫

⟨ ( ) ( )⟩

Applications of WKT:

Position autocorrelation of the Brownian oscillator: Starting from the result obtained by my friend Anish Bhardwaj in his mid-semester presentation on Brownian oscillator, its velocity autocorrelation function,

⟨ ( ) ( )⟩

⁄ ( ( )

( )) ( )

where is given by,

(

)

⁄

I would try to find the autocorrelation function of the position of the oscillator. The Fourier transform of the autocorrelation function in W-K Theorem can be inverted to get,

⟨ ( ) ( )⟩ ∫

( ) ( ) ( )

Differentiating both sides with respect to and in succession we get,

⟨ ( ) ( )⟩ ∫

( ) ( )

Now setting and we get,

⟨ ( ) ( )⟩ ∫

( ) ( )

But from the inverted W-K theorem equation ( ) for we also have,

⟨ ( ) ( )⟩ ∫

( ) ( )

Comparing ( ) and ( ) we have, ( ) ( ) ( )

I intend to apply this equation to the Brownian oscillator taking ( ) ( ) and hence ( ) ( ). Using result ( ) for the velocity auto correlation function of the oscillator in the formula ( ) it can be shown that its power spectral density (PSD) is given by,



( )

{

( )

}

So the PSD of the position using ( ) is given by,

( )

{

( )

}

Using the inverted form of the Wiener Khinchin theorem, equation ( ) we have,

⟨ ( ) ( )⟩ ∫

( )

Using the results of contour integration the above integration was evaluated to give,

⟨ ( ) ( )⟩

{

⁄ (

)

⁄ (

)

Observe that it is indeed an even function of , as required of the autocorrelation function of a (one component) stationary random process.

Works Cited

1. Balakrishnan, V. Elements of Nonequilibrium Statistical Mechanics. Madras : Ane Books Pvt. Ltd., 2008. pp. 223-228, 237-238, 281-282. 9789380156255.

a discussion on “wiener – khinchin theorem and its applications”

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