Spectral concentration problem

Introduction

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Consider the Discrete Fourier transform (DFT) $U(f)$ of a finite series $w_{t}$, $t=1,2,3,4,...,T$ defined by $U(f)=\sum _{t=1}^{T}w_{t}e^{-2\pi ift}$. The sampling interval $\triangle t=1$ and hence the frequency interval will be taken as $f\varepsilon \lbrack {\frac {-1}{2}},{\frac {1}{2}}\rbrack$. $U(f)$ is a periodic function with a period $1.$

For a given frequency $W$ such that $0<W<1/2$, the spectral concentration $\lambda (T,W)$ of $U(f)$ on the interval $\lbrack -W,W\rbrack$ is defined as the ratio of power of $U(f)$ contained in the frequency band $\lbrack -W,W\rbrack$ to the power of $U(f)$ contained in the entire frequency band $\lbrack {\frac {-1}{2}},{\frac {1}{2}}\rbrack$. That is, $\lambda (T,W)={\frac {\int _{-W}^{W}{\|U(f)\|}^{2}\,df}{\int _{-1/2}^{1/2}{\|U(f)\|}^{2}\,df}}$

It can be shown that $U(f)$ has only isolated zeros and hence $0<\lambda (T,W)<1$ (See [1]). Thus, the spectral concentration is strictly less than one, and there is no finite sequence $w_{t}$ for which the DFT can be confined to a band $\lbrack -W,W\rbrack$ and made to vanish outside this band.

Concentration Problem

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Among all sequences $\lbrace w_{t}\rbrace$ for a given $T$ and $W$, is there a sequence for which the spectral concentration is maximum? In other words, is there a sequence for which the sidelobe energy outside a frequency band $\lbrack -W,W\rbrack$ is minimum?

The answer is yes and such a sequence indeed exists and can be found by optimizing $\lambda (T,W)$. Thus maximising the power $\int _{-W}^{W}{\|U(f)\|}^{2}\,df$ subject to the constraint that the total power is fixed (say $\int _{-1/2}^{1/2}{\|U(f)\|}^{2}\,df=1$), leads to the following equation satisfied by the optimal sequence $w_{t}$:

$\sum _{t^{'}=1}^{T}{\frac {\sin 2\pi W(t-t^{'})}{\pi (t-t^{'})}}w_{t^{'}}=\lambda w_{t^{'}}$

This is an eigen value equation for a symmetric matrix given by $M_{t,t^{'}}=\sin 2\pi W(t-t^{'})/\pi (t-t^{'})$. It can be shown that this matrix is positive-definite, hence all the eigen values of this matrix lie between 0 and 1. The largest eigen value of the above equation, corresponds to the largest possible spectral concentration; the corresponding eigen vector is the required optimal sequence $w_{t}$. This sequence is called a $zero^{th}$--order Slepian sequence (also known by Discrete Prolate Spheroidal sequence), which is a unique taper with maximally suppressed sidelobes.

It turns out that the number of dominant eigenvalues of the matrix $M$ that are close to 1, corresponds to $N=2WT$ called as Shannon number. If we arrange the eigen vectors in the decreasing order of $\lambda$ (i.e, $\lambda _{1}>\lambda _{2}>\lambda _{3}>...>\lambda _{N}$), then the eigen vector corresponding to $\lambda _{n+1}$ is called $n^{th}-$order Slepian sequence (DPSS) $(0\leq n\leq N-1)$. This $n^{th}$-- order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders $(0,1,2,3....,n-1)$. These lower order Slepian sequences form the basis for spectral estimation by Multitaper method.

References

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[1] Partha Mitra and Hemant Bokil. Observed Brain Dynamics, Oxford University Press, USA (2007).

[2] Donald. B. Percival and Andrew. T. Walden. Spectral Analysis for Physical Applications: Multitaper andConventional Univariate Techniques, Cambridge University Press, UK (2002).

See Also

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Multitaper

Fourier_transform

Discrete_Fourier_transform

Shannon number

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External links

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http://www.mast.queensu.ca/~djt/

--Chronuxwiki (talk) 12:49, 18 September 2008 (UTC)