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[[Image:DPSS figure.JPG|thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. Note that each higher order sequence has an extra zero crossing.]]


The '''spectral concentration problem''' in [[Fourier analysis]]  refers to finding a time sequence whose [[discrete Fourier transform]] is maximally localized on a given [[frequency]] interval, as measured by the spectral concentration.


==Spectral concentration==
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The [[discrete-time Fourier transform]] (DTFT) ''U''(''f'') of a finite series <math>w_t</math>, <math>t = 1,2,3,4,...,T</math> is defined as
 
:<math>U(f) = \sum_{t=1}^{T}w_t e^{-2\pi ift}.</math>
 
In the following, the [[sampling interval]] will be taken as Δ''t'' = 1, and hence the frequency interval as
''f'' ∈ [-½,½]. ''U''(''f'') is a [[periodic function]] with a period 1.
 
For a given frequency ''W'' such that 0<''W''<½, the spectral concentration <math>\lambda(T,W)</math> of ''U''(''f'') on the interval [-''W'',''W''] is defined as the ratio of power of ''U''(''f'') contained in the [[frequency band]] [-''W'',''W''] to the power of ''U''(''f'') contained in the entire frequency band [-½,½]. That is,
 
:<math>\lambda(T,W) = \frac{\int_{-W}^{W} {\|U(f)\|}^2 \,df} {\int_{-1/2}^{1/2} {\|U(f)\|}^2 \,df}.</math>
 
It can be shown that ''U''(''f'') has only isolated zeros and hence <math>0<\lambda(T,W)<1</math> (see [1]).  Thus, the spectral concentration is strictly less than one, and there is no finite sequence <math>w_t</math> for which the DTFT can be confined to a band [-''W'',''W''] and made to vanish outside this band.
 
==Statement of the problem==
 
Among all [[sequences]] <math>\lbrace w_t \rbrace</math> 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 [-''W'',''W''] is minimum?
 
The answer is yes; such a sequence indeed exists and can be found by optimizing <math>\lambda(T,W)</math>. Thus maximising the power
 
:<math>\int_{-W}^{W} {\|U(f)\|}^2 \,dfe total power is fixed, say
 
:<math>\int_{-1/2}^{1/2} {\|U(f)\|}^2 \,df=1,</math>
 
leads to the following equation satisfied by the optimal sequence <math>w_t</math>:
 
:<math>\sum_{t^{'}=1}^{T} \frac{\sin2\pi W(t-t^{'})} {\pi(t-t^{'})}w_{t^{'}} = \lambda w_{t}.</math>
 
This is an [[eigenvalue]] equation for a [[symmetric]] [[Matrix (mathematics)|matrix]] given by
 
:<math>M_{t,t^{'}} = \frac{\sin2\pi W(t-t^{'})}{\pi(t-t^{'})}.</math>
 
It can be shown that this matrix is [[Positive-definite matrix|positive-definite]], hence all the eigenvalues of
this matrix lie between 0 and 1. The largest eigenvalue of the above equation corresponds to the largest possible spectral concentration; the corresponding eigenvector is the required optimal sequence <math>w_t</math>. This sequence is called a 0<sup>''th''</sup>–order Slepian sequence (also known as a discrete prolate spheroidal sequence, or DPSS), 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 [[No the other Shannon number|Shannon number]]. If the eigenvalues <math>\lambda</math> are arranged in decreasing order (i.e., <math>\lambda_{1}>\lambda_{2}>\lambda_{3}>...>\lambda_{N}</math>), then the eigenvector corresponding to <math>\lambda_{n+1}</math> is called ''n<sup>th''</sup>–order Slepian sequence (DPSS) (0≤''n''≤''N''-1). This ''n<sup>th''</sup>–order taper also offers the best sidelobe suppression and is pairwise [[orthogonal]] to the Slepian sequences of previous orders <math>(0,1,2,3....,n-1)</math>. These lower order Slepian sequences form
the basis for [[spectral estimation]] by [[multitaper]] method.
 
Not limited to time series, the spectral concentration problem can be reformulated to apply on the surface of the sphere by using [[spherical harmonics]], for applications in [[geophysics]] and [[cosmology]] among others.
 
==See also==
*[[Multitaper]]
*[[Fourier transform]]
*[[Discrete Fourier transform]]
*[[Shannon number]]
 
==References==
* Partha Mitra and Hemant Bokil.  ''Observed Brain Dynamics'', Oxford University Press, USA (2007), [http://www.us.oup.com/us/catalog/general/subject/Medicine/Neuroscience/?view=usa&ci=9780195178081 Link for book]
* Donald. B. Percival and Andrew. T. Walden. ''Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques'', Cambridge University Press, UK (2002).
* Partha Mitra and B. Pesaran, "Analysis of Dynamic Brain Imaging Data." The Biophysical Journal, Volume 76 (1999), 691-708, [http://arxiv.org/abs/q-bio/0309028 arxiv.org/abs/q-bio/0309028 ]
* F. J. Simons, M. A. Wieczorek and F. A. Dahlen. ''Spatiospectral concentration on a sphere''. SIAM Review, 2006, {{doi|10.1137/S0036144504445765}}
 
[[Category:Fourier analysis]]
[[Category:Signal processing]]

Latest revision as of 02:00, 25 September 2014


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