Spectral concentration problem explained

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

Spectral concentration

The discrete Fourier transform (DFT) U(f) of a finite series

w_t

t=1,2,3,4,...,T

is defined as

U(f)=

	T
\sum
	t=1

w_te^-2\pi.

In the following, the sampling interval will be taken as Δt = 1, and hence the frequency interval as f ∈ [-{{sfrac|1|2}},{{sfrac|1|2}}]. U(f) is a periodic function with a period 1.

For a given frequency W such that 0<W<, the spectral concentration

λ(T,W)

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 [-{{sfrac|1|2}},{{sfrac|1|2}}]. That is,

λ(T,W)=

	W
\int		{\\|U(f)\\|
	-W

df}

	1/2
{\int
	-1/2

{\|U(f)\|}²df}.

It can be shown that U(f) has only isolated zeros and hence

0<λ(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 DTFT can be confined to a band [-''W'',''W''] and made to vanish outside this band.

Statement of the problem

\lbracew_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 [-''W'',''W''] is minimum?

The answer is yes; such a sequence indeed exists and can be found by optimizing

λ(T,W)

. Thus maximising the power

	W
\int
	-W

{\|U(f)\|}²df

subject to the constraint that the total power is fixed, say

	1/2
\int
	-1/2

{\|U(f)\|}²df=1,

leads to the following equation satisfied by the optimal sequence

w_t

	T
\sum
	t'=1

	\sin2\piW(t-t')
	\pi(t-t')

w_t'=λw_t.

This is an eigenvalue equation for a symmetric matrix given by

M_t,t'=

	\sin2\piW(t-t')
	\pi(t-t')

It can be shown that this matrix is positive-definite, hence all the eigenvalues ofthis 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

w_t

. This sequence is called a 0^th–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 the Shannon number. If the eigenvalues

are arranged in decreasing order (i.e.,

λ₁>λ₂>λ₃>...>λ_N

), then the eigenvector corresponding to

λ_n+1

is called n^th–order Slepian sequence (DPSS) (0≤n≤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 formthe basis for spectral estimation by multitaper method.

Not limited to time series, the spectral concentration problem can be reformulated to apply in multiple Cartesian dimensions^[1] and on the surface of the sphere by using spherical harmonics,^[2] for applications in geophysics and cosmology^[3] among others.

References

Partha Mitra and Hemant Bokil. Observed Brain Dynamics, Oxford University Press, USA (2007), 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, arxiv.org/abs/q-bio/0309028

Notes and References

Simons . F. J. . Wang . D. V. . Spatiospectral concentration in the Cartesian plane . Int. J. Geomath . 2011 . 2 . 1–36 . 10.1007/s13137-011-0016-z. .
Simons . F. J. . Dahlen . F. A. . Wieczorek . M. A. . 10.1137/S0036144504445765 . Spatiospectral Concentration on a Sphere . SIAM Review . 48 . 3 . 504–536 . 2006 . math/0408424 . 2006SIAMR..48..504S .
Dahlen . F. A. . Simons . F. J. . 10.1111/j.1365-246X.2008.03854.x . Spectral estimation on a sphere in geophysics and cosmology . Geophysical Journal International . 174 . 3 . 774–807 . 2008 . free . 0705.3083 . 2008GeoJI.174..774D .

Spectral concentration problem explained