Spectral concentration problem

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.

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

For a given frequency W such that 0

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 [-⁠1/2⁠,⁠1/2⁠].

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

Thus, the spectral concentration is strictly less than one, and there is no finite sequence

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

Thus maximising the power subject to the constraint that the total power is fixed, say leads to the following equation satisfied by the optimal sequence

The largest eigenvalue of the above equation corresponds to the largest possible spectral concentration; the corresponding eigenvector is the required optimal sequence

This sequence is called a 0th–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.

is called nth–order Slepian sequence (DPSS) (0≤n≤N-1).

This nth–order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders

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 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.