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Slepian function

Mathematical function From Wikipedia, the free encyclopedia

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Slepian functions are a class of spatio-spectrally concentrated functions[1][2] (that is, space- or time-concentrated while spectrally bandlimited, or spectral-band-concentrated while space- or time-limited) that form an orthogonal basis for bandlimited or spacelimited spaces.[3][4][5][6] They are widely used as basis functions for approximation[7] and in linear inverse problems,[8][9] and as apodization tapers or window functions[10] in quadratic problems[11] of spectral density estimation.[12][13]

Slepian function constructions exist in discrete (regular[14] and irregular[15]) and continuous[16][17][18] varieties, in one, two, and three dimensions[19], in Cartesian and spherical geometry, on surfaces and in volumes,[20] on graphs,[21] and in scalar, vector,[22] and tensor forms.[23]

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General setting and operator formalism

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Without reference to any of these particularities,[24] let be a square-integrable function of physical space, and let represent Fourier transformation, such that and . Let the operators and project onto the space of spacelimited functions, , and the space of bandlimited functions, , respectively, whereby is an arbitrary nontrivial subregion of all of physical space, and an arbitrary nontrivial subregion of spectral space. Thus, the operator acts to spacelimit, and the operator acts to bandlimit the function .

Slepian's quadratic spectral concentration problem aims to maximize the concentration of spectral power to a target region , for a function that is spatially limited to a target region . Conversely, Slepian's spatial concentration problem maximizes the spatial concentration to of a function bandlimited to . Using for the inner product both in the space and the spectral domains, both problems are stated equivalently using Rayleigh quotients in the form

The equivalent spectral-domain and spatial-domain eigenvalue equations are

and

given that and are each others' adjoints, and that and are self-adjoint and idempotent.

The Slepian functions are solutions to either of these types of equations with positive-definite kernels, that is, they are bandlimited functions , concentrated to the spatial domain within , or spacelimited functions of the form , concentrated to the spectral domain within .

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Scalar Slepian functions in one dimension

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(a) Slepian functions in the time domain. (b) Slepian functions in the frequency domain. Shown is the square of the absolute value of the Fourier transform of the Slepian functions shown in (a). (c) Concentration factors associated with the successive Slepian functions shown in (a). (d) Cumulative energy by summation the square of the Slepian functions shown in (a).

Let and its Fourier transform be strictly bandlimited in angular frequency between . Attempting to concentrate in the time domain, to be contained within the time interval , amounts to maximizing

which is equivalent to solving either, in the frequency domain, the convolutional integral eigenvalue (Fredholm) equation

,

or the time- or space-domain version

.

Either of these can be transformed and rescaled to the dimensionless

.

The trace of the positive definite kernel is the sum of the infinite number of real and positive eigenvalues,

that is, the area of the concentration domain in time-frequency space (a time-bandwidth product).

One-dimensional scalar Slepian functions or tapers[25] are the workhorse of the Thomson multitaper method of spectral density estimation.

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Scalar Slepian functions in two Cartesian dimensions

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Slepian functions concentrated to a cat-like spatial (top row; rank and concentration eigenvalue ) and a duck-like spectral domain (bottom row; shown is the square of the absolute value of the Fourier transform of the functions shown in the top row).

We use and its Fourier transform to denote a function that is strictly bandlimited to , an arbitrary subregion of the spectral space of spatial wave vectors.[26] Seeking to concentrate into a finite spatial region , of area , we must find the unknown functions for which

Maximizing this Rayleigh quotient requires solving the Fredholm integral equation

The corresponding problem in the spatial domain is

Concentration to the disk-shaped spectral band allows us to rewrite the spatial kernel as

with a Bessel function of the first kind, from which we may derive that

in other words, again the area of the concentration domain in space-frequency space (a space-bandwidth product).

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Scalar Slepian functions on the surface of a sphere

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Spherical Slepian functions of spherical-harmonic bandwidth 18, and of spherical-harmonic order 0 (that is, only made of zonal spherical harmonics), either very well (top row) or very poorly (bottom row) concentrated, as indicated by the concentration ratio to the North-polar cap of opening angle 40.

We denote a function on the unit sphere and its spherical harmonic transform coefficient at the degree and order , respectively,[24] and we consider bandlimitation to spherical harmonic degree , that is, . Maximizing the quadratic energy ratio within the spatial subdomain via

amounts in the spectral domain to solving the algebraic eigenvalue equation

,

with the spherical harmonic at degree and order . The equivalent spatial-domain equation, is a homogeneous Fredholm integral equation of the second kind, with a finite-rank, symmetric, separable kernel.

The last equality is a consequence of the spherical harmonic addition theorem which involves , the Legendre polynomial. The trace of this kernel is given by

,

that is, once again a space-bandwidth product, of the dimension of and the fractional area of on the unit sphere , namely .

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References

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