Slepian Spatial-Spectral Concentration on the Ball

We formulate and solve the Slepian spatial-spectral concentration problem on the three-dimensional ball. Both the standard Fourier-Bessel and also the Fourier-Laguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact computation). The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the space-bandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used be represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeeds permits a sparse representation of concentrated signals.Comment: 33 pages, 10 figure

Multitaper estimation on arbitrary domains

Multitaper estimators have enjoyed significant success in estimating spectral densities from finite samples using as tapers Slepian functions defined on the acquisition domain. Unfortunately, the numerical calculation of these Slepian tapers is only tractable for certain symmetric domains, such as rectangles or disks. In addition, no performance bounds are currently available for the mean squared error of the spectral density estimate. This situation is inadequate for applications such as cryo-electron microscopy, where noise models must be estimated from irregular domains with small sample sizes. We show that the multitaper estimator only depends on the linear space spanned by the tapers. As a result, Slepian tapers may be replaced by proxy tapers spanning the same subspace (validating the common practice of using partially converged solutions to the Slepian eigenproblem as tapers). These proxies may consequently be calculated using standard numerical algorithms for block diagonalization. We also prove a set of performance bounds for multitaper estimators on arbitrary domains. The method is demonstrated on synthetic and experimental datasets from cryo-electron microscopy, where it reduces mean squared error by a factor of two or more compared to traditional methods.Comment: 28 pages, 11 figure

A Universal Framework for Holographic MIMO Sensing

This paper addresses the sensing space identification of arbitrarily shaped continuous antennas. In the context of holographic multiple-input multiple-output (MIMO), a.k.a. large intelligent surfaces, these antennas offer benefits such as super-directivity and near-field operability. The sensing space reveals two key aspects: (a) its dimension specifies the maximally achievable spatial degrees of freedom (DoFs), and (b) the finite basis spanning this space accurately describes the sampled field. Earlier studies focus on specific geometries, bringing forth the need for extendable analysis to real-world conformal antennas. Thus, we introduce a universal framework to determine the antenna sensing space, regardless of its shape. The findings underscore both spatial and spectral concentration of sampled fields to define a generic eigenvalue problem of Slepian concentration. Results show that this approach precisely estimates the DoFs of well-known geometries, and verify its flexible extension to conformal antennas

Slepian concentration problem for polynomials on the unit Ball

We study the Slepian spatiospectral concentration problem for the space of multi-variate polynomials on the unit ball in $\mathbb{R}^d$. We will discuss the phenomenon of an asymptotically bimodal distribution of eigenvalues of the spatiospectral concentration operators of polynomial spaces equipped with two different notions of bandwidth: (a) the space of polynomials with a fixed maximal overall polynomial degree, (b) the space of polynomials separated into radial and spherical contributions, with fixed but separate maximal degrees for the radial and spherical contributions, respectively. In particular, we investigate the transition position of the bimodal eigenvalue distribution (the so-called Shannon number) for both setups. The analytic results are illustrated by numerical examples on the 3-D ball