Prolate Spheroidal Wave Functions Research Articles

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1].To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

Grid-free compressive beamforming for arbitrary linear microphone arrays

Grid-free compressive beamforming is a new attractive method for acoustic source identification based on compressive sensing theory, which extracts the source information from the microphone array measurements without discretizing the target source region. In the framework of linear microphone array measurements, many studies have been carried out around this method. They have strict requirements on the distribution of microphones. Specifically, only the uniform arrays with equally spaced microphones and sparse arrays consisting of a selection of microphones from uniform arrays are available. In this paper, based on the prolate spheroidal wave functions, we propose a grid-free compressive beamforming method compatible with arbitrary linear microphone arrays. Some examples are conducted to demonstrate the correctness and superiority of the proposed method. Plenty of Monte Carlo simulations are performed to reveal the effects of source coherence, source separation, noise, and number of snapshots. This study helps to promote the popularization and application of grid-free compressive beamforming method and offers the possibility to improve the acoustic source identification performance by optimizing microphone distribution.

Sound Field Reconstruction Using Prolate Spheroidal Wave Functions and Sparse Regularization.

Near-field acoustic holography (NAH) based on compressing sensing (CS) theory enables accurate reconstruction of sound fields using a limited number of sampling points. However, the successful implementation of this technique depends on two crucial factors: (1) the appropriate selection or construction of the spatial basis and (2) an effective sparse regularization process. To enhance reconstruction performance for elongated sound sources, this paper proposes a novel sound field reconstruction method that combines prolate spheroidal wave functions (PSWFs) with the orthogonal matching pursuit (OMP) algorithm. In this method, PSWFs serve as a sparse spatial basis for representing the radiated sound field. The sparse coefficients are determined by the OMP algorithm in a linear subspace composed of basic functions that best match the residual error. The OMP algorithm effectively identifies significant components before potentially selecting incorrect ones by setting an appropriate stopping rule. Numerical simulations are conducted using a line-array source model. The results show that the proposed method can accurately reconstruct the sound pressures of the elongated source model using a relatively small number of samplings. In addition, the proposed method exhibits robustness across a wide frequency range, diverse array configurations and various sampling numbers. The experimental results further validate the feasibility and reliability of the proposed method.

A fast procedure for the construction of quadrature formulas for bandlimited functions

We introduce an efficient scheme for the construction of quadrature rules for bandlimited functions. While the scheme is predominantly based on well-known facts about prolate spheroidal wave functions of order zero, it has the asymptotic CPU time estimate O(nlog⁡n) to construct an n-point quadrature rule. Moreover, the size of the “nlog⁡n” term in the CPU time estimate is small, so for all practical purposes the CPU time cost is proportional to n. The performance of the algorithm is illustrated by several numerical examples.

Statistically Optimized Near-Field Acoustic Holography Using Prolate Spheroidal Wave Functions

Near-field acoustic holography (NAH) is an effective tool for realizing accurate sound field reconstruction in three-dimensional space on the prerequisite that appropriate elementary wave functions are selected or constructed to match the characteristics of the sound sources. However, for elongated sources, common wave functions, i.e., plane, cylindrical, or spherical waves, sometimes do not perform well during the sound field projections. To solve this problem, statistically optimized near-field acoustical holography combined with prolate spheroidal wave functions is proposed. In the approach, the sound field is expanded by a series of prolate spheroidal wave functions, whose wavefronts can be set nearly conformal to the elongated sources. Based on these wave functions, fewer expansion terms are required to model the sound field, and the need for regularization can be reduced during the inverse solving process. Therefore, the accuracy of the reconstruction results can be further improved. Numerical simulations are conducted by two types of elongated source models, namely, spatially separated and extended. The results show that the proposed method can effectively reconstruct the sound pressures of elongated sources and perform robustly across a wide frequency range. Simultaneously, a designed experiment is carried out in an anechoic chamber, which demonstrates the feasibility of the proposed method.

PSWF-based decoupled atomic norm minimization for DOD and DOA estimation in MIMO radar with arbitrary linear arrays

Decoupled atomic norm minimization (D-ANM) is a computationally-efficient gridless two-dimensional parameter estimation method via dividing the two-level Toeplitz matrix into two one-level matrices, and can be applied to the one-snapshot direction-of-departure (DOD) and direction-of-arrival (DOA) estimation in bistatic multiple-input multiple-output (MIMO) radar with uniform linear arrays (ULAs). Nevertheless, in actual applications, the transmitting and receiving arrays are not always ULAs, which leads to the Toeplitz structure not satisfied and the regular D-ANM algorithm not available. In this work, the one-snapshot DOD and DOA estimation in bistatic MIMO radar with arbitrary linear arrays (ALAs) is investigated. The Prolate spheroidal wave functions (PSWFs) are utilized to express such a D-ANM problem followed by a semi-definite programming (SDP) solution. The PSWF-based method can extend the application of D-ANM from ULAs to ALAs, and avoid the grid partition of the conventional sparse representation methods. The numerical experiments are conducted to verify the proposed method.

Prolate spheroidal wave functions associated with the canonical Fourier–Bessel transform and uncertainty principles

The aim of this paper is to establish an extension of the prolate spheroidal wave functions which is the eigenfunction of the finite Bessel type of linear canonical transform (LCT) so‐called finite canonical Fourier–Bessel transform. We study the concentration problem, uncertainty principles about the essential supports, and signal recovery related to the canonical Fourier–Bessel transform on the space of band‐limited functions.

Experimental validation of a phaseless, non‐redundant planar spiral scanning for antenna characterisation

AbstractAn effective, probe‐compensated, phaseless Near‐Field‐Far‐Field (NF–FF) transformation technique with planar spiral scanning is presented. The phase retrieval problem is solved by a quadratic inverse process, whose data are the voltage squared amplitudes of the NF samples acquired on two scanning surfaces. The available information on the antenna under test and on the scanning geometry are used to provide efficient representations for both the unknowns and the data, thus improving the accuracy and reliability of the technique. Furthermore, disconnected apertures are dealt with by adopting two different sets of prolate spheroidal wave functions to represent the aperture fields. Experimental results are reported to assess the effectiveness of the proposed approach. A drastic reduction (about 90%) of the required NF samples, as compared to standard λ/4 measurements (λ being the wavelength), is observed. Aggregating the measurements on the two scanning surfaces, the technique uses less samples even when compared to the case of complex, λ/2 measurements performed on a single surface. Our approach exhibits an accuracy comparable to that achieved when adopting complex measurements with non‐redundant sampling or classical, λ/2 plane‐rectangular NF–FF transformations.

Estimation of encountered wave elevation sequences based on response measurements in multi-directional seas

This paper presents a new approach for estimating encountered wave elevation sequences by use of measured ship responses, where wind-waves and swell may come from different directions, i.e. bi-directional waves. The main assumption of the approach, making use of Prolate Spheroidal Wave Functions (PSWF), is that the wave field is represented by multi-directional irregular waves. Thus, combining available measured responses, the phases and amplitudes of the multi-directional irregular waves are derived as the solution, by which the wave profiles can be estimated. Numerical investigations using artificially generated response measurements (sway, heave, pitch, vertical bending moment) of a bulk carrier in uni-directional and bi-directional long-crested as well as short-crested sea states are made. It is shown that the present approach can accurately estimate wave elevation sequences in such sea states.

Time-Limited Waveforms With Minimum Time Broadening for the Nonlinear Schrödinger Channel

Simple fiber optic communication systems can be implemented using energy modulation of isolated time-limited pulses. Fundamental solitons are one possible solution for such pulses which offer a fundamental advantage: they preserve their shape along the channel. Furthermore, a simple energy detector can be used at the receiver to detect the transmitted information. However, systems based on energy modulation of solitons are not competitive in terms of data rates. This is partly due to the fact that the effective time duration of a soliton depends on its chosen amplitude. In this paper, we propose to replace fundamental solitons by new time-limited waveforms that can be detected using an energy detector, and that are immune to fiber distortions. Our proposed solution relies on the prolate spheroidal wave functions and a numerical optimization routine. Time-limited waveforms that undergo minimum time broadening along an optical fiber are obtained and shown to outperform fundamental solitons. In the case of binary transmission and a single span of fiber, we report rate increases of 33.8% and 12% over lossy and lossless fibers, respectively. Furthermore, we show that the transmission rate of the proposed system increases as the number of used energy levels increases, which is not the case for fundamental solitons due to their effective time-amplitude constraint. For example, rate increases of 164% and 70% over lossy and lossless fibers respectively are reported when using four energy levels.

Perspective method for reconfigurable intelligent surface design for NOMA transmission: Theoretical study

The method (paradigm) presented here was inspired by two basic ideas: the assumption of the MIMO-RIS-NOMA communication channel as a dispersive channel with fading, and the representation of the latter by means of the generalized Kronecker channel model (GKCM), applying the “orthogonalization approach” based on utilization of the “universal” eigen basis in the form of prolate spheroidal wave functions (PSWFs). In other words, the essence of the approach is representing the MIMO-RIS-NOMA channel by means of a finite set of artificially created orthogonal trajectories for wave propagation, which are connected by the coupling matrix (CM) of the RIS. The goal for RIS (through the selection of specific CM elements) is to provide a decoding system for the NOMA users with the required SNR values for each user to guarantee successful decoding processing. A theoretical analysis of this paradigm is presented.

Non-asymptotic bounds for discrete prolate spheroidal wave functions analogous with prolate spheroidal wave function bounds

Non-asymptotic bounds for discrete prolate spheroidal wave functions analogous with prolate spheroidal wave function bounds

A new approach to generate diffuse sound pressure fields

If the acoustic wavelength is small compared to the characteristic size of an enclosed space, the sound field in that space is often modeled as diffuse. A diffuse sound field is conventionally defined as a zero-mean circularly-symmetric complex Gaussian random field. A more recent, generalized definition is that of a sound field having mode shapes that are diffuse in the conventional sense, and eigenfrequencies that conform to the Gaussian Orthogonal Ensemble. Such a generalized diffuse sound field can represent a random ensemble of sound fields that share gross features such as modal density and reverberation time, but otherwise have any possible arrangement of local wave scattering features. In this contribution, realizations of a conventional diffuse sound field or, equivalently, of the mode shapes of a generalized diffuse sound field, are generated in a Monte Carlo framework. As a discrete decomposition is numerically expensive when the sound pressures at many locations are of interest, a fast analytical decomposition based on prolate spheroidal wave functions is developed. The approach is numerically validated by comparison with a detailed room model, where random wave scatterers are explicitly modeled, and good correspondence is observed. Applications involving correlated sound sources and sound-structure interaction are presented.

High Dynamic Range Solar Radio Imaging Based on Deconvolution Using Prolate Spheroidal Wave Functions

利用综合孔径射电望远镜对太阳进行观测时，通过对图像中存在的明亮扩展源进行准确建模并移除，可以更好地观测视场内的微弱源并提高图像的动态范围。在射电天文领域，主要利用CLEAN算法对图像中的明亮源进行移除，以显示微弱的背景。然而，使用图像像素作为基函数的CLEAN算法的固有限制导致其对扩展源的建模效果较差。为了克服这种限制，将基于长椭球面波函数（Prolate Spheroidal Wave Functions, PSWF）的去卷积方法应用于太阳射电成像。PSWF最优正交基由脏图中的感兴趣区域（Region of Interest, ROI）和UV覆盖共同决定。为了验证该方法的有效性，基于PSWF正交基对均匀圆环阵观测的太阳射电图像进行去卷积，并从动态范围和保真度两个方面定量化对比了CLEAN算法和基于PSWF正交基方法的性能。基于PSWF正交基去卷积方法剩余脏图中的微弱源更接近真实情况且动态范围更高。

High Dynamic Range Solar Radio Imaging Based on Deconvolution Using Prolate Spheroidal Wave Functions

利用综合孔径射电望远镜对太阳进行观测时，通过对图像中存在的明亮扩展源进行准确建模并移除，可以更好地观测视场内的微弱源并提高图像的动态范围。在射电天文领域，主要利用CLEAN算法对图像中的明亮源进行移除，以显示微弱的背景。然而，使用图像像素作为基函数的CLEAN算法的固有限制导致其对扩展源的建模效果较差。为了克服这种限制，将基于长椭球面波函数（Prolate Spheroidal Wave Functions, PSWF）的去卷积方法应用于太阳射电成像。PSWF最优正交基由脏图中的感兴趣区域（Region of Interest, ROI）和UV覆盖共同决定。为了验证该方法的有效性，基于PSWF正交基对均匀圆环阵观测的太阳射电图像进行去卷积，并从动态范围和保真度两个方面定量化对比了CLEAN算法和基于PSWF正交基方法的性能。基于PSWF正交基去卷积方法剩余脏图中的微弱源更接近真实情况且动态范围更高。

Slepian eigenvalues as tunnelling rates

We calculate the eigenvalues of an integral operator associated with Prolate Spheroidal Wave Functions (or Slepian functions) by interpreting them as tunnelling probabilities in an analogous quantum problem. Doing so allows us to extend a well-known approximation due to Slepian so that it applies outside an important transition region where these eigenvalues pass from being near unity to being near zero. Study of the eigenvalues has traditionally been associated with problems arising in signal analysis and optics but have more recently found relevance in quantifying the channel strengths available to Multiple-In Multiple-Out (MIMO) radio communication. The approach presented promises easier generalisation to the broader range of geometries possible in the latter context.

Pulses With Minimum Residual Intersymbol Interference for Faster Than Nyquist Signaling

Faster than Nyquist signaling increases the spectral efficiency of pulse amplitude modulation by accepting intersymbol interference, where an equalizer is needed at the receiver. Since the complexity of an optimal equalizer increases exponentially with the number of the interfering symbols, practical truncated equalizers assume shorter memory. The power of the resulting residual interference depends on the transmit filter and limits the performance of truncated equalizers. In this paper, we use numerical optimizations and the prolate spheroidal wave functions to find optimal time-limited pulses that achieve minimum residual interference. Compared to root raised cosine pulses, the new pulses decrease the residual interference by an order of magnitude, for example, a decrease by 32 dB is achieved for an equalizer that considers four interfering symbols at 57% faster transmissions. As a proof of concept, for the 57% faster transmissions of binary symbols, we showed that using the new pulse with a 4-state equalizer has better bit error rate performance compared to using a root raised cosine pulse with a 128-state equalizer.

Gaussian Bounds for the Heat Kernel Associated to Prolate Spheroidal Wave Functions with Applications

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in $${\mathbb R}^d$$ . Further, we develop the related to the PSWFs of order zero smooth functional calculus, which provides the necessary groundwork in developing the theory of Besov and Triebel–Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel–Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel–Lizorkin spaces generated by the Legendre operator.

On the numerical evaluation of the prolate spheroidal wave functions of order zero

We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct asymptotic expansions of the prolate spheroidal wave functions, our algorithm operates by numerically solving the Riccati equation satisfied by their logarithms. Its running time grows much more slowly with bandlimit and characteristic exponent than standard algorithms. We illustrate this and other properties of our algorithm with numerical experiments.