Spheroidal Wave Functions Research Articles

Clutter Rank Estimation Method for Bistatic Radar Systems Based on Prolate Spheroidal Wave Functions

Bistatic radar exhibits spatial isomerism and diverse configurations, leading to unique clutter characteristics distinct from those of monostatic radar. The clutter rank serves as a pivotal indicator of clutter characteristics, enabling the quantification of clutter severity. Space-time adaptive processing (STAP) is a critical technique to detect moving targets, and clutter rank determines the number of independent and identically distributed (IID) training samples and the degree of freedom (DOF) for effective suppression of clutter that STAP requires. Therefore, the accurate estimation of clutter rank for bistatic radar can provide a crucial indicator for designing and constructing STAP processors, thereby facilitating fast and efficient clutter suppression in bistatic radar systems. This study is based on the idea that clutter rank is the number of prolate spheroidal wave function (PSWF) orthogonal bases utilized for approximating the clutter signal. Firstly, the challenge of utilizing PSWF orthogonal bases for approximating the clutter signal in bistatic radar is elucidated. This pertains to the fact that, unlike monostatic radar clutter, bistatic radar clutter is not capable of being expressed as a single-frequency signal. The clutter rank estimation for bistatic radar is thus derived as the frequency bandwidth estimation. Secondly, to achieve this estimation, the frequency distribution of each individual scattering unit is investigated, thereby determining their extending frequency broadening (EFB) as compared to that of single-frequency. Subsequently, the integral average of EFB across the entire range bin is computed, ultimately enabling the acquisition of bistatic radar’s frequency bandwidth. Finally, the estimation method is extended to non-side-looking mode and limited observation areas with pattern modulation. Simulation experiments confirm that our proposed method provides accurate clutter rank estimations, surpassing 99% proportions of large eigenvalues across various bistatic configurations, observation modes, and areas.

Analytical solution for acoustic radiation force and torque on a spheroid near a rigid or free planar boundary.

An analytical solution is developed for the acoustic radiation force and torque caused by an arbitrary sound field that is incident on a compressible spheroid of any size near a planar boundary that is either rigid or pressure release. The analysis is an extension of a recent solution for a compressible sphere near a planar boundary [Simon and Hamilton, J. Acoust. Soc. Am. 153, 627-642 (2023)]. Approximations that account for a boundary formed by a two-fluid interface may be incorporated as in the previous analysis for a sphere. The present solution is based on expansions of the total acoustic pressure field in spheroidal wave functions and the use of addition theorems. Verification of the solution is accomplished by comparison with a finite element model. Examples are presented for incident fields that are either plane or spherical waves. Effects resulting from the presence of the boundary are studied by comparing the full theory with a simplified model in which multiple scattering is neglected. Numerical implementation of the proposed solution is also discussed.

Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

Orthogonal waveform-based backscattering interference suppression technique for underwater optical wireless communication

The backscattering effects degrade the detection performance in a full-duplex underwater optical wireless communication (UOWC) system when the power of received backscattering interference is comparable to that of a target signal. In this paper, the orthogonal waveforms with the Hermite–Gaussian function and prolate spheroidal wave function are first introduced for a pulse position modulation based dynamic UOWC system to alleviate the backscattering effects. Then, a joint maximum likelihood detection scheme is proposed accordingly. Experimental results verify the superiority of the proposed method with orthogonal waveforms over the scheme without orthogonal pulse shaping in a dynamic underwater bubbly channel.

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 new practical approach to light scattering by spheroids with the use of spheroidal and spherical function bases

The approximation of real non-spherical scatterers by spheroids is used in various applications. It often requires fast massive calculations of the optical properties of spheroids. The most powerful approach to that is known to be a field expansion in terms of spheroidal wave functions, in particular, in the way suggested by Farafonov over 30 years ago. We improve the main shortcomings of such an approach. Our solution is formulated in terms of normalised spheroidal functions, and firstly for their definition given by Meixner & Schäfke, which is computationally favourable and is required by the unique subroutines recently created to compute these functions. By means of T-matrix transformations we solve a long-standing major problem of Farafonov’s version, namely the accuracy and time losses for one kind (TE mode) of the incident wave polarisation. Apart from this and other improvements of this solution, for the first time we relate its single particle spheroidal T-matrix to the standard spherical one which is widely employed for particle ensembles. The constructed algorithm has been extensively numerically tested. It is found to be very accurate for dielectric spheroids with the aspect ratio a/b reaching 100 and the diffraction (size) parameter xa=2πa/λ as large as 300, where a and b are the major and minor semi-axes, respectively, λ is the wavelength. The algorithm is supplied with a program interface to the free package CosTuuM to perform parallel computations of various optical properties for ensembles of spheroids with different distributions over orientation (alignment) and shape.

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.

Partial response continuous phase modulation with a prolate spheroidal wave function for compass system in the S-band

Continuous phase modulation (CPM) has the characteristics of high power efficiency, spectral efficiency, and less out-of-band radiation, which is very suitable for the Compass S-band (2483.5–2500 MHz) with limited power and bandwidth. However, as more and more navigation systems share the S-band, the mutual interference between different systems is gradually increasing, and the compatibility of CPM signals with rectangular or raised cosine pulses needs to be further improved. To enhance the navigation performance and compatibility of current CPM candidate signals, we propose to apply continuous phase modulation with prolate spheroidal wave function (CPM-PSWF) to S-band navigation. The proposed modulation scheme selects PSWF with excellent time-frequency energy aggregation and flexible time-bandwidth product as the frequency pulse function of CPM. Then, the influence of key modulation parameters such as M, L, h, and C on CPM-PSWF power spectral density is analyzed, and a specific partial-response CPM-PSWF signal is proposed as the S-band candidate navigation signal. Finally, the navigation performance evaluation criteria are derived, and the proposed signal scheme and existing candidate schemes are comprehensively evaluated. Theoretical analysis and simulation results indicate that compared with CPM with BM2RC(8), cos-phase binary offset carrier (BOCc(4,4)), sin-phase binary offset carrier (BOCs(4,4)), minimum shift keying-BOCs (MSK-BOCs(4,4)) and binary offset carrier (BPSK(8)), the proposed signal scheme not only delivers exceptional spectral performance but also exhibits superior performance in code tracking accuracy and multipath suppression. In addition, in the acquisition and tracking phase, compared with other candidate signals except for BOCc(4,4), the mutual interference between the proposed signal scheme and most S-band signals is the lowest, which is easier to realize the compatibility of the Compass system with IRNSS, Galileo, and Globalstar systems.

Spectral triples and $\zeta$-cycles

We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S . We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S . Then we use these functions to condition the canonical spectral triple of the circle of length L = 2\log (S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence that, when one varies L , the low lying spectrum of the perturbed spectral triple resembles the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. This fact is tested numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros obtained by the first author.

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.

Acoustic radiation force and torque on an object near a penetrable interface or impedance boundary

An analytical approach for calculating radiation force and torque on a sphere or spheroid near a rigid or pressure release boundary due to an arbitrary incident acoustic field was developed recently by the authors [Proc. Meet. Acoust. 48, 045004 (2022)]. In that approach, the linear scattering problem is solved using the method of images along with expansions of the acoustic fields in spherical or spheroidal wave functions. The expansion coefficients can then be substituted into an existing expression to obtain the radiation force and torque on the object. In the present work, the aforementioned theory is extended to include penetrable interfaces and impedance boundaries using an approximation of the interfacial boundary conditions. For fluid-fluid interfaces, the object can be placed on either side of the interface with respect to the incident field. This approximate analytical approach is computationally efficient and compares well with the results from an independent finite element model. In addition, an experiment was conducted with a plastic sphere submerged in water and positioned near a boundary. Results from the experiment are discussed in relation to the analytical model. [BES is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]