Spheroidal Wave Functions Research Articles

Short-time FORM analysis for extreme roll motion prediction in beam seas

This paper presents an efficient computational method to predict extreme roll motions by the First Order Reliability Method (FORM). The method facilitates accurate estimation of the reliability index distribution by short time domain calculations. The calculations are based on the transfer function of roll motion together with an equivalent linear damping for the estimation of the most probable largest roll motion using nonlinear time domain simulations. The linearization of the damping gives rise to an inconsistency (a “time shift”) in the time series evaluation. Thus, in order to account for this nonlinear effect, when evaluating the most probable largest roll motions, a method to correct the time shift from the linear solution is proposed. Two different container ships in beam seas and zero forward speed are used to demonstrate the effectiveness of the method. It is also shown that a further reduction of computational efforts is attained by using the Karhunen-Loeve wave representation based on Prolate Spheroidal Wave Functions (PSWF).

Efficient realization of on-demand functional ultrasonic fields based on prolate spheroidal wave functions from sampling theorem.

The past decades have witnessed great efforts in the on-demand ultrasonic field design in which the time reversal technology was widely used in the whole-space acoustic hologram. In practice, the acoustic field of interest is usually bounded in a finite region with flexible distribution. Here, the use of prolate spheroidal wave functions to generate an arbitrary ultrasonic field in a finite region is proposed. The prolate spheroidal functions, which form a complete set of band limited functions and are orthogonal in the infinite and finite regions, can be efficiently reconstructed by the sampling theorem. To display the validation of the proposed method, two types of functional ultrasonic fields are numerically simulated. One type is the ultrasound standing wave field for which six nodes and two nodes are separately realized for two different types of standing waves in the limited range of (-2λ, 2λ). In addition, a composite standing wave field is stimulated with more complicated nodal distributions. The other type is the ultrasound focusing field, where three focal spots with the mainlobe sizes of λ, 0.5λ, and 0.35λ are demonstrated. It is worth noting that the nontrivial side lobes for super-oscillation focusing are designed to be about 3λ away from the central focal spot (the mainlobe size 0.35λ). This work has much significance in the applications of acoustic tweezing, ultrasonic imaging, and treatment.

Spheroidal magnetic stars rotating in vacuum

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasis is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Results. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving (a/rL)2 and (a/R)2 where a is the stellar oblateness or prolateness, R the smallest star radius, and rL the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

Electrostatic interaction between two spheroidal particles at large separations

Asymptotic expressions for the electrostatic interaction energy between two weakly charged spheroidal particles (prolate and oblate) with uniform surface potential or surface charge density at large separations in an electrolyte solution are derived on the basis of the linear superposition approximation. The electrostatic interaction between two spheroids is found to be the screened Coulomb interaction between two hypothetical point particles with orientation-dependent charges. Explicit interaction energy equations, which are expressed in terms of spheroidal wave functions, are given for four particular configurations of two similar spheroids, that is, side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates. Comparison is also made with results obtained using Derjaguin’s approximation. Electrostatic interaction between two spheroids • Asymptotic expressions for the electrostatic interaction energy between two spheroidal particles are derived. • Constant surface potential and charge density cases are considered. • Side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates are considered.

On the sound fields of oblate and prolate hemispheroids in infinite baffles for directivity control.

The hemispheroid is presented as an apodized form for controlling the beam width of a sound source by varying its height-to-radius ratio. Directivity patterns, on-axis responses, and radiation impedances are calculated for various height-radius ratios of the oblate hemispheroid using spheroidal wave functions. It turns out that, for smaller angles at least, there is a direct relationship between the internal angle of the semi-elliptic cross section and the half-cone angle within which the far-field pressure is largely contained at high frequencies. The hemispheroid is compared with both a spherical cap, which produces a much less regular response, and a high-frequency asymptotic approximation. The high-frequency asymptotic approximation is in the form of a flat circular radiator with a delay that increases radially from the center to the perimeter, as used in some electrostatic loudspeakers. With the almost complete absence of lobes, this appears to be an effective alternative means of apodization to a shaded array and is more efficient because, unlike a shaded array, constant axial velocity is maintained over the whole surface. A high-frequency approximation is also derived for a prolate hemispheroid. Since this may be formed from a planar array, a beam steering option is added.

Multi-carrier modulation scheme based on prolate spheroidal wave functions with generalized index modulation

Multi-carrier modulation (MCM) based on prolate spheroidal wave functions (PSWFs) with generalized index modulation (GIM) was proposed in this paper. The proposed method relaxes the restriction on the number of the activated PSWFs signals, effectively increases the number of modulation symbol combinations and improves the spectral efficiency (SE) of the system. It solves the problem that the SE of the MCM-PSWFs with signal grouping optimization (MCM-PSWFs-SGO) is limited by the fixed number of the activated signals. At transmitter, a novel power control scheme of modulation signal based on fixed power gain is proposed. By introducing fixed power gain, the complexity of GIM transmitter is effectively reduced, and the minimum Euclidean distance between modulation symbols is increased. At receiver, based on the fact that the activation states of different PSWFs signals are independent of each other, a novel signal index detection method based on local optimization is proposed, which converts the signal index detection to the activation state detection of single branch signals. Under the same bit error rate (BER) performance, the complexity of signal index detection is reduced from $O(mMgn)$ of classical GIM to $O(gn)$. Theoretical and simulation results show that the proposed method can achieve improvement of SE. For example, when BER is $10^{-5}$, compared with MCM-PSWFs-SGO, the SE of the proposed method increases by 14.5%, and the required signal to noise ratio decreases by 2.73 dB. Compared with classical GIM, the SE increases by 2.9% and the required signal to noise ratio decreases by 1.92 dB.

Spheroidal ambisonics: Spatial audio in spheroidal bases.

Ambisonics is an established framework to capture and reproduce spatial sound fields based on the spherical harmonics representation [Gerzon (1973). J. Audio Eng. Soc. 21(1), 2-10]. A generalization-spheroidal ambisonics-based on spheroidal wave functions is proposed for use with spheroidal microphone arrays. Analytical conversion from spheroidal ambisonics to spherical ambisonics are derived to ensure compatibility with the existing ambisonics ecosystem. Numerical experiments verify spheroidal ambisonics encoding and transcoding for spatial sound field recording. The sound field reconstructed from the transcoded coefficients has a zone of accurate reconstruction which is prolonged towards the long axis of a prolate spheroidal microphone array.

Method of Moments Based Coronagraph Pupil Design for Exoplanet Exploration

Telescope coronagraph is an instrument to observe faint exoplanets located at very close proximities to their star by shielding the direct starlight. In this paper, the coronagraph pupil-mask system is described through a series of Fourier transforms, which leads to an eigenvalue system described by an integral equation. The method of moments (MoM) is applied to solve such an eigenvalue problem to obtain the eigen modes that represent an optimal set of refractive index distributions on the pupil. Due to the application of the MoM, the shape of coronagraph pupil can be arbitrary. A numerical example considering a circular pupil is given to demonstrate capability of the proposed method. The accuracy of the MoM solution is validate by comparing the numerical solutions with the analytical solutions given by the generalized prolate spheroidal wave functions.

Estimates for the SVD of the Truncated Fourier Transform on $$L^2(\cosh (b|\cdot |))$$ and Stable Analytic Continuation

The Fourier transform truncated on \([-c,c]\) has for a long time been analyzed as acting on \(L^2(-1/b,1/b)\) into \(L^2(-1,1)\) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator defined on the larger space \(L^2(\cosh (b|\cdot |))\) on which it remains injective. The main purpose is (1) to provide nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c and (2) to derive nonasymptotic upper bounds on the sup-norm of the right-singular functions. Finally, we propose a numerical method to compute the SVD. These are fundamental results for Gaillac and Gautier (Adaptive estimation in the linear random coefficients model when regressors have limited variation, Bernoulli, arXiv:1905.06584). This paper considers as an illustrative example analytic continuation of nonbandlimited functions when the function is observed with error on an interval.

Weil positivity and trace formula the archimedean place

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of Connes (Sel Math (NS) 5(1):29–106, 1999). We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for \(\Lambda =1\). We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH.

Multi-carrier modulation scheme based on prolate spheroidal wave functions with signal grouping optimization

Multi-carrier modulation scheme based on prolate spheroidal wave functions with signal grouping optimization

Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in ℓ2(Z) which are strictly bandlimited to a frequency band [−W,W] and maximally concentrated in a time interval {0,…,N−1}. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in CN whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band [−W,W]. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior – slightly fewer than 2NW eigenvalues are very close to 1, slightly fewer than N−2NW eigenvalues are very close to 0, and very few eigenvalues are not near 1 or 0. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near 0 or 1. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1−ϵ. Also, we obtain bounds detailing how close the first ≈2NW eigenvalues are to 1 and how close the last ≈N−2NW eigenvalues are to 0. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1−ϵ.

Monodromy in prolate spheroidal harmonics

AbstractWe show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators onhave a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

Acoustic radiation torque on a compressible spheroid.

Starting with the theoretical framework for calculating the acoustic radiation force on a compressible spheroid [Jerome et al., J. Acoust. Soc. Am. 148, 2403-2415 (2020)], the present work develops a model for the corresponding acoustic radiation torque. A general result is obtained that may be applied to an object of arbitrary size, shape, and impedance in an arbitrary incident sound field. Like for the radiation force, the general result for the radiation torque is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. For the compressible spheroid under consideration, spheroidal wave expansions are employed to satisfy the boundary conditions on the surface of the spheroid to obtain the scattering coefficients. Results are presented for the radiation torque exerted on a compressible spheroid by a progressive or standing incident plane wave. The results illustrate the dependence of the radiation torque on the size, aspect ratio, and impedance of the spheroid and on its orientation with respect to the incident wave field.

Non-asymptotic behavior of the spectrum of the sinc-kernel operator and related applications

Prolate spheroidal wave functions have recently attracted much attention in applied harmonic analysis, signal processing, and mathematical physics. They are eigenvectors of the sinc-kernel operator Qc: the time- and band-limiting operator. The corresponding eigenvalues play a key role, and the aim of this paper is to obtain precise non-asymptotic estimates with explicit constants related to the spectrum of Qc. This issue is rarely studied in the literature, while the asymptotic behavior of the spectrum of Qc has been well established from the 1960s. However, many recent applications require such non-asymptotic behavior. As applications of our non-asymptotic estimates, we first provide estimates for the constants appearing in the Remez- and Turàn–Nazarov-type concentration inequalities. Then, we give a non-asymptotic upper bound for the gap probability of the sinc determinantal point process. Consequently, one gets a non-asymptotic estimate for the hole probability, associated with bulk scaled asymptotics of a random matrix from the Gaussian unitary ensemble. This last result can be considered as a complement of the various and more involved asymptotic counterparts of this estimate.

Acoustic scattering by two fluid confocal prolate spheroids

The exact spheroidal-function series solution for the time-harmonic acoustic scattering of a plane wave by two fluid confocal prolate spheroids is developed and a numerical implementation is formulated and validated by independent methods. The two spheroids define three regions in which the acoustic fields are expanded in terms of spheroidal wave functions multiplied by unknown coefficients. These expansions are forced to satisfy the boundary conditions and by using the orthogonality properties of the involved functions an infinite matricial system for the coefficients is obtained. The resulting system is then solved through a truncation procedure. The implementation has no limitations regarding the sound speed and density of the three media involved or in the incidence frequency.

Sampling the Flow of a Bandlimited Function

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states f_t=phi _t*f resulting from the convolution with a kernel phi _t. It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function widehat{f} away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

Sparse Bayesian Learning Based Direction-of-Arrival Estimation under Spatially Colored Noise Using Acoustic Hydrophone Arrays

Direction-of-arrival (DOA) estimation in a spatially isotropic white noise background has been widely researched for decades. However, in practice, such as underwater acoustic ambient noise in shallow water, the ambient noise can be spatially colored, which may severely degrade the performance of DOA estimation. To solve this problem, this paper proposes a DOA estimation method based on sparse Bayesian learning with the modified noise model using acoustic vector hydrophone arrays. Firstly, an applicable linear noise model is established by using the prolate spheroidal wave functions (PSWFs) to characterize spatially colored noise and exploiting the excellent performance of the PSWFs in extrapolating band-limited signals to the space domain. Then, using the proposed noise model, an iterative method for sparse spectrum reconstruction is developed under a sparse Bayesian learning (SBL) framework to fit the actual noise field received by the acoustic vector hydrophone array. Finally, a DOA estimation algorithm under the modified noise model is also presented, which has a superior performance under spatially colored noise. Numerical results validate the effectiveness of the proposed method.

A method to calculate acoustic radiation modes based on spheroidal wave functions.

Acoustic radiation modes (ARMs) have been widely used in noise control engineering. For an arbitrarily shaped radiator, ARMs are usually obtained by using the boundary element method (BEM). For high frequencies and large structures, the BEM calculation is usually time consuming. An alternative method was recently developed in which the ARMs are obtained by using generalized singular value decomposition and spherical harmonic functions. In this paper, spheroidal harmonic functions are used instead of spherical harmonic functions. The proposed method is potentially more efficient for ARM calculations for structures with one dimension significantly larger than the other two.

Real-time deterministic prediction of wave-induced ship responses based on short-time measurements

This paper studies real-time deterministic prediction of wave-induced ship motions using the autocorrelation functions (ACFs) from short-time measurements, namely the instantaneous ACFs. The Prolate Spheroidal Wave Functions (PSWF) are introduced to correct the large lag time errors in the instantaneous sample ACF, together with a modification of the autocorrelation (AC) matrix for ensuring its positive definiteness. The validity of the PSWF-based ACFs is first examined by using the ship motion measurements from model experiment under stationary wave excitations. It is shown that the use of PSWF-based ACFs leads to better prediction accuracy than direct use of sample ACFs. The validation is then extended to ship motion prediction using in-service data from a container ship, and an improvement of the prediction accuracy by PSWF-based ACFs is again found. Finally, the effectiveness of use of the instantaneous ACFs for non-stationary wave-induced responses is highlighted by comparing with the prediction results based on the ACFs from long-time measurements.