Spheroidal Wave Functions Research Articles

Quantum limited superresolution of extended sources in one and two dimensions

We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed mid-point and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wavefunctions (PSWFs) on a centered line segment and its generalized version on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we subsequently calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.

Aliasing error of the exp⁡(β1−z2) kernel in the nonuniform fast Fourier transform

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel ϕ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new “exponential of semicircle” kernel ϕ ( z ) = e β 1 − z 2 , for z ∈ [ − 1 , 1 ] , zero otherwise, whose Fourier transform ϕ ˆ is unknown analytically. We place this kernel on a rigorous footing by proving an aliasing error estimate which bounds the error of the one-dimensional NUFFT of types 1 and 2 in exact arithmetic. Asymptotically in the kernel width measured in upsampled grid points, the error is shown to decrease with an exponential rate arbitrarily close to that of the popular Kaiser–Bessel kernel. This requires controlling a conditionally-convergent sum over the tails of ϕ ˆ , using steepest descent, other classical estimates on contour integrals, and a phased sinc sum. We also draw new connections between the above kernel, Kaiser–Bessel, and prolate spheroidal wavefunctions of order zero, which all appear to share an optimal exponential convergence rate.

Approximate analytic expressions for the electrophoretic mobility of spheroidal particles

Approximate analytic expressions are derived for the electrophoretic mobility of spheroidal particles (prolate and oblate) carrying low zeta potential in an electrolyte solution under an applied tangential or transverse electric field. The present approximation method, which is based on the observation that the electrophoretic mobility of a particle is determined mainly by the distortion of the applied electric field by the presence of the particle. The exact expression for the equilibrium electric potential distribution around the particle, which can be expressed as an infinite sum of spheroidal wave functions, is not needed in the present approximation. The electrophoretic mobility values calculated with these approximate expressions for spheroidal particles with constant surface potential or constant surface charge density are in excellent agreement with the exact numerical results of previous reports with the relative errors less than about 4%.

Acoustic radiation force on a compressible spheroid

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions. There is no restriction on the size or acoustic impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The solution is the same as that developed previously for the radiation force on an elastic sphere, which is expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii etal., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Spheroidal wave expansions are used to satisfy the boundary conditions on the compressible spheroid, and far-field asymptotes are used to relate the spheroidal and spherical wave scattering coefficients analytically. Explicit expressions for the scattering coefficients are available for spheroids with rigid or free surfaces. Results are compared with available analytical solutions for various limiting cases. The theoretical framework may be employed for objects of arbitrary shape provided the spherical wave expansion of the scattered field is known. [T.S.J. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.] a)Deceased.

Acoustic radiation force on a compressible spheroid.

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions that are matched analytically in the far field. There is no restriction on the size or impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The form of the solution is the same as that developed previously for the radiation force on an elastic sphere, which is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. Spheroidal wave expansions are employed to satisfy the boundary conditions and obtain the scattering coefficients. While the scattering coefficients must be obtained numerically for compressible spheroids, explicit expressions in terms of radial wave functions are available for spheroids with rigid or free surfaces. Results are compared with available analytical expressions for various limiting cases. The theoretical framework may be extended to objects of arbitrary shape.

An Optimized Design Method of PSWFs Pulse Group Based on DFT Precoding

Aiming at the problems of non-sinusoidal time-domain multiple quadrature modulation signals based on Prolate Spheroidal Wave Functions (PSWFs), as the number of pulses increases, the bandwidth efficiency of the modulation signal frequency band decreases, the Peak-to-Average Power Ratio (PAPR) and the implementation complexity are high, a method for optimal design of PSWFs pulse groups based on DFT precoding is proposed. The PSWFs pulse signal frequency domain generation method is adopted. By introducing the precoding matrix, the multi-channel PSWFs pulse group signals are optimized and designed, and then the input data is redistributed to obtain new PSWFs pulse group signals. Experimental results show that this method can further improve the power efficiency of the modulated signal and effectively reduce the peak-to-average power ratio of the modulated signal without changing the orthogonality of the original PSWFs subcarriers. As the number of carriers increases, the suppression effect on the peak-to-average power ratio becomes more obvious.

Quantum limited source localization and pair superresolution in two dimensions under finite-emission bandwidth

Optically localizing a single quasi-monochromatic source to sub-diffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width, $w$, of the optical system's point-spread function (PSF) and the sought localization precision, $d$, i.e., as $\alpha(w/d)^2$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\alpha$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\alpha$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.

Estimation of autocorrelation function and spectrum density of wave-induced responses using prolate spheroidal wave functions

Predicting the wave-induced response in the near-future is of importance to ensure safety of ships. To achieve this target, a possible method for deterministic and conditional prediction of future responses utilizing measured data from the most recent past has been developed. Herein, accurate derivation of the autocorrelation function (ACF) is required. In this study, a new approach for deriving ACFs from measurements is proposed by introducing the Prolate Spheroidal Wave Functions (PSWF). PSWF can be used in two ways: fitting the measured response itself or fitting the sample ACF from the measurements. The paper contains various numerical demonstrations, using a stationary heave motion time series of a containership, and the effectiveness of the present approach is demonstrated by comparing with both a non-parametric and a parametric spectrum estimation method; in this case, Fast Fourier Transformation (FFT) and an Auto-Regressive (AR) model, respectively. The present PSWF-based approach leads to two important properties: (1) a smoothed ACF from the measurements, including an expression of the memory time, (2) a high frequency resolution in power spectrum densities (PSDs). Finally, the paper demonstrates that a fitting of the ACF using PSWF can be applied for deterministic motion predictions ahead of current time.

Time-Limited Codewords over Band-Limited Channels: Data Rates and the Dimension of the W-T Space

We consider a communication system whereby T-seconds time-limited codewords are transmitted over a W-Hz band-limited additive white Gaussian noise channel. In the asymptotic regime as , it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of . We use prolate spheroidal wave functions to obtain an information lossless equivalent discrete formulation and then we apply Polyanskiy’s results on coding in the finite block-length regime. We derive upper and lower bounds on the achievable rates and the corresponding degrees of freedom and we numerically evaluate them for sample values of . The bounds are asymptotically tight and numerical computations show the gap between them decreases as increases. Additionally, the possible decrease from in the available degrees of freedom is upper-bounded by a logarithmic function of .

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Extreme value prediction of nonlinear ship loads by FORM using Prolate Spheroidal Wave Functions

In this study, a method for predicting the extreme value distribution of the Vertical Bending Moment (VBM) in a flexible ship under a given short-term sea state is presented. The First Order Reliability Method (FORM) is introduced to evaluate the Probability of Exceedances (PoEs) of extreme VBM levels. The Karhunen-Loeve (KL) representation of stochastic ocean wave is adopted in lieu of the normal wave representation using the trigonometric components, by introducing the Prolate Spheroidal Wave Functions (PSWFs) to formulate the wave elevations. By this means, reduction of the number of stochastic variables to reproduce ocean wave is expected, which in turn the number of computations required during FORM based prediction phases is significantly reduced. In this study, the Reduced Order Model (ROM), which was developed in our previous studies, is used to yield the time-domain VBMs along with the hydroelastic (whipping) component in a ship. Two different short-term sea states, moderate and severe ones, are assumed. The FORM based predictions using PSWF for normal wave-induced VBM are then validated by comparing with those using the normal trigonometric wave representation and Monte Carlo Simulations (MCSs). Through a series of numerical demonstrations, the computational efficiency of the FORM based prediction using PSWF is presented. Then, the validation is extended to the severe sea state where the whipping vibration contributes to the extreme VBM level to a large degree, and finally the conclusions are given.

Strict parity symmetric prolate spheroidal wave functions signal construction and low complexity detection method

Aiming at solving the problem of high signal processing and detection complexity of prolate spheroidal wave functions (PSWFs), we propose a time domain strict parity symmetry PSWFs signal construction method, based on the characteristics of PSWFs with the same parity symmetry after linear operation. According to the characteristics that the PSWFs of the same parity symmetry have the same orthogonality in the half-symbol period and the whole symbol period, we propose a PSWFs signal detection method based on the parity characteristics, which groups processing of odd symmetric and even symmetric signals and detects PSWFs signal in the half-symbol period signal to reduce the number of signal solutions for participating in the operation. Compared with the coherent detection method, theoretical and numerical analyses show that the proposed signal detection method can reduce the computational complexity from $O(NQ)$ to $O(NQ/2)$ (i.e., about 50%) without reducing the system error performance.

Discrete Prolate Spheroidal Wave Functions: Further Spectral Analysis and Some Related Applications

For fixed $W\in \big(0,\frac{1}{2}\big)$ and positive integer $N\geq 1,$ the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N,$ $0\leq k\leq N-1$ form the set of the eigenfunctions of the positive and finite rank integral operator $\widetilde Q_{N,W},$ defined on $L^2(-1/2,1/2),$ with kernel $K_N(x,y)=\frac{\sin(N\pi(x-y))}{\sin(\pi(x-y))}\, \mathbf 1_{[-W,W]}(y).$ It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of their eigenvalues $\widetilde \lambda_{k,N}(W).$ In his pioneer work \cite{Slepian}, D. Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as the asymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give further properties as well as new non-asymptotic decay rates of the spectrum of the operator $\widetilde Q_{N,W}.$ In particular, we show that each eigenvalue $\widetilde \lambda_{k,N}(W)$ is up to a small constant bounded above by the corresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs). Then, based on the well established results concerning the distribution and the decay rates of the eigenvalues associated with the PSWFs, we extend these results to the eigenvalues $\widetilde \lambda_{k,N}(W)$. Also, we show that the DPSWFs can be used for the approximation of classical band-limited functions and they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Almost everywhere convergence of prolate spheroidal series

In this paper, we show that the expansions of functions from L p -Paley–Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for 1 < p < ∞ , even in the cases when they might not converge in L p -norm. We thereby consider the classical Paley–Wiener spaces P W c p ⊂ L p ( R ) of functions whose Fourier transform is supported in [ − c , c ] and Paley–Wiener-like spaces B α , c p ⊂ L p ( 0 , ∞ ) of functions whose Hankel transform H α is supported in [ 0 , c ] . As a side product, we show the continuity of the projection operator P c α f : = H α ( χ [ 0 , c ] ⋅ H α f ) from L p ( 0 , ∞ ) to L q ( 0 , ∞ ) , 1 < p ≤ q < ∞ .

Frequency Domain Multi-Carrier Modulation Based on Prolate Spheroidal Wave Functions

A novel frequency domain multi-carrier modulation (MCM-FD) scheme is proposed based on prolate spheroidal wave functions (PSWFs) for reducing the high complexity in the conventional time domain PSWFs multi-carrier modulation (MCM-PSWFs-TD) schemes. By constructing the relationship between discrete representation and exponential function representation of MCM-PSWFs-TD signals, it can be observed that the orthogonality and parity symmetry of the waveforms of PSWFs in the frequency domain are the same as that in the time domain. Thus, the PSWFs signal can be divided into two groups based on their parity symmetry, while these two groups of PSWFs signals can be processed simultaneously. Based on this concept, signal waveforms with only half spectrum range are invoked in the process of information loading and signal detection for reducing the number of sampling points participated in the signal operation. Compared to the MCM-PSWFs-TD scheme, the proposed MCM-PSWFs-FD scheme is capable of significantly reducing the computational complexity without severely degrading the system performance, such as spectral efficiency (SE), bit error rate performance, signal energy concentration and peak-to-average power ratio (PAPR). Furthermore, the cyclic-prefix orthogonal frequency division multiplexing (CP-OFDM), OFDM with weighted overlap and add (WOLA-OFDM), filter OFDM (F-OFDM), universal filtered multi-carrier (UFMC), as well as the filter bank multi-carrier with offset quadrature amplitude modulation (FBMC-OQAM) are also demonstrated as benchmarks. Simulation results are provided for illustrating that the proposed MCM-PSWFs-FD scheme is capable of striking a favorable tradeoff between the computational complexity and the system performance (i.e. SE, out-of-band energy leakage, adjacent frequency band interference, and PAPR), while the signal waveform design of the MCM-PSWFs-FD scheme is also more concise and flexible than the benchmarks.

Sparse Signal Reconstruction Using Blind Super-Resolution With Arbitrary Sampling

The problem of blind super-resolution of sparse signals using arbitrary sampling and the atomic lift is studied. Using the Prolate Spheroidal Wave Functions (PSWFs) and matrix atomic norm, a new Semi-Definite Program (SDP) is proposed. Unlike the previous results, the proposed new version of SDP can localize spikes with higher precision and no need for the magnitude recovery and integer sampling schemes. Numerical simulations show that our method outperforms the recent techniques with generating a lower Normalized Mean Square Error (NMSE).

Sparse deconvolution via off-grid T.V minimization

We present a grid-less solution to the total variation minimization approach to the sparse deconvolution problem. Unlike the existing methods [1, 2], our algorithm works directly with time domain data, and does not suffer from the consequences of leakage errors incurred while transforming the time domain measurements to frequency domain. The total variation minimization problem requires us to optimize the total variation of a function. This is an infinite dimensional problem. To deal with the underlying infinite dimensionality, we consider its finite dimensional semi-infinite duals with point-wise constraints. We provide a finite parameterization to handle the point-wise constraints in a computationally tractable manner. This parameterization exploits the underlying bandlimitedness of the convolution kernel by employing Prolate Spheroidal Wave Functions. Consequently, we are able to reduce the total variation minimization problem into a semidefinite program, which can be solved in polynomial time. We demonstrate its utility via numerical simulation studies as well as real world signal reconstruction examples.

Extrapolation Theorem for Bandlimited Signals Associated with the Offset Linear Canonical Transform

The offset linear canonical transform (OLCT) is a novel and effective method in signal processing and optics. Many important properties and results of the OLCT have been well studied and published. However, the extrapolation theorem of the OLCT bandlimited signals still remains unknown. In this short paper, by using the generalized prolate spheroidal wave functions (GPSWFs), the extrapolation theorem for the OLCT bandlimited signals has been provided. Firstly, the concept of the GPSWFs in the OLCT domain has been proposed. Subsequently, the dual orthogonality and completeness of the GPSWFs associated with the OLCT have been derived based on the definition. In addition, the extrapolation theorem of the OLCT bandlimited signals has been obtained by using the orthogonality and completeness properties of the GPSWFs. Finally, the simulation results are presented to show the usefulness and effectiveness of the derived theorems.

Mean convergence of prolate spheroidal series and their extensions

The aim of this paper is to establish the range of p's for which the expansion of a function f∈Lp in a generalized prolate spheroidal wave function (PSWFs) basis converges to f in Lp. Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases cover the classical PSWFs for which the corresponding results have been previously established by Barceló and Cordoba.To establish those results, we prove a general result that allows to extend mean convergence in a given basis (e.g. Jacobi polynomials or Bessel basis) to mean convergence in a second basis (here the generalized PSWFs).

Exact greybody factors for the brane scalar field of five-dimensional rotating black holes

We study scalar perturbations of the five-dimensional rotating black holes and find an exact solution giving exact description of the Hawking radiation. Mathematically, the full solution for this spin-zero field is expressed in terms of the prolate spheroidal wave function with complex parameters. By using the spheroidal joining factor, we write the corresponding boundary condition and greybody factors. We also check that the exact result reproduces the low frequency limit of the greybody factor and shows good agreement with the known numerical results.