Prolate Spheroidal Wave Functions Research Articles

Frame properties of shifts of prolate spheroidal wave functions

We provide conditions on a shift parameter and number of basic prolate spheroidal wave functions with a fixed bandwidth and time concentrated to a fixed duration such that the shifts of the basic prolates form a frame or a Riesz basis for the Paley–Wiener space consisting of all square integrable functions with the given bandlimit.

Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels

In this paper, we employ the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels to solve eigenproblems of the prolate spheroidal resonators. To detect the spurious eigenvalues and the corresponding occurring mechanisms which are common issues while utilizing the boundary element method or the BIEM, we use angular prolate spheroidal wave functions and triangular functions to expand boundary densities. In this way, the boundary integral of a prolate spheroidal surface is exactly determined, and eigenequations are analytically derived. It is revealed that the spurious eigenvalues depend on the integral, representations and the shape of the inner boundary. Furthermore, it is interesting to find that some roots of the confocal prolate spheroidal resonator are double roots no matter that they are true or spurious eigenvalues. Illustrative examples include confocal prolate spheroidal resonators of various boundary conditions. To validate these findings and accuracy of the present approach, the commercial finite-element software ABAQUS is also applied to perform acoustic analyses. Good agreement is obtained between the acoustic results obtained by the null-field BIEM and those provided by the commercial finite-element software ABAQUS.

Coherence extrapolation for underwater ambient noise.

This paper considers extrapolation of the vertical coherence of surface-generated oceanic ambient noise to simulate measurements made on a longer sensor array. The extrapolation method consists of projecting the noise coherence measured with a limited aperture array into the domain spanned by prolate spheroidal wave functions, which are an orthogonal basis defined by array parameters and the noise frequency. Using simulated data corresponding to selected multi-layered seabeds as ground truth, the performance of the extrapolation method is explored. Application of the technique is also demonstrated on experimental data.

On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). A weak version of Hardy's uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.

A numerical study of the Legendre-Galerkin method for the evaluation of the prolate spheroidal wave functions

We conduct a numerical study of the Legendre-Galerkin method for the evaluation of the prolate spheroidal wave functions (PSWFs) viewed as the eigenfunctions of the prolate differential operator with boundary conditions of continuity. Our experiments indicate that the minimal dimension N of the Legendre-Galerkin matrix for the evaluation of the nth prolate with precision ? is O n + nc $\mathcal {O}( n + \sqrt {nc} ) $ , as n , c ? ? $n, c \rightarrow \infty $ , where c>0 is the bandwidth parameter. The behavior of N, when either c or n is held constant, is also examined. As a consequence, we obtain an upper bound on the complexity of the evaluation of the prolates. We also study the condition number of the approximate Legendre coefficients, computed as an eigenvector of the Legendre-Galerkin matrix. We observe experimentally that for fixed precision ?, an error estimate based on this condition number is O n + c $\mathcal {O}( n + c ) $ as n , c ? ? $n, c \rightarrow \infty $ . We conclude that the Legendre-Galerkin method is accurate for fairly large values of n and c.

Asynchronous processing of sparse signals

Unlike synchronous processing, asynchronous processing is more efficient in biomedical and sensing networks applications as it is free from aliasing constraints and quantization error in the amplitude, it allows continuous-time processing and more importantly data is only acquired in significant parts of the signal. We consider signal decomposers based on the asynchronous sigma delta modulator (ASDM), a non-linear feedback system that maps the signal amplitude into the zero-crossings of a binary output signal. The input, the zero-crossings and the ASDM parameters are related by an integral equation making the signal reconstruction difficult to implement. Modifying the model for the ASDM, we obtain a recursive equation that permits to obtain the non-uniform samples from the zero-time crossing values. Latticing the joint time-frequency space into defined frequency bands, and time windows depending on the scale parameter different decompositions are possible. We present two cascade low- and high-frequency decomposers, and a bank-of-filters parallel decomposer. This last decomposer using the modified ASDM behaves like a asynchronous analog to digital converter, and using an interpolator based on Prolate Spheroidal Wave functions allows reconstruction of the original signal. The asynchronous approaches proposed here are well suited for processing signals sparse in time, and for low-power applications.

On hp-convergence of prolate spheroidal wave functions and a new well-conditioned prolate-collocation scheme

The first purpose of this paper is to provide further illustrations, from both theoretical and numerical perspectives, for the nonconvergence of h-refinement in hp-approximation by the prolate spheroidal wave functions (PSWFs), a surprising convergence property that was first discovered by Boyd et al. (2013) [3]. The second purpose is to offer a new basis that leads to prolate-collocation systems with condition numbers independent of (c,N), the intrinsic bandwidth parameter and the number of collocation points. We highlight that the collocation scheme together with a very practical rule for pairing up (c,N) significantly outperforms the Legendre polynomial-based method (and likewise other Jacobi polynomial-based methods) in approximating highly oscillatory bandlimited functions.

Uniform bounds of prolate spheroidal wave functions and eigenvalues decay

The prolate spheroidal wave functions (PSWFs) form a set of special functions with remarkable properties. They are defined on [−1,1] as the bounded eigenfunctions ψn,c of a Sturm–Liouville differential operator Lc as well as the eigenfunctions of the linear integral operator Qc with kernel sin(c(x−y))π(x−y). We give new bounds for the values ψn,c(0), ψn,c′(0) and ψn,c(1), which allow us to obtain estimates for the Lp norms of the PSWFs and for eigenvalues of Lc and Qc. We get in particular an almost sharp exponential lower decay rate of the eigenvalues of Qc.

A generalization of the prolate spheroidal wave functions with applications to sampling

The goal of this paper is to use the theory of reproducing-kernel Hilbert spaces to obtain a generalization of the prolate spheroidal wave functions (PSWFs). We then employ this generalization to obtain a sampling formula for a general class of bandlimited functions. As a special case, we obtain a sampling formula for bandlimited signals in N variables that includes a generalization of Walter and Shen's result on sampling with the PSWFs. Another special case of our result is to show that the N-dimensional generalized PSWFs obtained by Slepian can also be used to obtain a sampling series for functions bandlimited to the unit ball in ℝN. The truncation errors of these sampling series are also investigated.

Analysis of power spectral density of ultra wideband waveforms for pulse shape modulation

Choice of ultra wideband (UWB) pulse shapes has become an interesting challenge such that the spectral limits of the frequency band approved by the Federal Communications Commission (FCC) can be utilised efficiently. In this paper, we compare the efficiency of various UWB pulse shapes based upon their power spectral density (PSD) and autocorrelation functions (ACF). We propose a set of prolate spheroidal wave functions (PSWF) for a typical M-ary pulse shape modulation (PSM) scheme. In the process, we show closed form expressions for the ACF of PSWF-based UWB pulse shapes of different orders.

Efficient Solution of Time-Domain Surface-Wire Integral Equation for Predicting Electromagnetic Responses of Complex Structures

An efficient marching-on-in-time (MOT) scheme is proposed to solve the time-domain surface-wire integral equation (TD-SWIE) for complex structures. A suitable set of approximate prolate spheroidal wave functions (APSWFs) is chosen as the temporal basis functions. Moreover, a novel integration approach for handling weakly singular inner integrals, along with some new smoothing techniques for treating the outer surface integrals, is applied to the TD-SWIE solver. The proposed algorithm is successfully implemented for a variety of problems involving surface-wire antennas on platforms.

Channel interpolation for LTE uplink systems with high mobility using Slepian sequences

Single Carrier Frequency Division Multiple Access (SC-FDMA) has been adopted for 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) uplink communication since it combines the low peak to average power ratio feature of single-carrier transmission systems with the multipath resistance and flexible frequency allocation of orthogonal frequency-division multiple access (OFDMA). Unfortunately, SC-FDMA transmission over time-varying multipath fading channels for very high speed environments the subcarrier orthogonality is destroyed giving rise to intercarrier interference (ICI) due to channel variation within an SC-FDMA symbol. This paper is, therefore, focused on the use of time-domain channel interpolator for LTE uplink systems in large Doppler spread scenarios to improve channel tracking performance. The resulting interpolation algorithm is based on the discrete prolate spheroidal wave functions (DPSWF) which is particularly well suited to represent the rapidly time-varying fading channel due to optimum finite time and frequency support feature. The variations of the channel taps are tracked first by a Kalman filter in time domain during training symbols. Then, a DPSWF based channel interpolator is applied to recover the time variation of channel taps between training symbols within two consecutive slots in a single subframe. The performance of the proposed interpolator is compared with a polynomial interpolator whose order is adjusted adaptively to achieve the best fit under particular user mobilities. Moreover, the analytical mean square error (MSE) performance of DPSWF is derived and compared with the simulations. The results show that the DPSWF represents the time-varying fading channel very effectively and the proposed algorithm has excellent channel estimation performance in high SNR even with a very small number of channel interpolation parameters employed in the algorithm.

DOA estimation for wideband signals based on sparse signal reconstruction using prolate spheroidal wave functions

The idea of block-sparse signal reconstruction, as an alternative perspective compared with the conventional approach, is exploited to formulate the problem of direction-of-arrival (DOA) estimation for wideband signals. Prolate spheroidal wave functions (PSWFs) are used to form the block-wise bases for this problem, due to its excellent performance in extrapolating bandlimited signals, and the block orthogonal matching pursuit (BOMP) algorithm is directly employed to verify its efficiency. Simulation results show that the proposed method yields better performance when the number of samples is highly limited.

Improved eigenanalysis canceler based on data-independent clutter subspace estimation for space-time adaptive processing

The eigenanalysis canceler (EC) method which is suitable for nonhomogeneous clutter can suppress clutter effectively by discarding the eigenvectors of small eigenvalues, which is a well-known subspace-based space-time adaptive processing (STAP) method. However, the computational complexity of conventional EC STAP method is huge due to the eigenvalue decomposition. Moreover, the corresponding performance would be degraded significantly by the subspace leakage phenomenon, since the clutter subspace is not strictly confined to a low-rank subspace any more. Therefore, an improved EC STAP method based on the data-independent clutter subspace estimation is proposed to reduce the computational complexity, where the clutter subspace is rapidly constructed by sampling the prolate spheroidal wave functions (PSWF) non-uniformly. In order to deal with the subspace leakage phenomenon, the proposed EC-PSWF STAP method is modified based on the covariance matrix taper (CMT) to obtain the covariance matrix by re-establishing the noise floor. The corresponding performance of proposed method is evaluated by using the numerical simulation.

Q-Prolate spheroidal wave functions and q-sampling theorem

In this paper we introduce a q-version of the prolate spheroidal wave functions which is the eigenfunction of the “concentration problem” related to sampling and interpolation of q-Hankel bandlimited functions. As application we give a q-sampling theorem which extrapolates functions in L2 space with {qn,n∈Z} as sampling points.

A finite difference construction of the spheroidal wave functions

A finite difference construction of the spheroidal wave functions

On the Irregular Nonbinary QC-LDPC-Coded Hybrid Multidimensional OSCD-Modulation Enabling Beyond 100 Tb/s Optical Transport

As a solution to limited bandwidth and high energy consumption of information infrastructure as well as heterogeneity of optical networks, a hybrid multidimensional coded modulation scheme is proposed employing all available electrical and optical degrees of freedom. Electrical basis functions are based on either modified orthogonal polynomials or prolate spheroidal wave functions, while optical ones on polarization and spatial mode states. The proposed scheme is both multi-Tb/s and beyond 100 Tb/s enabling technology. Additionally, the proposed scheme provides the adaptive, software-defined, and dynamic allocation of bandwidth with fine granularity. The adaptive coding is based on irregular nonbinary quasi-cyclic LDPC codes, providing record net coding gains.

Multidimensional Vector Quantization-Based Signal Constellation Design Enabling Beyond 1 Pb/s Serial Optical Transport Networks

In this paper, we propose a hybrid multidimensional coded-modulation (CM) scheme based on a new multidimensional signal constellation design as a solution to limited bandwidth and high energy consumption of information infrastructure. This multidimensional signal constellation, herewith called vector-quantization-based signal constellation design (VQ-SCD), is based on the vector quantization theory. The proposed scheme employs both the electrical basis functions (in the form of the prolate spheroidal wave functions) and the optical basis functions (in the form of polarization and spatial mode states) as optical basis functions. The proposed coded VQ-SCD scheme is the enabling technology for optical serial transport with bit rates exceeding 1000 Tb/s (1 Pb/s). In addition, the CM scheme we proposed allows for the adaptive, elastic, and dynamic allocation of the bandwidth. This fine granularity bandwidth manipulation is envisioned as a part of future software-defined optical networking.

Algorithm for Prolate Spheroidal Wave Functions Based on State Transition Matrix Approximation

该文针对现有椭圆球面波函数(PSWFs)求解算法存在的效率低、硬件实现复杂度高，尤其是精度不可控的问题，结合线性时变系统理论，提出一种基于微分方程状态转移矩阵逼近的PSWFs求解方法。该算法通过求解小区间上的状态转移矩阵来逼近整个时间区间上的状态转移矩阵，进而求得离散时间点上的系统运动轨迹，即PSWFs数值解。理论推导了求解误差并修正了算法，修正后算法具有简明的误差表达式，与Parr算法和Legendre多项式逼近算法在求解精度和复杂度上进行了对照分析。结果表明，该文算法求解精度高且可控，时间和空间复杂度低，易于硬件实现。

On the evaluation of prolate spheroidal wave functions and associated quadrature rules

As demonstrated by Slepian et al. in a sequence of classical papers (see Slepian (1983) [33], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35], Slepian (1965) [36]), prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. Recently, PSWFs have been becoming increasingly popular in various areas in which such functions occur – this includes physics (e.g. wave phenomena, fluid dynamics), engineering (signal processing, filter design), etc.To use PSWFs as a computational tool, one needs fast and accurate numerical algorithms for the evaluation of PSWFs and related quantities, as well as for the construction of corresponding quadrature rules, interpolation formulas, etc. During the last 15 years, substantial progress has been made in the design of such algorithms – see, for example, Xiao et al. (2001) [40] (see also Bowkamp (1947) [6], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35] for some classical results).The complexity of many of the existing algorithms, however, is at least quadratic in the band limit c. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires O(c2+n2) operations (see e.g. Xiao et al. (2001) [40]); the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires O(c3) operations (see e.g. Cheng et al. (1999) [8]). Therefore, while the existing algorithms are satisfactory for moderate values of c (e.g. c⩽103), they tend to be relatively slow when c is large (e.g. c⩾104).In this paper, we describe several numerical algorithms for the evaluation of PSWFs and related quantities, and design a class of PSWF-based quadratures for the integration of bandlimited functions. While the analysis is somewhat involved and will be published separately (currently, it can be found in Osipov and Rokhlin (2012) [27]), the resulting numerical algorithms are quite simple and efficient in practice. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires O(n+c⋅logc) operations; the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires O(c) operations. All algorithms described in this paper produce results essentially to machine precision. Our results are illustrated via several numerical experiments.