Multivariate Padé-approximants

AbstractFor an operator F: Rn → R, analytic in the origin, the notion of (abstract multivariate Padé-approximant (APA) is introduced, by making use of abstract polynomials. The classical Padé-approximant (n = 1) is a special case of the multivariate theory and many interesting properties of classical Padé-approximants remain valid such as covariance properties and the block-structure [Annie A. M. Cuyt, J. Oper. Theory6 (2) (1981), 207–209] of the Padé-table. Also a projection-property for multivariate Padé-approximants is proved

Parametric spectral analysis: scale and shift

We introduce the paradigm of dilation and translation for use in the spectral analysis of complex-valued univariate or multivariate data. The new procedure stems from a search on how to solve ambiguity problems in this analysis, such as aliasing because of too coarsely sampled data, or collisions in projected data, which may be solved by a translation of the sampling locations. In Section 2 both dilation and translation are first presented for the classical one-dimensional exponential analysis. In the subsequent Sections 3--7 the paradigm is extended to more functions, among which the trigonometric functions cosine, sine, the hyperbolic cosine and sine functions, the Chebyshev and spread polynomials, the sinc, gamma and Gaussian function, and several multivariate versions of all of the above. Each of these function classes needs a tailored approach, making optimal use of the properties of the base function used in the considered sparse interpolation problem. With each of the extensions a structured linear matrix pencil is associated, immediately leading to a computational scheme for the spectral analysis, involving a generalized eigenvalue problem and several structured linear systems. In Section 8 we illustrate the new methods in several examples: fixed width Gaussian distribution fitting, sparse cardinal sine or sinc interpolation, and lacunary or supersparse Chebyshev polynomial interpolation

Abstract Padé-approximants for the solution of a sytem of nonlinear equations

AbstractLet F: Rq → Rq and let x* be a simple root of the system of nonlinear equations F(x) = 0.We will construct several iterative methods, based on the (n, m)-APA (abstract Padé-approximant) or the (n, m)-ARA (abstract rational approximant)[4] for either F (direct one-point interpolation) or its inverse operator G (inverse one-point interpolation).The following methods are special cases: n = 1, m = 0: Newton-iteration (via direct and via inverse interpolation); inverse interpolation with n = 2, m = 0: improvement of the Newton-iteration as indicated by Ehrmann[7]; direct interpolation with n = 1, m = 1: method of tangent hyperbolas[10], under certain conditions for the ARA.Among other new methods an interesting third-order iterative procedure is constructed via inverse interpolation with n = 1, m = 1: Xi+1=Xi+ai2ai+12F′i−1F″iai2 with F′i the 1st Fréchet-derivative of F at xi, ai=−F′−1iFi the Newton-correction, F″i the 2nd Fréchet-derivative of F at xi and component-wise multiplication and division in Rq.This method is to be preferred to the method of tangent hyperbolas, which is also of third order, since it requires less numerical calculations. In general, the methods derived from the use of the (n, m)-APA or (n, m)-ARA with m ⩾ 1 are preferable when F or G have singularities in the neighborhood of x* or 0 respectively

Validated exponential analysis for harmonic sounds

In audio spectral analysis, the Fourier method is popular because of its stability and its low computational complexity. It suffers however from a time-frequency resolution trade off and is not particularly suited for aperiodic signals such as exponentially decaying ones. To overcome their resolution limitation, additional techniques such as quadratic peak interpolation or peak picking, and instantaneous frequency computation from phase unwrapping are used. Parametric methods on the other hand, overcome the time frequency trade off but are more susceptible to noise and have a higher computational complexity. We propose a method to overcome these drawbacks: we set up regularized smaller sized independent problems and perform a cluster analysis on their combined output. The new approach validates the true physical terms in the exponential model, is robust in the presence of outliers in the data and is able to filter out any non-physical noise terms in the model. The method is illustrated in the removal of electrical humming in harmonic sounds

A short-time Prony method for the detection of transients

Many parametric spectral methods are based on the classical algorithm of the French engineer G. de Prony for exponential analysis. A drawback of this method is that it cannot take into consideration any discontinuities due to the starting and ending of the exponential components at different instants. We introduce a short-time Prony method that allows to extract the characteristics from such a signal and we illustrate the new method on a number of power system signals. All parameters in the signals can be extracted with high accuracy and we show how to monitor the occurrence of the transients dynamically

Sparse multidimensional exponential analysis with an application to radar imaging

We present a d-dimensional exponential analysis algorithm that offers a range of advantages compared to other methods. The technique does not suffer the curse of dimensionality and only needs O((d + 1)n) samples for the analysis of an n-sparse expression. It does not require a prior estimate of the sparsity n of the d-variate exponential sum. The method can work with sub-Nyquist sampled data and offers a validation step, which is very useful in low SNR conditions. A favourable computation cost results from the fact that d independent smaller systems are solved instead of one large system incorporating all measurements simultaneously. So the method also lends itself easily to a parallel execution. Our motivation to develop the technique comes from 2D and 3D radar imaging and is therefore illustrated on such examples