Froissart Doublets Research Articles

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

ЗАДАЧА ЭКСПОНЕНЦИАЛЬНОГО АНАЛИЗА – ПРИМЕНЕНИЯ, МЕТОДЫ, ПРОБЛЕМЫ

This review article presents the problem of exponential analysis, its applications, limitations and problems that are relevant despite its more than two hundred years of history. During this time, many methods for solving the problem have been developed, but the greatest popularity and efficiency are demonstrated by Prony-like methods: the Padé–Laplace method, Prony method and the matrix pencil method, which are the subject of research in this work. The purpose of the study: to present the current state of the problem of exponential analysis, its main problems and methods for solving them. The main problems of these methods are: the issue of determining the number of exponents, choosing the optimal signal sampling frequency and reducing computational costs. The article presents solutions to these problems. Materials and methods. To solve problems, an analysis of scientific literature was carried out and linear algebra methods were used. Results. Modifications of the methods to solve their problems are described. The algorithm for calculating the Padé approximation with a minimum degree denominator solves the problem of Froissart doublets in the Padé–Laplace method. To increase the accuracy of calculations of Taylor coef-ficients, splines are used in the same method. To select the optimal signal sampling frequency, use the estimation of the matrix condition number in the Prony method. For the matrix pencil method, the problem of computational costs is solved due to its recurrent and multichannel versions. Conclusion. Various formulations of this problem are presented, which find their numerous applications. The limitations, problems of the exponential analysis problem, and their solutions are described. Three parametric methods are considered: Prony, Padé–Laplace and the matrix pencil method.

Removing the Froissart Doublets in a Rational Interpolation Based on Loewner Matrix

In order to implement wide band frequency sweeping, the S-parameters can be fitted with an adaptive rational interpolation based on Loewner matrix. However, the errors in the sampling data may lead to Froissart doublets, which look like spikes in the curve. In this paper, a novel technique is proposed to remove these doublets. At first, the rational expression is converted into the sum of partial fractions by solving two generalized eigenvalue problems. After that, the partial fraction term with the smallest imaginary part of the pole and relatively large absolute value is considered to generate the doublets. Removing this term results in a smooth rational polynomial, which is validated by the example of a passive circuit simulated by finite element method(FEM).

On rational functions without Froissart doublets

In this paper we consider the problem of working with rational functions in a numeric environment. A particular problem when modeling with such functions is the existence of Froissart doublets, where a zero is close to a pole. We discuss three different parameters which allow one to monitor the absence of Froissart doublets for a given general rational function. These include the euclidean condition number of an underlying Sylvester-type matrix, a parameter for determining coprimeness of two numerical polynomials and bounds on the spherical derivative. We show that our parameters sharpen those found in a previous paper by two of the authors.

Encoded in vivo time signals from the ovary in magnetic resonance spectroscopy: poles and zeros as the cornerstone for stability of response functions of systems to external perturbations

In order to handle encoded data from magnetic resonance spectroscopy (MRS), advanced signal processing methods are vital. This is presently carried out using the fast Padé transform (FPT) applied to in vivo MRS time signals encoded from the ovary. We examine the essential features of the response function, namely the spectral poles and zeros, as the key to stability of the system to external excitations. Noise is separated from signal by reliance upon the multi-level signature of Froissart doublets. Our focus is upon eliminating the oversensitivity to alterations in model order K, through systematic examination of poles and zeros, as well as Padé-reconstructed total shape spectra, spectral parameters and component shape spectra. This comprehensive examination of convergence of all variables under study includes investigation of the combined role of spectra averaging and time signal extrapolation. Comparisons are made throughout between the results for six model orders (K = 575, 585,ldots , 625) with an increment of ten and eleven model orders (K = 575, 580,ldots , 625) with an increment of 5. It is demonstrated that for the reconstructed poles and zeros, as well as for magnitudes and phases, spectra averaging and Padé-based extrapolation of time signals are essential for the stability of the system and for the accurate retrieval of resonances. Full convergence is achieved when spectra averaging and extrapolation are applied together. Spectra averaging and extrapolation are also shown to be needed to obtain stabilized results to the level of stochasticity for the component spectra for the six and the eleven model orders. Without spectra averaging and extrapolation, there were noticeable variances for the six and the eleven model orders with regard to all the variables under study. The present analysis and results have important implications for expediting the robust quantification by the FPT. All the analysis herein was applied to in vivo MRS data encoded on a 3 T scanner from a borderline serous cystic ovarian tumor. This clinical problem has been chosen in light of the urgent need to develop effective methods for early detection of ovarian cancer. The overriding motivation is to improve survival for women afflicted with this malignancy. The reported results further hone the Padé-designed methodology for practical applications of in vivo MRS and, therefore, are anticipated to help in achieving the stated goal.

A Numerical Test of Padé Approximation for Some Functions with Singularity

The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.

Beyond the Fourier Transform: Signal Symmetry Breaking in the Complex Plane

In this letter, we invert the ordinary point of view in the analysis of noisy data by treating the signal as a perturbation of the noise. The generating function of pure noise is represented, in the Complex Plane, by poles and zeros (Froissart doublets) having a universal, isotropic statistical distribution. The presence of a signal breaks this rotational symmetry. This allows to detect signals deeply embedded in noise that traditional methods cannot reach.

Magnetic resonance spectroscopy with high-resolution and exact quantification in the presence of noise for improving ovarian cancer detection

Ovarian cancer is the most common cause of death from gynecological malignancies in many developed countries. When confined to the ovary, the 5-year survival rates are over 90%, but detection methods are insufficiently accurate for widespread application. Consequently, ovarian cancer is typically diagnosed late, with poor prognosis. Magnetic resonance spectroscopy (MRS) could potentially improve ovarian cancer diagnosis, but is currently hampered by poor resolution for this small, moving organ. Advanced signal processing methods through the fast Pade transform (FPT) can improve resolution and provide quantitative metabolic information. We applied the FPT to noise-corrupted time signals generated according to in vitro MRS data as encoded from malignant ovarian cyst fluid. In the presence of background noise, the FPT converged using merely 1/8 of the full signal length N = 1024, amounting to some 128 data points. This number of time signal points permits reconstruction of altogether 128 spectral parameters for 64 ensuing resonances. Each resonance is quantified by 2 spectral parameters, the complex-valued frequencies and amplitudes. The FPT accurately reconstructed the spectral parameters for all twelve genuine resonances from which the input time signal is intrinsically built. Thereby, in the presence of noise, the FPT provided fully reliable estimates of metabolite concentrations characteristic of malignant ovarian cyst fluid. Through the practical concept of signal-noise separation by means of so-called pole-zero cancellations (Froissart doublets), the remaining 52 resonances from the total of 64 resonances were unequivocally identified as spurious and, as such, could confidently be discarded. Given that magnetic resonance-based modalities entail no exposure to ionizing radiation, if their diagnostic accuracy were improved, magnetic resonance imaging and spectroscopy could have broader applications in screening surveillance for early ovarian cancer detection, especially among women at high risk. The present results suggest that Pade-optimized MRS could help achieve that goal.

An algorithm for computing a Padé approximant with minimal degree denominator

In this paper, a new definition of a reduced Padé approximant and an algorithm for its computation are proposed. Our approach is based on the investigation of the kernel structure of the Toeplitz matrix. It is shown that the reduced Padé approximant always has nice properties which the classical Padé approximant possesses only in the normal case. The new algorithm allows us to avoid the appearance of Froissart doublets induced by computer roundoff in the non-normal Padé table.

Padé–Froissart exact signal-noise separation in nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance spectroscopy is one of the key methods for studying the structure of matter on different levels (sub-nuclear, nuclear, atomic, molecular, cellular, etc). Its overall success critically depends upon reliable mathematical analysis and interpretation of the studied data. This is especially aided by parametric signal processing with the ensuing data quantification, which can yield the abundance or concentrations of the constituents in the examined matter. The sought reliability of signal processing rests upon the possibility of an accurate solution of the quantification problem alongside the unambiguous separation of true from false information in the spectrally analysed data. We presently demonstrate that the fast Padé transform (FPT), as the unique ratio of two polynomials for a given Maclaurin series, can yield exact signal-noise separation for a synthesized free induction decay curve built from 25 molecules. This is achieved by using the concept of Froissart doublets or pole–zero cancellations. Unphysical/spurious (noise or noise-like) resonances have coincident or near-coincident poles and zeros. They possess either zero- or near-zero-valued amplitudes. Such spectral structures never converge due to their instability against even the smallest perturbations. By contrast, upon convergence of the FPT, physical/genuine resonances are identified by their persistent stability against external perturbations, such as signal truncation or addition of random noise, etc. In practice, the computation is carried out by gradually and systematically increasing the common degree of the Padé numerator and denominator polynomials in the diagonal FPT. As this degree changes, the reconstructed parameters and spectra fluctuate until stabilization occurs. The polynomial degree at which this full stabilization is achieved represents the sought exact number of resonances. An illustrative set of results is reported in this work to show the exact separation of genuine from spurious information by reliance upon Froissart doublets and stabilization of reconstructions. The FPT for optimal quantification of the physical constituents of the studied matter and the denoising Froissart filter for unequivocal signal-noise separation is expected to significantly aid nuclear magnetic resonance spectroscopy in achieving the most reliable data analysis and interpretation.

Fast Padé transform in the theory of resonances: exact solution of the harmonic inversion problem

We present the fast Padé transform (FPT) as a polynomial quotient for the Green or response function from signal processing, spectroscopy and resonant scattering theory. Specific illustrations are given for nuclear magnetic resonance spectroscopy within the problem of harmonic inversion, quantification or spectral analysis. Here, the input time signal points or auto-correlation functions are given via measurements or computations, and the task is to reconstruct the unknown components as the harmonic variables in terms of the fundamental complex frequencies and amplitudes. The FPT solves the harmonic inverse problem exactly by retrieving the true number of resonances with all their proper spectral parameters. This output list is finalized by means of Froissart doublets or pole-zero confluences for unequivocal disentangling of the physical/genuine from unphysical/spurious contents of the analysed time signal. Stability of investigated systems under external perturbations is especially challenged by the presence of noise. The FPT can assess the system's stability through determining the locations and distributions of spectral poles and zeros in the complex frequency plane. This permits identification of the regions that are void of noise, and gives the possibility for improved system performance under more stable conditions with full signal-noise separation. The FPT can provide a number of important biophysical and chemical quantities, including the density of states and abundance or concentrations of all the physical constituents from the investigated substance.

Unequivocal disentangling genuine from spurious information in time signals: clinical relevance in cancer diagnostics through magnetic resonance spectroscopy

Pole-zero cancellation in the polynomial quotient of the fast Pade transform (FPT) is shown to clearly and unequivocally distinguish spurious resonances from the true metabolites, in the presence of random Gauss-distributed zero mean noise in a magnetic resonance (MR) spectrum. Thus, by applying the principle of Froissart doublets within the FPT, noise as typically encountered in clinically encoded MR-time signals, is completely separated from the genuine metabolic information. The basis for the unprecedented algorithmic success of the FPT for processing of MR signals is explained within the framework of quantum mechanics. The clear, direct and immediate importance of these findings is reviewed with respect to clinical oncology. Besides confirming the high resolution and stability of the FPT in general studies of MR total shape spectra, this superior resolution performance of the FPT has also been confirmed with respect to data directly derived from malignant and benign ovarian samples. Not only does the FPT markedly enhance resolution of MR spectra compared to the conventional Fourier analysis, but it also yields the unequivocal, exact parametric data needed to reconstruct the metabolite concentrations which characterize ovarian cancer and distinguish this from non-malignant lesions. These features of the FPT are deemed to be of critical benefit to ovarian cancer diagnostics via MRS, in particular for early detection, a goal which has thus far been elusive, but achievement of which would definitely confer a major survival advantage. It is anticipated that MRS via Pade processing will reduce the false positive rates of MR-based modalities and, moreover, will further improve the sensitivity of these methods. Once this is achieved, and given that all MR-based diagnostic methods are free from ionizing radiation, new possibilities for cancer screening and early detection would emerge, especially for risk groups, e.g., the application of Pade-optimized MRS in younger women at high risk for breast and/or ovarian cancer.

The general concept of signal–noise separation (SNS): mathematical aspects and implementation in magnetic resonance spectroscopy

Magnetic resonance spectroscopy (MRS) and spectroscopic imaging (MRSI) are increasingly recognized as potentially key modalities in cancer diagnostics. It is, therefore, urgent to overcome the shortcomings of current applications of MRS and MRSI. We explain and substantiate why more advanced signal processing methods are needed, and demonstrate that the fast Pade transform (FPT), as the quotient of two polynomials, is the signal processing method of choice to achieve this goal. In this paper, the focus is upon distinguishing genuine from spurious (noisy and noise-like) resonances; this has been one of the thorniest challenges to MRS. The number of spurious resonances is always several times larger than the true ones. Within the FPT convergence is achieved through stabilization or constancy of the reconstructed frequencies and amplitudes. This stabilization is a veritable signature of the exact number of resonances. With any further increase of the partial signal length N, towards the full signal length N, i.e., passing the stage at which full convergence has been reached, it is found that all the fundamental frequencies and amplitudes “stay put”, i.e., they still remain constant. Moreover, machine accuracy is achieved here, proving that when the FPT is nearing convergence, it approaches straight towards the exact result with an exponential convergence rate (the spectral convergence). This proves that the FPT is an exponentially accurate representation of functions customarily encountered in spectral analysis in MRS and beyond. The mechanism by which this is achieved, i.e., the mechanism which secures the maintenance of stability of all the spectral parameters and, by implication, constancy of the estimate for the true number of resonances is provided by the so-called pole-zero cancellation, or equivalently, the Froissart doublets. This signifies that all the additional poles and zeros of the Pade spectrum will cancel each other, a remarkable feature unique to the FPT. The FPT is safe-guarded against contamination of the final results by extraneous resonances, since each pole due to spurious resonances stemming from the denominator polynomial will automatically coincide with the corresponding zero of the numerator polynomial, thus leading to the pole-zero cancellation in the polynomial quotient of the FPT. Such pole-zero cancellations can be advantageously exploited to differentiate between spurious and genuine content of the signal. Since these unphysical poles and zeros always appear as pairs in the FPT, they are viewed as doublets. Therefore, the pole-zero cancellation can be used to disentangle noise as an unphysical burden from the physical content in the considered signal, and this is the most important usage of the Froissart doublets in MRS. The general concept of signal–noise separation (SNS) is thereby introduced as a reliable procedure for separating physical from non-physical information in MRS, MRSI and beyond.

Modelling room transfer functions using the decimated Padé approximant

The numerical issues involved in modelling measured room transfer functions (RTFs) are examined. This is explored using a nonlinear parametric estimation technique known as the decimated Padé approximant (DPA). The DPA combines the well-known methodology of Padé rational polynomial approximation with beamspace windowing. This combination helps to overcome the severe numerical instabilities encountered in calculations with large data records. The aim is to accurately extract the parameters that reliably quantify the analytic structure of signals composed of decaying sinusoidal oscillations. DPA parameter estimation provides the ability to construct a high-resolution spectral estimate of such signals for either specific spectral regions or the entire Nyquist interval. As demonstrated in the authors' previous work (O'Sullivan and Cowan, 2006), this technique, developed in quantum chemistry, readily cross-fertilises to the field of acoustics, where it can fully reconstruct the complicated spectra of experimental RTFs. A noise-filtering technique using the removal of Froissart doublets to obtain an irreducible rational model is investigated. This noise filtering can be used to find an order of the parametric model that is inherent in the data. Additionally, an example is shown suggesting that such a process may be useful in room spatialisation problems.

Froissart doublets in Padé approximation in the case of polynomial noise

First, we study the relation between the zeros of random polynomials Rn+1 and the zeros and poles of their Padé approximants [n/n]Rn+1. Next, we consider the distribution of zeros and poles of Padé approximants to the geometric series perturbed by a random polynomial noise. We observe numerically interesting connections between two above problems. Some numerical observations on the Froissart doublets have been also made.

Padé approximants and noise: rational functions

We continue a study of Padé approximants (PA) for a series perturbed by random noise – this time we consider more general rational functions. We begin with the simple case of a sum of two geometric series, and then show how these considerations can be extended to a general rational function. We do not study the most general case, but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered. We show that Froissart doublets are a universal feature and we construct an analog of the Froissart polynomial introduced in the earlier paper.

Padé approximants and noise: A case of geometric series

We study effects, on Padé approximants (PA), of a random noise added to coefficients of a power series. Numerical experiments performed during the last 30 years have shown that such noise results in the appearance of, the so-called, Froissart doublets (FD) — pairs of zeros and poles separated by a distance of a scale of the noise. For the first time we can prove that this effect actually takes place for geometric series. We can also show what happens with noise-induced poles or zeros that do not form FD. We construct a polynomial that has as its roots mean positions of FD. Coefficients of this polynomial are expressed by random perturbations of coefficients of the series. We study numerically this polynomial for low orders to show where (in these orders) FD coalesce. Numerical study of this polynomial, to see a behaviour of FD for large orders, is much easier than a study of the entire PA. We also express a deviation of a pole of PAs approximating the true pole of 1 (1 − z) , from 1, by perturbations of coefficients and show numerically that when order of PAs increases, this deviation drops down.