Prony-like Method Research Articles

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

Hybrid Sparse Expansion and Separable Hybrid Prony Method

Signal sparsity plays an essential role in signal compression and reconstruction. Prony-like methods are widely used in the relevant applications e.g., the recovery of signals with finite rate of innovation (FRI). However, these methods are usually limited to the structured function expansions of a single family, i.e., the considered structured functions belonging to the eigenfunctions of a single operator. In this paper, we investigate hybrid Prony models (HPMs), wherein functions can be expanded sparsely by the combination of eigenfunctions of two different operators $\mathcal {L}_1$ and $\mathcal {L}_2$ . Although the Prony-like methods cannot directly deal with HPMs, we show that HPMs can be directly converted to the generalized Prony model recently considered by Peter and Plonka when the operator pair $\lbrace \mathcal {L}_1, \mathcal {L}_2\rbrace$ satisfies certain conditions; we refer to these functions as separable HPMs (SHPMs). Similar to the Prony method, the problem of nonlinear parameter estimation in SHPMs can be converted to the problem of finding the roots of polynomials. Examples and numerical results are given to illustrate SHPMs.

Distortion Characterization of Exponential Signal Reconstructed by Low-Chirp Signal

This paper is an extension of the procedure pointed out to evaluate the distortion of the exponential signal, modeled as multiexponential signal with different amplitudes and time constants. This procedure evaluates the distortion by applying the Prony-like method to the signal reconstructed by zero-crossing time evaluated on the difference between the exponential original signal and the reference sinusoidal one. The extension proposed in this paper concerns with the use of the low-chirp signal as reference instead of the sinusoidal signal. The low-chirp signal permits to increase the number of the zero-crossing time in the section of exponential signal with the higher slope and decreasing in the section with the lower slope. This setting permits to estimate the distortion of the exponential signal with higher accuracy and precision. The parameters’ amplitude, phase, and start frequency of the reference low-chirp signal are used to reconstruct the exponential signal in the zero-crossing time in order to overcome the problem of the high-resolution sampling. Their estimation is carried out by the time-varying sine-fitting algorithm that processes the resulting signal that is the difference between the exponential original signal and the low-chirp signal. Advantages of the proposed extended procedure regard the direct estimation of the reference low-chirp signal parameters from the resulting signal over-sampled at low resolution, and the use of only one input acquisition channel that permits to overcome the synchronization problem between the reference and the resulting signal. The numerical and experimental tests validate the extended procedure.

Spherical Bessel transform via exponential sum approximation of spherical Bessel function

A new algorithm for numerical evaluation of spherical Bessel transform is proposed in this paper. In this method, the spherical Bessel function is approximately represented as an exponential sum with complex parameters. This is obtained by expressing an integral representation of spherical Bessel function in complex plane, and discretizing contour integrals along steepest descent paths and a contour path parallel to real axis using numerical quadrature rule with the double-exponential transformation. The number of terms in the expression is reduced using the modified balanced truncation method. The residual part of integrand is also expanded by exponential functions using Prony-like method. The spherical Bessel transform can be evaluated analytically on arbitrary points in half-open interval.

Multivariate exponential analysis from the minimal number of samples

The problem of multivariate exponential analysis or sparse interpolation has received a lot of attention, especially with respect to the number of samples required to solve it unambiguously. In this paper we show how to bring the number of samples down to the absolute minimum of (d + 1)n where d is the dimension of the problem and n is the number of exponential terms. To this end we present a fundamentally different approach for the multivariate problem statement. We combine a one-dimensional exponential analysis method such as ESPRIT, MUSIC, the matrix pencil or any Prony-like method, with some linear systems of equations because the multivariate exponents are inner products and thus linear expressions in the parameters.

Signal recovery by discrete approximation and a Prony-like method

We introduce an algorithm which combines ideas of Prony’s approach to recover signals from given samples with approximation methods. We solve two overdetermined systems of linear equations with linear programming methods and calculate the zeros of a suitable ‘Prony-like’ polynomial. We get the bandwidth m , the frequencies as well as the amplitudes and some other characteristics of the signal. Especially, it is reconstructed if sufficient many (at least 2 m ) samples are given. If we have too few samples or if they are too much noised, we get an approximation of the original noiseless signal. Even if we have sufficient many but possible erroneous samples the obtained signal interpolates at least m of them (usually the low-noised or noiseless ones) and, of course, all samples if the signal is recovered. The described method behaves well to moderate sampling errors and is resistant to outliers in the samples which can be detected, filtered off or corrected during the calculations to improve the quality of the computed signal.

Модификация метода матричных пучков, использующая совместное оценивание полюсов сигнала и обратных к ним

The Prony-like parametric method of signal processing, namely the Matrix Pencil Method, is considered in the paper. The method is able to find frequencies, damping factors, phases and amplitudes of a sinusoids sum. A modification of the method using a combined evaluation of signal poles and their inverses is proposed. This modification is able to solve the problem of true/false poles separation. Algorithms of the modified and classical Matrix Pencil Methods are given and compared on the example of signal detection in noise. It is shown that the classical method is not able to detect the time of arrival of the signal since it fits an exponentials sum to the noise. The modified method can detect both the time of arrival and the signal frequency.

Reconstruction of sparse Legendre and Gegenbauer expansions

We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples. Using asymptotic properties of Legendre polynomials, this reconstruction is based on Prony-like methods. The method proposed is robust with respect to noisy sampled data. Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order.

Distortion measurement of exponential signal

Exponential voltage was proposed as a stimulus in the analog-to-digital converter (ADC) test because it is much more similar to the theoretical one than the sinusoidal signal. For the correct execution of the ADC test, the distortion of the exponential stimulus must be assessed by evaluating the amplitude and the time coefficient of the fundamental and the superimposed exponential components that best fit the real exponential voltage. To achieve this goal, the Prony-like method is proposed in this paper. Two problems arise: the a priori evaluation of the degree of polynomial function that best fits the input samples, and the number of exponential components that have to be evaluated. To overcome these problems, two proper sub-routines are pointed out allowing operation without a priori information and in accordance with the physical model of the signal. The results of experimental tests are shown in order to assess the correctness and the accuracy of the proposed procedure.

Sparse polynomial interpolation in Chebyshev bases

We study the problem of reconstructing a sparse polynomial in a basis of Chebyshev polynomials (Chebyshev basis in short) from given samples on a Chebyshev grid of [−1, 1]. A polynomial is called M-sparse in a Chebyshev basis, if it can be represented by a linear combination of M Chebyshev polynomials. For a polynomial with known and unknown Chebyshev sparsity, respectively, we present efficient reconstruction methods, where Prony-like methods are used. The reconstruction results are mainly presented for bases of Chebyshev polynomials of first and second kind, respectively. But similar issues can be obtained for bases of Chebyshev polynomials of third and fourth kind, respectively.

Structural Characterization of Bitumen System by Prony-Like Method Applied to NMR and Rheological Relaxation Data

In order to characterize colloidal structures, T2-nuclear magnetic resonance (NMR) and rheological relaxation times are used. NMR Car–Purcell sequence and mechanical stress relaxation experiments have been performed on a bitumen system at different temperatures. The rheological relaxation times spectra are fingerprints of the aggregates that constitute the system. These typical relaxation times have been obtained from an exponential fitting of the experimental data, based on a Prony-like method. The unknown parameters are estimated on the basis of a linear regression equation that uses altered signals obtained directly from the NMR and rheological measurements. This approach uses the derivative method in the frequency domain, yielding exact formula in terms of multiple integrals of the signal, when placed in the time domain. These integrals are explicitly solved by projecting signal on some set of orthogonal basis functions or, more in general, by using a polynomial that fits data in the least-squares sense. The method is able to deal with the case of nonuniform sampled signal.

Parameter estimation for nonincreasing exponential sums by Prony-like methods

Let zj:=efj with fj∈(-∞,0]+i[-π,π) be distinct nodes for j=1,…,M. With complex coefficients cj≠0, we consider a nonincreasing exponential sum h(x):=c1ef1x+⋯+cMefMx(x⩾0). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if 2,N sampled values h(k) (k=0,…,2N-1; N⩾M) are given. This parameter estimation problem is a nonlinear inverse problem. For noiseless sampled data, we describe the close connections between Prony-like methods, namely the classical Prony method, the matrix pencil method, and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

A Prony-like method for non-uniform sampling

In this paper a Prony-like method for non-uniform sampling is proposed. The unknown parameters are estimated on the base of a new linear regression equation which uses filtered signals obtained directly from the measurements. The approach uses the derivative method in the frequency domain yielding exact formula in terms of multiple integrals of the signal, when placed in the time domain. These integrals are explicitly solved by projecting signal on some set of orthogonal basis functions or, more in general, by using a polynomial that fits data in the least-squares sense. The effectiveness of the proposed approach is shown by simulated experiments.

Fast algorithms for the analysis of single and double exponential decay curves with a background term. Application to time‐resolved imaging microscopy

Computer programs have been developed to determine decay time constants from a temporal sequence of digitized images with decaying intensities characterized by either single or double exponentials plus a constant background term. The very fast algorithms are evaluated at every pixel position. A non-iterative Prony-like method provides high quality initial estimates that are used for the subsequent non-linear least-squares procedure in which the normal equations are solved directly. Error analysis routines enable a pixel-by-pixel estimation of the quality of the experimental data. The stand-alone programs were fully integrated into a commercial image-processing environment (SCIL-Image) for a convenient and optimal display of the decay analysis. The features of the programs are illustrated by the analysis of simulated image data. With current workstations, the fitting routines (including reading of data, initial estimate and error analysis) require 0.13 ms/pixel using the single exponential algorithm applied to 50 time points, and 1.37 ms/pixel for 100 time points and the double exponential algorithm. The programs are of general applicability and have been used to analyse data from time-resolved fluorescence, phosphorescence, and photobleaching-based microscopy. Two examples of the latter case are shown, illustrating the utility of the programs for the quantitative evaluation of spatially resolved fluorescence resonance energy transfer (FRET) and for generating contrast by allocating specific cellular structures to particular decay components in a fluorescence image.