Problem Of Spectral Analysis Research Articles

A Quantitative Analysis of Virtual Endmembers' Increased Impact on the Collinearity Effect in Spectral Unmixing

In the past decades, spectral unmixing has been studied for deriving the fractions of spectrally pure materials in a mixed pixel. However, limited attention has been given to the collinearity problem in spectral mixture analysis. In this paper, quantitative analysis and detailed simulations are provided, which show that the high correlation between the endmembers, including the virtual endmembers introduced in a nonlinear model, has a strong impact on unmixing errors through inflating the Gaussian noise. While distinctive spectra with low correlations are often selected as true endmembers, the virtual endmembers formed by their product terms can be highly correlated. It is found that a virtual-endmember-based nonlinear model generally suffers more from collinearity problems compared to linear models and may not perform as expected when the Gaussian noise is high, despite its higher modeling power. Experiments were conducted on a set of in situ measured data, and the results show that the linear mixture model performs better in 61.5% of the cases.

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.

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.

Sampling of Spectrally Correlated Processes

The problem of sampling continuous-time spectrally correlated (SC) processes is addressed. SC processes have Loève bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. This class of nonstationary processes extends that of the almost-cyclostationary processes and occurs in wideband mobile communications and spectral analysis with nonuniform spectral resolution. The class of the discrete-time SC processes is introduced and characterized. It is shown that such processes can be obtained by uniformly sampling the continuous-time SC processes. Sampling theorems are presented and sufficient conditions to avoid aliasing are provided. Applications are illustrated with reference to the problem of cross spectral analysis with nonuniform frequency resolution and the propagation of communications signals through MIMO multipath Doppler channels.

Reconstruction of Sturm-Liouville differential operators on A-graphs

We study the inverse problem of spectral analysis for Sturm-Liouville operators on A-graphs. We obtain a constructive procedure for solving the inverse problem of reconstruction of coefficients of differential operators from spectra and prove the uniqueness of the solution.

Spectral analysis on commutative hypergroups

The problem of spectral analysis is formulated on commutative hypergroups and is solved for finite dimensional varieties.

Inverse problem of spectral analysis of conflict dynamical systems

For conflict dynamical systems, we study the problem of the existence and description of initial measures that converge to measures with given spectral distributions.

An inverse problem for Sturm–Liouville differential operators on A-graphs

An inverse problem of spectral analysis is studied for Sturm–Liouville differential operators on a A-graph with the standard matching conditions for internal vertices. The uniqueness theorem is proved, and a constructive solution for this class of inverse problems is obtained.

Parameter estimation for exponential sums by approximate Prony method

The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES

We consider nonparametric estimation of spectral densities of stationary processes, a fundamental problem in spectral analysis of time series. Under natural and easily verifiable conditions, we obtain consistency and asymptotic normality of spectral density estimates. Asymptotic distribution of maximum deviations of the spectral density estimates is also derived. The latter result sheds new light on the classical problem of tests of white noises.

Recovering Sturm-Liouville operators from spectra on a graph with a cycle

An inverse problem of spectral analysis is studied for Sturm-Liouville differential operators on a graph with a cycle and with generalized matching conditions at the internal vertex. Theorems on the unique recovery of operators from a system of spectra are proved, and a constructive solution is obtained for this class of inverse problems.Bibliography: 26 titles.

Spectral synthesis problems on locally compact groups

Spectral analysis and spectral synthesis problems are formulated on noncommutative locally compact groups and solved on compact groups.

Inverse spectral problems for Sturm–Liouville differential operators on a finite interval

A short review is given on inverse problems of spectral analysis for the Sturm–Liouville differential operator on a finite interval. Using this operator as a model we present the main ideas and methods in the inverse problem theory, and describe the main results for this class of inverse problems.

A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family

Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramer-Rao bound, the Bobrovsky-MayerWolf-Zakai bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakai bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakai. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakai bound, and the Bayesian Cramer-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.

A Wegner-type Estimate for Correlated Potentials

We propose a fairly simple and natural extension of Stollmann’s lemma to correlated random variables. This extension allows to obtain Wegner-type estimates even in various problems of spectral analysis of random operators where the original Wegner’s lemma is inapplicable, e.g., for correlated random potentials with singular marginal distributions and for multi-particle Hamiltonians.

Numerical Methods for Solving Inverse Sturm–Liouville Problems

A short review on numerical methods for solving inverse problems of spectral analysis for Sturm–Liouville differential operators is presented; moreover, a new approach based on ideas of the method of spectral mappings is given. Results of numerical experiments are also presented.

The inverse spectral problem for pencils of differential operators

The inverse problem of spectral analysis for a quadratic pencil of Sturm-Liouville operators on a finite interval is considered. A uniqueness theorem is proved, a solution algorithm is presented, and sufficient conditions for the solubility of the inverse problem are obtained.Bibliography: 31 titles.

New methods for fitting multiple sinusoids from irregularly sampled data

A novel framework is proposed for the estimation of multiple sinusoids from irregularly sampled time series. This spectral analysis problem is addressed as an under-determined inverse problem, where the spectrum is discretized on an arbitrarily thin frequency grid. As we focus on line spectra estimation, the solution must be sparse, i.e. the amplitude of the spectrum must be zero almost everywhere. Such prior information is taken into account within the Bayesian framework. Two models are used to account for the prior sparseness of the solution, namely a Laplace prior and a Bernoulli–Gaussian prior, associated to optimization and stochastic sampling algorithms, respectively. Such approaches are efficient alternatives to usual sequential prewhitening methods, especially in case of strong sampling aliases perturbating the Fourier spectrum. Both methods should be intensively tested on real data sets by physicists.

Asymptotic spectral theory for nonlinear time series

We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.

Peculiarities and distinctions between matrix algorithms for fast Fourier and Hartley transforms in the “running” spectral analysis problems

The paper describes the unified matrix approach to FFT algorithms, described in [1], for synthesis of algorithms permitting to realize the operation of “running” (or “jumping”) spectral analysis. Here we maximally use information about the spectrum of the preceding step for each new position of the time window. A formula is derived for calculation of Hartley’s transform for a sequence of data expressed in matrix form. The formula represents an extension, to an arbitrary base, of the known Hartley formula using the partition of the initial sequence in two ones—with odd and even numbers.