arXiv subjects

Filip Elvander

Publications and source records attributed to Filip Elvander.

26 records · Page 2Linked to original sources

arXiv2021

Mixed-Spectrum Signals -- Discrete Approximations and Variance Expressions for Covariance Estimates

The estimation of the covariance function of a stochastic process, or signal, is of integral importance for a multitude of signal processing applications. In this work, we derive closed-form expressions for the variance of covariance estimates for mixed-spectrum signals, i.e., spectra containing both absolutely continuous and singular parts. The results cover both finite-sample and asymptotic regimes, allowing for assessing the exact speed of convergence of estimates to their expectations, as well as their limiting behavior. As is shown, such covariance estimates may converge even for non-ergodic processes. Furthermore, we consider approximating signals with arbitrary spectral densities by sequences of singular spectrum, i.e., sinusoidal, processes, and derive the limiting behavior of covariance estimates as both the sample size and the number of sinusoidal components tend to infinity. We show that the asymptotic regime variance can be described by a time-frequency resolution product, with dramatically different behavior depending on how the sinusoidal approximation is constructed. In a few numerical examples we illustrate the theory and the corresponding implications for direction of arrival estimation.

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arXiv2020

Defining Fundamental Frequency for Almost Harmonic Signals

In this work, we consider the modeling of signals that are almost, but not quite, harmonic, i.e., composed of sinusoids whose frequencies are close to being integer multiples of a common frequency. Typically, in applications, such signals are treated as perfectly harmonic, allowing for the estimation of their fundamental frequency, despite the signals not actually being periodic. Herein, we provide three different definitions of a concept of fundamental frequency for such inharmonic signals and study the implications of the different choices for modeling and estimation. We show that one of the definitions corresponds to a misspecified modeling scenario, and provides a theoretical benchmark for analyzing the behavior of estimators derived under a perfectly harmonic assumption. The second definition stems from optimal mass transport theory and yields a robust and easily interpretable concept of fundamental frequency based on the signals' spectral properties. The third definition interprets the inharmonic signal as an observation of a randomly perturbed harmonic signal. This allows for computing a hybrid information theoretical bound on estimation performance, as well as for finding an estimator attaining the bound. The theoretical findings are illustrated using numerical examples.

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arXiv2019

On Harmonic Approximations of Inharmonic Signals

In this work, we present the misspecified Gaussian Cramér-Rao lower bound for the parameters of a harmonic signal, or pitch, when signal measurements are collected from an almost, but not quite, harmonic model. For the asymptotic case of large sample sizes, we present a closed-form expression for the bound corresponding to the pseduo-true fundamental frequency. Using simulation studies, it is shown that the bound is sharp and is attained by maximum likelihood estimators derived under the misspecified harmonic assumption. It is shown that misspecified harmonic models achieve a lower mean squared error than correctly specified unstructured models for moderately inharmonic signals. Examining voices from a speech database, we conclude that human speech belongs to this class of signals, verifying that the use of a harmonic model for voiced speech is preferable.

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arXiv2019

Mismatched Estimation of Polynomially Damped Signals

In this work, we consider the problem of estimating the parameters of polynomially damped sinusoidal signals, commonly encountered in, for instance, spectroscopy. Generally, finding the parameter values of such signals constitutes a high-dimensional problem, often further complicated by not knowing the number of signal components or their specific signal structures. In order to alleviate the computational burden, we herein propose a mismatched estimation procedure using simplified, approximate signal models. Despite the approximation, we show that such a procedure is expected to yield predictable results, allowing for statistically and computationally efficient estimates of the signal parameters.

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arXiv2019

Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport

In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes.

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arXiv2019

Multi-Marginal Optimal Mass Transport with Partial Information

During recent decades, there has been a substantial development in optimal mass transport theory and methods. In this work, we consider multi-marginal problems wherein only partial information of each marginal is available, which is a setup common in many inverse problems in, e.g., imaging and spectral estimation. By considering an entropy regularized approximation of the original transport problem, we propose an algorithm corresponding to a block-coordinate ascent of the dual problem, where Newton's algorithm is used to solve the sub-problems. In order to make this computationally tractable for large-scale settings, we utilize the tensor structure that arises in practical problems, allowing for computing projections of the multi-marginal transport plan using only matrix-vector operations of relatively small matrices. As illustrating examples, we apply the resulting method to tracking and barycenter problems in spatial spectral estimation. In particular, we show that the optimal mass transport framework allows for fusing information from different time steps, as well as from different sensor arrays, also when the sensor arrays are not jointly calibrated. Furthermore, we show that by incorporating knowledge of underlying dynamics in tracking scenarios, one may arrive at accurate spectral estimates, as well as faithful reconstructions of spectra corresponding to unobserved time points.

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arXiv2018

Non-Coherent Sensor Fusion via Entropy Regularized Optimal Mass Transport

This work presents a method for information fusion in source localization applications. The method utilizes the concept of optimal mass transport in order to construct estimates of the spatial spectrum using a convex barycenter formulation. We introduce an entropy regularization term to the convex objective, which allows for low-complexity iterations of the solution algorithm and thus makes the proposed method applicable also to higher-dimensional problems. We illustrate the proposed method's inherent robustness to misalignment and miscalibration of the sensor arrays using numerical examples of localization in two dimensions.

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arXiv2017

Designing Sampling Schemes for Multi-Dimensional Data

In this work, we propose a method for determining a non-uniform sampling scheme for multi-dimensional signals by solving a convex optimization problem reminiscent of the sensor selection problem. The resulting sampling scheme minimizes the sum of the Cramér-Rao lower bound for the parameters of interest, given a desired number of sampling points. The proposed framework allows for selecting an arbitrary subset of the parameters detailing the model, as well as weighing the importance of the different parameters. Also presented is a scheme for incorporating any imprecise a priori knowledge of the locations of the parameters, as well as defining estimation performance bounds for the parameters of interest. Numerical examples illustrate the efficiency of the proposed scheme.

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