Parameter Estimation for Superimposed Weighted Exponentials

The approach of modeling measured signals as superimposed exponentials in white Gaussian noise is popular and effective. However, estimating the parameters of the assumed model is challenging, especially when the data record length is short, the signal strength is low, or the parameters are closely spaced. In this dissertation, we first review the most effective parameter estimation scheme for the superimposed exponential model: maximum likelihood. We then provide a historical review of the linear prediction approach to parameter estimation for the same model. After identifying the improvements made to linear prediction and demonstrating their weaknesses, we introduce a completely tractable and statistically sound modification to linear prediction that we call iterative generalized least squares. It is shown, that our algorithm works to minimize the exact maximum likelihood cost function for the superimposed exponential problem and is therefore, equivalent to the previously developed maximum likelihood approach. However, our algorithm is indeed linear prediction, and thus revives a methodology previously categorized as inferior to maximum likelihood. With our modification, the insight provided by linear prediction can be carried to actual applications. We demonstrate this by developing an effective algorithm for deep level transient spectroscopy analysis. The signal of deep level transient spectroscopy is not a straight forward superposition of exponentials. However, with our methodology, an estimator, based on the exact maximum likelihood cost function for the actual signal, is quickly derived. At the end of the dissertation, we verify that our estimator extends the current capabilities of deep level transient spectroscopy analysis

Approximate maximum likelihood estimation of two closely spaced sources

The performance of the majority of high resolution algorithms designed for either spectral analysis or Direction-of-Arrival (DoA) estimation drastically degrade when the amplitude sources are highly correlated or when the number of available snapshots is very small and possibly less than the number of sources. Under such circumstances, only Maximum Likelihood (ML) or ML-based techniques can still be effective. The main drawback of such optimal solutions lies in their high computational load. In this paper we propose a computationally efficient approximate ML estimator, in the case of two closely spaced signals, that can be used even in the single snapshot case. Our approach relies on Taylor series expansion of the projection onto the signal subspace and can be implemented through 1-D Fourier transforms. Its effectiveness is illustrated in complicated scenarios with very low sample support and possibly correlated sources, where it is shown to outperform conventional estimators

Multipath Parameter Estimation from OFDM Signals in Mobile Channels

We study multipath parameter estimation from orthogonal frequency division multiplex signals transmitted over doubly dispersive mobile radio channels. We are interested in cases where the transmission is long enough to suffer time selectivity, but short enough such that the time variation can be accurately modeled as depending only on per-tap linear phase variations due to Doppler effects. We therefore concentrate on the estimation of the complex gain, delay and Doppler offset of each tap of the multipath channel impulse response. We show that the frequency domain channel coefficients for an entire packet can be expressed as the superimposition of two-dimensional complex sinusoids. The maximum likelihood estimate requires solution of a multidimensional non-linear least squares problem, which is computationally infeasible in practice. We therefore propose a low complexity suboptimal solution based on iterative successive and parallel cancellation. First, initial delay/Doppler estimates are obtained via successive cancellation. These estimates are then refined using an iterative parallel cancellation procedure. We demonstrate via Monte Carlo simulations that the root mean squared error statistics of our estimator are very close to the Cramer-Rao lower bound of a single two-dimensional sinusoid in Gaussian noise.Comment: Submitted to IEEE Transactions on Wireless Communications (26 pages, 9 figures and 3 tables

Variational Bayesian Inference of Line Spectra

In this paper, we address the fundamental problem of line spectral estimation in a Bayesian framework. We target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous-valued, i.e., not restricted to a grid; and the coefficients are governed by a Bernoulli-Gaussian prior model turning model order selection into binary sequence detection. Unlike earlier works which retain only point estimates of the frequencies, we undertake a more complete Bayesian treatment by estimating the posterior probability density functions (pdfs) of the frequencies and computing expectations over them. Thus, we additionally capture and operate with the uncertainty of the frequency estimates. Aiming to maximize the model evidence, variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters. We propose an accurate representation of the pdfs of the frequencies by mixtures of von Mises pdfs, which yields closed-form expectations. We define the algorithm VALSE in which the estimates of the pdfs and parameters are iteratively updated. VALSE is a gridless, convergent method, does not require parameter tuning, can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified. Simulation results show that accounting for the uncertainty of frequency estimates, rather than computing just point estimates, significantly improves the performance. The performance of VALSE is superior to that of state-of-the-art methods and closely approaches the Cram\'er-Rao bound computed for the true model order.Comment: 15 pages, 8 figures, accepted for publication in IEEE Transactions on Signal Processin

Nonuniform array processing via the polynomial approach

Caption title.Includes bibliographical references (leaf 19).Research supported by the National Science Foundation. ECS-8312921 Research supported by the Army Research Office. DAAL03-86-K-1071 DAAG-84-K-0005 Research supported by a Rothschild fellowship.by Anthony J. Weiss, Alan S. Willsky, Bernard C. Levy

D3^3PO - Denoising, Deconvolving, and Decomposing Photon Observations

The analysis of astronomical images is a non-trivial task. The D3PO algorithm addresses the inference problem of denoising, deconvolving, and decomposing photon observations. Its primary goal is the simultaneous but individual reconstruction of the diffuse and point-like photon flux given a single photon count image, where the fluxes are superimposed. In order to discriminate between these morphologically different signal components, a probabilistic algorithm is derived in the language of information field theory based on a hierarchical Bayesian parameter model. The signal inference exploits prior information on the spatial correlation structure of the diffuse component and the brightness distribution of the spatially uncorrelated point-like sources. A maximum a posteriori solution and a solution minimizing the Gibbs free energy of the inference problem using variational Bayesian methods are discussed. Since the derivation of the solution is not dependent on the underlying position space, the implementation of the D3PO algorithm uses the NIFTY package to ensure applicability to various spatial grids and at any resolution. The fidelity of the algorithm is validated by the analysis of simulated data, including a realistic high energy photon count image showing a 32 x 32 arcmin^2 observation with a spatial resolution of 0.1 arcmin. In all tests the D3PO algorithm successfully denoised, deconvolved, and decomposed the data into a diffuse and a point-like signal estimate for the respective photon flux components.Comment: 22 pages, 8 figures, 2 tables, accepted by Astronomy & Astrophysics; refereed version, 1 figure added, results unchanged, software available at http://www.mpa-garching.mpg.de/ift/d3po

Recursive estimation of exponential signals in impulsive noise using M-estimation

The 47th Midwest Symposium on Circuits and Systems Conference, Salt Lake City, Utah, USA, 25-28 July 2004This paper proposes a robust method for recursive estimating the frequencies and amplitudes of an exponential signal model under impulsive noise. Using the concept of M-estimation, a recursive algorithm based on the recursive least M-estimate (RLM) is developed. Simulation results show that the proposed method performs better than that of the conventional least square method under impulsive noise environment. The algorithm also possesses low arithmetic complexity due to its recursive nature.published_or_final_versio