Bayesian Sparse Fourier Representation of Off-Grid Targets

We consider the problem of estimating a finite sum of cisoids via the use of a sparsifying Fourier dictionary (problem that may be of use in many radar applications). Numerous signal sparse representation (SSR) techniques can be found in the literature regarding this problem. However, they are usually very sensitive to grid mismatch. In this paper, we present a new Bayesian model robust towards grid mismatch. Synthetic and experimental radar data are used to assess the ability of the proposed approach to robustify the SSR towards grid mismatch

Gridless Evolutionary Approach for Line Spectral Estimation with Unknown Model Order

Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic $l_0$ norm (i.e., the continuous analog of $l_0$ norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of atomic $l_0$ norm, such as nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However, the relaxations give rise to a resolution limit, subsequently leading to biased model order and convergence error. To overcome the above shortcomings of relaxation, we propose a novel idea of simultaneously estimating the frequencies and model order by means of the atomic $l_0$ norm. To accomplish this idea, we build a multiobjective optimization model. The measurment error and the atomic $l_0$ norm are taken as the two optimization objectives. The proposed model directly exploits the model order via the atomic $l_0$ norm, thus breaking the resolution limit. We further design a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations. One is a variable-length coding and search strategy. It flexibly codes and interactively searches diverse solutions with different model orders. These solutions act as steppingstones that help fully exploring the variable and open-ended frequency search space and provide extensive potentials towards the optima. Another innovation is a model order pruning mechanism, which heuristically prunes less contributive frequencies within the solutions, thus significantly enhancing convergence and diversity. Simulation results confirm the superiority of our approach in both frequency estimation and model order selection.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

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

Unambiguous Sparse Recovery of Migrating Targets with a Robustified Bayesian Model

The problem considered is that of estimating unambiguously migrating targets observed with a wideband radar. We extend a previously described sparse Bayesian algorithm to the presence of diffuse clutter and off-grid targets. A hybrid-Gibbs sampler is formulated to jointly estimate the sparse target amplitude vector, the grid mismatch and the (assumed) autoregressive noise. Results on synthetic and fully experimental data show that targets can be actually unambiguously estimated even if located in blind speeds

Fast Algorithms for Sampled Multiband Signals

Over the past several years, computational power has grown tremendously. This has led to two trends in signal processing. First, signal processing problems are now posed and solved using linear algebra, instead of traditional methods such as filtering and Fourier transforms. Second, problems are dealing with increasingly large amounts of data. Applying tools from linear algebra to large scale problems requires the problem to have some type of low-dimensional structure which can be exploited to perform the computations efficiently. One common type of signal with a low-dimensional structure is a multiband signal, which has a sparsely supported Fourier transform. Transferring this low-dimensional structure from the continuous-time signal to the discrete-time samples requires care. Naive approaches involve using the FFT, which suffers from spectral leakage. A more suitable method to exploit this low-dimensional structure involves using the Slepian basis vectors, which are useful in many problems due to their time-frequency localization properties. However, prior to this research, no fast algorithms for working with the Slepian basis had been developed. As such, practitioners often overlooked the Slepian basis vectors for more computationally efficient tools, such as the FFT, even in problems for which the Slepian basis vectors are a more appropriate tool. In this thesis, we first study the mathematical properties of the Slepian basis, as well as the closely related discrete prolate spheroidal sequences and prolate spheroidal wave functions. We then use these mathematical properties to develop fast algorithms for working with the Slepian basis, a fast algorithm for reconstructing a multiband signal from nonuniform measurements, and a fast algorithm for reconstructing a multiband signal from compressed measurements. The runtime and memory requirements for all of our fast algorithms scale roughly linearly with the number of samples of the signal.Ph.D