Sparse Modeling of Grouped Line Spectra

This licentiate thesis focuses on clustered parametric models for estimation of line spectra, when the spectral content of a signal source is assumed to exhibit some form of grouping. Different from previous parametric approaches, which generally require explicit knowledge of the model orders, this thesis exploits sparse modeling, where the orders are implicitly chosen. For line spectra, the non-linear parametric model is approximated by a linear system, containing an overcomplete basis of candidate frequencies, called a dictionary, and a large set of linear response variables that selects and weights the components in the dictionary. Frequency estimates are obtained by solving a convex optimization program, where the sum of squared residuals is minimized. To discourage overfitting and to infer certain structure in the solution, different convex penalty functions are introduced into the optimization. The cost trade-off between fit and penalty is set by some user parameters, as to approximate the true number of spectral lines in the signal, which implies that the response variable will be sparse, i.e., have few non-zero elements. Thus, instead of explicit model orders, the orders are implicitly set by this trade-off. For grouped variables, the dictionary is customized, and appropriate convex penalties selected, so that the solution becomes group sparse, i.e., has few groups with non-zero variables. In an array of sensors, the specific time-delays and attenuations will depend on the source and sensor positions. By modeling this, one may estimate the location of a source. In this thesis, a novel joint location and grouped frequency estimator is proposed, which exploits sparse modeling for both spectral and spatial estimates, showing robustness against sources with overlapping frequency content. For audio signals, this thesis uses two different features for clustering. Pitch is a perceptual property of sound that may be described by the harmonic model, i.e., by a group of spectral lines at integer multiples of a fundamental frequency, which we estimate by exploiting a novel adaptive total variation penalty. The other feature, chroma, is a concept in musical theory, collecting pitches at powers of 2 from each other into groups. Using a chroma dictionary, together with appropriate group sparse penalties, we propose an automatic transcription of the chroma content of a signal

Exploiting Sparse Structures in Source Localization and Tracking

This thesis deals with the modeling of structured signals under different sparsity constraints. Many phenomena exhibit an inherent structure that may be exploited when setting up models, examples include audio waves, radar, sonar, and image objects. These structures allow us to model, identify, and classify the processes, enabling parameter estimation for, e.g., identification, localisation, and tracking.In this work, such structures are exploited, with the goal to achieve efficient localisation and tracking of a structured source signal. Specifically, two scenarios are considered. In papers A and B, the aim is to find a sparse subset of a structured signal such that the signal parameters and source locations maybe estimated in an optimal way. For the sparse subset selection, a combinatorial optimization problem is approximately solved by means of convex relaxation, with the results of allowing for different types of a priori information to be incorporated in the optimization. In paper C, a sparse subset of data is provided, and a generative model is used to find the location of an unknown number of jammers in a wireless network, with the jammers’ movement in the network being tracked as additional observations become available

Sparse Localization of Harmonic Audio Sources

In this paper, we propose a novel method for estimating the locations of near- and/or far-field harmonic audio sources impinging on an arbitrary, but calibrated, sensor array. Using a joint pitch and location estimation formed in two steps, we first estimate the fundamental frequencies and complex amplitudes under a sinusoidal model assumption, whereafter the location of each source is found by utilizing both the difference in phase and the relative attenuation of the magnitude estimates. As audio recordings often consist of multi-pitch signals exhibiting some degree of reverberation, where both the number of pitches and the source locations are unknown, we propose to use sparse heuristics to avoid the necessity of detailed a priori assumptions on the spectral and spatial model orders. The method’s performance is evaluated using both simulated and measured audio data, with the former showing that the proposed method achieves near-optimal performance, whereas the latter confirms the method’s feasibility when used with real recordings