Modeling and Sampling of Spectrally Structured Signals
Abstract: This thesis consists of five papers concerned with the modeling of stochastic signals, as well as deterministic signals in stochastic noise, exhibiting different kinds of structure. This structure is manifested as the existence of finite-dimensional parameterizations, and/or in the geometry of the signals' spectral representations. The two first papers of the thesis, Papers A and B, consider the modeling of differences, or distances, between stochastic processes based on their second-order statistics, i.e., covariances. By relating the covariance structure of a stochastic process to spectral representations, it is proposed to quantify the dissimilarity between two processes in terms of the cost associated with morphing one spectral representation to the other, with the cost of morphing being defined in terms of the solutions to optimal mass transport problems. The proposed framework allows for modeling smooth changes in the frequency characteristics of stochastic processes, which is shown to yield interpretable and physically sensible predictions when used in applications of temporal and spatial spectral estimation. Also presented are efficient computational tools, allowing for the framework to be used in high-dimensional problems.Paper C considers the modeling of so-called inharmonic signals, i.e., signals that are almost, but not quite, harmonic. Such signals appear in many fields of signal processing, not least in audio. Inharmonicity may be interpreted as a deviation from a parametric structure, as well as from a particular spectral structure. Based on these views, as well as on a third, stochastic interpretation, Paper C proposes three different definitions of the concept of fundamental frequency for inharmonic signals, and studies the estimation theoretical implications of utilizing either of these definitions. Paper D then considers deliberate deviations from a parametric signal structure arising in spectroscopy applications. With the motivation of decreasing the computational complexity of parameter estimation, the paper studies the implications of estimating the parameters of the signal in a sequential fashion, starting out with a simplified model that is then refined step by step.Lastly, Paper E studies how parametric descriptions of signals can be leveraged as to design optimal, in an estimation theoretical sense, schemes for sampling or collecting measurements from the signal. By means of a convex program, samples are selected as to minimize bounds on estimator variance, allowing for efficiently measuring parametric signals, even when the parametrization is only partially known.
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