Parameter Estimation - In Sparsity We Trust
ABSTRACT This thesis is based on nine papers, all concerned with parameter estimation. The thesis aims at solving problems related to real-world applications such as spectroscopy, DNA sequencing, and audio processing, using sparse modeling heuristics. For the problems considered in this thesis, one is not only concerned with finding the parameters in the signal model, but also to determine the number of signal components present in the measurements. In recent years, developments in sparse modeling have allowed for methods that jointly estimate the parameters in the model and the model order. Based on these achievements, the approach often taken in this thesis is as follows. First, a parametric model of the considered signal is derived, containing different parameters that capture the important characteristics of the signal. When the signal model has been determined, an optimization problem is formed aimed at finding the parameters in the model as well as the model order. An important aspect when formulating the optimization problem is to include the characteristics and properties inherent in the signal model. For instance, if we know that the true set of parameters are smooth, this should also be a requirement reflected in the optimization problem. In the ideal case, the optimization problem is convex, in which case powerful solvers exist that may be used for finding the solution. In many cases, however, the original optimization problem is rather complex and definitely not convex. In this case, a common approach is to use a convex relaxation that approximates the original problem. In papers A, B, C, E, F, and H, this approach is utilized, however in different variations and for different applications. Paper A deals with estimation of periodic signals in symbolic sequences used in DNA sequences, paper B looks at the estimation of multi-dimensional sinusoids for NMR data, paper C considers the estimation of an unknown number of chirps for audio signals, papers E and F study pitch estimation, where the first paper considers online estimation and where the second paper proposes an off-grid method. Paper D proposes a generalization of a popular estimation method, whereas paper G introduces a new approach to frequency estimation. Paper I investigates how to sample a partially know signal to minimize the number of samples needed given a lower bound on the desired estimation performance. In all papers, the proposed methods are examined using simulated and/or measured data and compared to competing state-of-the-art methods.