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Parameter Estimation - in sparsity we trust

På svenska

Author

  • Johan Swärd

Summary, in English

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.

Department/s

Publishing year

2017

Language

English

Document type

Dissertation

Publisher

Mathematical Statistics, Centre for Mathematical Sciences, Lund University

Topic

  • Probability Theory and Statistics
  • Signal Processing

Keywords

  • Parameter estimation
  • Sparse models
  • Convex optimization
  • Symbolic Periodicity
  • Alternating direction method of multipliers (ADMM)
  • Covariance fitting
  • multi-pitch estimation problem
  • Off-grid estimation
  • Dictionary learning
  • Atomic norm
  • Sampling schemes

Status

Published

Research group

  • Statistical Signal Processing Group

ISBN/ISSN/Other

  • ISBN: 978-91-7753-354-2
  • ISBN: 978-91-7753-353-5

Defence date

15 September 2017

Defence time

09:15

Defence place

lecture hall MH:Rieszsalen, Centre for Mathematical Sciences, Sölvegatan 18, Faculty of Engineering LTH, Lund University, Lund

Opponent

  • Emil Björnsson (Doktor)