Using Local State Space Model Approximation for Fundamental Signal Analysis Tasks - Ren, Elizabeth

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Résumé :
With increasing availability of computation power, digital signal analysis algorithms have the potential of evolving from the common framewise operational method to samplewise operations which offer more precision in time. This thesis discusses a set of methods with samplewise operations: local signal approximation via Recursive Least Squares (RLS) where a mathematical model is fit to the signal within a sliding window at each sample. Thereby both the signal models and cost windows are generated by Autonomous Linear State Space Models (ALSSMs). The modeling capability of ALSSMs is vast, as they can model exponentials, polynomials and sinusoidal functions as well as any linear and multiplicative combination thereof. The fitting method offers efficient recursions, subsample precision by way of the signal model and additional goodness of fit measures based on the recursively computed fitting cost. Classical methods such as standard Savitzky-Golay (SG) smoothing filters and the Short-Time Fourier Transform (STFT) are united under a common framework. First, we complete the existing framework. The ALSSM parameterization and RLS recursions are provided for a general function. The solution of the fit parameters for different constraint problems are reviewed. Moreover, feature extraction from both the fit parameters and the cost is detailed as well as examples of their use. In particular, we introduce terminology to analyze the fitting problem from the perspective of projection to a local Hilbert space and as a linear filter. Analytical rules are given for computation of the equivalent filter response and the steady-state precision matrix of the cost. After establishing the local approximation framework, we further discuss two classes of signal models in particular, namely polynomial and sinusoidal functions. The signal models are complementary, as by nature, polynomials are suited for time-domain description of signals while sinusoids are suited for the frequency-domain. For local approximation of polynomials, we derive analytical expressions for the steady-state covariance matrix and the linear filter of the coefficients based on the theory of orthogonal polynomial bases. We then discuss the fundamental application of smoothing filters based on local polynomial approximation. We generalize standard SG filters to any ALSSM window and introduce a novel class of smoothing filters based on polynomial fitting to running sums.

Biographie:
Elizabeth Ren was born in Zurich, Switzerland in 1993, where she lived until 1997. She then lived in Toronto, Canada between 1999 and 2003. From 2003 onwards, she returned to Switzerland. She attended high school at Gymnasium Kirschgarten in Basel-Stadt, where she earned the Swiss Matura in 2011, graduating in the top ten students of her grade. Subsequently, she enrolled at ETH Zurich, Switzerland, where she obtained her B.Sc. and M.Sc. degrees in Information Technology and Electrical Engineering in 2014 and 2017, respectively. During her Master?s studies, in 2016, she did a half-year internship with Siemens Building Technologies, Zug. She continued on at Siemens with her Master?s thesis, which was co-supervised by the Signal and Information Processing Laboratory (ISI) at ETH Zurich. This was followed by a three-month internship at Siemens in 2017. Thereafter, in the Summer of 2017, she was a visiting scholar at the Department of Mechanical and Mechatronics Engineering at University of Waterloo. Since August 2017, she has been a PhD candidate and a full research assistant at ISI. Her focus is on model-based signal processing and machine learning.