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Series in Signal and Information Processing, Vol. 37
edited by Hans-Andrea Loeliger

Rui Xing Elizabeth Ren

Using Local State Space Model Approximation

for Fundamental Signal Analysis Tasks

1st Edition 2023. XVIII, 268 pages. € 64,00.

ISBN 978-3-86628-792-1

Abstract

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. The properties of the smoothing

filters are derived and compared.

Finally, we discuss local sinusoidal approximation. A versatile set of

tools is introduced which can be combined to extend the local fitting

to signal analysis of locally periodic signals. The tools comprise timefrequency

representations of the signals, novel spectrograms based on

goodness of fit measures, as well as basic methods for detection of onsets

after periods of noise, time shift computation from the model fits and

phase and frequency tracking. Our time-frequency representations can be

understood as the generalization of the standard STFT to any ALSSM

window. The use of the toolbox is demonstrated via several real-world

applications. We discuss the estimation of the interaural time delay with a

method inspired by the psychoacoustical precedence effect. Furthermore,

we propose a scheme for clock synchronization based on our frequency

tracking tool. Finally, all the tools are combined for acoustic scene

analysis in recordings of killer whale vocalizations.

Keywords: Linear state space models; recursive least squares;
event detection and estimation; smoothing filters;
time-frequency representation; acoustic scene analysis.

About the author:

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.

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