Using Local State Space Model Approximation for Fundamental Signal Analysis Tasks

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 time-frequency 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.