Using Local State Space Model Approximation for Fundamental Signal Analysis Tasks

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With the growing availability of computational power, digital signal analysis algorithms can transition from framewise to samplewise operations, enhancing precision. This thesis presents methods utilizing samplewise operations, specifically local signal approximation through Recursive Least Squares (RLS). Here, a mathematical model is fitted to the signal within a sliding window at each sample, generating signal models and cost windows via Autonomous Linear State Space Models (ALSSMs). ALSSMs can model a wide range of functions, including exponentials, polynomials, and sinusoids, allowing for efficient recursions and subsample precision. Classical methods like Savitzky-Golay smoothing filters and Short-Time Fourier Transform are integrated into a unified framework. The thesis elaborates on ALSSM parameterization and RLS recursions, reviews fitting parameter solutions for various constraints, and details feature extraction from fit parameters and costs. It introduces a perspective for analyzing fitting problems and provides analytical rules for computing filter responses and precision matrices. The discussion extends to polynomial and sinusoidal signal models, deriving expressions for steady-state covariance matrices and smoothing filters. A novel class of smoothing filters based on polynomial fitting is introduced, alongside tools for local sinusoidal approximation, time-frequency representations, and onset detection. Real-wor