Adaptive additive modeling with continuous parameter trajectories

This paper investigates the estimation of time varying amplitude and phase trajectories of sinusoidal signal components. The new algorithm adaptively optimizes the parameters of a smoothly connected piecewise polynomial trajectory model. A mathematical analysis is presented that relates the user-selected meta parameters of the trajectory model (polynomial order, segment size, and smoothness at the junctions) to the analysis properties of the adaptive algorithm. It reveals new insights into the relationships between the meta parameters and the resulting time/frequency resolution of the estimate. Moreover, it is shown that for efficient optimization, the phase trajectory needs to be represented in a specific form. A new approach to address the bias/variance tradeoff of the polynomial phase trajectory model by means of regularization is presented and a complete adaptive analysis/synthesis system for sinusoidal sound components is proposed. The adaptive analysis system is investigated by means of simple tracking experiments to demonstrate the effect of the smoothness constraints and compare the results with a standard short-time Fourier transformation (STFT) base frequency estimation technique and known Cramer-Rao bounds. The potential of the adaptive strategy for the modeling of sinusoidal transients is discussed and it is shown that it achieves similar transient quality as a previously proposed method, however, with considerably lower model error. Two examples for modeling real-world signals are discussed