Time-variant modal decomposition and uses in transient modeling

Harvey Thornburg $<$harv23 at ccrma$>$ (EE)

My talk focuses on a "piecewise-slowly-nonstationary" parametric model for tracking multiple sinusoids. The ultimate goal of the model is an analysis-synthesis method for transient sounds, where the assumption of local stationarity (so important to frequency-domain methods) fails to hold.

In the piecewise model, we assume sinusoidal parameters vary smoothly except on a finite set of points where abrupt changes occur. After we segment according to boundaries of abrupt change, we constrain AR parameters to vary according to a basis where the number of functions indicates the degree of smoothness. Then order selection criteria may be applied, to choose both the number of components and their degree of smoothness. To extract sinusoidal parameters from this "AR" representation, we use Kamen's time-variant cascade decomposition, then show exactly how this can be used for the modal decomposition. The modal decomposition is followed by a process of state estimation for amplitude/phase information.

However, the modal decomposition does not guarantee that "smooth" or "slow" AR parameter variations result in this same behavior for the sinusoids. Fortunately, I can show in the case of convergent AR parameters, corresponding sinusoidal parameters will have limit cycles which trace out a convex path, thus enabling a feedback stabilization of Kamen's recursions which yields at least an asymptotically smooth trajectory for the sinusoids. However, the stabilization fails in the case of real-valued modes.

In the talk, I review the entire system (both segmentation and modeling tasks), then focus on the specifics of time-variant mode extraction as it relates to the problem of smoothness.