A scalar homotopy method for parallel and robust tracking of line spectral pairs

We present an adaptive path-following method based on the technique of homotopy, which efficiently computes the line spectral pairs by exploiting their natural ordering and low frame-to-frame variation. We first define continuous paths from known roots of the LSP polynomials of a prior speech frame to the unknown roots of the next frame in the sequence. A gradient-search based numerical predictor-corrector procedure is then used for tracing these paths in order to compute the unknown roots. This method uses only scalar operations and allows for all paths to be tracked independently. Conditions guaranteeing the existence of continuously differentiable paths are established by using a transformation between the s and z domains, and simulation results are presented to verify these conditions. Finally, we suggest guidelines for algorithm parameter selection as well as for reducing the computational complexity