Application of Homotopy Continuation Methods to Computation of Line Spectral Pairs
Usha Pillai · 1994
Line spectral pairs (LSPs) provide an alternate parametrization of the analysis and synthesis filters used in linear predictive coding (LPC) of speech. Effective use of LSPs in low bit-rate transmission of speech requires fast and accurate computation of these parameters. Special properties make LSFs highly amenable to computation by a path-tracking method based on the technique of homotopy, which is presented in this report. The system polynomial for the analysis filter is converted to two even-order symmetric polynomials whose roots give the line spectral pairs. The proposed method adaptively computes LSPs of incoming speech data by first defining continuous paths from known roots of the LSP polynomials of a speech frame to unknown roots of the next frame in the sequence. A gradient-search based numerical predictor-corrector procedure, having been initialized appropriately, is then used for tracking these paths in order to to compute the unknown roots. Conditions guaranteeing the existence of continuous paths are established by using a transformation between the s and z planes to translate the rules of root-locus construction in feedback control systems. Finally, simulation results are presented which verify the derived conditions and attest the applicability of the homotopy method for real-time computation of LSPs.