A Novel Two-Level Method for the Computation of the LSP Frequencies Using a Decimation-in-Degree Algorithm

A novel two-level method is proposed in this study for rapidly and accurately computing the line spectrum pair (LSP) frequencies. An efficient decimation-in-degree (DID) algorithm is also proposed in the first level, which can transform any symmetric or antisymmetric polynomial with real coefficients into the other polynomials with lower degrees and without any transcendental functions. The DID algorithm not only can avoid prior storage or large calculation of transcendental functions but can also be easily applied toward those fast root-finding methods. In the second level, if the transformed polynomial is of degree 4 or less, employing closed-form formulas is the fastest procedure of quite high accuracy. If it is of a higher degree, a modified Newton-Raphson method with cubic convergence is applied. Additionally, the process of the modified Newton-Raphson method can be accelerated by adopting a deflation scheme along with Descartes rule of signs and the interlacing property of LSP frequencies for selecting the better initial values. Besides this, Homer's method is extended to efficiently calculate the values of a polynomial and its first and second derivatives. A few conventional numerical methods are also implemented to make a comparison with the two-level method. Experimental results indicate that the two-level method is the fastest one. Furthermore, this method is more advantageous under the requirement of a high level of accuracy.