The computation of line spectrum pair frequencies using Tschirnhaus transform

• Published in: 2009 IEEE International Symposium on Circuits and Systems (ISCAS) pp. 2333 - 2336
• Main Authors: Shi-Huang Chen, Yaotsu Chang, Chang Jian Yu Syuan
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01-05-2009
• Subjects: Cities and towns; Computational complexity; Computer science; Discrete cosine transforms; Equations; Frequency; Linear predictive coding; Mathematics; Newton method; Polynomials
• ISBN: 1424438276; 9781424438273
• ISSN: 0271-4302; 2158-1525
• DOI: 10.1109/ISCAS.2009.5118267

In this paper, a new algorithm based on the Tschirnhaus transforms is developed to reduce the computation complexity of the 10-order line spectrum pairs (LSP) frequencies. The first step of the proposed algorithm is to derive a quartic equation from the 1st derivative of the given 5-degree LSP polynomial. Then the extremes of the 5-degree LSP polynomial can be found by applying the Tschirnhaus transform to the above quartic equation. By the use of these extremes as the initial approximations, one can easily solve the roots of the 5-degree LSP polynomial via the Newton's method and get the accurate LSP frequencies. One of the main advantages of the proposed algorithm is the rapid root determination of a quartic equation without complex number operations and resulting in considerable computational saving. Compared to other methods, the proposed algorithm can determine the precise LSP frequencies with the lowest computational complexity.