The Use of k-Means Algorithm to Compute the Line Spectrum Pair Frequencies with Tschirnhaus Transform

This paper proposes a fast 10-order line spectrum pair (LSP) frequencies calculation method using the k-means algorithm and the Tschirnhaus transforms. The first step of the proposed method 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 k-means algorithm, the proposed method can build up precise cosine lookup tables with the minimum memory size for Tschirnhaus transform. Finally these extremes could be used as the initial approximations to solve the roots of the 5-degree LSP polynomial via the Newton method and get the accurate LSP frequencies. One of the main advantages of the proposed method 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 as well as memory usage.