一种新的基于矩的改进离散余弦变换及其反变换快速算法

Novel approach for fast computation of MDCT and IMDCT based on moments

为了提高离散余弦变换(MDCT)及其反变换(IMDCT)的计算效率,提出一种新的基于一维离散矩的快速算法.首先把MDCT和IMDCT的核函数映射到另外一个集合进行合并化简,再用三角函数泰勒级数展开的方法,将MDCT和IMDCT的计算用有限项的一维离散矩的线性加权和近似.一维离散矩的快速计算可以采用p+1维的矢量加法结构进行,用加法运算代替乘法运算,有效地减少了乘法的运算量.该算法的乘法计算量仅为O(Nlog2N/log2log2N),少于通常快速算法所需的O(Nlog2N),可以有效地降低运算时间.理论分析和实验结果都表明:用一维矩近似的方法计算MDCT和IMDCT的结果精度很高,运行速度比较快,能够很好地满足实际计算的要求.

For fast computation of the modified discrete cosine transform （MDCT） and its inverse MDCT （IMDCT）, a novel approach based on one-dimensional discrete moments is proposed. By using the Taylor expansion of trigonometric function, both MDCT and IMDCT are approximated by a linear sum of a finite sequence of one-dimensional discrete moments after the kernel modular mapping and simplification. Then one-dimensional discrete moments can be calculated by an addition map of p ＋ 1 dimensional vector, which replaces most multiplication with simple additions. The number of multiplicative operations in the method is only O（Nlog2N/log2log2N） that is superior to O（Nlog2N） needed in the existing fast algorithms, and thus the operation can be speeded up. Theoretical analyses and experimental results show that the proposed method ensures high accuracy and fast computation, and it can satisfy the accuracy requirements of most applications.