王雷， 郭迎^*

（  中南大学信息科学与工程学院；  ）

摘要： 本文将现有的Haar变换扩展到任意阶的Haar-Jacket变换。具体从以下几个方面展开：1、运用Jacket理论的思想，结合Haar矩阵的定义，严格定义Jacket-Haar变换。提出构造任意阶Jacket-Haar矩阵的方法。从最低阶的2阶Jacket-Haar矩阵出发，给出了2阶Jacket-Haar矩阵的一般形式。分N为奇数和N为偶数两种情况，提出由低阶Jacket-Haar矩阵构造出高阶Jacket-Haar矩阵的方法。从而成功构造出任意阶的Jacket-Haar矩阵。2、提出Jacket-Haar变换的快速算法。相比直接计算，这种方法可以显著减少运算量，提升计算效率。3、以心电图信号分析为例，说明了Jacket-Haar变换在信号分析中的应用。
关键词： Jacket 变换；Haar 变换；Jacket-Haar 变换；快速算法

Wang Lei， Guo Ying^*

（  School of Information Science&Engineering, Centre South University, Changsha 410083；  ）

Abstract： In this paper, Haar transform is generalized into the Jacket-Haar transform with any size. This paper is organized as follows. Firstly, by using the idea of Jacket transform theory and according to the definition of Haar transform, a new transform denoted as "Jacket-Haar transform" is mathematically defined. Methods to construct Jacket-Haar transform with any size is given. The general form of Jacket-Haar transform with size 2 is showed, followed by the "folding extension algorithm" that can be used to extend Jacket-Haar transform with the smaller size to one with the larger size. With the proposed algorithms, we can derive Jacket-Haar matrix with any size. Furthermore, fast implementation algorithms of Jacket-Haar matrix is exploited, compared with the direct computation, which is less computation and more efficiently. Moreover, the electrocardiogram (ECG) signal analysis is shown as an example of the applications of Jacket-Haar matrix.

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