任意阶 Jacket-Haar 变换

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.