Jacket Haar transform

Abstract

As the Walsh (Hadamard) transform can be generalized into the Jacket transform, in this paper, we generalize the Haar transform into the Jacket-Haar transform. The entries of the Jacket-Haar transform are 0 and ±2 k . Compared with the original Haar transform, the basis of the Jacket-Haar transform is general and more suitable for signal processing. Furthermore, with the proposed generalization algorithm, it is possible to define the N-point Jacket-Haar transform, where N is not a power of 2. From our simulations, the proposed Jacket Haar transform has better performance in ECG signal analysis.

Fast parametric reciprocal-orthogonal jacket transforms

“…Hence we obtain the proposed PROJT involving matrix [ J] 8 (see (18)) http://asp.eurasipjournals.com/content/2014/1/149…”

“…A lot of services of these transforms are mainly due to their practical usefulness and the existence of fast and efficient algorithms for their computation. However, since each of the well-known transforms [4][5][6][7][8][9], for example DFT, DHT, WHT, DCT, etc., is fixed without any parameters, each single transform only deals with its special area of applications. In order to best match the given input signal class or the application, many parametric transforms [10][11][12][13][14][15] with matrices associating a set of parameters are presented to fit the desirable signal by choosing appropriate parameters.…”

Fast Jacket-Haar Transform with Any Size

“…Haar matrix, which is useful for localized signal analysis [19], edge detection [20], OFDM, and filter design and electrocardiogram (ECG) analysis, has been generalized for Jacket-Haar matrix [21], whose entries are 0 and ±2 compared with entries of the original Haar matrix being 1, −1, and 0. Although the 2 -point Jacket-Haar matrices are successfully proposed in [21], there is still a problem on how to construct the arbitrary-length Jacket-Haar transform as the arbitrary-length Walsh-Jacket transform has already done with high efficiency [22]. Unfortunately, until this paper, the method to solve this problem is preliminary and not comprehensive except that some original results are shown [23].…”

A novel r-circulant block Jacket transform with fast algorithms

“…From the aspect of the matrix transform theory [3], Jacket transforms, such as co-cyclic Jacket transform [4], block Jacket transform [5], center weighted block Jacket transform [6], blind-block parametric Jacket transform [7], Jacket harr transform [8], have been presented and investigated one after another, while from the aspect of the practical applications, there exist literatures relevant with signal processing [9], image compression [10], quantum information system for quantum coding [11], wireless mobile communications for pre-coding, coding and decoding [12] [13] [14], new emerging 3G and 4G MIMO communication [15], encryption and decryption [16], and so on. Furthermore, lots of widely used transforms, such as WHT [2], DFT [17], DCT, HWT [8], slant transform all belong to the Jacket transform family. In other words, applications that have adopted the above-mentioned transforms can theoretically employ the corresponding Jacket transform.…”