A fast hybrid Jacket–Hadamard matrix based diagonal block-wise transform

Abstract

In this paper, based on the block (element)-wise inverse Jacket matrix, a unified fast hybrid diagonal block-wise transform (FHDBT) algorithm is proposed. A new fast diagonal block matrix decomposition is made by the matrix product of successively lower order diagonal Jacket matrix and Hadamard matrix. Using a common lower order matrix in the form of 1 1, a fast recursive structure can be developed in the FHDBT, which is able to convert a newly developed discrete cosine transform (DCT)-II, discrete sine transform (DST)-II, discrete Fourier transform (DFT), and Haar-based wavelet transform (HWT). Since these DCT-II, DST-II, DFT, and HWT are widely used in different areas of applications, the proposed FHDBT can be applied to the heterogeneous system requiring several transforms simultaneously. Comparing with pre-existing DCT-II, DST-II, DFT, and HWT, it is shown that the proposed FHDBT exhibits less the complexity as its matrix size gets larger. The proposed algorithm is also well matched to circulant channel matrix. From the numerical experiments, it is shown that a better performance can be achieved by the use of DCT/DST-II compression scheme compared with the DCT-II only compression method. Signal Processing: Image CommunicationThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. , a fast recursive structure can be developed in the FHDBT, which is able to convert a newly developed discrete cosine transform (DCT)-II, discrete sine transform (DST)-II, discrete Fourier transform (DFT), and Haar-based wavelet transform (HWT). Since these DCT-II, DST-II, DFT, and HWT are widely used in different areas of applications, the proposed FHDBT can be applied to the heterogeneous system requiring several transforms simultaneously. Comparing with pre-existing DCT-II, DST-II, DFT, and HWT, it is shown that the proposed FHDBT exhibits less the complexity as its matrix size gets larger. The proposed algorithm is also well matched to circulant channel matrix. From the numerical experiments, it is shown that a better performance can be achieved by the use of DCT/ DST-II compression scheme compared with the DCT-II only compression method.

Fast Jacket-Haar Transform with Any Size

“…x [7] x [8] x [9] x [10] x [11] x [12] x [1] x [5] x [2] x [4] x [ problem of the traditional constraint that the number of the points is the power of 2. With the proposed fast generation algorithms, the arbitrary length Jacket-Haar transform can be derived in a successive fashion.…”

“…Jacket transform, motivated by the center weighted Hadamard transform [5], is a special transform with its inverse transform matrix being determined by the element or block-wise inverse of an original matrix [6]. Jacket transform has been extensively applied in many fields, such as signal and data processing [1], digital wireless communications [7], cryptography [8], and encoding designs [9]. Meanwhile, several interesting matrices, such as Hadamard matrices and DFT matrices, belong to the Jacket transform matrix family.…”

“…In the later aspect, block Jacket transforms have been tentatively applied to quantum signal processing [10], Big-Data processing [11], emerging new-generation mobile communication, and so on. In addition, new orthogonal transforms, such as complex Hadamard transform [12][13][14], fractional Hadamard or Jacket transforms [15], parametric transforms [16][17][18], hybrid transforms [7], and the generalized orthogonal transforms [14], have been gradually proposed while enriching the orthogonal transform family. Particularly, with advancement of digital systems and widespread availability in the recent few decades, there exist some urgent demands for seeking a scheme to achieve compromise between the generalization efficiencies of Hadamard or Haar transforms and their the extended transforms to adaptively meet the practical implementation requirements.…”

Signal flow graph approach to efficient and forward stable DST algorithms

“…Among applications of the DFT; sine and cosine waves of the DFT with different frequencies are used to classify the traffic monitoring sites into different seasonal patterns [35], DST has been identified as the method which generates better results for noise estimation as compared with Discrete Cosine Transform (DCT) and the DFT [10], discrete fractional sine transform has identified as the method for generating fingerprint templates with high recognition accuracy [49], DCT, DEST, and DFT can be approximated to the Karhunen Loeve Transformation (KLT) and the connection of KLT to the color image compression [6,22,31,32], DST can be used to analyze image reconstruction via signal transition through a square-optical fiber lenses [47], spectral interference and additive wideband noise on the accuracy of the normalized frequency estimator can be investigated through discrete-time sine-wave [1], to mention a few. Together with the above, the engagement of DCT and DST in image processing, signal processing, finger print enhancement, quick response code (QR code), and multimode interface can also be seen in e.g., [2,7,11,12,16,18,19,20,21,23,24,29,30,39,40,43,44,45].…”

Fast circulant block Jacket transform based on the Pauli matrices

“…Hadamard matrices and their generalisations are orthogonal matrices that play an important roles in the signal sequence transform and data processing (Guo et al, 2011). Jacket matrices motivated by the centre weighted Hadamard matrices (Lee, 1989), whose inverse can be simply obtained by their element-wise (Lee et al, 2013;Jiang et al, 2011), have been extensively investigated and applied in many fields, such as signal processing (Lee et al, 2013), encoding design (Jiang et al, 2011), wireless communication (Lee and Guo, 2012), image compression (Lee et al, 2014), watermarking (Ajay et al, 2010) and cryptography (Ma, 2004;Venkata Kishore and JayaVani, 2011). Particularly, some significant matrices, such as Hadamard, Harr, DFT and slant matrices, all belong to the Jacket matrix family (Song et al, 2010;Dr and Vaishali, 2011).…”