Arikan and Alamouti matrices based on fast block-wise inverse Jacket transform

Abstract

Recently, Lee and Hou (IEEE Signal Process Lett 13: 461-464, 2006) proposed one-dimensional and two-dimensional fast algorithms for block-wise inverse Jacket transforms (BIJTs). Their BIJTs are not real inverse Jacket transforms from mathematical point of view because their inverses do not satisfy the usual condition, i.e., the multiplication of a matrix with its inverse matrix is not equal to the identity matrix. Therefore, we mathematically propose a fast block-wise inverse Jacket transform of orders N = 2 k , 3 k , 5 k , and 6 k , where k is a positive integer. Based on the Kronecker product of the successive lower order Jacket matrices and the basis matrix, the fast algorithms for realizing these transforms are obtained. Due to the simple inverse and fast algorithms of Arikan polar binary and Alamouti multiple-input multiple-output (MIMO) non-binary matrices, which are obtained from BIJTs, they can be applied in areas such as 3GPP physical layer for ultra mobile broadband permutation matrices design, first-order q-ary Reed-Muller code design, diagonal channel design, diagonal subchannel decompose for interference alignment, and 4G MIMO long-term evolution Alamouti precoding design.

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).…”

“…In recent years, block Jacket matrices with their elements substituted by common matrices or block matices, have been introduced and extensively investigated (Lee et al, 2013;Zeng and Lee, 2008;Jiang and Lee, 2007;Khan et al, 2013). The fast algorithms for one-dimensional and two-dimensional block centre weight Hadamard transform (BCWHT) and block inverse jacket transform are respectively obtained based on the sparse matrix factorisation and the Kroneker products in Lee and Zhang (2007), and Lee and Hou (2006).…”

“…Instead of the conventional block-wise inverse Jacket matrix (BIJM), in Mao et al (2008), the cocyclic block-wise inverse Jacket matrix (CBIJM) was introduced, and corresponding fast algorithms were given subsequently. Besides the afore-mentioned researches emphasising on theoretical aspect of the block Jacket transforms, there also exist another attempts concentrating on the practical applications of the block Jacket transform, such as non-binary LDPC codes design (Jiang and Lee, 2007), MIMO communication system (Khan et al, 2013), Arikan and Alamouti precoding design (Lee et al, 2013), etc.…”

A novel r-circulant block Jacket transform with fast algorithms

“…Jacket transform and its corresponding matrices have been extensively investigated. 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.…”

A New Space Vector Modulation Strategy Based on MC

“…the matrix converter has been present for decades [1][2], and research has made great progress. Matrix converter circuit topology is different to the traditional pay -direct -converter.…”