A Generalized Reverse Block Jacket Transform

• Published in: IEEE transactions on circuits and systems. I, Regular papers Vol. 55; no. 6; pp. 1589 - 1600
• Main Authors: Guihua Zeng, Guihua Zeng, Moon Ho Lee, Moon Ho Lee
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2008; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algebra; Block Jacket transform; Blocking; Construction; Cryptography; Decomposition; Error correction; Error correction codes; fast matrices construction and decomposition; Image coding; Information theory; Mathematical analysis; Matrices; Matrix decomposition; Matrix methods; Pauli matrices; Recursive; Signal generators; Signal processing; Signal processing algorithms; signal sequence transforms; fast matrices construction and decomposition; Block Jacket transform; Pauli matrices; signal sequence transforms
• ISSN: 1549-8328; 1558-0806
• DOI: 10.1109/TCSI.2008.916620

Jacket matrices motivated by the center weight Hadamard matrices have played important roles in signal processing, communication, image compression, cryptography, etc. In this paper we propose a notation called block Jacket matrix which substitutes elements of the matrix into common matrices or even block matrices. Employing the well-known Pauli matrices which are very important in many subjects, block Jacket matrices with any size are investigated in detail, and some recursive relations for fast construction of the block Jacket matrices are obtained. Based on the general recursive relations, several special block Jacket matrices are constructed. To decompose high order block Jacket matrices, a fast decomposition algorithm for the factorable block Jacket matrices is suggested. After that some properties of the block Jacket matrices are investigated. Finally, several remarks are presented. These remarks are associated with comparisons between the Clifford algebra and the block Jacket matrices, generations of orthogonal and quasi-orthogonal sequences, and relations of the block Jacket matrices to the orthogonal transforms for signal processing. Since the Pauli matrices are actually infinitesimal generators of SU (2) group, the proposed construction and decomposition algorithms for the block Jacket matrices are available in the signal processing, communication, quantum signal processing and information theory.