Decomposition of Binary Matrices and Fast Hadamard Transforms

Binary matrices or (+/- 1)-matrices have numerous applications in coding, signal processing, and communications. In this paper, a general and efficient algorithm for decomposition of binary matrices and the corresponding fast transform is developed. As a special case, Hadamard matrices are considere...

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Bibliographic Details
Published in:Circuits, systems, and signal processing Vol. 24; no. 4; pp. 385 - 400
Main Authors: Sarukhanyan, Hakob, Agaian, Sos, Astola, Jaakko, Egiazarian, Karen
Format: Journal Article
Language:English
Published: Heidelberg Springer 01.08.2005
Springer Nature B.V
Subjects:
Algorithms
Applied sciences
Binary system
Decomposition
Exact sciences and technology
Information, signal and communications theory
Mathematical methods
Numerical analysis
Signal processing
Telecommunications and information theory
Performance evaluation
Williamson type matices
Matrix decomposition
Matrix element
Hadamard matrix
1-mitrices
Hadamard transform. -1
Algorithm performance
Signal processing
Hadamard transformation
Numerical simulation
Fast algorithm
shift/add architechture
ISSN:0278-081X, 1531-5878
Online Access:Get full text
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Summary:Binary matrices or (+/- 1)-matrices have numerous applications in coding, signal processing, and communications. In this paper, a general and efficient algorithm for decomposition of binary matrices and the corresponding fast transform is developed. As a special case, Hadamard matrices are considered. The difficulties of the construction of 4n-point Hadamard transforms are related to the Hadamard problem: the question of the existence of Hadamard matrices. (It is not known whether for every integer n, there is an orthogonal 4n X 4n matrix with elements +/- 1.) In the derived fast algorithms, the number of real operations is reduced from O(N2) to O(N log N) compared to direct computation. The proposed scheme requires no zero padding of the input data. Comparisions revealing the efficiency of the proposed algorithms with respect to the known ones are given. In particular, it is demonstrated that, in typical applications, the proposed algorithm is significantly more efficient than the conventional Walsh-Hadamard transform. Note that for Hadamard matrices of orders >/= 96 the general algorithm is more efficient than the classical Walsh-Hadamard transform whose order is a power of 2. The algorithm has a simple and symmetric structure. The results of numerical examples are presented.[PUBLICATION ABSTRACT]
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ISSN:0278-081X
1531-5878
DOI:10.1007/s00034-004-0811-y