Decomposition of Binary Matrices and Fast Hadamard Transforms

Abstract

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 × 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.