Decomposition of the Hadamard matrices and fast Hadamard transform

Decomposition of Binary Matrices and Fast Hadamard Transforms

“…In this section, a fast algorithm for more general Hadamard transforms will be derived based on the multiplicative theorem [2], [14], [15], [17] given below. A Hadamard matrix H n of order n is an n × n matrix with elements equal to ±1 and satisfying the condition H n H T n = H T n H n = n I n , where T denotes transpose and I n is the identity matrix of order n. A Hadamard matrix of order n of the form…”

“…Note that if H n is an A(n, k)-matrix, then n ≡ 0(mod 2k) (see, e.g., [15]). From the definition of A(n, k)-matrices, the Kronecker product of A(n, k)-and A(m, r )-matrices is an A(mn, kr)-matrix.…”

“…Based on Theorem 3.1, we now derive a fast Hadamard transform algorithm for the cases when the order of the Hadamard matrix is not a power of 2 [13], [15], [16].…”

“…Recently, Hadamard transforms and their variations have been used in audio and video processing [4]- [6], [8], [11], [18]. For efficient computation of these transforms, fast algorithms have been developed [2], [3], [15], [16], [19]. These algorithms require only N log 2 N addition and subtraction operations (N = 2 k , N = 12 · 2 k , N = 4 k and some others) [1], [6], [19].…”

“…The origin of Hadamard matrices goes back at least to 1867 to Sylvester. There are many approaches to the construction of Hadamard matrices [1], [15], [17]. The survey by Seberry and Yamada [17] indicates that progress has been made in the past 100 years on this problem.…”