In mathematics, a Walsh matrix is a specific square matrix with dimensions of some power of 2, entries of +1 or −1, and the property that the dot product of any two distinct rows (or columns) is zero. The Walsh matrix was proposed by Joseph L. Walsh in 1923. Each row of a Walsh matrix corresponds to a Walsh function.
The natural ordered Hadamard matrix is defined by the recursive formula below, and the sequency ordered Hadamard matrix is formed by rearranging the rows so that the number of sign-changes in a row is in increasing order. Confusingly, different sources refer to either matrix as the Walsh matrix.
The Hadamard matrices of dimension 2k for k ∈ N are given by the recursive formula
The lowest order of Hadamard matrix is 2
and in general
for 2 ≤ k ∈ N, where denotes the Kronecker product.
rearrange the rows of Matrix according the number of sign change of each row. for example:
the successive rows have 0, 3, 1, and 2 sign changes, then we rearrange the rows in sequency ordering.
where the successive rows have 0, 1, 2, 3, and 4 sign changes.
Alternative Forms of the Walsh Matrix
where the successive rows have 0, 1, 2, 3, 4, 5, 6, and 7 sign changes.
where the successive rows have 0, 1, 3, 2, 7, 6, 4, and 5 sign changes.
where the successive rows have 0, 7, 3, 4, 1, 6, 2, and 5 sign changes.
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