Walsh matrix
The Hadamard matrices of dimension 2"k" for "k" ∈ "N" are given by the recursive formula (the lowest order of Hadamard matrix is 2):
formula_1
and in general
formula_2
for 2 ≤ "k" ∈ "N", where ⊗ denotes the Kronecker product. Rearrange the rows of the matrix according to the number of sign change of each row. For example, in
formula_3
the successive rows have 0, 3, 1, and 2 sign changes. If we rearrange the rows in sequency ordering:
formula_4
then the successive rows have 0, 1, 2, and 3 sign changes. The sequency ordering of the rows of the Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray-code permutation:
formula_5
where the successive rows have 0, 1, 2, 3, 4, 5, 6, and 7 sign changes. formula_6 where the successive rows have 0, 1, 3, 2, 7, 6, 4, and 5 sign changes. formula_7 where the successive rows have 0, 7, 3, 4, 1, 6, 2, and 5 sign changes.
Formula
In mathematics, a Walsh matrix is a specific square matrix of dimensions 2, where "n" is some particular natural number. Each row of a Walsh matrix corresponds to a Walsh function. The Walsh matrices are a special case of Hadamard matrices. The "naturally 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.