Walsh matrix

In mathematics, a Walsh matrix is a square matrix, with dimensions a power of 2, the entries of which are +1 or -1, defined by the recursive formula below. The Walsh matrix can be obtained from a Hadamard matrix by rearranging the rows so that the number of sign-changes is in increasing order. Since a Walsh matrix can be obtained from Hadamard matrix solely by exchanging rows it retains the property that the dot product of any two distinct rows (or columns) is zero. Each row of a Walsh Matrix corresponds to a Walsh function.

The Walsh matrix (and Walsh functions) are used in computing the Walsh transform and have application in efficient implementation of certain signal processing operations.

Formula

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H(1) = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}

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H(k) = \begin{bmatrix} H(k-1) & H(k-1)\\ H(k-1) & -H(k-1)\end{bmatrix} <math>