In mathematics, a Walsh matrix is a square matrix, with dimensions a power of 2, the entries of which are +1 or -1. The Walsh matrix can be obtained from a Hadamard matrix (which is defined by the recursive formula below) of the same dimension by rearranging the rows so that the number of sign-changes is in increasing order. This is called sequency ordering. 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 applications in the efficient implementation of certain signal processing operations.

Formula

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The Hadamard matrices of dimension $2^k$ for $k\; \backslash in\; N$ are given by the recursive formula

$$

H(1) = \begin{bmatrix} 1 \end{bmatrix},

$$

H(2) = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix},

and

$$

H(2^k) = \begin{bmatrix} H(2^{k-1}) & H(2^{k-1})\\ H(2^{k-1}) & -H(2^{k-1})\end{bmatrix} = H(2)\otimes H(2^{k-1}),

for $2\; \backslash le\; k\; \backslash in\; N$, where $\backslash otimes$ denotes the Kronecker product.

Sequency Ordering

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The 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.

References

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Yuen, C. 1972. Remarks on the Ordering of Walsh Functions. IEEE Transactions on Computers. C-21: 1452.

Matrices