Walsh matrix

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.[1] Each row of a Walsh matrix corresponds to a Walsh function.

Walsh matrix of order 16 multiplied with a vector

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.[1] Confusingly, different sources refer to either matrix as the Walsh matrix.

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

The Hadamard matrices of dimension 2^k for k ∈ N are given by the recursive formula

The lowest order of Hadamard matrix is 2

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

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

and in general

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

for 2 ≤ k ∈ N, where \otimes denotes the Kronecker product.

Sequency ordering

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.[2]

e.g.

\( W(4) = \begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\\ 1 & -1 & 1 & -1\\ \end{bmatrix}

where the successive rows have 0, 1, 2, and 3 sign changes.
See also

Haar wavelet
Quincunx matrix
Hadamard transform
Code division multiple access
OEIS A228539 (OEIS A228540) - rows of the (negated) binary Walsh matrices read as reverse binary numbers
OEIS A197818 - antidiagonals of the negated binary Walsh matrix read as binary numbers

Notes

Kanjilal, P.P. (1995). Adaptive Prediction and Predictive Control. Stevenage: IET. p. 210. ISBN 0-86341-193-2.
Yuen, C.-K. (1972). "Remarks on the Ordering of Walsh Functions". IEEE Transactions on Computers 21 (12): 1452. doi:10.1109/T-C.1972.223524.