Walsh matrix

In mathematics, a Walsh matrix is a specific square matrix of dimensions 2ⁿ, where n are some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns are orthogonal, i.e. dot product 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

Naturally ordered Hadamard matrix permuted into sequency-ordered Hadamard matrix. The number of sign changes per row in the naturally ordered matrix is (0, 15, 7, 8, 3, 12, 4, 11, 1, 14, 6, 9, 2, 13, 5, 10), in the sequency-ordered matrix the number of sign changes is consecutive.

LDU decomposition of a Walsh matrix. The ones in the triangular matrices form Sierpinski triangles. The entries of the diagonal matrix are values from Gould's sequence, with the minus signs distributed like the ones in Thue–Morse sequence.

Binary Walsh matrix as a matrix product. The binary matrix (white 0, red 1) is the result with operations in F₂. The gray numbers show the result with operations in R.

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.[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):

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

and in general

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

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

Permutation

Rearrange the rows of the matrix according to the number of sign change of each row. For example, in

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

the successive rows have 0, 3, 1, and 2 sign changes. If we rearrange the rows in sequency ordering:

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

then the successive rows have 0, 1, 2, and 3 sign changes.

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]

W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\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, 3, 4, 5, 6, and 7 sign changes.

Dyadic ordering

W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&1&1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&-1&1&-1&-1&1&-1&1\\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, 3, 2, 7, 6, 4, and 5 sign changes.

Natural ordering

W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&1&1&1&-1&-1&-1&-1\\1&-1&1&-1&-1&1&-1&1\\1&1&-1&-1&-1&-1&1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},

where the successive rows have 0, 7, 3, 4, 1, 6, 2, and 5 sign changes.

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.

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[kanjilal-1] Kanjilal, P. P. (1995). Adaptive Prediction and Predictive Control. Stevenage: IET. p. 210. ISBN 0-86341-193-2.

[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.