Walsh matrix

Hadamard matrix of order 16 multiplied with a vector

Naturally ordered Hadamard matrix permuted into sequency-ordered Walsh 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 Hadamard 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 Hadamard 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

\mathbb {N}

.

In mathematics, a Walsh matrix is a specific square matrix of dimensions 2ⁿ, where n is some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns are orthogonal. The Walsh matrix was proposed by Joseph L. Walsh in 1923.^[1] Each row of a Walsh matrix corresponds to a Walsh function.

The Walsh matrices are a special case of Hadamard matrices where the rows are rearranged so that the number of sign changes in a row is in increasing order. In short, a Hadamard matrix is defined by the recursive formula below and is naturally ordered, whereas a Walsh matrix is a sequency-ordered.^[1]^[2] 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 $k\in \mathbb {N}$ 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

We can obtain a Walsh matrix from a Hadamard matrix. For that, we first generate the Hadamard matrix for a given dimension. Then, we count the number of sign changes of each row. Finally, we re-order the rows of the matrix according to the number of sign changes in ascending order.

For example, let us assume that we have a Hadamard matrix of dimension $2^{2}$

H(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, 3, 1, and 2 sign changes (we count the number of times we switch from a positive 1 to a negative 1, and vice versa). If we rearrange the rows in sequency ordering, we obtain:

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.

Alternative forms of the Walsh matrix

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:^[3]

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

H(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 (Hadamard matrix).

References

1 2 Kanjilal, P. P. (1995). Adaptive Prediction and Predictive Control. Stevenage: IET. p. 210. ISBN 0-86341-193-2.
↑ Da Fontoura Costa, Luciano (2021). "A Compact Guide to the Hadamard and Walsh Matrices (CDT-64)". doi:10.13140/RG.2.2.22899.86561. {{cite journal}}: Cite journal requires |journal= (help)
↑ 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

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

[2] Da Fontoura Costa, Luciano (2021). "A Compact Guide to the Hadamard and Walsh Matrices (CDT-64)". doi:10.13140/RG.2.2.22899.86561. {{cite journal}}: Cite journal requires |journal= (help)

[3] 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.

[1]

[2]

[3]

Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices