Walsh matrix

In

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 ${\displaystyle 2^{2}}$

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

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:

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

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:^[2]

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

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

Dyadic ordering

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

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

Natural ordering

${\displaystyle H(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&1&1&1&-1&-1&-1&-1\\1&-1&1&-1&-1&1&-1&1\\1&1&-1&-1&-1&-1&1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}$

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

See also

Wikimedia Commons has media related to Walsh matrix.

• 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

References