Editing Walsh matrix (section)

===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|Gray-code]] [[permutation]]:<ref>{{cite journal |last=Yuen |first=C.-K. |year=1972 |title=Remarks on the Ordering of Walsh Functions |journal=IEEE Transactions on Computers |volume=21 |issue=12 |page=1452 |doi=10.1109/T-C.1972.223524 }}</ref>

:<math>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},</math>

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

====Dyadic ordering====
:<math>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},</math>

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

====Natural ordering====
:<math>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},</math>

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