Walsh Matrix

A Walsh matrix is a square matrix whose rows are sampled Walsh functions. For order , with a nonnegative integer, it can be obtained by permuting the rows of the Sylvester Hadamard matrix. In the sequency-ordered convention, the rows are arranged in increasing order of the number of sign changes between consecutive entries. For example,

Its rows have 0, 1, 2, and 3 sign changes, respectively. Other conventions use the natural Hadamard order or a different ordering of the same Walsh functions.

The rows are orthogonal, and

where is the identity matrix and denotes the transpose. Consequently, is an orthogonal matrix. Multiplication by a Walsh matrix implements a discrete Walsh transform, with the normalization depending on the transform convention.