Walsh functions

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Walsh functions A complete set of functions that form an orthonormal basis for Walsh analysis: they take only the values +1 and –1, and are defined on a set of 2n points for some n. For purposes of computer representation, and also for their use in coding, it is usual to represent “+1” by “0”, and “–1” by “1”. As an example, the 8-point Walsh functions are then as follows: wal(8,0) = 00000000 wal(8,1) = 11110000 wal(8,2) = 00111100 wal(8,3) = 11001100 wal(8,4) = 10011001 wal(8,5) = 01101001 wal(8,6) = 01011010 wal(8,7) = 10101010

Note that the Walsh functions (usually denoted wal) consist alternatively of even and odd functions (usually denoted cal and sal by analogy with cos and sin). Furthermore, within the set of 2n functions there is one function of zero sequency, one of (normalized) sequency 2n–1, and one pair (odd and even) of each (normalized) sequency from 1 to 2n–1 – 1.

A set of Walsh functions corresponds, with some permutation of columns, to a Reed–Muller code and, with a column deleted, to a simplex code. See also Hadamard matrices.