## Quick Reference

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 2* ^{n}* 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 2

*functions there is one function of zero sequency, one of (normalized) sequency 2*

^{n}*, and one pair (odd and even) of each (normalized) sequency from 1 to 2*

^{n−1}*− 1.*

^{n−1}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

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.

*Subjects:*
Computing.