Walsh function
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In mathematical analysis, the set of Walsh functions form an orthogonal basis of the square-integrable functions on the unit interval. The functions take the values -1 and 1 only, on sub-intervals defined by dyadic fractions. They are useful in electronics, and other engineering applications.
The Walsh functions are related to the Haar functions; both form a complete orthogonal system.
The order of the function is 2s, where s is an integer, meaning that there is 2s (time-)intervals in which the value is -1 or 1.
2s Potential function
1 ---------------- 2 --------________ 3 ----________---- 4 ----____----____ 5 --____----____-- 6 --____--__----__ 7 --__--____--__-- 8 --__--__--__--__
Table of the first eight orthogonal functions from the Walsh basis set.
One way to define Walsh functions is using the binary digit representations of reals and integers. For an integer k consider the binary digit representation
- k = k0 + k12+...+km2m,
for some integer m, and with ki equal to 0 or 1. Then the k-th Walsh function at a point x, with 0 ≤ x < 1, is
- walk(x) = (-1)(k0x0+...kmxm)</sub>,
if
- x = x0/2+ x1/22 + x2/23+...,
where again xi is 0 or 1 (only finitely often 1, if x is a rational number).
Walsh functions can be interpreted as the characters of
- (Z2)N,
the group of series over Z2; using this viewpoint, several generalizations have been defined.
Applications (in mathematics) can be found wherever digit representations are used, e.g. in the analysis of digital quasi-Monte Carlo methods.