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 Haar function system may on the one hand be preferable because of its wavelet properties (e.g. localization), on the other hand the Walsh functions are bounded (in fact of modulus 1 everywhere).

The order of the function is 2^s, where s is an integer, meaning that there is 2^s (time-)intervals in which the value is -1 or 1.

2^s Potential function

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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+...+k_m2^m,

for some integer m, and with k_i equal to 0 or 1. Then the k-th Walsh function at a point x, with 0 ≤ x < 1, is

wal_k(x) = (-1)^{(k₀x₀+...k_mx_m),}

x = x₀/2+ x₁/2² + x₂/2³+...,

where again x_i 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 sequences 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.

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