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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 orthogonal Walsh functions are used to perform the Hadamard transform, which is very similar to the way the orthogonal sinusoids are used to perform the Fourier transform.
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 2s, where s is an integer, meaning that there are 2s (time‑)intervals in which the value is -1 or 1.
A list of the 2s Walsh functions make a Hadamard matrix.
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 if k is the Gray code transform of j-1, the j-th Walsh function at a point x, with 0 ≤ x < 1, is
- wal j(x) = (-1)(k0x0+...kmxm),
- x = x0/2+ x1/22 + x2/23+...,
where again xi is 0 or 1 (only finitely often 1, if x is a dyadic number).
Walsh functions can be interpreted as the characters of
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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