Editing Walsh function (section)

{{Short description|Concept in mathematics}}
[[File:Natural and sequency ordered Walsh 16.svg|thumb|480px|Natural ordered [[Hadamard matrix]] (middle matrix) of order 16 that is sequency ordered to output a  [[Walsh matrix]] (right matrix).<br>Both contain the 16 Walsh functions of order 16 as rows (and columns).<br>In the right matrix, the number of sign changes per row is consecutive.]]

In [[mathematics]], more specifically in [[harmonic analysis]], '''Walsh functions''' form a [[Complete orthogonal system|complete orthogonal set]] of [[function (mathematics)|function]]s that can be used to represent any discrete function—just like [[trigonometric functions]] can be used to represent any [[continuous function]] in [[Fourier analysis]].<ref>{{harvnb|Walsh|1923}}.</ref>  They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the [[unit interval]]. But unlike the [[sine and cosine]] functions, which are continuous, Walsh functions are piecewise [[constant function|constant]]. They take the values −1 and +1 only, on sub-intervals defined by [[dyadic fraction]]s.

The system of Walsh functions is known as the '''Walsh system'''. It is an extension of the [[Rademacher system]] of orthogonal functions.<ref>{{harvnb|Fine|1949}}.</ref>

Walsh functions, the Walsh system, the Walsh series,<ref>{{harvnb|Schipp|Wade|Simon|1990}}.</ref> and the [[fast Walsh–Hadamard transform]] are all named after the American mathematician [[Joseph L. Walsh]]. They find various applications in [[physics]] and [[engineering]] when [[Digital signal processing|analyzing digital signals]].

Historically, various [[numeration]]s of Walsh functions have been used; none of them is particularly superior to another. This articles uses the ''Walsh–Paley numeration''.