Walsh function

In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis. 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. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions.

The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions.

Walsh functions, the Walsh system, the Walsh series, 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 analyzing digital signals.

Historically, various numerations of Walsh functions have been used; none of them is particularly superior to another. In this article, we use the Walsh–Paley numeration.

We define the sequence of Walsh functions ${\displaystyle W_{k}:[0,1]\rightarrow \{-1,1\}}$, ${\displaystyle k\in \mathbb {N} _{0}}$ as follows.

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