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.
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 , as follows.
For any , let
such that there are only finitely many non-zero kj and no trailing xj all equal to 1, be the canonical binary representations of integer k and real number x, correspondingly. Then, by definition
In particular, everywhere on the interval.
Notice that is precisely the Rademacher function rm. Thus, the Rademacher system is a subsystem of the Walsh system. Moreover, every Walsh function is a product of Rademacher functions:
Comparison between Walsh functions and trigonometric functions
Walsh functions and trigonometric functions are systems that both form a complete, orthonormal set of functions, an orthonormal basis in Hilbert space of the square-integrable functions on the unit interval. Both are systems of bounded functions, unlike, say, the Haar system or the Franklin system.
Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line . Furthermore, both Fourier analysis on the unit interval (Fourier series) and on the real line (Fourier transform) have their digital counterparts defined via Walsh system, the Walsh series analogous to the Fourier series, and the Hadamard transform analogous to the Fourier transform.
Walsh system is an orthonormal basis of Hilbert space . Orthonormality means
and being a basis means that if, for every , we set then
It turns out that for every , the series converge to for almost every .
The Walsh system (in Walsh-Paley numeration) forms a Schauder basis in , . Note that, unlike the Haar system, and like the trigonometric system, this basis is not unconditional, nor is the system a Schauder basis in .
Let be the compact Cantor group endowed with Haar measure and let be its discrete group of characters. Elements of are readily identified with Walsh functions. Of course, the characters are defined on while Walsh functions are defined on the unit interval, but since there exists a modulo zero isomorphism between these measure spaces, measurable functions on them are identified via isometry.
Then basic representation theory suggests the following broad generalization of the concept of Walsh system.
For an arbitrary Banach space let be a strongly continuous, uniformly bounded faithful action of on X. For every , consider its eigenspace . Then X is the closed linear span of the eigenspaces: . Assume that every eigenspace is one-dimensional and pick an element such that . Then the system , or the same system in the Walsh-Paley numeration of the characters is called generalized Walsh system associated with action . Classical Walsh system becomes a special case, namely, for
where is addition modulo 2.
In the early 1990s, Serge Ferleger and Fyodor Sukochev showed that in a broad class of Banach spaces (so called UMD spaces ) generalized Walsh systems have many properties similar to the classical one: they form a Schauder basis  and a uniform finite dimensional decomposition  in the space, have property of random unconditional convergence. One important example of generalized Walsh system is Fermion Walsh system in non-commutative Lp spaces associated with hyperfinite type II factor.
Fermion Walsh system
The Fermion Walsh system is a non-commutative, or "quantum" analog of the classical Walsh system. Unlike the latter, it consists of operators, not functions. Nevertheless, both systems share many important properties, e.g., both form an orthonormal basis in corresponding Hilbert space, or Schauder basis in corresponding symmetric spaces. Elements of the Fermion Walsh system are called Walsh operators.
The term Fermion in the name of the system is explained by the fact that the enveloping operator space, the so-called hyperfinite type II factor , may be viewed as the space of observables of the system of countably infinite number of distinct spin fermions. Each Rademacher operator acts on one particular fermion coordinate only, and there it is a Pauli matrix. It may be identified with the observable measuring spin component of that fermion along one of the axes in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.
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Romanuke showed that Walsh functions can be generalized to binary surfaces in a particular case of function of two variables. There also exist eight Walsh-like bases of orthonormal binary functions, whose structure is nonregular (unlike the structure of Walsh functions). These eight bases are generalized to surfaces (in the case of the function of two variables) also. It was proved that piecewise-constant functions can be represented within each of nine bases (including the Walsh functions basis) as finite sums of binary functions, when weighted with proper coefficients.
For example, the fast Walsh–Hadamard transform (FWHT) may be used in the analysis of digital quasi-Monte Carlo methods. In radio astronomy, Walsh functions can help reduce the effects of electrical crosstalk between antenna signals. They are also used in passive LCD panels as X and Y binary driving waveforms where the autocorrelation between X and Y can be made minimal for pixels that are off.
- Discrete Fourier transform
- Fast Fourier transform
- Harmonic analysis
- Orthogonal functions
- Walsh matrix
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