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.[1] 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.

Natural ordered and sequency ordered Hadamard matrix of order 16.
Especially the former is usually called Walsh matrix.
Both contain the 16 Walsh functions of order 16 as rows (and columns).
In the right matrix, the number of sign changes per row is consecutive.

The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions.[2]

Walsh functions, the Walsh system, the Walsh series,[3] 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.

Definition

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

For any natural number k, and real number $x\in [0,1]$ , let

k_{j}

be the jth bit in the binary representation of k, starting with

k_{0}

as the least significant bit, and

x_{j}

be the jth bit in the binary representation of x, starting with

x_{1}

as the most significant fractional bit.

Then, by definition

W_{k}(x)=(-1)^{\sum _{j=0}^{\infty }k_{j}x_{j+1}}

In particular, $W_{0}(x)=1$ everywhere on the interval, since all bits of k are zero.

Notice that $W_{2^{m}}$ is precisely the Rademacher function r_m. Thus, the Rademacher system is a subsystem of the Walsh system. Moreover, every Walsh function is a product of Rademacher functions:

W_{k}(x)=\prod _{j=0}^{\infty }r_{j}(x)^{k_{j}}

Comparison between Walsh functions and trigonometric functions

Walsh functions and trigonometric functions are both systems that form a complete, orthonormal set of functions, an orthonormal basis in Hilbert space $L^{2}[0,1]$ 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 $\mathbb {R}$ . 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.

Properties

The Walsh system $\{W_{k}\},k\in \mathbb {N} _{0}$ is a commutative multiplicative discrete group isomorphic to $\coprod _{n=0}^{\infty }\mathbb {Z} /2\mathbb {Z}$ , the Pontryagin dual of Cantor group $\prod _{n=0}^{\infty }\mathbb {Z} /2\mathbb {Z}$ . Its identity is $W_{0}$ , and every element is of order two (that is, self-inverse).

The Walsh system is an orthonormal basis of Hilbert space $L^{2}[0,1]$ . Orthonormality means

\int _{0}^{1}W_{k}(x)W_{l}(x)dx=\delta _{kl}

,

and being a basis means that if, for every $f\in L^{2}[0,1]$ , we set $f_{k}=\int _{0}^{1}f(x)W_{k}(x)dx$ then

\int _{0}^{1}(f(x)-\sum _{k=0}^{N}f_{k}W_{k}(x))^{2}dx{\xrightarrow[{N\rightarrow \infty }]{}}0

It turns out that for every $f\in L^{2}[0,1]$ , the series $\sum _{k=0}^{\infty }f_{k}W_{k}(x)$ converge to $f(x)$ for almost every $x\in [0,1]$ .

The Walsh system (in Walsh-Paley numeration) forms a Schauder basis in $L^{p}[0,1]$ , $1<p<\infty$ . Note that, unlike the Haar system, and like the trigonometric system, this basis is not unconditional, nor is the system a Schauder basis in $L^{1}[0,1]$ .

Generalizations

Walsh-Ferleger systems

Let $\mathbb {D} =\prod _{n=1}^{\infty }\mathbb {Z} /2\mathbb {Z}$ be the compact Cantor group endowed with Haar measure and let ${\hat {\mathbb {D} }}=\coprod _{n=1}^{\infty }\mathbb {Z} /2\mathbb {Z}$ be its discrete group of characters. Elements of ${\hat {\mathbb {D} }}$ are readily identified with Walsh functions. Of course, the characters are defined on $\mathbb {D}$ 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 $(X,||\cdot ||)$ let $\{R_{t}\}_{t\in \mathbb {D} }\subset Aut(X)$ be a strongly continuous, uniformly bounded faithful action of $\mathbb {D}$ on X. For every $\gamma \in {\hat {\mathbb {D} }}$ , consider its eigenspace $X_{\gamma }=\{x\in X:R_{t}x=\gamma (t)x\}$ . Then X is the closed linear span of the eigenspaces: $X={\overline {\operatorname {Span} }}(X_{\gamma },\gamma \in {\hat {\mathbb {D} }})$ . Assume that every eigenspace is one-dimensional and pick an element $w_{\gamma }\in X_{\gamma }$ such that $||w_{\gamma }||=1$ . Then the system $\{w_{\gamma }\}_{\gamma \in {\hat {\mathbb {D} }}}$ , or the same system in the Walsh-Paley numeration of the characters $\{w_{k}\}_{k\in {\mathbb {N} }_{0}}$ is called generalized Walsh system associated with action $\{R_{t}\}_{t\in \mathbb {D} }$ . Classical Walsh system becomes a special case, namely, for

R_{t}:x=\sum _{j=1}^{\infty }x_{j}2^{-j}\mapsto \sum _{j=1}^{\infty }(x_{j}\oplus t_{j})2^{-j}

where $\oplus$ 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 [4]) generalized Walsh systems have many properties similar to the classical one: they form a Schauder basis [5] and a uniform finite dimensional decomposition [6] in the space, have property of random unconditional convergence.[7] One important example of generalized Walsh system is Fermion Walsh system in non-commutative L^p 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 ${\mathcal {R}}$ , may be viewed as the space of observables of the system of countably infinite number of distinct spin ${\frac {1}{2}}$ 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 $\{x,y,z\}$ in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.

Binary Surfaces

Romanuke showed that Walsh functions can be generalized to binary surfaces in a particular case of function of two variables.[8] There also exist eight Walsh-like bases of orthonormal binary functions,[9] 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.[10]

Applications

Applications of the Walsh functions can be found wherever digit representations are used, including speech recognition, medical and biological image processing, and digital holography.

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.

References

Ferleger, Sergei V. (March 1998). RUC-Systems In Non-Commutative Symmetric Spaces (Technical report). MP-ARC-98-188.CS1 maint: ref=harv (link)

Ferleger, Sergei V.; Sukochev, Fyodor A. (March 1996). "On the contractibility to a point of the linear groups of reflexive non-commutative Lp-spaces". Mathematical Proceedings of the Cambridge Philosophical Society. 119 (3): 545–560. doi:10.1017/s0305004100074405.CS1 maint: ref=harv (link)

Fine, N.J. (1949). "On the Walsh functions". Trans. Amer. Math. Soc. 65 (3): 372–414. doi:10.1090/s0002-9947-1949-0032833-2.CS1 maint: ref=harv (link)

Pisier, Gilles (2011). Martingales in Banach Spaces (in connection with Type and Cotype). Course IHP (PDF).CS1 maint: ref=harv (link)

Romanuke, V. V. (2010a). "On the Point of Generalizing the Walsh Functions to Surfaces".CS1 maint: ref=harv (link)

Romanuke, V. V. (2010b). "Generalization of the Eight Known Orthonormal Bases of Binary Functions to Surfaces".CS1 maint: ref=harv (link)

Romanuke, V. V. (2010c). "Equidistantly Discrete on the Argument Axis Functions and their Representation in the Orthonormal Bases Series".CS1 maint: ref=harv (link)

Schipp, Ferenc; Wade, W.R.; Simon, P. (1990). Walsh series. An introduction to dyadic harmonic analysis. Akadémiai Kiadó.CS1 maint: ref=harv (link)

Sukochev, Fyodor A.; Ferleger, Sergei V. (December 1995). "Harmonic analysis in (UMD)-spaces: Applications to the theory of bases". Mathematical Notes. 58 (6): 1315–1326. doi:10.1007/bf02304891.CS1 maint: ref=harv (link)

Walsh, J.L. (1923). "A closed set of normal orthogonal functions". Amer. J. Math. 45 (1): 5–24. doi:10.2307/2387224. JSTOR 2387224.CS1 maint: ref=harv (link)

External links

"Walsh functions". MathWorld.

"Walsh functions". Encyclopedia of Mathematics.

"Walsh system". Encyclopedia of Mathematics.

"Walsh functions". Stanford Exploration Project.

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[1] Walsh 1923.

[2] Fine 1949.

[3] Schipp, Wade & Simon 1990.

[4] Pisier 2011.

[5] Sukochev & Ferleger 1995.

[6] Ferleger & Sukochev 1996.

[7] Ferleger 1998.

[8] Romanuke 2010a.

[9] Romanuke 2010b.

[10] Romanuke 2010c.