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.
Contents
- Definition
- Comparison between Walsh functions and trigonometric functions
- Properties
- Walsh Ferleger systems
- Fermion Walsh system
- Binary Surfaces
- Applications
- References
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.
Definition
We define the sequence of Walsh functions
For any
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,
Notice that
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
Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line
Properties
The Walsh system
Walsh system is an orthonormal basis of Hilbert space
and being a basis means that if, for every
It turns out that for every
The Walsh system (in Walsh-Paley numeration) forms a Schauder basis in
Walsh-Ferleger systems
Let
Then basic representation theory suggests the following broad generalization of the concept of Walsh system.
For an arbitrary Banach space
where
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
Binary Surfaces
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.
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.