Wikipedia / Walsh function
Wikipedia / Walsh function
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functionss 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. This articles uses the Walsh–Paley numeration.
Definition
as follows.
For any natural number k, and real number, let
be the jth bit in the binary representation of k as the least significant bit, and
be the j as the most significant fractional bit.
Then, by definition
everywhere on the interval, since all bits of k are zero.
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 both systems that form a complete, orthonormal set of functions, an orthonormal basis in the 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.
Properties
is an abelian multiplicative discrete group isomorphic, the Pontryagin dual of the Cantor group. Its identity, and every element is of order two (that is, self-inverse).
. Orthonormality means
,
then
The Walsh system (in Walsh-Paley numeration) forms a Schauder basis. Note that, unlike the Haar system, and like the trigonometric system, this basis is not unconditional.
Generalizations
Walsh-Ferleger systems
be the compact Cantor group endowed with Haar measure be its discrete group of characters 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 be a strongly continuous, uniformly bounded faithful action on X, consider its eigenspace. Then X. Assume that every eigenspace is one-dimensional. Classical Walsh system becomes a special case, namely, for
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 in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.
Vilenkin system
of integers endowed with the product topology can be associated with the real number
and the unit interval. It also defines a norm which generates the topology where
is called generalized Rademacher system. The Vilenkin system is the group of (complex and
where
is the (real-valued) Walsh-Paley system.
and forms a Schauder basis.
Nonlinear Phase Extensions
Nonlinear phase extensions of discrete Walsh-Hadamard transform were developed. It was shown that the nonlinear phase basis functions with improved cross-correlation properties significantly outperform the traditional Walsh codes in code division multiple access (CDMA) communications.
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.