Walsh function

[1] They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval.

They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions.

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.

Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line.

is an abelian multiplicative discrete group isomorphic to

The Walsh system is an orthonormal basis of the Hilbert space

The Walsh system (in Walsh-Paley numeration) forms a Schauder basis in

be the compact Cantor group endowed with Haar measure and let

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.

be a strongly continuous, uniformly bounded faithful action of

Assume that every eigenspace is one-dimensional and pick an element

Classical Walsh system becomes a special case, namely, for where

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 Lp spaces associated with hyperfinite type II factor.

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.

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

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

Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.

endowed with the product topology and the normalized Haar measure.

can be associated with the real number This correspondence is a module zero isomorphism between

[8] 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.

[9] 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.