Natural ordered
Hadamard matrix (middle matrix) of order 16 that is sequency ordered to output a
Walsh matrix (right 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.
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
We define the sequence of Walsh functions , as follows.
be the jth bit in the
binary representation of k, starting with as the least significant bit, and
be the jth bit in the fractional binary representation of , starting with as the most significant fractional bit.
Then, by definition
In particular, everywhere on the interval, since all bits of k are zero.
Notice that is precisely the
Rademacher functionrm.
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
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.
The Walsh system is an orthonormal basis of the Hilbert space . Orthonormality means
,
and being a basis means that if, for every , we set then
It turns out that for every , the
seriesconverges 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 .
Generalizations
Walsh-Ferleger systems
Let be the
compactCantor 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
faithfulaction 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
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.
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
spinfermions. 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.
Vilenkin system
Fix a sequence of
integers with and let endowed with the
product topology and the normalized Haar measure. Define and . Each can be associated with the real number
This correspondence is a module zero isomorphism between and the unit interval. It also defines a norm which generates the
topology of . For , let where
The set is called generalized Rademacher system. The Vilenkin system is the
group of (
complex-valued) characters of , which are all finite products of . For each non-negative integer there is a unique sequence such that and
Then where
In particular, if , then is the Cantor group and is the (real-valued) Walsh-Paley system.
The Vilenkin system is a complete orthonormal system on and forms a
Schauder basis in , .[8]
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.[9]
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.
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.
Bibcode:
1996MPCPS.119..545F.
doi:
10.1017/s0305004100074405.
S2CID119786894.
Schipp, Ferenc; Wade, W.R.; Simon, P. (1990). Walsh series. An introduction to dyadic harmonic analysis. Akadémiai Kiadó.
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.
S2CID121256402.