Walsh function
Encyclopedia
In mathematical analysis
, the set of Walsh functions form an orthogonal basis
of the square-integrable function
s on the unit interval
. The functions take the values -1 and +1 only, on sub-intervals defined by dyadic fractions. They are useful in electronics
, and other engineering applications.
The orthogonal Walsh functions are used to perform the Hadamard transform
, which is very similar to the way the orthogonal sinusoids are used to perform the Fourier transform.
The Walsh functions are related to the Haar function
s; both form a complete orthogonal system.
The Haar function system may on the one hand be preferable
because of its wavelet properties (e.g. localization), on the
other hand the Walsh functions are bounded (in fact of modulus
1 everywhere).
The order of the function is 2s, where s is an integer
, meaning that there are 2s (time-)intervals in which the value is -1 or 1.
A list of the 2s Walsh functions make a Hadamard matrix
.
One way to define Walsh functions is using the binary digit representations of reals and integers. For an integer k consider the binary digit representation
for some integer m, and with ki equal to 0 or 1. Then if k is the Gray code
transform of j-1,
the j-th Walsh function at a point x, with 0 ≤ x < 1, is
if
where again xi is 0 or 1 (only finitely often 1, if x is a dyadic number).
Walsh functions can be interpreted as the characters of
N,
the group of sequences over Z2; using this viewpoint, several generalizations have been defined.
Applications (in mathematics) can be found wherever digit representations are used, e.g. in the analysis of digital quasi-Monte Carlo method
s.
Walsh functions are used in Radio Astronomy
to reduce the effects of electrical crosstalk between antenna signals.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, the set of Walsh functions form an orthogonal basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
of the square-integrable function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s on the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
. The functions take the values -1 and +1 only, on sub-intervals defined by dyadic fractions. They are useful in electronics
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, and other engineering applications.
The orthogonal Walsh functions are used to perform the Hadamard transform
Hadamard transform
The Hadamard transform is an example of a generalized class of Fourier transforms...
, which is very similar to the way the orthogonal sinusoids are used to perform the Fourier transform.
The Walsh functions are related to the Haar function
Haar wavelet
In mathematics, the Haar wavelet is a certain sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis...
s; both form a complete orthogonal system.
The Haar function system may on the one hand be preferable
because of its wavelet properties (e.g. localization), on the
other hand the Walsh functions are bounded (in fact of modulus
1 everywhere).
The order of the function is 2s, where s is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
, meaning that there are 2s (time-)intervals in which the value is -1 or 1.
2s Potential function 1 ---------------- 2 --------________ 3 ----________---- 4 ----____----____ 5 --____----____-- 6 --____--__----__ 7 --__--____--__-- 8 --__--__--__--__ Table of the first eight orthogonal functions from the Walsh basis set. |
A list of the 2s Walsh functions make a Hadamard matrix
Hadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...
.
One way to define Walsh functions is using the binary digit representations of reals and integers. For an integer k consider the binary digit representation
- k = k0 + k12+...+km2m,
for some integer m, and with ki equal to 0 or 1. Then if k is the Gray code
Gray code
The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. It is a non-weighted code....
transform of j-1,
the j-th Walsh function at a point x, with 0 ≤ x < 1, is
- wal j(x) = (-1)(k0x0+...kmxm),
if
- x = x0/2+ x1/22 + x2/23+...,
where again xi is 0 or 1 (only finitely often 1, if x is a dyadic number).
Walsh functions can be interpreted as the characters of
N,
the group of sequences over Z2; using this viewpoint, several generalizations have been defined.
Applications (in mathematics) can be found wherever digit representations are used, e.g. in the analysis of digital quasi-Monte Carlo method
Quasi-Monte Carlo method
In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral that is based on low-discrepancy sequences...
s.
Walsh functions are used in Radio Astronomy
Radio astronomy
Radio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The initial detection of radio waves from an astronomical object was made in the 1930s, when Karl Jansky observed radiation coming from the Milky Way. Subsequent observations have identified a number of...
to reduce the effects of electrical crosstalk between antenna signals.