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Walsh Function


WalshFunctions

The Walsh functions consist of trains of square pulses (with the allowed states being -1 and 1) such that transitions may only occur at fixed intervals of a unit time step, the initial state is always +1, and the functions satisfy certain other orthogonality relations. In particular, the 2^n Walsh functions of order n are given by the rows of the Hadamard matrix H_(2^n) when arranged in so-called "sequency" order (Thompson et al. 1986, p. 204; Wolfram 2002, p. 1073). There are 2^n Walsh functions of length 2^n, illustrated above for n=1, 2, and 3.

Walsh functions were used by electrical engineers such as Frank Fowle in the 1890s to find transpositions of wires that minimized crosstalk and were introduced into mathematics by Walsh (1923; Wolfram 2002, p. 1073).

Amazingly, concatenating the Walsh functions W(n-1,[2^n/3]) (while simultaneously replacing -1s by 0s), where [x] is the ceiling function, gives the Thue-Morse sequence (Wolfram 2002, p. 1073). The values of [2^n/3] are given explicitly by (3+2^(n+1)+(-1)^n)/6, and the first few are 1, 2, 3, 6, 11, 22, 43, 86, 171, ... (OEIS A005578).

WalshFunctionOrderings

Walsh functions can be ordered in a number of ways, illustrated above (Wolfram 2002, p. 1073). The sequency k of a Walsh function is defined as half the number of zero crossings in one cycle of the time base. In sequency order (left figure), each row has one more color change than the preceding row. In natural (or Hadamard) order (middle figure), the Walsh functions display a nested structure. Dyadic (or Paley) order (right figure) is related to Gray code reordering of the rows (Wolfram 2002, p. 1073).

Walsh functions with nonidentical sequencies are orthogonal, as are the functions W(n,2k) and W(n,2k+1). The product of two Walsh functions is also a Walsh function.

Harmuth (1969) designates the even Walsh functions Cal(k) and the odd Walsh functions Sal(k),

Cal(n,k)=W(n,2k+1)
(1)
Sal(n,k)=W(n,2k),
(2)

where k is the sequency.

Taking the matrix product of a set of two-dimensional data (represented as a square matrix with size a power of two) with a corresponding array of Walsh functions is known as the Walsh transform (Wolfram 2002, p. 1073). Walsh transforms can be performed particular efficiently, resulting in the so-called fast Walsh transform.


See also

Boxcar Function, Cal, Fast Walsh Transform, Hadamard Matrix, Heaviside Step Function, Rectangle Function, Sal, Sequency, Square Wave, Walsh Transform

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References

Beauchamp, K. G. Walsh Functions and Their Applications. London: Academic Press, 1975.Harmuth, H. F. "Applications of Walsh Functions in Communications." IEEE Spectrum 6, 82-91, 1969.Sloane, N. J. A. Sequence A005578/M0788 in "The On-Line Encyclopedia of Integer Sequences."Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, p. 204, 1986.Tzafestas, S. G. Walsh Functions in Signal and Systems Analysis and Design. New York: Van Nostrand Reinhold, 1985.Walsh, J. L. "A Closed Set of Normal Orthogonal Functions." Amer. J. Math. 45, 5-24, 1923.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 573 and 1072-1073, 2002.

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Walsh Function

Cite this as:

Weisstein, Eric W. "Walsh Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WalshFunction.html

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