Walsh Function
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).
