Walsh functions in signal processing and electronic music
J. E. Lay · The Journal of the Acoustical Society of America · 1975
The class of functions, wal(k,t), of index k ⩾ 0 and argument t represents a complete set of orthogonal functions of the Hilbert space L2(−12,12). A signal function f εL2(−12,12) can consequently be represented by a generalized Walsh-Fourier series, with coefficients obtained from the usual criterion of minimum mean-square error. The theory of Walsh-Fourier series has a long tradition in mathematics, but it is the advance of semiconductor technology that has produced the first really new, useful set of orthogonal functions. Their importance is now being recognized in many applications, including the generation of periodic waveforms and envelope shapes for additive synthesis in electronic music. Walsh functions have many interesting properties. Seen mathematically, they represent in the main the characters of a specific locally compact Abelian group. However, from a pedagogical point of view, they can very simply be introduced through a discussion of orthogonal functions. For use in waveform synthesis, Walsh functions may easily be generated by means of bistable dividers and exclusive-or gates. For a given f(t), the series coefficients are obtainable from a fast Walsh transform algorithm.