Symmetry of Walsh Functions

by Steven H. Cullinane

Walsh functions have an inherent symmetry that is best seen by regarding the non-constant Walsh functions as hyperplanes in finite geometry. Such symmetry is exhibited by the 15 affine hyperplanes of the 4-space over the 2-element field.

For a definition of Walsh functions, see The first sixteen Walsh functions.

(The Rao book referred to there is Piecewise Constant Orthogonal Functions and their Application to Systems and Control, by G. P. Rao, Springer-Verlag, 1983.)

Note that these are isomorphic to the 16 parts of the figure in the following research note:

For more on symmetry properties in finite geometry, see Diamond Theory.

For another picture showing the equivalence of nonconstant Walsh functions with the 15 affine hyperplanes that account for 4x4 symmetry in diamond theory, see the 15 "stencils" in Fig. VIII-8 of Shift Register Sequences, by Solomon W. Golomb, Holden-Day, 1967 (Revised edition published by Aegean Park Press, 1982).

For a bibliography of the theory of Walsh functions, click here.

For a bibliography of the applications of Walsh functions, click here.

Page last maintained Sept. 18, 2001; created August 31, 2001. Hits: 73 E-mail: m759@post.harvard.edu.