Upper Bounds on Walsh Transforms

Abstract

Many applications of Walsh functions employ Walsh series expansions. To obtain the advantage of data compression many of the Walsh coefficients must be negligible. It is thus desirable to have a scheme by which their magnitudes can be estimated with little computation. We found that the Walsh transform Fi of a function that can be differentiated by an arbitrary number of times is expressible as a weighted average of one of its derivatives. The mean and maximum of the weighting function is readily expressed in terms of i, from which we derived two upper bounds on Fi depending only on i and the maximum or mean magnitude of the derivative. As the bounds decrease rapidly with increasing degree and rank, Walsh transforms with high degree and rank are negligible. We also proved that discrete Walsh transforms calculated using sampled values obey the same bounds. While bounds can also be found for transforms of functions that can be differentiated by only a finite number of times, they tend to be larger and less dependent on rank. As most autocorrelation functions have singularities in second derivative, bounds on Walsh power spectra depend only on degree. Computed examples show that the behavior of Walsh spectra closely follow that of their upper bounds.