Walsh Spectral Analysis

Abstract

<p> Walsh functions are defined both by recursive and non-r~cursive equations. A synopsis is given of the properties of Walsh functions relevant to this thesis. Two algorithms for simple evaluation of an arbitrary point on a Walsh function that use only the binary codes for the parameters of the Walsh function result from the non-r~cursive definitions. Direct hardware implementation of the evaluation algorithms yields programmable digital Halsh function generators. One of the generators, which produces functions that are free of hazards or ambigious states, is modified to produce a parallel array of Walsh functions. This generator is used in a Walsh Spectral Analyzer that evaluates simultaneously several Walsh series coefficients of an input signal. </p> <p> Walsh series analysis and the concepts of the design of a digital Walsh Spectral Analyzer are discussed. The equation that is used to determine a Walsh series coefficient is modified so that each portion of the equation can be manipulated conveniently by a digital instrument. Although the instrument was designed primarily to analyze periodic waves, extensions to the design can be made to accommodate aperiodic signals. Signals with frequencies from the audio range downwards can be analyzed by the Walsh Spectral Analyzer. </p> <p> Walsh series to Fourier series conversion is dealt with. It has been found that the Fourier coefficients of signals that are limited either in frequency or in sequency can be evaluated precisely using a finite number of Walsh coefficients of the same signal. A dual relationship holds for Fourier to Walsh series conversion. The Fourier series coefficients of Walsh functions comprise part of the conversion relationships. The Fourier transforms of Walsh functions, from which the above coefficients can be obtained, are derived in non-recursive form. </p>