Sinusoids versus Walsh functions

In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.