Synthesis of Periodic Sinusoids from Walsh Waves

The Walsh expansion of a periodic sinusoid is an infinite series; however, if one synthesizes a sinusoid from a truncated Walsh expansion, the result is a stepped approximation to the sinusoid that has minimum mean-square error. The first 2m nonzero terms in the Walsh expansion yield a wave that has 2m steps per quarter-cycle. A notable feature of this wave is the wide separation between its harmonic pairs. For example, eight nonzero Walsh terms yield a wave whose only nonzero harmonics are the 31st and 33rd, the 63rd and 65th, etc., harmonic magnitudes being inversely proportional to frequency. Waves of this type are readily synthesized using a synchronous Walsh array generator with true and complement outputs, a level converter system, and a summing circuit.