A synthesizer based on frequency-phase analysis and square waves

• Authors: Sossio Vergara
• Venue: Digital Signal Processing
• Year: 2012

The generation of synthetic signals has been one of the first applications of computers. As a matter of fact the earliest electronic computers were analog and their output was, in effect, a signal. With the advent of digital electronic the initial method employed has been the application of the Fourier Theorem, generating signals as series of sinusoids, a technique deserving a name by its own: ''additive synthesis''. The advantage of the technique is the great control on the parameters of the generated wave. The main disadvantage is the complexity of the computation involved, namely for each component many samples of sinusoid need to be computed, and this usually requires a special hardware to be performed in real time. The reason is that the sine wave, although being natural for physical linear systems, is very complex in the digital domain. This article introduces an effective generalization of the polar flavor of the Fourier Theorem based on a new method of analysis. Under the premises of the new theory an ample class of functions become viable as bases, with the further advantage of using the same basis for analysis and reconstruction. In fact other tools, like the wavelets, admit specially built nonorthogonal bases but require different bases for analysis and reconstruction (biorthogonal and dual bases) and vectorial coordinates; this renders those systems unintuitive and computing intensive. As an example of the advantages of the new generalization of the Fourier Theorem, this paper introduces a novel method for the synthesis that is based on frequency-phase series of square waves (the equivalent of the polar Fourier Theorem but for nonorthogonal bases). The resulting synthesizer is very efficient needing only few components, frugal in terms of computing needs, and viable for many applications.