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.
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