A Synthesizer based on square waves

One of the most widely employed technique for the sound synthesis is based on the Fourier theorem that states that any signal can be obtained as a sum of sinusoids. Unfortunately this algorithm, when applied to synthesizers, requires some peculiar operations, as the addressing of a Look Up Table, that are not easily built-in in standard processors, thus requiring specially designed architectures. The aim of this paper is to show that, when using a new method for the analysis and polar coordinates, a much broader class of functions can be employed as a basis, and it turns out that the square wave is just one of such functions. When the synthesis of signals is carried out by summing square waves, the additive synthesizer architecture results much more simplified, allowing for example to synthesize complex signals simply in software, using general purpose microprocessors, even in real-time. Firstly it will be proven that when using a novel method for the analysis phase, the L2 function space admits a broad class of functions as bases, not necessarily orthogonal. The requirements for a function, in order to be a basis, will be defined. A straightforward and computationally simple algorithm for the analysis will be proposed. The end result is in effect the generalization of the Fourier Theorem to the case of nonorthogonal bases. It will be shown that the given method is more powerful and efficient than the wavelets and frames techniques.