2 research outputs found
A Synthesizer Based on Frequency-Phase Analysis and Square Waves
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.Comment: 9 pages. Digital Signal Processing Journal 22 (2012
Nonorthogonal Bases and Phase Decomposition: Properties and Applications
In a previous paper [1] it was discussed the viability of functional analysis
using as a basis a couple of generic functions, and hence vectorial
decomposition. Here we complete the paradigm exploiting one of the analysis
methodologies developed there, but applied to phase coordinates, so needing
only one function as a basis. It will be shown that, thanks to the novel
iterative analysis, any function satisfying a rather loose requisite is
ontologically a basis. This in turn generalizes the polar version of the
Fourier theorem to an ample class of nonorthogonal bases. The main advantage of
this generalization is that it inherits some of the properties of the original
Fourier theorem. As a result the new transform has a wide range of applications
and some remarkable consequences. The new tool will be compared with wavelets
and frames. Examples of analysis and reconstruction of functions using the
developed algorithms and generic bases will be given. Some of the properties,
and applications that can promptly benefit from the theory, will be discussed.
The implementation of a matched filter for noise suppression will be used as an
example of the potential of the theory.Comment: 11 page