Nonorthogonal Bases and Phase Decomposition: Properties and Applications
For signal processing and functional analysis researchers, this work provides a theoretical generalization of Fourier analysis to nonorthogonal bases, but the results are incremental as they build on prior work and lack concrete performance numbers.
The paper extends a previously developed functional analysis paradigm to phase coordinates, showing that any function satisfying a loose condition can serve as a basis, generalizing the polar Fourier theorem to nonorthogonal bases. The new transform is compared with wavelets and frames, and a matched filter for noise suppression demonstrates its potential.
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.