Nonorthogonal bases and phase decomposition: Properties and applications

• A novel iterative methodology for function decomposition is discussed.
• The methodology is applied to generalize the polar Fourier theorem to nonorthogonal bases.
• The advantages of nonorthogonal phase decomposition, compared to the usual tools as wavelets and frames, are discussed.
• Some of the properties and applications of the new transform are derived.

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.

The generalized phase–frequency transform applied to the separation of signals.Figure optionsDownload as PowerPoint slide