A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ten lectures on wavelets
Advances in Nonlinear Signal and Image Processing
Advances in Nonlinear Signal and Image Processing
On generic frequency decomposition. Part 1: Vectorial decomposition
Digital Signal Processing
Matching pursuits with time-frequency dictionaries
IEEE Transactions on Signal Processing
A synthesizer based on frequency-phase analysis and square waves
Digital Signal Processing
The wavelet transform, time-frequency localization and signal analysis
IEEE Transactions on Information Theory
Common Waveform Analysis: A New And Practical Generalization of Fourier Analysis
Common Waveform Analysis: A New And Practical Generalization of Fourier Analysis
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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.