On generic frequency decomposition. Part 1: vectorial decomposition
It provides a theoretical generalization of Fourier analysis for signal processing researchers, but the results are incremental as it builds on existing frameworks without demonstrating practical advantages.
This paper extends Fourier analysis to nonorthogonal bases by keeping the reconstruction algorithm but changing the analysis method, enabling generic frequency decomposition. Examples of analysis and reconstruction using nonorthogonal bases are provided.
The famous Fourier theorem states that, under some restrictions, any periodic function (or real world signal) can be obtained as a sum of sinusoids, and hence, a technique exists for decomposing a signal into its sinusoidal components. From this theory an entire branch of research has flourished: from the Short-Time or Windowed Fourier Transform to the Wavelets, the Frames, and lately the Generic Frequency Analysis. The aim of this paper is to take the Frequency Analysis a step further. It will be shown that keeping the same reconstruction algorithm as the Fourier Theorem but changing to a new computing method for the analysis phase allows the generalization of the Fourier Theorem to a large class of nonorthogonal bases. New methods and algorithms can be employed in function decomposition on such generic bases. It will be shown that these algorithms are a generalization of the Fourier analysis, i.e. they are reduced to the familiar Fourier tools when using orthogonal bases. The differences between this tool and the wavelets and frames theories will be discussed. Examples of analysis and reconstruction of functions using the given algorithms and nonorthogonal bases will be given. In this first part the focus will be on vectorial decomposition, while the second part will be on phased decomposition. The phased decomposition thanks to a single function basis has many interesting consequences and applications.