On generic frequency decomposition. Part 1: Vectorial decomposition

• Authors: Sossio Vergara
• Venue: Digital Signal Processing
• Year: 2007

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 generic frequency analysis a step further. When a broader paradigm of functional analysis is defined, a much larger class of functions than those initially introduced in [Y. Wei, N. Chen, Square wave analysis, J. Math. Phys. 39 (8) (1998); Y. Wei, Frequency analysis based on general periodic functions, J. Math. Phys. 40 (7) (1999); Y. Wei, Frequency analysis based on easily generated functions, Appl. Comput. Harmon. Anal. (2000)] can be used to assemble bases in L^2. 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 the generic frequency analysis and the wavelets and frames theories will be discussed. Examples of analysis and reconstruction of functions using the given algorithms and generic bases will be given. In this first part the focus will be on vectorial decomposition, while the second part will be on polar decomposition with its consequences and applications.