A signal-processing tool adapted to the periodic biphasic phenomena: the Dynalet transform

The linear functional analysis, historically founded by Fourier and Legendre (Fourier's supervisor), has provided an original vision of the mathematical transformations between functional vector spaces. Fourier, and later Laplace and Wavelet transforms, respectively, defined using the simple and damped pendulum have been successfully applied in numerous applications in Physics and engineering problems. However, the classical pendulum basis may not be the most appropriate in several problems, such as biological ones, where the modelling approach is not linked to the pendulum. Efficient functional transforms can be proposed by analyzing the links between the physical or biological problem and the orthogonal (or not) basis used to express a linear combination of elementary functions approximating the observed signals. In this study, an extension of the Fourier point of view called Dynalet transform is described. The approach provides robust approximated results in the case of relaxation signals of periodic biphasic organs in human physiology.