The Atomic Decompositions of Signals

It is one of the applications of waveforms theory, which corresponds to the transformation by the “Pursuit” algorithm with adaptive window. This technique will be applied in this chapter to a stock index, i.e. the French stock index: Cac40. We know that this transformation decomposes the signal in a time–frequency plane, the analyzing function is usually Gaussian of a variable width. The variable elements in the decomposition are the frequency, the position of the window and the size of the window, and we know that these three elements are independent. This transformation is particularly adapted to the strongly non-stationary signals which contain very different components. The algorithm “Pursuit” seeks the best “accord” (i.e. concordance) for each component of the signal rather than for the entire signal. The encoding or the decomposition of a non-stationary signal with “Pursuit” is concise and invariant by translation.