Stochastic Signal Representation

The representation of signals in the time domain requires the decomposition of functions of time into linear combinations of a finite number of basis functions. The purpose of this paper is to introduce a stochastic approximation algorithm, which, given an ensemble of signals , selects from a set that subset of basis functions that best represents the elements of the ensemble. In this context the optimum representation is defined as the basis set , which minimizes the expected value of a suitable performance index , defined as a function of both the basis and the random signal . In particular, is here assumed to be the least-square error that may be achieved when the signal is approximated by a linear combination of the elements of . The procedure is analyzed in detail for the case when the basis functions are one-sided complex exponentials. The convergence properties of the algorithm are discussed for this case. An illustrative example of application of the proposed method is also presented.