A CLASS OF 2ND-ORDER STATIONARY SELF-SIMILAR PROCESSES FOR 1 F PHENOMENA/

We propose a class of statistically self-similar processes and outline an alternative mathematical framework for the modeling and analysis o f 1/f phenomena. The foundation of the proposed class is based on the extensions of the basic concepts of classical time series analysis, in particular, an the notion of stationarity, We consider a class of sto chastic processes whose second-order structure is invariant with respe ct to time scales, i.e., E[X(t)X(lambda t)] = t(2H)lambda(H)R(lambda), lambda, t > 0 for some -x < H < x. For H = 0, we refer to these proce sses as wide sense scale stationary, We show that any self-similar pro cess can be generated from scale stationary processes. We establish a relationship between linear scale-invariant system theory and the prop osed class that leads to a concrete analysis framework. We introduce n ew concepts, such as periodicity, autocorrelation, and spectral densit y functions, by which practical signal processing schemes can be devel oped, We give several examples of scale stationary processes including Gaussian, non-Gaussian, covariance, and generative models, as well as fractional Brownian motion as a special case. In particular, we intro duce a class of finite parameter self-similar models that are similar in spirit to the ordinary ARMA models by which an arbitrary self-simil ar process can be approximated. Results from our study suggest that th e proposed self-similar processes and the mathematical formulation pro vide an intuitive, general, and mathematically simple approach to 1/f signal processing.