B. Yazici et Rl. Kashyap, A CLASS OF 2ND-ORDER STATIONARY SELF-SIMILAR PROCESSES FOR 1 F PHENOMENA/, IEEE transactions on signal processing, 45(2), 1997, pp. 396-410
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.