Linear Scale-invariant Systems Research Articles

On the Fractional Linear Scale Invariant Systems

The linear scale invariant systems are introduced for both integer and fractional orders. They are defined by the generalized Euler-Cauchy differential equation. The quantum fractional derivatives are suitable for dealing with this kind of systems, allowing us to define impulse response and transfer function with the help of the Mellin transform. It is shown how to compute the impulse responses corresponding to the two half plane regions of convergence of the transfer function.

A class of second-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 of 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, on the notion of stationarity. We consider a class of stochastic processes whose second-order structure is invariant with respect to time scales, i.e., E[X(t)X(/spl lambda/t)]=t/sup 2H//spl lambda//sup H/R(/spl lambda/), t>0 for some -x<H</spl infin/. For H=0, we refer to these processes as wide sense scale stationary. We show that any self-similar process can be generated from scale stationary processes. We establish a relationship between linear scale-invariant system theory and the proposed class that leads to a concrete analysis framework. We introduce new concepts, such as periodicity, autocorrelation, and spectral density functions, by which practical signal processing schemes can be developed. 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 introduce a class of finite parameter self-similar models that are similar in spirit to the ordinary ARMA models by which an arbitrary self-similar process can be approximated. Results from our study suggest that the proposed self-similar processes and the mathematical formulation provide an intuitive, general, and mathematically simple approach to 1/f signal processing.

The scale representation

The authors considers a physical attribute of a signal and develop its properties. He presents an operator which represents scale and study its characteristics and representation. This allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal. He obtains explicit expressions for the mean scale, scale bandwidth, instantaneous scale, and scale group delay. Furthermore, he derives expressions for mean time, mean frequency, duration, frequency bandwidth in terms of the scale variable. The short-time transform is defined and used to obtain the conditional value of scale for a given time. He shows that as the windows narrows one obtains instantaneous scale. Convolution and correlation theorems for scale are derived. A formulation is devised for studying linear scale-invariant systems. He derives joint representations of time-scale and frequency-scale, General classes for each are presented using the same methodology as for the time-frequency case. As special cases the joint distributions of Marinovich-Altes (1978, 1986) and Bertrand-Bertrand (1984) are recovered. Also, joint representations of the three quantities, time-frequency-scale are devised. A general expression for the local scale autocorrelation function is given. Uncertainty principles for scale and time and scale and frequency are derived. >