A LEVEL-CROSSING-BASED SCALING DIMENSIONALITY TRANSFORM APPLIED TO STATIONARY GAUSSIAN-PROCESSES

The scaling dimensionality transform D(a)(r, theta) of stochastic processes is introduced as a generalization of the fractal dimension concept over an infinite range of time scales. It is based on the expected number of crossings of a constant level a, and is a function of two variables: the scaling factor r and the sampling time theta. General properties of this transform are discussed, whereby D(a)(1, theta) emerges as the fundamental transform. Results for stationary Gaussian processes, calculable from Rice's formula, are applied to signals with asymptotic f(-beta) spectra and to the problem of adjusting amplitude quantization to the sampling period in discrete signal representations.