A level-crossing-based scaling dimensionality transform applied to stationary Gaussian processes

The scaling dimensionality transform D/sub 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/sub a/(1, theta ) emerges as the fundamental transform. Results for stationary Gaussian processes, calculable from Rice's formula (1945) are applied to signals with asymptotic f/sup - beta / spectra and to the problem of adjusting amplitude quantization to the sampling period in discrete signal representations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>