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# Fractal Modulation and Other Applications from a Theory of the Statistics of Dimension

Conference paper

## Abstract

This paper considers a model and the applications of a non-stationary stochastic field where r is a constant and

*u(x,t)*using a fractional partial differential equation of (time dependent) order*q(t)*given by$$ \left[ {\frac{{{\partial ^2}}}{{\partial {x^2}}} - {r^{q(t)}}\frac{{{\partial ^{q(t)}}}}{{\partial {t^{q(t)}}}}} \right]u(x,t) = - F(x,t),{\kern 1pt} - \infty < q(t) < \infty ,{\kern 1pt} \forall t $$

*F*and*q*are stochastic functions. Numerical algorithms for computing this solution are introduced and results presented to illustrate its characteristics which depend on the random behaviour of*q(t)*. An important aspect of this approach is concerned with the effect of changing the statistics [i.e. the Probability Density Function (PDF) of*q(t)]*used to ‘drive’ the solution. Since*q*is a dimension (the ‘Fourier dimension’ which is related to the fractal dimension), the introduction of a PDF associated with*q(t)*leads directly to the notion of the ‘statistics of dimension’. We address the inverse problem in which a discrete solution is found for*q(t)*from a known stochastic field*u(x,t)*when x→0. The application of fractal modulation, where*q(t)*is assigned just two states for all*t*as well as other applications are considered.## Keywords

Fractal Dimension Probability Density Function Discrete Fourier Transform Fractal Modulation Power Spectral Density Function
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## Copyright information

© Springer-Verlag New York, Inc. 2002