# On the Statistics of Dimension: Multi-fractalsand Fractal Modulation

Friday, January 19, 2001 - 11:30am - 12:30pm

Keller 3-180

Martin Turner (De Montfort University)

Below is a TeX version

We consider a new approach to modelling a non-stationary or multi-fractal stochastic field $u(x,t)$ using a fractional partial differential equation of (time dependent) order $q(t)$ given by

$$left [{partial^{2}overpartial x^{2}}-tau^{q(t)}{partial ^{q(t)} over partial t^{q(t)}}right ]u(x,t)=-F(x,t), h -infty

where $tau$ is a constant and $F$ and $q$ are stochastic functions. The theoretical background and ideas upon which this equation is based are presented. This includes a brief overview of random fractal walks, L'evy flights, fractional calculus and fractional dynamics. A general solution to this equation is then considered using a Green's function method from which an asymptotic solution is derived for $xrightarrow 0$. Numerical algorithms for computing this solution are introduced and results presented to illustrate its characteristics which depend on the random behaviour of $q(t)$. One important and interesting 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 lq drive' the solution. Since $q$ is a dimension (the lq 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 lq statistics of dimension'. v In the context of $q(t)$ being a random variable, the asymptotic solution for $u(x,t)$ when $xrightarrow 0$ yields special behaviour when $q=2$. This is characterised by events that are analogous to L'evy-type flights which we call lq Brownian transients'. We also address the inverse problem in which a discrete solution is found for $q(t)$ from a known stochastic field $u(x,t)$ when $xrightarrow 0$. A single application is considered - Fractal Modulation - where $q(t)$ is assigned just two states for all $t$. This approach is analogous to Frequency Modulation but where a bit stream is forced to lq look like' background fractal noise for applications in (covert) digital (multimedia or otherwise) communication systems.

We consider a new approach to modelling a non-stationary or multi-fractal stochastic field $u(x,t)$ using a fractional partial differential equation of (time dependent) order $q(t)$ given by

$$left [{partial^{2}overpartial x^{2}}-tau^{q(t)}{partial ^{q(t)} over partial t^{q(t)}}right ]u(x,t)=-F(x,t), h -infty

where $tau$ is a constant and $F$ and $q$ are stochastic functions. The theoretical background and ideas upon which this equation is based are presented. This includes a brief overview of random fractal walks, L'evy flights, fractional calculus and fractional dynamics. A general solution to this equation is then considered using a Green's function method from which an asymptotic solution is derived for $xrightarrow 0$. Numerical algorithms for computing this solution are introduced and results presented to illustrate its characteristics which depend on the random behaviour of $q(t)$. One important and interesting 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 lq drive' the solution. Since $q$ is a dimension (the lq 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 lq statistics of dimension'. v In the context of $q(t)$ being a random variable, the asymptotic solution for $u(x,t)$ when $xrightarrow 0$ yields special behaviour when $q=2$. This is characterised by events that are analogous to L'evy-type flights which we call lq Brownian transients'. We also address the inverse problem in which a discrete solution is found for $q(t)$ from a known stochastic field $u(x,t)$ when $xrightarrow 0$. A single application is considered - Fractal Modulation - where $q(t)$ is assigned just two states for all $t$. This approach is analogous to Frequency Modulation but where a bit stream is forced to lq look like' background fractal noise for applications in (covert) digital (multimedia or otherwise) communication systems.