fractal dimensions of communication signal don't change after FIR channel?

Martin Eisenberg wrote:
> Andor wrote:
> > Martin Eisenberg wrote:
> >> Andor wrote:
>
> >> Actually, the paper itself states that Hausdorff dimension is
> >> not equivalent to [Minkowski and box counting dimension].
> >
> > I was mislead by a paragraph. The last two are equivalent, but
> > they "coincide (in the continuous case) with [the Hausdorff
> > dimension] in many cases of practical interest". I wonder why
> > they insist on mentioning the "continuous case". For discrete
> > sets, as you already noted, all definitions return the
> > topological dimension of the set.
>
> Not every countable set is discrete -- intuitively, the infimum of
> distances among any two members may be zero as opposed to positive.
> (But see http://mathworld.wolfram.com/DiscreteSet.html for the
> general definition.) I imagine that's the reason.

Yes, for example the rational numbers in the interval [0,1] form a
countable set with dimension 1. However, the input data to the
estimation algorithm is always a finite set (therefore discrete). I'm
glad estimating fractal dimension isn't my business :-).

Regards,
Andor

Martin Eisenberg wrote:
> Andor wrote:
> > Martin Eisenberg wrote:
> >> Andor wrote:
>
> >> Actually, the paper itself states that Hausdorff dimension is
> >> not equivalent to [Minkowski and box counting dimension].
> >
> > I was mislead by a paragraph. The last two are equivalent, but
> > they "coincide (in the continuous case) with [the Hausdorff
> > dimension] in many cases of practical interest". I wonder why
> > they insist on mentioning the "continuous case". For discrete
> > sets, as you already noted, all definitions return the
> > topological dimension of the set.
>
> Not every countable set is discrete -- intuitively, the infimum of
> distances among any two members may be zero as opposed to positive.
> (But see http://mathworld.wolfram.com/DiscreteSet.html for the
> general definition.) I imagine that's the reason.
>
> >> > interpolation function with arbitrary fractal dimension
>
> >> I shall try those for controllable pseudo-random modulation
> >> trajectories in musical DSP sometime. Thanks for digging the
> >> paper up!
> >
> > Sounds interesting - let us know when you have some listening
> > samples available :-).
>
> It's in the queue but the stress is on "sometime"...
>
>
> Martin
>
> --
> Quidquid latine scriptum sit, altum viditur.


    Thanks all.
    I have checked the BCD of Weierstrass function ,and the BCD of the
result of convolving  Weierstrass function and an FIR filter
(representing an FIR channel).They are about equal,notwithstanding the
BCDs are both biased compared to its BCD in theory. I applied      the
FRACTAL LAB Toolbox for MATLAB (www.inria.fr) to calculate the BCDs.
    While I turned to communication signal, the change always existed
between the signal before and after the FIR channel. I guess the reason
is that communication signal is not a strictly fractal,with a low
degree of similarity.Maybe I should try other ways to approximate  the
dimension.    


  youyou