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

Hi all. i am told frctal dimension is invariant under bi-Lipschitz
condition,say linear transform.So i guess the fractal dimensions of
communication signal don't change after it goes through a FIR channel.
i want to check if it's ture.i had done some simulation with
box-counting dimension,but dimensions of signals before channel and
after channel always dont be equal,changes unommitable. it seems the
theory dont hold,then what property of signal dont change after go
throgh channel?
confused.anyone have done related work?any comments and advice is
welcom.

"youyou" <readleister@gmail.com> wrote in message
news:1162021678.149256.178390@m73g2000cwd.googlegroups.com...
> Hi all. i am told frctal dimension is invariant under bi-Lipschitz
> condition,

Use plain English please....



-- 
Posted via a free Usenet account from http://www.teranews.com

Major Misunderstanding wrote:
> "youyou" <readleister@gmail.com> wrote in message
> news:1162021678.149256.178390@m73g2000cwd.googlegroups.com...

>> Hi all. i am told frctal dimension is invariant under bi-Lipschitz
>> condition,
> 
> Use plain English please....

I think the original post was just fine. What's your problem with the 
quoted sentence?


Martin

-- 
Let others praise ancient times;
I am glad I was born in these.
--Ovid

Major Misunderstanding wrote:
> "youyou" <readleister@gmail.com> wrote in message
> news:1162021678.149256.178390@m73g2000cwd.googlegroups.com...
> > Hi all. i am told frctal dimension is invariant under bi-Lipschitz
> > condition,
>
> Use plain English please....
>
>
>
> --
> Posted via a free Usenet account from http://www.teranews.com

  I just want to know whether the fractal dimension of communication
signal change after it was transmitted  throgh an FIR channel.I do it
by calculating the box-counting dimension of the transmitted passband
signal wave,and that of the wave after channel.I am disappinted to find
both dimensions dont be equal and the change always exist.
  I am trying to find a characteristic of signals which is invariant
after the channel.

  youyou

Let's see if a wider audience can help. Meanwhile, despite being 
clueless about your topic myself, some questions pop into my mind.

First of all, the question is underspecified. You mention FIR 
filtering which implies a discrete-time signal. Taken as a discrete 
subset of the time-value plane, its box-counting dimension (BCD) 
would seem always to be 0. On the other hand, taking the time series 
to specify a bandlimited function means imposing a smoothness 
condition that might just suffice (but I'm only guessing) to make the 
BCD always 1. So what are you actually doing?

Then there are the implementation issues. How do you handle 
computers' inability to carry out the limit process defining the BCD? 
Have you checked your approximation on signals of known BCD? Have you 
convinced yourself that roundoff errors in computing the FIR don't 
mess up bi-Lipschitzness enough to invalidate your results?


Martin

youyou wrote in comp.dsp:

> Hi all. i am told frctal dimension is invariant under
> bi-Lipschitz condition,say linear transform.So i guess the
> fractal dimensions of communication signal don't change after it
> goes through a FIR channel. i want to check if it's ture.i had
> done some simulation with box-counting dimension,but dimensions
> of signals before channel and after channel always dont be
> equal,changes unommitable. it seems the theory dont hold,then
> what property of signal dont change after go throgh channel?
> confused.anyone have done related work?any comments and advice
> is welcom.

-- 
Quidquid latine scriptum sit, altum viditur.

Martin Eisenberg wrote:
> Let's see if a wider audience can help. Meanwhile, despite being
> clueless about your topic myself, some questions pop into my mind.
>
> First of all, the question is underspecified. You mention FIR
> filtering which implies a discrete-time signal. Taken as a discrete
> subset of the time-value plane, its box-counting dimension (BCD)
> would seem always to be 0. On the other hand, taking the time series
> to specify a bandlimited function means imposing a smoothness
> condition that might just suffice (but I'm only guessing) to make the
> BCD always 1. So what are you actually doing?

Yes, interesting remark. Upon reading the original post, I searched
around a bit and found an article [1] concerned with estimating the
fractal dimension of continuous curves from a finite set of samples. It
goes on to define the Hausdorff dimension of a curve in the plane
(which is what I learned was the fractal dimension per definition of a
set), and also introduces the Minkowski dimension and the box counting
dimension. All three definitions are equivalent.

The problem is as follows: assume you have just a finite set of samples
from a (continuous) fractal curve, and you want to estimate the fractal
dimension from these samples. This looks to me a bit hopeless, because
you can never really know if the samples are sufficient to represent
the geometric character of the curve - there could always be finer
twists in the curve which can only be guessed at with higher
resolution.

I don't think it exists, but the fractal estimators are in dire need of
a kind of sampling theorem, which tells them when a set of samples is
sufficiently dense to estimate what lies in between the points. In that
sense, the fractal dimension is a similar parameter like bandwidth -
the higher, the more samples you need. If you don't know it in advance,
you never know if you have enough samples.

As it turns out, the Minkowski definition of fractal dimension is
easily extendable into an algorithm than can be applied to discrete
data. Its performance is checked against some standard fractals, and
considered "good enough". The authors then introduce an iterative
procedure to further estimate the fractal curve from the estimated
dimension and the data set. Convergence is guaranteed by a fixed-point
theorem for the space of continuous interpolating functions (similar
like in the proof of the theorem of Picard-Lindel=F6f).

The obvious problem of guessing what lies in between the sampled points
must also lie in between the lines of the paper, because I didn't see
it addressed anywhere. Or perhaps I just missed it. What I didn't miss
was a construction of a fractal interpolation function with arbitrary
fractal dimension: for every discrete data set, there exists a
continuous curve with any given fractal dimension (between 1 and 2, of
course). On what grounds one then can estimate the fractal dimension
from the point set eludes me. Furthermore, the OPs claim that the
fractal dimension doesn't change after FIR filtering seems ridiculous,
from that point of view.

Regards,
Andor

PS: Apperantly Mandelbrot once visited the ETH in Zurich, which has a
nice view of the Alps. When enjoying this view, he estimated the
fractal dimension of the alps at 1.4.

Introduction to fractal dimension of curves in the plane:
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

[1] Maragos and Sun:
"Measuring the Fractal Dimension of Signals: Morphological Covers and
Iterative Optimization",
IEEE Trans. on Signal Proc. , Vol. 41, No. 1, Jan 1993.
http://cvsp.cs.ntua.gr/publications/jpubl+bchap/MaragosSun_FrDimMorfCov_iee=
etSP1993.pdf

youyou wrote:
>   I just want to know whether the fractal dimension of communication
> signal change after it was transmitted  throgh an FIR channel.I do it
> by calculating the box-counting dimension of the transmitted passband
> signal wave,and that of the wave after channel.I am disappinted to find
> both dimensions dont be equal and the change always exist.

Sometimes, long-range correlated time series are called fractal because
of the self-similar statistics (Browninan motion, fractional Brownian
motion and fractional Gaussian noise) or the definition of the process
(fractional ARIMA). Their typical characteristic is the 1/|omega|^beta
power spectrum close to DC. Unless the FIR has a zero of the
appropriate order at DC, it cannot change this characteristic, and
therefore the fractional parameter (dimension?) of the time series.
There are several methods to estimate the fractional parameter of a
time series, one is the obvious test via the power spectrum.

I'm not sure how or if at all this is related to estimating the fractal
dimension of planar curves from discrete data sets (which you seem to
be implying).

Regards,
Andor

>   I am trying to find a characteristic of signals which is invariant
> after the channel.
> 
>   youyou

Andor wrote:

> http://cvsp.cs.ntua.gr/publications/jpubl+bchap/
> MaragosSun_FrDimMorfCov_ieeetSP1993.pdf 

> Upon reading the original post, I searched around a bit and found
> an article [1] concerned with estimating the fractal dimension of
> continuous curves from a finite set of samples. It goes on to
> define the Hausdorff dimension of a curve in the plane (which is
> what I learned was the fractal dimension per definition of a set),
> and also introduces the Minkowski dimension and the box counting
> dimension. All three definitions are equivalent. 

Actually, the paper itself states that Hausdorff dimension is not 
equivalent to the other two.

> The obvious problem of guessing what lies in between the sampled
> points must also lie in between the lines of the paper, because
> I didn't see it addressed anywhere.

Looks like it comes down to the assumption that the fractal dimension 
of the signal under scrutiny is constant over all scales so that the 
absolute sampling interval doesn't matter, and sweeping aliasing 
errors off the table 'cause it's just an approximation anyway. The 
authors don't state that assumption up front but it figures in the 
paragraph halfway into the second column on p. 115 (8 in the pdf). 
Also, the definition of discrete-time Hausdorff distance in eq. 44 
seems to use the "structuring function" g sort of analogously to a 
cardinal function in interpolation -- notice how eq. 15, 16 remind of 
convolutions?

> Or perhaps I just missed it. What I didn't miss was a construction
> of a fractal interpolation function with arbitrary fractal
> dimension: for every discrete data set, there exists a continuous
> curve with any given fractal dimension (between 1 and 2, of
> course).

I shall try those for controllable pseudo-random modulation 
trajectories in musical DSP sometime. Thanks for digging the paper 
up!

> On what grounds one then can estimate the fractal dimension from
> the point set eludes me.

That arbitrary time series can be turned into fractals of a given 
dimension doesn't mean an arbitrary time series may be the result of 
sampling such an interpolant, but I guess you have a point.

> PS: Apperantly Mandelbrot once visited the ETH in Zurich, which
> has a nice view of the Alps. When enjoying this view, he
> estimated the fractal dimension of the alps at 1.4.

A keen eye is a researcher's greatest asset ;)


Martin

-- 
The drowning girl will remark how pretty the coral.
--Sara Swanson, Malignant

Martin Eisenberg wrote:
> Andor wrote:
>
> > http://cvsp.cs.ntua.gr/publications/jpubl+bchap/
> > MaragosSun_FrDimMorfCov_ieeetSP1993.pdf
>
> > Upon reading the original post, I searched around a bit and found
> > an article [1] concerned with estimating the fractal dimension of
> > continuous curves from a finite set of samples. It goes on to
> > define the Hausdorff dimension of a curve in the plane (which is
> > what I learned was the fractal dimension per definition of a set),
> > and also introduces the Minkowski dimension and the box counting
> > dimension. All three definitions are equivalent.
>
> Actually, the paper itself states that Hausdorff dimension is not
> equivalent to the other two.

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.

>
> > The obvious problem of guessing what lies in between the sampled
> > points must also lie in between the lines of the paper, because
> > I didn't see it addressed anywhere.
>
> Looks like it comes down to the assumption that the fractal dimension
> of the signal under scrutiny is constant over all scales so that the
> absolute sampling interval doesn't matter, and sweeping aliasing
> errors off the table 'cause it's just an approximation anyway.

I guess the hope is that the discrete contains at least two
self-similarity scales, from which one could deduce the fractal
dimension using pattern matching techniques.

> The
> authors don't state that assumption up front but it figures in the
> paragraph halfway into the second column on p. 115 (8 in the pdf).
> Also, the definition of discrete-time Hausdorff distance in eq. 44
> seems to use the "structuring function" g sort of analogously to a
> cardinal function in interpolation -- notice how eq. 15, 16 remind of
> convolutions?

Now that you mention it.

>
> > Or perhaps I just missed it. What I didn't miss was a construction
> > of a fractal interpolation function with arbitrary fractal
> > dimension: for every discrete data set, there exists a continuous
> > curve with any given fractal dimension (between 1 and 2, of
> > course).
>
> 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 :-).

Regards,
Andor

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.