Hi Frank,
I think the typo in your name might be deliberate but I'll point it
out anyway...
Fr**** McKenney wrote:
>> Like any integral transform, this is basically linear algebra.
>> The class of functions f whose CWT converges ...
> As I've wandered among various descriptions of the CWT I've seen
> much concern about psi() converging, and the integral of
> f()*psi() converging, but none about whether one might have to
> deal with an f() such that the CWT of f() didn't converge for
> _any_ psi(). Are there such? Or am I reading too much into
> this wording?
Given that psi itself has finite energy, I think the only way for the
CWT at particular (s,tau) to diverge would be a non-integrable pole
of f inside the sup****t. But in that case, with any psi goes (at
least) a finite region in the s-tau plane where the CWT in fact
diverges. (Unless a zero of psi happens to align with the pole, of
course, so divergence is only "almost everywhere" within the region.)
>> ... is a vector space, and you
>> project (inner product) f onto each of a one-parameter set B of
>> vectors. That parameter is s; and psi is supposed to make B a
>> basis for functions f.
>
> As in: B = { b(s) } for s in (-inf,+inf) ?
>
> Or a set of bases B(s) = { b(s,huh?) } ?
I meant the former. I must note that I misunderstood you, though --
for *fixed* s, tau is the parameter and B consists of translates,
B_s = {psi((u-tau)/s)/sqrt(|s|), tau in R}.
Anyway, it's an inner product ;)
>> Compare Fourier theory: frequency is the parameter and the
>> complex sinusoids do form an (orthonormal) basis of L2(R).
> L2(R): the set of functions which are square-integrable over
> R.
> Hm? I thought the whole point of Fourier theory was to deal with
> periodic -- a.k.a. infinite -- functions. Which was why one
> needed sinusoids (also infinite) as bases.
Since the FT drops the time dimension the prototype basis function
must catch a blip at any finite lag, so it must have infinite sup****t
even within L2. Of course we ususally deal with divergent Fourier
integrals using Dirac's delta, but I'm unclear on how wavelets work
with distributions.
>> More to the point, you could say that your view leads to the
>> so- called filterbank approach to wavelets. In terms of basis
>> change this means that you make psi some appropriate bandpass
>> filter and usually take the scales s from a discrete set so the
>> representation won't be overcomplete.
>
> "Discrete"? Ack! I'm still working on continuous wavelets.
That's what I'm talking about ;) The DWT discretizes both time and
scale. By contrast, the CWT-as-filterbank just "elides" values of s
that would yield redundant information, given the bandpass shape of
psi. A geometric progression results, i.e., s comes from
{s_0*r^n, n in Z}.
>> In view of my previous paragraph, perhaps you'll understand the
>> scal2frq docs better: it takes all those bandpass filters,
>> finds each of their spectral peaks (which are related by
>> successive time dilation), and calls the result the bands'
>> "pseudo-frequencies".
>
> Which implies that anyone _sane_ who wants to use wavelets
> should limit his choice of wavelets to those with distinct --
> and unique -- spectral peaks (with respect to time). <grin!>
I don't know what you mean by "spectral peaks re time", but since psi
must have finite energy the spectral centroid is always well-defined.
Whether it's also appropriate...
> Knowing where to start looking seemed useful, hence the desire
> to find a 'scale' that matched '100Hz'.
For an initial guess you can divide the prototype psi's center
frequency by your target frequency, like scal2frq does.
Martin
--
There are two kinds of people -- those who do the
work and those who take the credit. Try to be in
the first group; there is less competition there.
--Indira Gandhi
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