Martin,
Thank you for jumping in.
On 22 Aug 2008 19:46:17 GMT, Martin Eisenberg <martin.eisenberg@[EMAIL PROTECTED]
>
wrote:
> Fr**** McKenney wrote:
>
>> CWT(psi, f, tau, s) = (1/sqrt(abs(s))) *
>> Integral(-inf,+inf, du,
>> ( f(u) * Conjugate( psi( (u-tau)/s ) ) ) )
>
>> First question: The CWT for any specific value of "s", that is,
>> for any single "horizontal" evaluation, this formula looks a lot
>> like the formula for a cross-correlation between f() and psi().
>> If so, this would mean that the CWT could be described in terms
>> of multiple cross-correlations between a given signal and one's
>> chosen "mother" wavelet.
>
> Like any integral transform, this is basically linear algebra. The
> class of functions f whose CWT converges ...
(Easy for _you_ to say that. <grin!>)
Forgive me for breaking this apart, but I'm not at a point where I
can absorb it as a single "chunk". (Somewhere the spirit of Bill
Smith is waggling a finger at me and reminding me that I should have
paid more attention in class. Sorry, Bill. <grin!>)
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? (Ignorance
provides a solid basis for paranoia; it's not clear whether it is
also orthogonal. <grin!>)
> ... 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'm not trying to be picky; It's just that it has been a while since
I worked with those particular terms, and I want to make sure I'm
not misunderstanding what you're saying. If I have, the fault is
likely mine.
> Compare Fourier theory: frequency is the parameter and the complex
> sinusoids do form an (orthonormal) basis of L2(R).
Pause for a bit of research...
L2(R): the set of functions which are square-integrable over R.
"Functions which, when multiplied by themselves and summed up, don't
explode in your face." Or, less formally, "stuff whose effects have
limits".
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. Sorry, I must have dropped
something in here somewhere.
>> This was puzzling, since none of the wavelet papers I have
>> scanned mentioned this way of looking at wavelets. Is it simply
>> considered so obvious that no-one bothers mentioning it?
>
> Papers from the mathematical side of the field certainly tend to
> presume the above understanding, but any book should say as much
> somewhere.
Okay. Thanks... I just wanted to make sure my mental model wasn't
falling apart.
> 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. But,
yes, I can see that, if I have a way of constructing a series of
filters, I can split up the frequency components.
> Which brings us to your third question about sinelike wavelets. You
> might make a bandpass from a lowpass by modulating the IR with a
> sinusoid. The classical Gabor analysis uses a Gaussian IR lowpass,
> for instance. Your "single sinelet" uses a boxcar filter whose slow
> spectral rolloff leads to bad band separation in the filterbank -- in
> other words, bad decorrelation between WT coefficients, which matters
> if you're going to modify and reconstruct.
I think I need to read this over a couple (more) times, and work my
way through it in detail. It's helpful in that it gives me some
concepts to match against any results I come up with.
>> Second question: I can "see" how the CWT transforms a
>> time-based signal into a time-scale plane, but I'm having
>> trouble seeing how one goes from that to a more useful -- to me,
>> anyway <grin!> -- time-frequency plane.
>
> 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!>
> Scale and frequency are similar in intent but different. I don't know
> the application you mentioned. What appeals to you about wavelets,
> and in turn about frequency?
The application? Take an audio signal, with embedded 100Hz-modulated
digital encoding and magically have the digital stuff pop out.
Strictly speaking, I suppose I don't need to _know_ which 'scale'
value corresponds to 100Hz; I can just look for "interesting
patterns" in the offset/scale plane. Knowing where to start looking
seemed useful, hence the desire to find a 'scale' that matched
'100Hz'.
In a way it's a chicken-or-egg situation. I wanted to understand
what wavelets offered, and it's difficult to know what kind of
places wavelets fit nicely into without at least a basic
understanding of them, so I picked something out of the air that
would use data I already have (WWV recordings). My hope is that, by
the time I finish beating my head against this particular project,
I'll at least know why it might be... um, "perhaps not the most
appropriate use of wavelets". <grin!>
Frank
--
"Give a man a fire, and he's warm for a day. Set him on fire,
and he's warm for the rest of his life"
-- Terry Pratchett/Jingo
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)
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