Trying to follow the math behind wavelets

Rune,

Sorry to take so long responding. Being confused is something I can
do immediately; putting my confusion into words in such a way that
exposes the specifics of my confusion to another person takes a bit
longer. <grin?>

On Sun, 24 Aug 2008 23:05:06 -0700 (PDT), Rune Allnor <allnor@tele.ntnu.no> wrote:
> On 24 Aug, 22:35, Frnak McKenney
><fr...@far.from.the.madding.crowd.com> wrote:
>> On Fri, 22 Aug 2008 12:18:05 -0700 (PDT), Rune Allnor <all...@tele.ntnu.no> wrote:
>> > On 22 Aug, 16:30, Frnak McKenney
>> ><fr...@far.from.the.madding.crowd.com> wrote:
> ...
>> > The texts don't use 'cross correlation' because it would be
>> > positively misleading. Correlation is a concept that is used
>> > in statistics. Wavelets is a decomposition technique which
>> > is analyzed by maths. Real Analysis.
>>
>> I hear the words you're using, but I may not be sufficiently versed in
>> the distinctions you're making to feel that I agree or disagree.
>
> As to *why* I make the distinction: Consider a motor. It's basically
> a device which takes power from some external power source and
> converts it to rotational energy by making a the drive axel
> (I don't know if this is the correct technical term in English;
> please correct me if I'm wrong) spin.

Note: "Drive axle", but we in the Colonies are notoriously bad
spellers. It seems to be a competitive thing with our former mother
country: they drop 'h's, we drop 'u's (e.g. colour=>color). <grin!>

> Suppose you first study a compustion motor and find the term
> 'crank shaft' used for the drive axel. Then you study the
> electric motor and all of a sudden you realize that 'Hey!
> This thing has a spinning part too! It's just like the crank
> shaft in the combustion engine!'

Yup. It goes around and around, and can push against other things
to make them turn, or move.

> It would be wrong to use the term 'crank shaft' about the
> drive axel in the electric motor. First of all, the people
> who deal with electrical motors would conider you a fool
> if they heard you. Second, the term 'crank shaft' brings
> with it a lot of excess luggage like the very close
> connections with terms like 'piston', 'cylinder', 'piston
> rod', none of which make sense in the context of electric
> motors.

Understood. I get that kind of reaction when I talk to my brother;
he's studying DSP with no electronics background, so I've had to
learn to avoid describing filters in terms of "capacitors" and
"inductors". <grin!>

> So instead of saying that 'The electric motor works with
> a crankshaft which lacks the piston and rods but has
> something completely different gadget attached', it is far
> better to cut to the core and say 'the electric motor
> contains a rotating axle with gadgets attached.'
>
> In the case of 'correlation' vs 'inner product' you might
> not see the difference up front, but 'correlation' brings
> with it all the excess luggage of statistics (which you
> only need in the context of statistics) whereas 'inner
> product' is the distilled generic term of a mathematical
> operation.

Ah. In terms of your example, think of me as someone who is
concentrating on "things that go around and push/drive other things"
who picked up on "crankshaft" rather than "drive axle" because it
"went around and pushed things" and because I ran across it first.
Thank you for correcting my usage (and again for talking the time to
explain the correction <grin!>).

>> I have been interpreting the values "returned" from the CWT at a point
>> (tau,s) as a "degree of match" between my original signal, on the one
>> hand, and the function psi() on the other, at that same point. Are you
>> saying that "correlation" in a bad way of describing this? Or that my
>> model (interpretation) is flawed? Or Something Completely Different?
>>
>> Sorry, but it was a nice model, and it seemed to work. My head is
>> reluctant to surrender it. <grin!>
>
> You were certainly on the right track. I am sorry if I confused you;
> maybe the explanation above might help you out.

It did. I get to keep my model of "how the parts act", but need to
change my descriptive terminology. Progress. <grin!>

--snip--
>> >> What I've read suggests that the relationship
>> >> between 'scale' and 'frequency' depends on the specific choice of
>> >> the wavelet function psi(),
>>
>> > The relation depends on both the wavelet function as such
>> > and the scale from the mother wavelet.
>>
>> But what would the process look like, in general terms?
>
> That's a very technical issue which was treated in the R&V tutorial.

R&V "Wavelets & Signal Processing" has phrases like "since two
scales a0 < a1 roughly correspond to two frequencies f0 > f1.."
(p.20). There may be more, but if so it's on one of the sections I
haven't been able to "unpack" yet.

However, I found the following in a 1992 Technical Report from
Hewlett-Packard(!):

An Introduction to Wavelets
http://www.hpl.hp.com/techreports/92/HPL-92-124.pdf

"Scale" is roughly "minus log frequency", in the following limited
sense. A single scale 's' contains information from a band of
frequencies. The width and the center frequency of the band are
both proportional to -log(s). Each scale's band has the same
ratio of bandwidth to center frequency, so scales correspond to a
set of constnat-Q filters."

It's not a complete answer for my purposes, but it does give me
arough idea of what direction I should go hunting in, and some hope
of finding results there.

>> >> but other than using the Matlab magic
>> >> 'scal2freq' function it's not clear to me how I would go about doing
>> >> it in practice.
>>
>> > There's lots of stuff going on in matlab that you
>> > should be very, very catious about.
>>
>> S'OK. *I'll be working with Scilab, so I'll have to be cautious about
>> other things. <grin!>
>
> Just be cautious. The fact that there is a function for it in *lab
> doesn't mean it can be done.

Or even should be done. <grin!>

> ...
>> > One is that what you write is not scalable.
>>
>> Apologies for my density, and perhaps I wrote it badly. *If I have a
>> single cycle of a sine wave, can't I stretch it into a longer-lasting
>> single cycle, or increase its amplitude?
>
> You can. But you might certain important key issues with wavelets.
> One basic limiting condition in signal processing is the Heissenberg
> inequality. It states that in order to fix a sinusoidal with
> a certain accuracy in frequency you need a recording of a certain
> duration in time. Conversely, if you want to locate a spike with
> a certain accuracy in time you need a spectrum of a certain
> bandwidth.

Several of the papers I've scanned mentioned this. It's not so much
that I'm deliberately ignoring it, as much as, for me, it's hanging
from a branch I can't quite reach yet. I can see its general shape,
but I can't quite reach it to add it to what I already know.

> Heissenberg's inequality inserts the numbers so that you can compute
> the required bandwidths and durations once your target accuracies are
> known.
>
> Wavelets are an attempt to close in on the Heissenberg limit. That
> is, the ideal wavelet represents a pulse in time by a spectrum
> no wider than the Heissenberg limit.
>
> So while your windowed sinusoidal at first glance might serve the
> purpose of a wavelet, it fails completely with this added (or maybe
> implicit) constraint about wavelets hovering around Heissenberg.

I read this, but couldn't make a "picture" out of it yet -- my
mental model needs a great deal of improvement <grin!> -- so I
decided to try to see what it would look like plotted using a really
simple wavelet (Haar). And I hit something that confused me even
more: it looks like the "shape" of a CWT, for a given 'scale', is
not what I've been assuming it would be.

Assumption: The CWT(g(),psi(),t,s) is, in some measure, related to
the Fourier "frequency" spectrum of f(). That is, for a fixed g()
and psi(), if the frequency spectrum of g() contains some frequency
f0 at time t0, then for some 'scale' s0 the results of the
CWT(g(),psi(),t0,s0) has some sort of "peak".

If so, and if I use (say) sin(t) as my g(), then I should see this
"peak" at _all_ values of 't', since the frequency is constant and
always present. In fact, if I can determine s0, then it feels like
CWT(sin(t), psi(), t, s0) should be... a constant?

Apparent contradiction: Using the Haar wavelet (since I can picture
that fairly easily), assume that for some scale s1 the "width" of
the Haar wavelet exactly matches one cycle of the sin() function.
When the left side of the Haar wavelet coincides, for some time
value t1, with the "beginning" of a sine cycle's period, then the
sin()*Haar() computation will give a nice positive, bounded value:
the +1 part of the Haar will multiply by the positive portion of the
sine wave, and the -1 portion of the Haar will multiply against the
negative-going half of the sine wave.

All well and good... until you shift the Haar slightly. As you shift
it, the integral of sin()*Haar() will drop to zero, go negative, and
then come back to zero and go positive. Gosh! This shape seems
familiar! <grin!>

But this says -- assuming I haven't dropped a radical or index or
something -- that, at scale s1, and for a function g() of "constant
frequency", the CWT _varies_. It's _not_ a nice, neat constant
"horizontal" value running along the time axis.

I realize that my question may be somewhat confusing -- it's
certainly confused <grin!> -- and a detailed breakdown might just
pass over my head at this point, but it might help me get my
bearings a bit if you can comment on what the "shape" of the CWT of
a simple, single-frequency sin() function should "look like".
Should it be "constant" for some scale and zero (or close to it)
elsewhere? This is also suggested by a graph in a paper I've
downloaded but haven't read yet:

Wavelet Analysis: Theory and Applications
http://www.hpl.hp.com/hpjournal/94dec/dec94a6.pdf

page 49, Example 1, Figures 10-12.)

Or is a varying result from that same CWT perfectly consistent with
wavelet theory, and the problem is that my notion of what 'scale'
represents is completely screwed up, and needs to be stretched to
accomodate this?

As always, any coments will be appreciated.


Frank
--
It's a damn poor mind that can only think of one way
to spell a word. -- Andrew Jackson
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

On 26 Aug 2008 20:06:02 GMT, Martin Eisenberg <martin.eisenberg@udo.edu> wrote:
> I think the typo in your name might be deliberate but I'll point it
> out anyway...

Back in the '70s, when computers were... larger... and Real
Terminals had golfballs with mathematical symbols on them (or at
least the APL subset), it was humorously observed that I had a
severe case of Keybroad Dylsexia; the humor was gentle, and I adopted
it. <grin!>

> Frnak McKenney wrote:
--snip--
>>> ... 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

Got it (I think). Thanks.

--snip--
>>> 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}.

I had to re-read that a couple of times. At first I thought you
were saying that the CWT _was_ "discretized" along the 'scale'
dimension, but I'm now reading it as saying that restricting 'scale'
to those values provides a complete set of "information". Yes?

Interesting.

>>> 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...

I wasn't imagining anything "sensible". I was just reminding myself
that saying that something is "finite" doesn't mean that it can't
also be "very, very big".

In theory, it would appear that a wavelet-candidate could have one
-- or a few -- clustered peaks:

-+|+- or -+-|||-+-

On the other hand, one accidentally purchased from the Department of
Mathematical Perversity could have lots of widely-dispersed,
equal-height peaks:

-+--------+--------+--------+--------+-

I tried to come up with a time-based signal that could use something
like that, but the best I could do was something like looking for
clusters of frequency bursts appearing in a regular pattern just
prior to a major 'quake. Or not. <grin!>

>> 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.

I think it's time to review all the stuff I've printed off to see
how to calculate a "center frequency". I know I saw it specified
for some of the Daubechies family, but I don't remember if it gave a
derivation or simply plunked numbers down.

I'm still an optimist. When I read (okay, scan) through these
papers most of the symbols seem to run together after a page or
three, but each time I read them, or beat my head against the
feedback that you and others have so generously provided, I think I
understand a little more. When I started out, all I could see was a
general shape, like seeing a tree from a distance. Now I can pick
out some of the fruit hanging from one or two of the branches, even
though the underlying trunk/limb structure still seems a bit dim and
out-of-focus.

So... time to go re-(re-re-)read some stuff.

Thanks again.


Frank
--
It worked! Now, if only I could remember what I did...
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

On Thu, 28 Aug 2008 00:43:41 -0400, Ben Bradley <ben_nospam_bradley@frontiernet.net> wrote:
> On Sun, 24 Aug 2008 15:33:35 -0500, Frnak McKenney
><frnak@far.from.the.madding.crowd.com> wrote:
>>Ah. I've seen "Strang & Nguyen" mentioned favorably here and
>>elsewhere. I noticed that Amazon.com has it listed for $80, but I
>>can save some money by buying it used for only... $80. <grin!>
>
> I really should stop telling others where I get books as cheaply as
> I do as it just adds to the competition among buyers and raises
> prices, but bookfinder.com finds a less expensive copy on half.com.
>
> And I'd be remiss not to mention this copy in which Strang has
> drawn his own personal wavelet:
> http://www.biblio.com/books/145751531.html

Ben,

Thanks for the pointers. I haven't gotten my hands on a copy of
"Wavelets and Filter Banks" yet, but it's nice to know where I can
pick up a copy once I have. I did note, with some amusements,
several >$100 copies out there. If one can obtain a "really,
_really_ new" copy from Amazon.com for $80, why would someone dream
that someone else might pay that kind of money for a used copy? Or
do books like that cost more when sold as textbooks, perhaps?

Oh, and I found a used copy of Hubbard's "The World According to
Wavelets" on Amazon.com. Maybe by the time it arrives I'll have dug
anough out of these papers to understand it. <grin!>

> And sorry if I'm not helpful, but this post pretty much exhausts my
> knowledge of wavelets.

S'OK. We're not far past my own limits. <grin!>


Frank
--
Books are the compasses and telescopes and sextants and charts
which other men have prepared to help us navigate the dangerous
seas of human life. --Jesse Lee Bennett
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

Frnak McKenney wrote:

> Back in the '70s, when computers were... larger... and Real
> Terminals had golfballs with mathematical symbols on them (or at
> least the APL subset), it was humorously observed that I had a
> severe case of Keybroad Dylsexia; the humor was gentle, and I
> adopted it. <grin!>

Shouldn't you be using "Farnk" then? That's sayable.

>> 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}.
>
> I had to re-read that a couple of times. At first I thought you
> were saying that the CWT _was_ "discretized" along the 'scale'
> dimension, but I'm now reading it as saying that restricting
> 'scale' to those values provides a complete set of
> "information". Yes?

Yes, because those discretely-indexed channels suffice to cover the
frequency line. However I answer "yes" to the first part too --
discretization *is* nothing but restriction to a discrete set, and
this may or may not lose real information in any given case. Witness
sampling above the Nyquist rate.

The point is that talk of "discrete wavelets" means discrete-time,
while discrete scale (aka the filterbank view) is almost universally
employed.

> In theory, it would appear that a wavelet-candidate could have
> one -- or a few -- clustered peaks:
>
> -+|+- or -+-|||-+-
>
> On the other hand, one accidentally purchased from the
> Department of Mathematical Perversity could have lots of
> widely-dispersed, equal-height peaks:
>
> -+--------+--------+--------+--------+-

But if the pulse train is long then you can shift it quite a bit and
still have significant overlap, so the transform's time variable
would get kinda mushy. More technically, that thing isn't even
concentrated in one of the two domains, time and spectral, which is
the whole idea of wavelets.

> I'm still an optimist. When I read (okay, scan) through these
> papers most of the symbols seem to run together after a page or
> three

Well, of course they will if you just scan Don't be shy to
allocate a day to a single page with that kind of material.

> When I started out, all I could see was a general shape, like
> seeing a tree from a distance. Now I can pick out some of the
> fruit hanging from one or two of the branches, even though the
> underlying trunk/limb structure still seems a bit dim and
> out-of-focus.

Some years ago I got interested in the inner workings of the music
software I used, so I started reading this group and devoured any
paper that might have to do with musical DSP. At first I naturally
understood zilch^(-1), but look how I can talk now

Perhaps you'll get something out of this read:
http://www.occampress.com/#mathgrades


Martin

--
We don't often think of the lavish jeweled scarabs that decorated
the mummies as representations of dung beetles who nourished their
/pupae/ in excrement. We have, however, appropriated this term
into English as /pupil/. --B. Hagens, Timbre of the Spheres

On 31 Aug, 20:02, Frnak McKenney
<fr...@far.from.the.madding.crowd.com> wrote:
> Rune,
>
> Sorry to take so long responding. *Being confused is something I can
> do immediately; putting my confusion into words in such a way that
> exposes the specifics of my confusion to another person takes a bit
> longer. *<grin?>

Don't worry; The confusing of others is a fine art it takes
ages to learn and aeons to perfect....

> On Sun, 24 Aug 2008 23:05:06 -0700 (PDT), Rune Allnor <all...@tele.ntnu.no> wrote:
> > On 24 Aug, 22:35, Frnak McKenney
> ><fr...@far.from.the.madding.crowd.com> wrote:
> >> On Fri, 22 Aug 2008 12:18:05 -0700 (PDT), Rune Allnor <all...@tele.ntnu.no> wrote:
> >> > On 22 Aug, 16:30, Frnak McKenney
> >> ><fr...@far.from.the.madding.crowd.com> wrote:
> > ...
> >> > The texts don't use 'cross correlation' because it would be
> >> > positively misleading. Correlation is a concept that is used
> >> > in statistics. Wavelets is a decomposition technique which
> >> > is analyzed by maths. Real Analysis.
>
> >> I hear the words you're using, but I may not be sufficiently versed in
> >> the distinctions you're making to feel that I agree or disagree.
>
> > As to *why* I make the distinction: Consider a motor. It's basically
> > a device which takes power from some external power source and
> > converts it to rotational energy by making a the drive axel
> > (I don't know if this is the correct technical term in English;
> > please correct me if I'm wrong) spin.
>
> Note: "Drive axle", but we in the Colonies are notoriously bad
> spellers. It seems to be a competitive thing with our former mother
> country: they drop 'h's, we drop 'u's (e.g. colour=>color). <grin!>

Thanks for the clarification.

> > Suppose you first study a compustion motor and find the term
> > 'crank shaft' used for the drive axel. Then you study the
> > electric motor and all of a sudden you realize that 'Hey!
> > This thing has a spinning part too! It's just like the crank
> > shaft in the combustion engine!'
>
> Yup. *It goes around and around, and can push against other things
> to make them turn, or move.

Exactly. And there are other variations of that same principle
as well, like turbines in all shapes and forms. So by distilling
out a fundamental principle we all of a sudden have a building
block that can be used in a lot of places and contexts.

> > It would be wrong to use the term 'crank shaft' about the
> > drive axel in the electric motor. First of all, the people
> > who deal with electrical motors would conider you a fool
> > if they heard you. Second, the term 'crank shaft' brings
> > with it a lot of excess luggage like the very close
> > connections with terms like 'piston', 'cylinder', 'piston
> > rod', none of which make sense in the context of electric
> > motors.
>
> Understood. *I get that kind of reaction when I talk to my brother;
> he's studying DSP with no electronics background, so I've had to
> learn to avoid describing filters in terms of "capacitors" and
> "inductors". *<grin!>

Your brother has my sympathies...

> > So instead of saying that 'The electric motor works with
> > a crankshaft which lacks the piston and rods but has
> > something completely different gadget attached', it is far
> > better to cut to the core and say 'the electric motor
> > contains a rotating axle with gadgets attached.'
>
> > In the case of 'correlation' vs 'inner product' you might
> > not see the difference up front, but 'correlation' brings
> > with it all the excess luggage of statistics (which you
> > only need in the context of statistics) whereas 'inner
> > product' is the distilled generic term of a mathematical
> > operation.
>
> Ah. *In terms of your example, think of me as someone who is
> concentrating on "things that go around and push/drive other things"
> who picked up on "crankshaft" rather than "drive axle" because it
> "went around and pushed things" and because I ran across it first.

Exactly.

> Thank you for correcting my usage (and again for talking the time to
> explain the correction <grin!>).

Y're welcome.

> >> I have been interpreting the values "returned" from the CWT at a point
> >> (tau,s) as a "degree of match" between my original signal, on the one
> >> hand, and the function psi() on the other, at that same point. Are you
> >> saying that "correlation" in a bad way of describing this? Or that my
> >> model (interpretation) is flawed? Or Something Completely Different?
>
> >> Sorry, but it was a nice model, and it seemed to work. My head is
> >> reluctant to surrender it. <grin!>
>
> > You were certainly on the right track. I am sorry if I confused you;
> > maybe the explanation above might help you out.
>
> It did. *I get to keep my model of "how the parts act", but need to
> change my descriptive terminology. Progress. <grin!>

Ah. Efforts well spent, then!

> --snip--
>
> >> >> What I've read suggests that the relationship
> >> >> between 'scale' and 'frequency' depends on the specific choice of
> >> >> the wavelet function psi(),
>
> >> > The relation depends on both the wavelet function as such
> >> > and the scale from the mother wavelet.
>
> >> But what would the process look like, in general terms?
>
> > That's a very technical issue which was treated in the R&V tutorial.
>
> R&V "Wavelets & Signal Processing" has phrases like "since two
> scales a0 < a1 roughly correspond to two frequencies f0 > f1.."
> (p.20). *There may be more, but if so it's on one of the sections I
> haven't been able to "unpack" yet.
>
> However, I found the following in a 1992 Technical Report from
> Hewlett-Packard(!):
>
> * An Introduction to Wavelets
> *http://www.hpl.hp.com/techreports/92/HPL-92-124.pdf
>
> * "Scale" is roughly "minus log frequency", in the following limited
> * sense. *A single scale 's' contains information from a band of
> * frequencies. *The width and the center frequency of the band are
> * both proportional to -log(s). *Each scale's band has the same
> * ratio of bandwidth to center frequency, so scales correspond to a
> * set of constnat-Q filters."
>
> It's not a complete answer for my purposes, but it does give me
> arough idea of what direction I should go hunting in, and some hope
> of finding results there.

Maybe you, after having encountered such linguistic
equilibristics understand why I don't want to discuss
scale vs frequency....

....
> >> > One is that what you write is not scalable.
>
> >> Apologies for my density, and perhaps I wrote it badly. *If I have a
> >> single cycle of a sine wave, can't I stretch it into a longer-lasting
> >> single cycle, or increase its amplitude?
>
> > You can. But you might certain important key issues with wavelets.
> > One basic limiting condition in signal processing is the Heissenberg
> > inequality. It states that in order to fix a sinusoidal with
> > a certain accuracy in frequency you need a recording of a certain
> > duration in time. Conversely, if you want to locate a spike with
> > a certain accuracy in time you need a spectrum of a certain
> > bandwidth.
>
> Several of the papers I've scanned mentioned this. *It's not so much
> that I'm deliberately ignoring it, as much as, for me, it's hanging
> from a branch I can't quite reach yet. *I can see its general shape,
> but I can't quite reach it to add it to what I already know.

You should make the effort to get to that branch, as this is
*the* key why wavelets emerged as a separate field. Virtually
everything about wavelets was well known long before somebody
started to collect the bits and pieces in a system. As far as
I understand, the ratinale behind wavelets is to come up with
a filter bank which has the shortest possible impulse responses.
The Heissenberg limit is what governs all that.

> > Heissenberg's inequality inserts the numbers so that you can compute
> > the required bandwidths and durations once your target accuracies are
> > known.
>
> > Wavelets are an attempt to close in on the Heissenberg limit. That
> > is, the ideal wavelet represents a pulse in time by a spectrum
> > no wider than the Heissenberg limit.
>
> > So while your windowed sinusoidal at first glance might serve the
> > purpose of a wavelet, it fails completely with this added (or maybe
> > implicit) constraint about wavelets hovering around Heissenberg.
>
> I read this, but couldn't make a "picture" out of it yet -- my
> mental model needs a great deal of improvement <grin!> -- so I
> decided to try to see what it would look like plotted using a really
> simple wavelet (Haar). *And I hit something that confused me even
> more: *it looks like the "shape" of a CWT, for a given 'scale', is
> not what I've been assuming it would be.

I can just say that the insights will come if you play with
these things. I don't have much hands/on experience with
wavelets, so I can't help you out with your specifics.

Rune

Frnak McKenney wrote:

> Assumption: The CWT(g(),psi(),t,s) is, in some measure, related
> to the Fourier "frequency" spectrum of f(). That is, for a
> fixed g() and psi(), if the frequency spectrum of g() contains
> some frequency f0 at time t0, then for some 'scale' s0 the
> results of the CWT(g(),psi(),t0,s0) has some sort of "peak".

It has power there but that doesn't say anything about the waveform
carrying it.

> If so, and if I use (say) sin(t) as my g(), then I should see
> this "peak" at _all_ values of 't', since the frequency is
> constant and always present. In fact, if I can determine s0,
> then it feels like CWT(sin(t), psi(), t, s0) should be... a
> constant?

No, it should be a waveform of constant power.

> But this says -- assuming I haven't dropped a radical or index
> or something -- that, at scale s1, and for a function g() of
> "constant frequency", the CWT _varies_. It's _not_ a nice, neat
> constant "horizontal" value running along the time axis.

And neither would the output of a DFT channel be, but the variation
would be all in the phase and none in the magnitude. The same is true
of the HP example which uses a complex wavelet (with a one-sided
spectrum). Your real Haar wavelet gives real output that has only one
dimension available to vary in, but its amplitude and waveshape
(hence effective value) are still constant for input with the same
properties.

> I realize that my question may be somewhat confusing -- it's
> certainly confused <grin!> -- and a detailed breakdown might
> just pass over my head at this point, but it might help me get
> my bearings a bit if you can comment on what the "shape" of the
> CWT of a simple, single-frequency sin() function should "look
> like".

You're filtering a sinusoid so a sinusoid will come out -- real if
the wavelet is real, complex otherwise. In the latter case, the
*scalogram* will show what you want it to (and that's what fig. 11 in
the HP paper is), just like a spectrogram would (but not the DFT
itself).


Martin

--
Quidquid latine scriptum est, altum videtur.

On Aug 22, 10:30*am, Frnak McKenney
<fr...@far.from.the.madding.crowd.com> wrote:
> After scanning a few 'web pages and online articles, it appeared
> that wavelets would be a good way of decomposing a complex, coded
> data stream like the (demodulated) audio from NIST's WWV 10MHz
> transmitter. *Given a collection of WAV files and the signal
> description from:
>
> * NIST Time and Frequency Services, Pub. 432
> *http://tf.nist.gov/timefreq/general/pdf/1383.pdf
>
> all I'd have to do would be feed the sampled signal int A Wavelet
> and, Hey! *Presto! *I'd have a 2D array telling me the exact pattern
> of the 100Hz "subcarrier" carrying the digital timecode. <grin!>
>

> Frank McKenney, McKenney Associates
> Richmond, Virginia / (804) 320-4887
> Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

Frank,

Returning to the application side for a minute, did you manage to
develop a method for demodulating WWVB waveforms using wavelets? Is
there a "better mousetrap" out there as you thought?

- Kenn

On Mon, 1 Sep 2008 01:28:28 -0700 (PDT), Rune Allnor <allnor@tele.ntnu.no> wrote:
> On 31 Aug, 20:02, Frnak McKenney
><fr...@far.from.the.madding.crowd.com> wrote:
>> On Sun, 24 Aug 2008 23:05:06 -0700 (PDT), Rune Allnor <all...@tele.ntnu.no> wrote:
>> > On 24 Aug, 22:35, Frnak McKenney
>> ><fr...@far.from.the.madding.crowd.com> wrote:

--snip--
>> > One basic limiting condition in signal processing is the Heissenberg
>> > inequality. It states that in order to fix a sinusoidal with
>> > a certain accuracy in frequency you need a recording of a certain
>> > duration in time. Conversely, if you want to locate a spike with
>> > a certain accuracy in time you need a spectrum of a certain
>> > bandwidth.
>>
>> Several of the papers I've scanned mentioned this. *It's not so much
>> that I'm deliberately ignoring it, as much as, for me, it's hanging
>> from a branch I can't quite reach yet. *I can see its general shape,
>> but I can't quite reach it to add it to what I already know.
>
> You should make the effort to get to that branch, as this is
> *the* key why wavelets emerged as a separate field. Virtually
> everything about wavelets was well known long before somebody
> started to collect the bits and pieces in a system. As far as
> I understand, the ratinale behind wavelets is to come up with
> a filter bank which has the shortest possible impulse responses.
> The Heissenberg limit is what governs all that.

Thanks. One of my projects for the weekend is to make up a list of
the topics that I need to dig more deeply into (a.k.a. "Stuff it
looks like I can't put off learning about" <grin!>). This is
definitely on the list.

--snip--
> I can just say that the insights will come if you play with
> these things. I don't have much hands/on experience with
> wavelets, so I can't help you out with your specifics.

What I have now is still a collection of "concepts". A lot of what
I'm taking from you and the other posters here is what amounts to
"faith": a belief that they really can fit together into a
framework that I can eventually "chunk" into useful way of looking
at the universe. And it _is_ helpful.


Frank
--
A science is any discipline in which a fool of this generation can
go beyond the point reached by a genius of the last generation.
-- Max Gluckman
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

On 31 Aug 2008 21:18:57 GMT, Martin Eisenberg <martin.eisenberg@udo.edu> wrote:
> Frnak McKenney wrote:
--snip--
> Shouldn't you be using "Farnk" then? That's sayable.

So is "Fr-nak". English is a versatile language with many
variations in spelling and pronunciation (see below <grin!>).

> The point is that talk of "discrete wavelets" means discrete-time,
> while discrete scale (aka the filterbank view) is almost universally
> employed.

Ah. Thank you for answering a terminology question I had forgotten
I wanted to ask.

>> In theory, it would appear that a wavelet-candidate could have
>> one -- or a few -- clustered peaks:
>>
>> -+|+- or -+-|||-+-
>>
>> On the other hand, one accidentally purchased from the
>> Department of Mathematical Perversity could have lots of
>> widely-dispersed, equal-height peaks:
>>
>> -+--------+--------+--------+--------+-
>
> But if the pulse train is long then you can shift it quite a bit and
> still have significant overlap, so the transform's time variable
> would get kinda mushy. More technically, that thing isn't even
> concentrated in one of the two domains, time and spectral, which is
> the whole idea of wavelets.

"Even if it gave any results, they wouldn't be very useful".

Okay. I'll send it back to the D.M.P. I only borrowed it for
discussion purposes. <grin!>

>> I'm still an optimist. When I read (okay, scan) through these
>> papers most of the symbols seem to run together after a page or
>> three
>
> Well, of course they will if you just scan Don't be shy to
> allocate a day to a single page with that kind of material.

You're right, it's time to settle down. The reason for scanning was
to try to try to find a "good match" between the papers and terms
and notation I (thought I) was familiar with; at this point, I'm
willing to concentrate on the best of what I;ve found and force
myself to accept the vanishingly small chance that I've overlooked
some paper that would Instantly Make Everything Clear.

Also, although my copy of Hubbard's World of Wavelets hasn't
arrived, I managed to find and borrow one from a local library. So
far it seems to be an annotated History of Wavelets, with an
extended technical "appendix", but its pace is slow enough I can
make sense of it. (I think. <grin!>)

> Some years ago I got interested in the inner workings of the music
> software I used, so I started reading this group and devoured any
> paper that might have to do with musical DSP. At first I naturally
> understood zilch^(-1), but look how I can talk now

So... you're saying that, over time and with great effort, I might
eventually be able to claim comprehension of... zilch(^-2)??

> Perhaps you'll get something out of this read:
> http://www.occampress.com/#mathgrades

Thanks for the pointer. I skimmed the first tow chapters, and I
hope to go back and re-read the material at some point in the
future.

I think I am in general agreement with the author's concern over
classifying student abilities based on limited data; on the other
hand, his comment that "those who do not have mathematical ability
have no choice but to go into worthless subjects" concerns me a bit.
I need to re-read that section, but -- admittedly taken out of
context -- it has echoes of a dichotomy between the "sciences" and
the "humanities" that has frequently bothered me, and that I had
hoped was finally out of fashion.

By the way: I'm still working on your other posting: I keep
revising what I've written, which may or may not be a good thing.
<grin!>


Frank
--
"We don't just borrow words; on occasion, English has pursued other
languages down alleyways to beat them unconscious and riffle their
pockets for new vocabulary." -- James D. Nicoll
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)

Kenn,

Thanks for joining in.

On Tue, 2 Sep 2008 06:09:10 -0700 (PDT), kennheinrich@sympatico.ca <kennheinrich@sympatico.ca> wrote:
> On Aug 22, 10:30*am, Frnak McKenney
><fr...@far.from.the.madding.crowd.com> wrote:
>> After scanning a few 'web pages and online articles, it appeared
>> that wavelets would be a good way of decomposing a complex, coded
>> data stream like the (demodulated) audio from NIST's WWV 10MHz
>> transmitter. *Given a collection of WAV files and the signal
>> description from:
>>
>> NIST Time and Frequency Services, Pub. 432
>> *http://tf.nist.gov/timefreq/general/pdf/1383.pdf
>>
>> all I'd have to do would be feed the sampled signal int A Wavelet
>> and, Hey! Presto! I'd have a 2D array telling me the exact pattern
>> of the 100Hz "subcarrier" carrying the digital timecode. <grin!>
--snip--

> Returning to the application side for a minute, did you manage to
> develop a method for demodulating WWVB waveforms using wavelets? Is
> there a "better mousetrap" out there as you thought?

My first attempt using Scilab clearly and thoroughly demonstrated my
lack of understanding of several fundamental concepts related to
wavelets. Worse, it didn't produce any useful results. <grin!>

Seriously, no, no results yet; the tried-and-true basic Fourier Trap
looks like it handles "stationary" rodents fairly well, but I'm
still learning the fundamentals of capturing the more active,
"non-stationary" variety. It's not so much that I believe that
understanding the basics of Wavelet Theory will keep me from making
silly implementation errors as a hope that it will improve my odds
of catching and correcting them after the fact.

My choice of a problem to apply "wavelet techniques" to was based
more on a belief that I understood -- had a good mental model of --
the data than by a need to solve an otherwise insoluble problem.
When you're testing a new navigation techinque it's considered
advisable to test it first in known waters. The wide-based
experience that will let you compare it with other methods comes
much later. <grin!>


Frank
--
I am ready to meet my Maker. Whether my Maker is prepared for
the ordeal of meeting me is another matter.
-- Winston Churchill
--
Frank McKenney, McKenney Associates
Richmond, Virginia / (804) 320-4887
Munged E-mail: frank uscore mckenney ayut mined spring dawt cahm (y'all)