ben wrote:> also, forgetting about if the book's incorrect or not for a minute, > about the fourier transform generally, from physical space to fourier > space, would a reasonable, very simple way of summing up what happens > in a fourier transform be: > > counts of occurances of similar values - so ending up with a set of > counts > > ? that is, in physical space say the value 5 occurs there and over > there and there etc, say 20 times in the space you're looking at, then > in fourier space the 5 marker/postion carries 20 values worth? > > how would anyone explain what goes on in a fourier transform to an 8 > year old?None of that. One way to look at a Fourier transform is as the solution of a set of simultaneous equations. Suppose that you have an arbitrary shape within some interval, and it's either true that the shape is repeated outside that interval, or assuming that it is doesn't interfere with what you want to find out. We want to approximate the shape as a sum of sines and cosines. Because of the actual or assumed periodicity, those sines and cosines will all be harmonics of 1/T, where T is the period of the curve. (It needn't be time. It can as well be distance.) So we have our standard series infinity sum[a_n*cos(2*pi*t/n*T) + b_n*sin(2*pi*t/n*T)]. n=0 To make the series fit the curve, we need to find appropriate values of a_n and b_n. Ten coefficients are needed to get an approximation with 5 harmonics. The math can get a bit hairy, but by they can be computed from any ten points along the curve. The math gets easier if those points are uniformly spaced along the curve. There are even easier ways to solve that particular problem. Sines and cosines form what's called an orthogonal set, which simply means that the integral of their products over a whole number of periods vanishes unless the multiplied elements are identical. That's what makes Fourier series easy enough to calculate to be useful. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
apparent contradiction between coefficient graph and text
Started by ●February 16, 2005
Reply by ●February 17, 20052005-02-17
Reply by ●February 17, 20052005-02-17
ben wrote: ...> ah, right -- that's what i was trying to get at earlier on asking are > the dots actually representing coefficients or not -- i suspected that > what you've just said was the case. that explains all my confusion. > that's why i was talking about, suggesting the situation of describing > a curve on a graph as a function. it's analogously (in a language use > way) the same as that. talking about the result of something as the > something. so pointing to the dots and calling the coefficients is > pretty darn misleading itself. those dots represent calculations that > just used the sodding coefficients, they're not the coefficients > themselves.The dots are a graphical representation of the coefficients' magnitudes. The plot is there to illustrate their evolution in time. All the dots on the line of a given number represent the coefficient of a single term in the series at one time or another. Connecting the dots does not create the curve. The terms in the series are distinct, and it would be meaningless to draw a curve through them. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 17, 20052005-02-17
In article <ZdKdna_TM4eKtIjfRVn-3A@rcn.net>, Jerry Avins <jya@ieee.org> wrote:> ben wrote: > > ... > > > ah, right -- that's what i was trying to get at earlier on asking are > > the dots actually representing coefficients or not -- i suspected that > > what you've just said was the case. that explains all my confusion. > > that's why i was talking about, suggesting the situation of describing > > a curve on a graph as a function. it's analogously (in a language use > > way) the same as that. talking about the result of something as the > > something. so pointing to the dots and calling the coefficients is > > pretty darn misleading itself. those dots represent calculations that > > just used the sodding coefficients, they're not the coefficients > > themselves. > > The dots are a graphical representation of the coefficients' magnitudes.i keep thinking i've understood at least one little point about that page but then i realise haven't. are the dots representing actual coefficients (but just a bit rounded off), not the results of calculations that use the coefficient(s) then? which of these is true: a single dot per coefficient value, or several dots per coefficient value, or dots represent the outcome of some calculations that involve coefficient values? or none of those? if i could get it reasonably clear and straight what the dots are representing, that'd probably go a long way to helping. possibly.> The plot is there to illustrate their evolution in time. All the dots on > the line of a given number represent the coefficient of a single term in > the series at one time or another."All the dots on the line of a given number represent the coefficient.." - you've used plural 'dots' and singular 'coefficient'. a single value called the coefficient is represented by a handful of dots (who obviously have different values as they're at different heights, or locations) -- that's quite confusing to me. how can it be like that? here's my guess at what dots are: the handful of dots on one verticle line are the result of the same number of calculations (one dot per calculation) and those calculations all use the same coefficient? (dots on the same verticle line that is). one coefficient per verticle line.> Connecting the dots does not create > the curve.that above sentance is interesting and possibly very illuminating.> The terms in the series are distinct, and it would be > meaningless to draw a curve through them.something new i've found confusing about that page (which is something i could probably keep up for eternity): the formula that's for calculating the various coeffecients at the time 0 doesn't even seem to use the heights (or values) of the waves! the formula is c_n(0) = 1 / sqrt(n) "where n is the wave number" (again see http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only information it seems to take from the original temperature measurements at time 0 are the number of waves -- not the hieghts/values of those waves. i just must be wrong in saying that. that'd be rediculous. the shape of the top right graph does reflect the shape of the top left graph, so, as i would expect, the process that goes from top left graph to top right graph must and does use the initial temperature time 0 measurements. but that's not what the formula indicates to me. am i misinterpretting the formula? thanks, ben
Reply by ●February 17, 20052005-02-17
ben wrote:> In article <ZdKdna_TM4eKtIjfRVn-3A@rcn.net>, Jerry Avins <jya@ieee.org> > wrote: > > >>ben wrote: >> >> ... >> >> >>>ah, right -- that's what i was trying to get at earlier on asking are >>>the dots actually representing coefficients or not -- i suspected that >>>what you've just said was the case. that explains all my confusion. >>>that's why i was talking about, suggesting the situation of describing >>>a curve on a graph as a function. it's analogously (in a language use >>>way) the same as that. talking about the result of something as the >>>something. so pointing to the dots and calling the coefficients is >>>pretty darn misleading itself. those dots represent calculations that >>>just used the sodding coefficients, they're not the coefficients >>>themselves. >> >>The dots are a graphical representation of the coefficients' magnitudes. > > > i keep thinking i've understood at least one little point about that > page but then i realise haven't. > > are the dots representing actual coefficients (but just a bit rounded > off), not the results of calculations that use the coefficient(s) then?Each vertical line is the locus on all possible coefficients of a particular term in the series whose sum is the curve. As the curve evolves over time, the coefficients change to reflect that. The dots represent the actual values of a coefficient at some particular time. I'm surprised that there are no negative values.> which of these is true: a single dot per coefficient value, or several > dots per coefficient value, or dots represent the outcome of some > calculations that involve coefficient values? or none of those?A single dot represents the value of a particular coefficient at a particular time.> if i could get it reasonably clear and straight what the dots are > representing, that'd probably go a long way to helping. possibly.Each curve is described by a series a_0*f_0(x) + a_1*f_1(x) + a_2*f_2(x) + ..., where the f_n(x) are the same for all curves, and the a_n are chosen to make the sum match the curve. Each a_n is a coefficient. Each curve has its own set of coefficients.>>The plot is there to illustrate their evolution in time. All the dots on >>the line of a given number represent the coefficient of a single term in >>the series at one time or another.> >> "All the dots on the line of a given number represent the > coefficient.." - you've used plural 'dots' and singular 'coefficient'. > a single value called the coefficient is represented by a handful of > dots (who obviously have different values as they're at different > heights, or locations) -- that's quite confusing to me. how can it be > like that?Each dot on the first line represents a coefficient of the first term in the summation for a particular curve. Each dot on the second line represents a coefficient of the second term in the summation for a particular curve. Each dot on the third line represents a coefficient of the third term in the summation for a particular curve.> here's my guess at what dots are: the handful of dots on one verticle > line are the result of the same number of calculations (one dot per > calculation) and those calculations all use the same coefficient? (dots > on the same verticle line that is). one coefficient per verticle line. > > >>Connecting the dots does not create >>the curve. > > > that above sentance is interesting and possibly very illuminating. > > >>The terms in the series are distinct, and it would be >>meaningless to draw a curve through them. > > > > > something new i've found confusing about that page (which is something > i could probably keep up for eternity): the formula that's for > calculating the various coeffecients at the time 0 doesn't even seem to > use the heights (or values) of the waves! the formula is > > c_n(0) = 1 / sqrt(n)That's a derived result for this particular case. Implicit in it is that the temperature at the hot end is normalized to 1. (The wavy curve has a value of zero there, proving that it's artificial.)> "where n is the wave number" (again see > http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only > information it seems to take from the original temperature measurements > at time 0 are the number of waves -- not the hieghts/values of those > waves. i just must be wrong in saying that. that'd be rediculous.Not ridiculous, just (for the moment) incomprehensible. Anyhow, I think you better nail down Fourier analysis before you tackle wavelets.> the shape of the top right graph does reflect the shape of the top left > graph,Coincidence, to the extent that any coincidence is possible in math.> so, as i would expect, the process that goes from top left graph > to top right graph must and does use the initial temperature time 0 > measurements. but that's not what the formula indicates to me. am i > misinterpretting the formula?In this example, the temperature at the left end of the bar is always unity. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 17, 20052005-02-17
In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> wrote:> > "where n is the wave number" (again see > > http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only > > information it seems to take from the original temperature measurements > > at time 0 are the number of waves -- not the hieghts/values of those > > waves. i just must be wrong in saying that. that'd be rediculous. > > Not ridiculous, just (for the moment) incomprehensible.> Anyhow, I think > you better nail down Fourier analysis before you tackle wavelets.(haven't digested this message nor another one of your messages yet - will do later - but just to say in response to the above): that's what all this is about -- fourier analysis. i'm still talking about the same page which is about fourier's work, not wavelets so far as i know.
Reply by ●February 17, 20052005-02-17
In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> wrote:> > c_n(0) = 1 / sqrt(n) > > That's a derived result for this particular case. Implicit in it is that > the temperature at the hot end is normalized to 1. (The wavy curve has a > value of zero there, proving that it's artificial.)is that 0 not representing the time: time 0?
Reply by ●February 17, 20052005-02-17
ben wrote:> In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> > wrote: > > > >>> c_n(0) = 1 / sqrt(n) >> >>That's a derived result for this particular case. Implicit in it is that >>the temperature at the hot end is normalized to 1. (The wavy curve has a >>value of zero there, proving that it's artificial.) > > > is that 0 not representing the time: time 0?The wavy curve seems to go through the origin: 0,0. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 18, 20052005-02-18
In article <65Wdnb98_6BU-4jfRVn-3Q@rcn.net>, Jerry Avins <jya@ieee.org> wrote:> ben wrote: > > In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> > > wrote: > > > > > > > >>> c_n(0) = 1 / sqrt(n) > >> > >>That's a derived result for this particular case. Implicit in it is that > >>the temperature at the hot end is normalized to 1. (The wavy curve has a > >>value of zero there, proving that it's artificial.) > > > > > > is that 0 not representing the time: time 0? > > The wavy curve seems to go through the origin: 0,0.maybe but in the text on the scanned page it says "the fourier coefficients at time 0, c_n(0) (where n is the ..." and it says "the coefficients at time t are computed with the formula c_n(t) = ...."
Reply by ●February 18, 20052005-02-18
ben wrote:> In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> > wrote: > > > >>> c_n(0) = 1 / sqrt(n) >> >>That's a derived result for this particular case. Implicit in it is that >>the temperature at the hot end is normalized to 1. (The wavy curve has a >>value of zero there, proving that it's artificial.) > > > is that 0 not representing the time: time 0?More specifically, the curves are all of temperature vs. distance at particular times. The curve for t = 0 is the wavy one we discussed in detail a long time ago. That should have been clear from the labels in the bottom set of curves on the left side of the page. I was evidently wrong about the initial conditions. I was thinking of another version of th problem. In this one, it seems that after the bar had reached equilibrium with the left end hot and the right end cold, the left end of the bar is set to zero at t = 0. The curves are the temperature along the bar at successive times. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 18, 20052005-02-18
In article <idCdnUTzJoViwIjfRVn-hQ@rcn.net>, Jerry Avins <jya@ieee.org> wrote:> ben wrote:> > something new i've found confusing about that page (which is something > > i could probably keep up for eternity): the formula that's for > > calculating the various coeffecients at the time 0 doesn't even seem to > > use the heights (or values) of the waves! the formula is > > > > c_n(0) = 1 / sqrt(n) > > That's a derived result for this particular case. Implicit in it is that > the temperature at the hot end is normalized to 1. (The wavy curve has a > value of zero there, proving that it's artificial.) > > > "where n is the wave number" (again see > > http://www.hdbatik.co.uk/temp/waveletsbookpage.html ). so the only > > information it seems to take from the original temperature measurements > > at time 0 are the number of waves -- not the hieghts/values of those > > waves. i just must be wrong in saying that. that'd be rediculous. > > Not ridiculous, just (for the moment) incomprehensible. Anyhow, I think > you better nail down Fourier analysis before you tackle wavelets.the direction and flow of the (i've thought all along; fourier, but even that's in question now) process goes top left graph, top right graph, bottom right graph, bottom left graph. the data in the line in the first graph doesn't seem to be used at all (not called upon, not read/accessed/looked at), and yet there it is in the final graph, so the data must be used somehow. it obviously isn't a concidence that the bendy line in the last graph is the same as the bendy line in the first graph. where does the final graph's bendy line data come from? how is it ending up there in the last graph (as the formula that starts the process doesn't seem to use that data at all)? putting the same thing above in another way: the text says about the bottom right graph "the information on space seesm to have dissapeeared" -- yes, but so has the values of the waves it seems. so all data seems to have dissapeared at that point -- no data, oh, apart from the number of waves. the number of waves is the only data that makes it out of the first graph. that just can't be right. can it?> > > the shape of the top right graph does reflect the shape of the top left > > graph, > > Coincidence, to the extent that any coincidence is possible in math.i see. so the coefficient has nothing to do with the wave's height? that seems somewhat, to say the least, contradictory to something that was established a few messages ago, something i thought i'd understood. (i'm drawing that statement from the fact(??) that the formula does not use the wave's heights, only the number of waves <<<<< is that correct?) it would appear right now, that the calculation that gives the coefficients don't use the heights of waves, only the number of waves in the data. is that correct? thanks, ben
