Rune Allnor wrote:> On 7 Mar, 20:28, Jerry Avins <j...@ieee.org> wrote: >> Rune Allnor wrote: >>> On 7 Mar, 18:30, Jerry Avins <j...@ieee.org> wrote: > ... >>>> If it were so obvious, it would have been widely used long before it >>>> actually was. Its introduction as an abstract tool for the likes of us >>>> -- excluding Gauss et al. -- began with the electrical engineer Oliver >>>> Heaviside's operational calculus, about a generation before your time. >>> Wrong. The one wikipedia link I've found that covers the >>> subject fairly well, is tis one: >>> http://en.wikipedia.org/wiki/Normal_mode >>> A 'normal mode' is an eigenfunction to a differential equation. >>> The complex exponential is the eigenfunction to the wave equation >>> in rectangular coordinates. As you can see on the list of related >>> subjects at the bottom of that page, 'normal modes' (and hence >>> generalized Fourier theory) covers lots of ground. >>> For your information, Sturm and Liouville, whose theory (based >>> on Fourier analysis) concerns one particular class of differential >>> equations, were both born some 200 years ago, in 1803 and 1809. >>> Hardly new nor groundbreakin stuff, any of this. Well, with the >>> possible exception of certain seismological phenomena... >> My point was that, until Heaviside, all that was considered the domain >> of higher mathematicians, not accessible to most engineers and >> physicists. > > Irrelevant. The same was true for computers some 60 years ago. > Does that mean it's acceptable for DSP practitioners to regard > computers as diabolic devices best handled with black arts > and voodoo? > >> You won't find transforms in J.C.Maxwell's /Treatise on >> Electricity and Magnetism/. (Nor complex numbers!) > > Do you mean that complex numbers are esoteric phenomena > which DSP practitioners need not understand or use?Of course not. Return to the point I was trying to illustrate. If the use of Laplace transforms is as simple and transparent as you suggest, they would have been a computational staple in engineering and physics much earlier than history shows is the case. High-school math is devilishly difficult for most people to teach well. That's because once most people understand it, they lose the ability to comprehend how anyone might _not_ understand it. I respectfully suspect that you are in the majority. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
The Fourier transform in nature not just time<>frequency domain
Started by ●March 6, 2009
Reply by ●March 8, 20092009-03-08
Reply by ●March 8, 20092009-03-08
Alun wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:wVzsl.78494$RJ7.4624@newsfe18.iad... >> My point was that, until Heaviside, all that was considered the domain of >> higher mathematicians, not accessible to most engineers and physicists. >> You won't find transforms in J.C.Maxwell's /Treatise on Electricity and >> Magnetism/. (Nor complex numbers!) >> > > Nor vectors, even. > > A truly amazing genius for having seen the wood through > the trees but without the benefit of modern concise notations. > > Imagine "del.E = rho" having to be expressed as an explosion > of individual symbols!Maxwell expressed the wave equations -- "Maxwell's equations" as paired triplets of integral equations. Vector analysis had not yet been invented, but quaternions had been. I suspect that Maxwell wrote in the form he did in order to make his treatise accessible to more than a mere dozen or so. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 8, 20092009-03-08
On 8 Mar, 18:04, Jerry Avins <j...@ieee.org> wrote:> Rune Allnor wrote: > > On 7 Mar, 20:28, Jerry Avins <j...@ieee.org> wrote: > >> Rune Allnor wrote: > >>> On 7 Mar, 18:30, Jerry Avins <j...@ieee.org> wrote:...> > Do you mean that complex numbers are esoteric phenomena > > which DSP practitioners need not understand or use? > > Of course not. Return to the point I was trying to illustrate. If the > use of Laplace transforms is as simple and transparent as you suggest, > they would have been a computational staple in engineering and physics > much earlier than history shows is the case.Well, a counterexample is linear algebra. Once you know how to use it, it's obvious that it is a powerful tool that will have a huge impact in DSP. Nevertheless, there are few textbooks that even attempt to phrase DSP in terms of linear algebra (I only know about two, the books by Therrien and Scharf, both dating from the early '90s), and very few practitioners use that approach. Proliferation is not a measure for usefulness.> High-school math is devilishly difficult for most people to teach well. > That's because once most people understand it, they lose the ability to > comprehend how anyone might _not_ understand it. I respectfully suspect > that you are in the majority.What majority? The poor teachers or those who forgot the difficulties? I can point to the very specfic occasion when I 'learned' maths. Not the date and time (I didn't realize its importance at the time, so I didn't make a note) but I remember the lesson well. It was at university, in an intro class on EM waves. Some old fox of an eremitus had been dug out of retirement to teach this class while the intended lecturer was away on sabbatical. Anyway, this guy derived some results based on Maxwell's equations, and rounded up by saying "note that I have said nothing about the function f(x). This result does not depend on the function being a sine or a cosine, but is valid for any waveform." A very simple statement, but it opened my eyes to maths. And it revealed that my previos maths teachers were very bad at their jobs, since they had not made that simple, obvious (at least in retrospect) remark years before. However, before that I had at least realized on my own that what we learn in maths class also is valid in physics class. I never understood why fellow students seemed to be at a loss when they faced problems in physics class they had learned how to solve in maths class. My point was that if somebody have been exposed to maths for decades without realizing - for whatever reason - that maths is universal and not dependant on context, there are serious flaws in the preparations for dealing with DSP. Of course, poor maths teachers are as likely a cause for this lack of insight as any, but that doesn't help much. Rune
Reply by ●March 8, 20092009-03-08
Rune Allnor wrote:> On 8 Mar, 18:04, Jerry Avins <j...@ieee.org> wrote:>> ... Return to the point I was trying to illustrate. If the >> use of Laplace transforms is as simple and transparent as you suggest, >> they would have been a computational staple in engineering and physics >> much earlier than history shows is the case. > > Well, a counterexample is linear algebra. Once you know how > to use it, it's obvious that it is a powerful tool that will > have a huge impact in DSP. Nevertheless, there are few textbooks > that even attempt to phrase DSP in terms of linear algebra (I only > know about two, the books by Therrien and Scharf, both dating from > the early '90s), and very few practitioners use that approach.I would think that's a supporting example. A useful technique waiting for rediscovery and widespread use.> Proliferation is not a measure for usefulness.It is a measure of recognition of usefulness.>> High-school math is devilishly difficult for most people to teach well. >> That's because once most people understand it, they lose the ability to >> comprehend how anyone might _not_ understand it. I respectfully suspect >> that you are in the majority. > > What majority? The poor teachers or those who forgot the > difficulties?Those who no longer recognize sources of confusion.> I can point to the very specfic occasion > when I 'learned' maths. Not the date and time (I didn't realize > its importance at the time, so I didn't make a note) but I > remember the lesson well. It was at university, in an intro > class on EM waves. Some old fox of an emereitus had been dug > out of retirement to teach this class while the intended > lecturer was away on sabbatical. Anyway, this guy derived > some results based on Maxwell's equations, and rounded up by > saying "note that I have said nothing about the function f(x). > This result does not depend on the function being a sine or > a cosine, but is valid for any waveform." A very simple > statement, but it opened my eyes to maths. And it revealed > that my previous maths teachers were very bad at their jobs, > since they had not made that simple, obvious (at least in > retrospect) remark years before.You were fortunate to have that experience. How many did not?> However, before that I had at least realized on my own that > what we learn in maths class also is valid in physics class. > I never understood why fellow students seemed to be at a > loss when they faced problems in physics class they had > learned how to solve in maths class.Perhaps that's why American university Engineering courses teach most of the needed math in the physics or engineering classes.> My point was that if somebody have been exposed to maths > for decades without realizing - for whatever reason - that > maths is universal and not dependant on context, there are > serious flaws in the preparations for dealing with DSP.Not just DSP. Life in general.> Of course, poor maths teachers are as likely a cause for > this lack of insight as any, but that doesn't help much.Pity the poor student who is told that the product of two negative numbers being positive is a logical law of nature. The smart students will try and fail to comprehend the reason, and conclude thereby that math is beyond their ability. How much better to be told that we choose to define it that way in order to simplify the overall scheme, eliminating special-case rules. The arbitrary is at least comprehensible. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 8, 20092009-03-08
Jerry Avins wrote:> Pity the poor student who is told that the product of two > negative numbers being positive is a logical law of nature. The > smart students will try and fail to comprehend the reason, and > conclude thereby that math is beyond their ability. How much > better to be told that we choose to define it that way in order > to simplify the overall scheme, eliminating special-case rules.I know you know how recent that insight is even within mathematics, never mind filtering through to general education! Martin -- Quidquid latine scriptum est, altum videtur.
Reply by ●March 8, 20092009-03-08
Martin Eisenberg wrote:> Jerry Avins wrote: > >> Pity the poor student who is told that the product of two >> negative numbers being positive is a logical law of nature. The >> smart students will try and fail to comprehend the reason, and >> conclude thereby that math is beyond their ability. How much >> better to be told that we choose to define it that way in order >> to simplify the overall scheme, eliminating special-case rules. > > I know you know how recent that insight is even within mathematics, > never mind filtering through to general education!I knew of it before 1950, when I graduated high school. A far cry from the elementary-school teacher who told me that long division works because when the divisor and the quotient are multiplied, the dividend is reformed. She couldn't understand that a proof (defective at that) that it works is not the same as a reason for doing it that way. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 9, 20092009-03-09
Rune Allnor <allnor@tele.ntnu.no> wrote:> However, before that I had at least realized on my own that > what we learn in maths class also is valid in physics class. > I never understood why fellow students seemed to be at a > loss when they faced problems in physics class they had > learned how to solve in maths class.This has been a problem forever, and likely still is, and even at the best schools. The math department teaches math, the physics deparment teaches physics, but expects to use the math from the math department. It is rare that there is any discussion between the two on how to teach them.> My point was that if somebody have been exposed to maths > for decades without realizing - for whatever reason - that > maths is universal and not dependant on context, there are > serious flaws in the preparations for dealing with DSP.I do remember doing a physics quiz requiring a solution to a differential equation the day before I learned that equation in the math class. And of course the math department rarely explains it in the way that is useful for solving physics equations. -- glen
