Rune Allnor wrote:> On 7 Mar, 14:34, makol...@yahoo.com wrote: >> On Mar 7, 4:13 am, Rune Allnor <all...@tele.ntnu.no> wrote: >> >> >> >> >> >>> On 7 Mar, 00:58, "Alun" <no.spam.thank....@invalid.invalid> wrote: >>>> "Rune Allnor" <all...@tele.ntnu.no> wrote in message >>>> news:9ab6ef27-b5fc-4f4a-8af7-1abe088fcf47@p11g2000yqe.googlegroups.com... >>>> On 6 Mar, 18:37, makol...@yahoo.com wrote: >>>>> I was enlightened to read here the other day that the Fourier >>>>> Transform also describes the relationship in electromagnetics >>>>> between the current distribution and the far field radiation >>>>> pattern. >>>> !The next obvious step on your path to Nirvana (or the >>>> !looney bin, whichever is closer) is to investigate why[*] >>>> !2+2=4 in such diverse fields as >>>> ! >>>> !- Maths >>>> !- Physics >>>> !- Economy >>>> !- Medicine >>>> !- and so on. >>>> ! >>>> !In fact, it't hard to come up with a single case >>>> !where 2+2=/=4. Now, *that's* food for tought. >>>> !Preferably somebody else's, but still. >>>> It is a great pity when those who would be the pundits of >>>> this NG seek personal satisfaction by pooh-poohing >>>> the expressed sense of academic excitement of a newbie. >>>> Such excitement is to be encouraged and nurtured. >>> I could agree with you in principle, but the timing is >>> wrong. I would assume one is at least a couple of years >>> into college or university levels before one gets to >>> know the FT; let's assume students to be of age at >>> least 20. >>> By that time, it ought to be clear to students that >>> mathemathichs is a universal too which does not depend >>> on context. Problem formulations and boundary conditions >>> vary, but maths doesn't. >>> The fact that the OP apparently realizes this general >>> validity of maths in the context of the FT - which means >>> he has some 15 years of previos exposure to maths - is >>> a sign that there is something seriously missing with >>> his insights into the basic tools of engineering. >>> Rune- Hide quoted text - >>> - Show quoted text - >> Hi Rune, >> >> YEP I'm an engineer and your estimate of my experience is off by a >> factor of 2, (I'll leave it to you to wonder which way). > > You're a wonder kid? Switch schools. The sooner the better. > >> But no I'm not a mathematician.... to me, math is a useful but >> sometimes cumbersome tool to describe what happens in nature... > > That doesn't excuse you from learning how to use it. > >> I found it interesting that the natural behavior of EM fields mimics >> the natural behavior of waveforms in the frequency and time domain... > > Could the fact that EM fields actually *are* waves that can > be analyzed in both time and frequency domains, have anything > to do with this shocking insight...? > >> I'm sorry if this was obvious to everyone else here. Interestingly, >> there are only a few other pairings mentioned so far... come on there >> must be more, maybe it's not so obvious after all... > > Right. Because *you* don't see the solution, none exists... > A one-way ticket to the looney bin, if there ever was one. > > There is a Norwedian proverb that "one can't see the forest > through the trees". Which is what you are up against: > Everything is out there in front of your eyse, but you > don't recognize it for what it is. > > The (generalized) Fourier transform is a standard tool for > solving linear Differential Equations, which in turn are > the backbone tools of mathematical physics. You'd know > that if you paid attention in maths class. > > Your earth-shattering revelation that the FT can be used > to describe several types of waves are caused by the > fact - prepare yourself for a surprise - that they solve > wave equations of the same form. Again, mathematics is > generally useful and does not depend on context. > > So for your remaining time in High School and throughout > college, pay attention when the teachers talk about abstract > concepts. The concepts are abstracted because they apply > generally, not only in specific contexts. It'a a part of > the engineering training to be able to distill any given > problem into its abstract form, and then find whatever > standard solution for this abstract core problem. > And vice versa. You have to know a set of standard problems > and their solutions, so you have a standard 'language' which > you can use to express problems so you don't have to derive > e.g. the FT from scratch in every context where you > encounter it.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. Heaviside's formulation (he invented the unit step, U(t)) although entirely accurate, lacked sufficient rigor to satisfy mathematicians, and some set out to supply the missing rigor. (The parallel with the early development of differential and integral calculus should be clear.) In the process, Fourier's forgotten work was rediscovered. I think it is largely through professional and academic jealousy that we speak of Laplace, rather than Heaviside, transforms. Heaviside (for whom the Heaviside Layer of the ionosphere is named) derived most of the ones we now tabulate and was the first to use them to reduce homogeneous linear differential equations with constant coefficients to algebra. Heaviside died in 1925. Nobody used Fourier transforms before he introduced them. Many things are obvious after they become known. 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 7, 20092009-03-07
Reply by ●March 7, 20092009-03-07
On 7 Mar, 18:30, Jerry Avins <j...@ieee.org> wrote:> Rune Allnor wrote: > > On 7 Mar, 14:34, makol...@yahoo.com wrote: > >> On Mar 7, 4:13 am, Rune Allnor <all...@tele.ntnu.no> wrote: > > >>> On 7 Mar, 00:58, "Alun" <no.spam.thank....@invalid.invalid> wrote: > >>>> "Rune Allnor" <all...@tele.ntnu.no> wrote in message > >>>>news:9ab6ef27-b5fc-4f4a-8af7-1abe088fcf47@p11g2000yqe.googlegroups.com... > >>>> On 6 Mar, 18:37, makol...@yahoo.com wrote: > >>>>> I was enlightened to read here the other day that the Fourier > >>>>> Transform also describes the relationship in electromagnetics > >>>>> between the current distribution and the far field radiation > >>>>> pattern. > >>>> !The next obvious step on your path to Nirvana (or the > >>>> !looney bin, whichever is closer) is to investigate why[*] > >>>> !2+2=4 in such diverse fields as > >>>> ! > >>>> !- Maths > >>>> !- Physics > >>>> !- Economy > >>>> !- Medicine > >>>> !- and so on. > >>>> ! > >>>> !In fact, it't hard to come up with a single case > >>>> !where 2+2=/=4. Now, *that's* food for tought. > >>>> !Preferably somebody else's, but still. > >>>> It is a great pity when those who would be the pundits of > >>>> this NG seek personal satisfaction by pooh-poohing > >>>> the expressed sense of academic excitement of a newbie. > >>>> Such excitement is to be encouraged and nurtured. > >>> I could agree with you in principle, but the timing is > >>> wrong. I would assume one is at least a couple of years > >>> into college or university levels before one gets to > >>> know the FT; let's assume students to be of age at > >>> least 20. > >>> By that time, it ought to be clear to students that > >>> mathemathichs is a universal too which does not depend > >>> on context. Problem formulations and boundary conditions > >>> vary, but maths doesn't. > >>> The fact that the OP apparently realizes this general > >>> validity of maths in the context of the FT - which means > >>> he has some 15 years of previos exposure to maths - is > >>> a sign that there is something seriously missing with > >>> his insights into the basic tools of engineering. > >>> Rune- Hide quoted text - > >>> - Show quoted text - > >> Hi Rune, > > >> YEP I'm an engineer and your estimate of my experience is off by a > >> factor of 2, (I'll leave it to you to wonder which way). > > > You're a wonder kid? Switch schools. The sooner the better. > > >> But no I'm not a mathematician.... to me, �math is a useful but > >> sometimes cumbersome tool to describe what happens in nature... > > > That doesn't excuse you from learning how to use it. > > >> I found it interesting that the natural behavior of EM fields �mimics > >> the natural behavior of waveforms in the frequency and time domain... > > > Could the fact that EM fields actually *are* waves that can > > be analyzed in both time and frequency domains, have anything > > to do with this shocking insight...? > > >> I'm sorry if this was obvious to everyone else here. �Interestingly, > >> there are only a few other pairings mentioned so far... �come on there > >> must be more, � maybe it's not so obvious after all... > > > Right. Because *you* don't see the solution, none exists... > > A one-way ticket to the looney bin, if there ever was one. > > > There is a Norwedian proverb that "one can't see the forest > > through the trees". Which is what you are up against: > > Everything is out there in front of your eyse, but you > > don't recognize it for what it is. > > > The (generalized) Fourier transform is a standard tool for > > solving linear Differential Equations, which in turn are > > the backbone tools of mathematical physics. You'd know > > that if you paid attention in maths class. > > > Your earth-shattering revelation that the FT can be used > > to describe several types of waves are caused by the > > fact - prepare yourself for a surprise - that they solve > > wave equations of the same form. Again, mathematics is > > generally useful and does not depend on context. > > > So for your remaining time in High School and throughout > > college, pay attention when the teachers talk about abstract > > concepts. The concepts are abstracted because they apply > > generally, not only in specific contexts. It'a a part of > > the engineering training to be able to distill any given > > problem into its abstract form, and then find whatever > > standard solution for this abstract core problem. > > And vice versa. You have to know a set of standard problems > > and their solutions, so you have a standard 'language' which > > you can use to express problems so you don't have to derive > > e.g. the FT from scratch in every context where you > > encounter it. > > 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... Rune
Reply by ●March 7, 20092009-03-07
Rune Allnor wrote:> On 7 Mar, 18:30, Jerry Avins <j...@ieee.org> wrote: >> Rune Allnor wrote: >>> On 7 Mar, 14:34, makol...@yahoo.com wrote: >>>> On Mar 7, 4:13 am, Rune Allnor <all...@tele.ntnu.no> wrote: >>>>> On 7 Mar, 00:58, "Alun" <no.spam.thank....@invalid.invalid> wrote: >>>>>> "Rune Allnor" <all...@tele.ntnu.no> wrote in message >>>>>> news:9ab6ef27-b5fc-4f4a-8af7-1abe088fcf47@p11g2000yqe.googlegroups.com... >>>>>> On 6 Mar, 18:37, makol...@yahoo.com wrote: >>>>>>> I was enlightened to read here the other day that the Fourier >>>>>>> Transform also describes the relationship in electromagnetics >>>>>>> between the current distribution and the far field radiation >>>>>>> pattern. >>>>>> !The next obvious step on your path to Nirvana (or the >>>>>> !looney bin, whichever is closer) is to investigate why[*] >>>>>> !2+2=4 in such diverse fields as >>>>>> ! >>>>>> !- Maths >>>>>> !- Physics >>>>>> !- Economy >>>>>> !- Medicine >>>>>> !- and so on. >>>>>> ! >>>>>> !In fact, it't hard to come up with a single case >>>>>> !where 2+2=/=4. Now, *that's* food for tought. >>>>>> !Preferably somebody else's, but still. >>>>>> It is a great pity when those who would be the pundits of >>>>>> this NG seek personal satisfaction by pooh-poohing >>>>>> the expressed sense of academic excitement of a newbie. >>>>>> Such excitement is to be encouraged and nurtured. >>>>> I could agree with you in principle, but the timing is >>>>> wrong. I would assume one is at least a couple of years >>>>> into college or university levels before one gets to >>>>> know the FT; let's assume students to be of age at >>>>> least 20. >>>>> By that time, it ought to be clear to students that >>>>> mathemathichs is a universal too which does not depend >>>>> on context. Problem formulations and boundary conditions >>>>> vary, but maths doesn't. >>>>> The fact that the OP apparently realizes this general >>>>> validity of maths in the context of the FT - which means >>>>> he has some 15 years of previos exposure to maths - is >>>>> a sign that there is something seriously missing with >>>>> his insights into the basic tools of engineering. >>>>> Rune- Hide quoted text - >>>>> - Show quoted text - >>>> Hi Rune, >>>> YEP I'm an engineer and your estimate of my experience is off by a >>>> factor of 2, (I'll leave it to you to wonder which way). >>> You're a wonder kid? Switch schools. The sooner the better. >>>> But no I'm not a mathematician.... to me, math is a useful but >>>> sometimes cumbersome tool to describe what happens in nature... >>> That doesn't excuse you from learning how to use it. >>>> I found it interesting that the natural behavior of EM fields mimics >>>> the natural behavior of waveforms in the frequency and time domain... >>> Could the fact that EM fields actually *are* waves that can >>> be analyzed in both time and frequency domains, have anything >>> to do with this shocking insight...? >>>> I'm sorry if this was obvious to everyone else here. Interestingly, >>>> there are only a few other pairings mentioned so far... come on there >>>> must be more, maybe it's not so obvious after all... >>> Right. Because *you* don't see the solution, none exists... >>> A one-way ticket to the looney bin, if there ever was one. >>> There is a Norwedian proverb that "one can't see the forest >>> through the trees". Which is what you are up against: >>> Everything is out there in front of your eyse, but you >>> don't recognize it for what it is. >>> The (generalized) Fourier transform is a standard tool for >>> solving linear Differential Equations, which in turn are >>> the backbone tools of mathematical physics. You'd know >>> that if you paid attention in maths class. >>> Your earth-shattering revelation that the FT can be used >>> to describe several types of waves are caused by the >>> fact - prepare yourself for a surprise - that they solve >>> wave equations of the same form. Again, mathematics is >>> generally useful and does not depend on context. >>> So for your remaining time in High School and throughout >>> college, pay attention when the teachers talk about abstract >>> concepts. The concepts are abstracted because they apply >>> generally, not only in specific contexts. It'a a part of >>> the engineering training to be able to distill any given >>> problem into its abstract form, and then find whatever >>> standard solution for this abstract core problem. >>> And vice versa. You have to know a set of standard problems >>> and their solutions, so you have a standard 'language' which >>> you can use to express problems so you don't have to derive >>> e.g. the FT from scratch in every context where you >>> encounter it. >> 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. You won't find transforms in J.C.Maxwell's /Treatise on Electricity and Magnetism/. (Nor complex numbers!) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 7, 20092009-03-07
Rune Allnor <allnor@tele.ntnu.no> writes:> [...] > 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.Hi Rune, Perhaps this is just semantics, but I would say Fourier analysis _falls out of_ the specific form of the Sturm-Liouville differential equation \begin{align} \frac{d^2y}{dx^2} + \lambda y = 0 \end{align} and not that the theory is "based on it." -- % Randy Yates % "My Shangri-la has gone away, fading like %% Fuquay-Varina, NC % the Beatles on 'Hey Jude'" %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Shangri-La', *A New World Record*, ELO http://www.digitalsignallabs.com
Reply by ●March 7, 20092009-03-07
On Mar 7, 8:31�pm, Randy Yates <ya...@ieee.org> wrote:> Rune Allnor <all...@tele.ntnu.no> writes: > > [...] > > 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. > > Hi Rune, > > Perhaps this is just semantics, but I would say Fourier analysis > _falls out of_ the specific form of the Sturm-Liouville differential > equation > > \begin{align} > � \frac{d^2y}{dx^2} �+ \lambda y = 0 > \end{align} > > and not that the theory is "based on it."i really agree with you, Randy. although, when i'm on USENET, i would express it with "ASCII-art" rather than LaTeX: (d^2)/(dx^2)y + lambda*y = 0 feh... doesn't look much better. maybe, someday, all of our news client programs and Google will have a setting where we can make it decode LaTeX. L8r, r b-j r b-j
Reply by ●March 8, 20092009-03-08
<makolber@yahoo.com> wrote in message news:22451a68-8601-417d-9901-233758a9f1a1@v1g2000prd.googlegroups.com...> The Fourier transform in nature not just time<>frequency domain > > Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain. > > I was enlightened to read here the other day that the Fourier > Transform also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. Yep, they have sidelobes just like in waveforms... > > So ... > > I thought it would be fun and interesting to ask the group to come up > with a list of other pairings in nature whose relationship can be > described by the FT. > > MarkHey Mark! Everything in the universe can be anylised by dividing things in two. Speaking as a biped male that needs two to multiply, and an egg that divides by two of course. Think of it as the start of thought, then apply atoms... Bingo! you've only just begun. Have fun, it'll never end. : )
Reply by ●March 8, 20092009-03-08
On 8 Mar, 02:31, Randy Yates <ya...@ieee.org> wrote:> Rune Allnor <all...@tele.ntnu.no> writes: > > [...] > > 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. > > Hi Rune, > > Perhaps this is just semantics, but I would say Fourier analysis > _falls out of_ the specific form of the Sturm-Liouville differential > equation > > \begin{align} > � \frac{d^2y}{dx^2} �+ \lambda y = 0 > \end{align} > > and not that the theory is "based on it."From a mathematical point of view you are right. I was talking from a historical point of view, where Sturm and Liouville refined and generalized Fourier's ideas. Rune
Reply by ●March 8, 20092009-03-08
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? Rune
Reply by ●March 8, 20092009-03-08
"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!
Reply by ●March 8, 20092009-03-08
Space/spacial frequency etc... I once thought that the oposite sides of the brain tend to solve problems which are fourier transforms of each other, explaining in some way the massive correlation/covolution potential of the minds eye (minds eye's lense??). Gussian identities occur in quantum mechanical waves. And in other types of phase summation integrals. (i.e. antenna field patterns) Heavyside is also quite interesting to look at. Although the maths might be a bit on the heavy side ;-) Differential forms and filter responses all benefit from FT or laplace (no negative infinity time) or Z transform (descrete laplace). cheers jacko
