Harmonics in oscillator-
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From: JEMebius on 29 Jan 2010 21:25 I.N. Galidakis wrote: > I am playing around with a specific oscillator. I have identified 5 modes of > oscillation, which correspond to 7 frequencies N_j, therefore the system phasor > is: > > p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) > > Do I need to also include all the harmonics > > exp(2*Pi*i*k*N_j*t), k=2,3,4,... > > in the phasor, in order to analyze it in the Fourier sense? > > In other words, is the system phasor p(t) or q(t), with: > > q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) > > Thanks, I guess you have identified 7 modes, not 5 modes of oscillation? Anyhow, one does not need the extra terms to obtain a decent Fourier analysis; (A) if all of the N_j are commensurable then one has almost by definition a classical Fourier analysis: let N_j = n_j * W, the numbers n_j are integers, the positive number W is the common measure of the N-j; in other words: all frequencies are integer multiples of the ground frequency W. then p(t) = Sum for j = 1...7 of exp(2pi*n_j*W)*t (B) if the N_j have no common measure, then one still has an almost-periodic function, i.e. a function f on the reals with the property that for any epsilon > 0 there is a translation number T such that |f(x -T) - f(x)| < epsilon for all x. Harald Bohr was the pioneer of the theory of almost-periodic functions. He proved the Main Theorem: almost-periodic functions are precisely the trigonometric polynomials and the convergent trigonometric series, convergent in the sense of the Hilbert space norm induced by the inner product (f, g) = lim for T->infinity of [Integral from -T to +T of f*(x).g(x).dx] / 2T (* = complex conjugation) In other words: generalised Fourier synthesis yields exactly the (complex-valued) almost-periodic functions. They form the simplest case of a Hilbert space with a non-countable orthonormal basis, a Hilbert space which is therefore non-separable. Ciao: Johan E. Mebius
From: JEMebius on 29 Jan 2010 21:26 I.N. Galidakis wrote: > I am playing around with a specific oscillator. I have identified 5 modes of > oscillation, which correspond to 7 frequencies N_j, therefore the system phasor > is: > > p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) > > Do I need to also include all the harmonics > > exp(2*Pi*i*k*N_j*t), k=2,3,4,... > > in the phasor, in order to analyze it in the Fourier sense? > > In other words, is the system phasor p(t) or q(t), with: > > q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) > > Thanks, I guess you have identified 7 modes, not 5 modes of oscillation? Anyhow, one does not need the extra terms to obtain a decent Fourier analysis; (A) if all of the N_j are commensurable then one has almost by definition a classical Fourier analysis: let N_j = n_j * W, the numbers n_j are integers, the positive number W is the common measure of the N-j; in other words: all frequencies are integer multiples of the ground frequency W. then p(t) = Sum for j = 1...7 of exp(2pi*n_j*W)*t (B) if the N_j have no common measure, then one still has an almost-periodic function, i.e. a function f on the reals with the property that for any epsilon > 0 there is a translation number T such that |f(x -T) - f(x)| < epsilon for all x. Harald Bohr was the pioneer of the theory of almost-periodic functions. He proved the Main Theorem: almost-periodic functions are precisely the trigonometric polynomials and the convergent trigonometric series, convergent in the sense of the Hilbert space norm induced by the inner product (f, g) = lim for T->infinity of [Integral from -T to +T of f*(x).g(x).dx] / 2T (* = complex conjugation) In other words: generalised Fourier synthesis yields exactly the (complex-valued) almost-periodic functions. They form the simplest case of a Hilbert space with a non-countable orthonormal basis, a Hilbert space which is therefore non-separable. Ciao: Johan E. Mebius
From: I.N. Galidakis on 30 Jan 2010 05:05 JEMebius wrote: > I.N. Galidakis wrote: >> I am playing around with a specific oscillator. I have identified 5 modes of >> oscillation, which correspond to 7 frequencies N_j, therefore the system >> phasor is: >> >> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >> >> Do I need to also include all the harmonics >> >> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >> >> in the phasor, in order to analyze it in the Fourier sense? >> >> In other words, is the system phasor p(t) or q(t), with: >> >> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >> >> Thanks, > Hi Johan, > I guess you have identified 7 modes, not 5 modes of oscillation? No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 vibrates at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest vibrate at one mode, total 7 frequencies. > Anyhow, one does not need the extra terms to obtain a decent Fourier analysis; The reason I asked is because everywhere I asked, they told me that any oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the harmonics. For my case, this complicates matters considerably, because according to Wiki, the harmonics fall as ~1/k^2 and I don't know how to model them using an appropriate function. It's easy to see that if an oscillator broadcasts at N_j, its spectrum is Dirac(x-N_j), but I have no idea how to model the harmonics with the Dirac. > (A) if all of the N_j are commensurable then one has almost by definition a > classical Fourier analysis: > > let N_j = n_j * W, the numbers n_j are integers, the positive number W is the > common measure of the N-j; in other words: all frequencies are integer > multiples of the ground frequency W. > then p(t) = Sum for j = 1...7 of exp(2pi*n_j*W)*t Yes. > (B) if the N_j have no common measure, then one still has an almost-periodic > function, i.e. a function f on the reals with the property that for any > epsilon > 0 there is a translation number T such that |f(x -T) - f(x)| < > epsilon for all x. > > > Harald Bohr was the pioneer of the theory of almost-periodic functions. He > proved the Main Theorem: almost-periodic functions are precisely the > trigonometric polynomials and the convergent trigonometric series, convergent > in the sense of the Hilbert space norm induced by the inner product > > (f, g) = lim for T->infinity of [Integral from -T to +T of f*(x).g(x).dx] / 2T > (* = complex conjugation) > > In other words: generalised Fourier synthesis yields exactly the > (complex-valued) almost-periodic functions. > > They form the simplest case of a Hilbert space with a non-countable > orthonormal basis, a Hilbert space which is therefore non-separable. Many thanks, > Ciao: Johan E. Mebius -- Ioannis
From: David Bernier on 30 Jan 2010 07:55 I.N. Galidakis wrote: > JEMebius wrote: >> I.N. Galidakis wrote: >>> I am playing around with a specific oscillator. I have identified 5 modes of >>> oscillation, which correspond to 7 frequencies N_j, therefore the system >>> phasor is: >>> >>> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >>> >>> Do I need to also include all the harmonics >>> >>> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >>> >>> in the phasor, in order to analyze it in the Fourier sense? >>> >>> In other words, is the system phasor p(t) or q(t), with: >>> >>> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >>> >>> Thanks, > > Hi Johan, > >> I guess you have identified 7 modes, not 5 modes of oscillation? > > No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 vibrates > at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest vibrate at one > mode, total 7 frequencies. > >> Anyhow, one does not need the extra terms to obtain a decent Fourier analysis; > > The reason I asked is because everywhere I asked, they told me that any > oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the > harmonics. For my case, this complicates matters considerably, because according > to Wiki, the harmonics fall as ~1/k^2 and I don't know how to model them using > an appropriate function. [...] From what I remember about the vibrating string (such as one idealized guitar string), the string end-points could be say at 0 centimeters and 48 centimeters. In the fundamental mode, the string goes up and down say 200 times a second. (pure 200 Hz tone). In the usual sense of harmonics, there can be discrete, finitely many points that don't move: 0 cm, 24 cm and 48 cm ("node" points or just "nodes" in physics) --> 2x 200 Hz = 400 Hz [I hope I've got the math/physics right here ...] fixed points at 0 cm, 16 cm, 32 cm and 48 cm: the distance between fixed points is 1/3 what it is for the fundamental mode, and the harmonic is at 3 x 200 Hz = 600 Hz, and so on giving harmonics at (200n) Hertz for n = 1, 2, 3, 4, ... n in N^* . Then 200 Hertz /2 = 100 Hertz could be called a "sub-harmonic". In the case of the usual standard theory of the vibrating string, I don't know enough to comment on how it might happen. As I understand your description, there could be a 200 cycles per second tone, and also 200/(k^2) cycles per second tones for all k in N* . So for example 200/(10^2) = 2 cycles per second is a possible tone or sub-harmonic ... From what quite basic things I know in Fourier analysis for periodic signals, I don't know the way to proceed. Wikipedia has something on subharmonics: < http://en.wikipedia.org/wiki/Subharmonic > regards, David Bernier
From: I.N. Galidakis on 30 Jan 2010 13:33
David Bernier wrote: > I.N. Galidakis wrote: >> JEMebius wrote: >>> I.N. Galidakis wrote: >>>> I am playing around with a specific oscillator. I have identified 5 modes >>>> of oscillation, which correspond to 7 frequencies N_j, therefore the system >>>> phasor is: >>>> >>>> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >>>> >>>> Do I need to also include all the harmonics >>>> >>>> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >>>> >>>> in the phasor, in order to analyze it in the Fourier sense? >>>> >>>> In other words, is the system phasor p(t) or q(t), with: >>>> >>>> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >>>> >>>> Thanks, >> >> Hi Johan, >> >>> I guess you have identified 7 modes, not 5 modes of oscillation? >> >> No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 >> vibrates at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest >> vibrate at one mode, total 7 frequencies. >> >>> Anyhow, one does not need the extra terms to obtain a decent Fourier >>> analysis; >> >> The reason I asked is because everywhere I asked, they told me that any >> oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the >> harmonics. For my case, this complicates matters considerably, because >> according to Wiki, the harmonics fall as ~1/k^2 and I don't know how to >> model them using an appropriate function. > [...] > > From what I remember about the vibrating string (such as one idealized > guitar string), the string end-points could be say at 0 centimeters > and 48 centimeters. In the fundamental mode, the string goes > up and down say 200 times a second. (pure 200 Hz tone). Hi David, Although vibrating strings are a good example of oscillators, I believe that the case of music is only a particular example of general oscillator theory. In any case, I have some experience with musical instruments and if the A0 piano chord vibrates at 440 Hz, it will also generate all the higher harmonics, at 880, 1760, ... n*440, etc., n=2,3,4,... This means that if I apply the damper ONLY on the A0 piano chord after it is struck, there will be some residual resonance at all the higher A's, like A1, A2, A3, up to the right limit of the piano. In other words, if I silence A0 with the damper, the rest of the A's will resonate for a short time. In THAT sense I am interested in what sense to include all the harmonics in my spectrum. I haven't heard of chords or instruments spontaneously vibrating at sub-harmonics. Thanks, > In the usual sense of harmonics, there can be discrete, finitely many points > that don't move: 0 cm, 24 cm and 48 cm ("node" points or just "nodes" > in physics) --> 2x 200 Hz = 400 Hz [I hope I've got the math/physics > right here ...] fixed points at 0 cm, 16 cm, 32 cm and 48 cm: > the distance between fixed points is 1/3 what it is for > the fundamental mode, and the harmonic is at 3 x 200 Hz = 600 Hz, > and so on giving harmonics at (200n) Hertz for n = 1, 2, 3, 4, ... n in N^* . [snip for brevity] > regards, > > David Bernier -- Ioannis |