Is it possible to generate harmonic distortion analytically?
in [DSP]
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From: Les Cargill on 8 Mar 2010 19:11 dvsarwate wrote: > On Mar 8, 4:19 pm, Les Cargill <lcargil...(a)comcast.net> wrote: >> Jerry Avins wrote: >>> Les Cargill wrote: >>>> robert bristow-johnson wrote: >>> ... >>>>> are you trying to introduce a specified amount of specific harmonics >>>>> to a pure sine wave? >>>> No, for any random PCM stream. >>> As I see it, the answer is a non-sequitur. >>> ... >>> Jerry >> How about "for any *arbitrary* PCM stream?" >> >> -- >> Les Cargill > > Perhaps the OP should be more clear as to what exactly he > wants. For a pure sinusoid, say s(t) = 3cos(wt), whether as an > analog signal or as a PCM stream, we can add in second > harmonic distortion by adding (say) 0.1s^2(t) to s(t) and > subtracting off the DC term thus introduced. But using > a cubic adds in a third harmonic and also changes the > amplitude of the fundamental. Is this OK? Or does > harmonic distortion in this instance mean 0.1cos(3wt) > gets added to s(t) with no change in the fundamental? > My understanding is that there's no change to the fundamental. You just calculate the interesting harmonics ( second, third, nth) and mix them in. But the original signal may be relatively harmonically complex to begin with. And by "harmonics", those will be vector Fourier quantities, not single sine tones. > Turning to more arbitrary signals, if s(t) = 3 sin(w1t) + 4 sin(w2t), > then adding in (say) 0.1s^2(t) to s(t) adds in second harmonics > of w1 and w2 but also intermodulation products at frequencies > w1+w2 and w1-w2. Is this OK? Or are looking to change > s(t) = 3 sin(w1t) + 4 sin(w2t) by adding in (say) > 0.1sin(2w1t) + 0.2sin(2w2t)? Pure harmonic distortion and > *no* intermodulation distortion? I would expect the difference tone signals to be part of this - just as if you opened the appropriate stop on an organ. But realize - I am not 100% sure what is possible - if you can quantify the difference tones, I'd think you could subtract those out as well. If it's really just a matter of banging the DFT at multiples of t, then that's what I'm looking for. > And what exactly does the > OP mean by harmonic distortion of a random or arbitrary > data stream which is not necessarily periodic at all. > Absent a fundamental, what is *harmonic* distortion? I suppose harmonics of any element of the Fourier series of the signal. Obviously, if it's white noise, the result will be white noise. > > --Dilip Sarwate -- Les Cargill
From: Les Cargill on 8 Mar 2010 20:20 Jerry Avins wrote: > Les Cargill wrote: >> Jerry Avins wrote: >>> Les Cargill wrote: >>>> robert bristow-johnson wrote: >>> >>> ... >>> >>>>> are you trying to introduce a specified amount of specific harmonics >>>>> to a pure sine wave? >>>> >>>> No, for any random PCM stream. >>> >>> As I see it, the answer is a non-sequitur. >>> >>> ... >>> >>> Jerry >> >> >> How about "for any *arbitrary* PCM stream?" > > That doesn't fix it. I wasn't picking nits. The question was, given any > arbitrary waveform, what harmonics do you want to add to it. The usual suspects - second, third, fifth. > An answer > pins down a process that applies to any waveform, whether it be random > or have a familiar name. > Ah! Thanks! > Jerry -- Les Cargill
From: Mark on 9 Mar 2010 09:58 On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote: > Jerry Avins wrote: > > Les Cargill wrote: > >> Jerry Avins wrote: > >>> Les Cargill wrote: > >>>> robert bristow-johnson wrote: > > >>> ... > > >>>>> are you trying to introduce a specified amount of specific harmonics > >>>>> to a pure sine wave? > > >>>> No, for any random PCM stream. > > >>> As I see it, the answer is a non-sequitur. > > >>> ... > > >>> Jerry > > >> How about "for any *arbitrary* PCM stream?" > > > That doesn't fix it. I wasn't picking nits. The question was, given any > > arbitrary waveform, what harmonics do you want to add to it. > > The usual suspects - second, third, fifth. > > > An answer > > pins down a process that applies to any waveform, whether it be random > > or have a familiar name. > > Ah! Thanks! > > > Jerry > > -- > Les Cargill- Hide quoted text - > > - Show quoted text - I may be missing something in the discussion but I think the simple answer to your question is: the second harmonic component is created by the A*x^2 term and the the third harmonic component is created by the B*x^3 term etc of the transfer function. The (complex) magnitude of the coefficents A and B etc determine the magnitude (and phase) of the varous harmonics. Is that what you were asking? Mark
From: Les Cargill on 9 Mar 2010 20:30 Mark wrote: > On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote: >> Jerry Avins wrote: >>> Les Cargill wrote: >>>> Jerry Avins wrote: >>>>> Les Cargill wrote: >>>>>> robert bristow-johnson wrote: >>>>> ... >>>>>>> are you trying to introduce a specified amount of specific harmonics >>>>>>> to a pure sine wave? >>>>>> No, for any random PCM stream. >>>>> As I see it, the answer is a non-sequitur. >>>>> ... >>>>> Jerry >>>> How about "for any *arbitrary* PCM stream?" >>> That doesn't fix it. I wasn't picking nits. The question was, given any >>> arbitrary waveform, what harmonics do you want to add to it. >> The usual suspects - second, third, fifth. >> >>> An answer >>> pins down a process that applies to any waveform, whether it be random >>> or have a familiar name. >> Ah! Thanks! >> >>> Jerry >> -- >> Les Cargill- Hide quoted text - >> >> - Show quoted text - > > I may be missing something in the discussion but I think the simple > answer to your question is: > > the second harmonic component is created by the A*x^2 term and the the > third harmonic component is created by the B*x^3 term etc of the > transfer function. The (complex) magnitude of the coefficents A and > B etc determine the magnitude (and phase) of the varous harmonics. > > Is that what you were asking? > > Mark > I believe it is. -- Les Cargill
From: robert bristow-johnson on 10 Mar 2010 00:23
On Mar 9, 11:30 pm, Jerry Avins <j...(a)ieee.org> wrote: > Les Cargill wrote: > > Mark wrote: > >> On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote: > >>> Jerry Avins wrote: > >>>> Les Cargill wrote: > >>>>> Jerry Avins wrote: > >>>>>> Les Cargill wrote: > >>>>>>> robert bristow-johnson wrote: > >>>>>> ... > >>>>>>>> are you trying to introduce a specified amount of specific > >>>>>>>> harmonics > >>>>>>>> to a pure sine wave? > >>>>>>> No, for any random PCM stream. > >>>>>> As I see it, the answer is a non-sequitur. > >>>>>> ... > >>>>>> Jerry > >>>>> How about "for any *arbitrary* PCM stream?" > >>>> That doesn't fix it. I wasn't picking nits. The question was, given any > >>>> arbitrary waveform, what harmonics do you want to add to it. > >>> The usual suspects - second, third, fifth. > > >>>> An answer > >>>> pins down a process that applies to any waveform, whether it be random > >>>> or have a familiar name. > >>> Ah! Thanks! > > >>>> Jerry > >>> -- > >>> Les Cargill- Hide quoted text - > > >>> - Show quoted text - > > >> I may be missing something in the discussion but I think the simple > >> answer to your question is: > > >> the second harmonic component is created by the A*x^2 term and the the > >> third harmonic component is created by the B*x^3 term etc of the > >> transfer function. The (complex) magnitude of the coefficents A and > >> B etc determine the magnitude (and phase) of the varous harmonics. > > >> Is that what you were asking? > > >> Mark > > > I believe it is. > > But be aware that the x^3 term can affect the amplitude of the > fundamental/ I need to think more about the phase. i don't think that memoryless mapping functions can do much about phase except maybe flip the polarity. now, for Les and Mark. yes, in general, an Nth-order polynomial will create, out of a single sinusoid, new frequencies that are harmonic up to the Nth harmonic. But the generation of harmonics is level dependent (as it is for any non-linear distortion). now, if you *know* or can assume the level of your sinusoid, you can create an Nth-order polynomial that will create a specified level of each harmonic up to the Nth harmonic. you define your Nth-order polynomial as a sum of Tchebyshev polynomials up to the Nth-order Tchebyshev. if a normalized sinusoid goes into a 5th-order Tchebyshev, what comes out is only the 5th harmonic and no others. now, does that help you, Les, regarding the "generate harmonic distortion analytically"? otherwise, i think you need to be more specific about the nature of your input signal. r b-j |