Is it possible to generate harmonic distortion analytically?
in [DSP]
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From: Robert Orban on 10 Mar 2010 18:58 In article <hmv572$ru8$2(a)news.eternal-september.org>, lcargill99(a)comcast.net says... > > >(in PCM streams, of course) > >What is a good book or website which can expose techniques which might >be of use to do this? > >Thanks in advance. I know how to create a nonlinear function that produces no harmonic distortion, only IM distortion, for arbitrarily complex waveforms and arbitrary amplitudes. However, I don't know how to do what you want, if what you want is to create harmonic distortion without IM distortion. The technique for producing only IM distortion is based on the idea of an RF clipper, where the signal is amplitude-modulated by a sinewave whose frequency in much higher than the highest frequency present in the signal, clipped, demodulated, and then lowpass-filtered. All of the clipper-induced harmonics appear as sidebands around harmonics of the carrier frequency, so the lowpass filter removes them after demodulation because they are much higher in frequency than the demodulated baseband signal. In the '70s, Michael Gerzon published a clever way of achieving the same end without modulation; he used Hilbert Transforms along with a x(t) / SQRT[x(t)^2 + (Hilbert(x(t))^2] operation. His method is better for DSP because it requires a lower sample rate than for a naive DSP implementation of the RF clipper. The bandwidth still increases because of the sum-frequency IMD created for anything other than a single sinewaves, but the system does not have to digitize a high frequency carrier.
From: Les Cargill on 10 Mar 2010 20:03 robert bristow-johnson wrote: > On Mar 10, 6:05 pm, Les Cargill <lcargil...(a)comcast.net> wrote: >> Jerry Avins wrote: >> > ... >>> Positive/negative symmetry in a transfer function precludes even >>> harmonics, > > i would say that even symmetry causes even harmonics and odd symmetry > generates only odd harmonics. > > not sure what you meant, Jerry, by positive/negative symmetry. (as > opposed to "same sign"? okay, you meant odd.) > > >> I am mostly experimenting. And rather than buy parts and >> assembling circuits, for some reason trying to do it DSP seemed >> appealing. >> >> I may also wander off and try the DFT approach. > > i wouldn't recommend it. > > > On Mar 10, 6:06 pm, Les Cargill <lcargil...(a)comcast.net> wrote: >> I also found that if you run a long enough string of 'em, >> they generate a sync pulse. Talk about generating an >> identity the hard way... > > do you mean > > { 1/N for 1<=n<=N > a[n] = { > { 0 otherwise? > > and then you input > > x(t) = (A-1)/2 + ((A+1/2)*cos(w0*t) ? > > let A >= 1. > > that's what you mean? > > r b-j Yarg... "sinc", not "sync". I used the T(a,sample) = cos(a*sample) form. If you apply V = T(1,V) + T(2,V)... T(n,V) where V is initially a sine wave, it produces a sinc function. The complexity of this is O(n*M), (where M is the number of samples) so it's very inefficient, but it's interesting. -- Les Cargill
From: Les Cargill on 10 Mar 2010 20:18 Robert Orban wrote: > In article <hmv572$ru8$2(a)news.eternal-september.org>, > lcargill99(a)comcast.net says... >> >> (in PCM streams, of course) >> >> What is a good book or website which can expose techniques which might >> be of use to do this? >> >> Thanks in advance. > > I know how to create a nonlinear function that produces no harmonic > distortion, only IM distortion, for arbitrarily complex waveforms and > arbitrary amplitudes. However, I don't know how to do what you want, if > what you want is to create harmonic distortion without IM distortion. > Hi, Robert. Yes, I've discovered that. I was surprised to find that IM and harmonic distortion are that related - as a consumer, they're stated as separate quantities in product specifications. > The technique for producing only IM distortion is based on the idea of > an RF clipper, where the signal is amplitude-modulated by a sinewave > whose frequency in much higher than the highest frequency present in > the signal, clipped, demodulated, and then lowpass-filtered. Something like a switchmode power supply, then? > All of the > clipper-induced harmonics appear as sidebands around harmonics of the > carrier frequency, so the lowpass filter removes them after > demodulation because they are much higher in frequency than the > demodulated baseband signal. > > In the '70s, Michael Gerzon published a clever way of achieving the same > end without modulation; he used Hilbert Transforms along with a > > x(t) / SQRT[x(t)^2 + (Hilbert(x(t))^2] > That looks familiar... > operation. His method is better for DSP because it requires a lower > sample rate than for a naive DSP implementation of the RF clipper. The > bandwidth still increases because of the sum-frequency IMD created for > anything other than a single sinewaves, but the system does not have to > digitize a high frequency carrier. > > Interesting. -- Les Cargill
From: Tauno Voipio on 11 Mar 2010 02:04 On 11.3.10 1:58 , Robert Orban wrote: > In article<hmv572$ru8$2(a)news.eternal-september.org>, > lcargill99(a)comcast.net says... >> >> >> (in PCM streams, of course) >> >> What is a good book or website which can expose techniques which might >> be of use to do this? >> >> Thanks in advance. > > I know how to create a nonlinear function that produces no harmonic > distortion, only IM distortion, for arbitrarily complex waveforms and > arbitrary amplitudes. However, I don't know how to do what you want, if > what you want is to create harmonic distortion without IM distortion. > > The technique for producing only IM distortion is based on the idea of > an RF clipper, where the signal is amplitude-modulated by a sinewave > whose frequency in much higher than the highest frequency present in > the signal, clipped, demodulated, and then lowpass-filtered. All of the > clipper-induced harmonics appear as sidebands around harmonics of the > carrier frequency, so the lowpass filter removes them after > demodulation because they are much higher in frequency than the > demodulated baseband signal. > > In the '70s, Michael Gerzon published a clever way of achieving the same > end without modulation; he used Hilbert Transforms along with a > > x(t) / SQRT[x(t)^2 + (Hilbert(x(t))^2] > > operation. His method is better for DSP because it requires a lower > sample rate than for a naive DSP implementation of the RF clipper. The > bandwidth still increases because of the sum-frequency IMD created for > anything other than a single sinewaves, but the system does not have to > digitize a high frequency carrier. In a RF clipper, you can get rid of half of the IM distortion, by using SSB instead of plain AM with both sidebands. In a DSP, the SSB modulation is quite easy if Hilbert is available. This method does not allow the Gerzon method, as in a SSB signal, the total amplitude cannot use as the detected RF signal. -- Tauno Voipio tauno voipio (at) ik fi
From: robert bristow-johnson on 11 Mar 2010 15:19
On Mar 10, 8:03 pm, Les Cargill <lcargil...(a)comcast.net> wrote: > robert bristow-johnson wrote: > > On Mar 10, 6:05 pm, Les Cargill <lcargil...(a)comcast.net> wrote: > >> Jerry Avins wrote: > > > ... > >>> Positive/negative symmetry in a transfer function precludes even > >>> harmonics, > > > i would say that even symmetry causes even harmonics and odd symmetry > > generates only odd harmonics. > > > not sure what you meant, Jerry, by positive/negative symmetry. (as > > opposed to "same sign"? okay, you meant odd.) > > >> I am mostly experimenting. And rather than buy parts and > >> assembling circuits, for some reason trying to do it DSP seemed > >> appealing. > > >> I may also wander off and try the DFT approach. > > > i wouldn't recommend it. > > > On Mar 10, 6:06 pm, Les Cargill <lcargil...(a)comcast.net> wrote: > >> I also found that if you run a long enough string of 'em, > >> they generate a sync pulse. Talk about generating an > >> identity the hard way... > > > do you mean > > > { 1/N for 1<=n<=N > > a[n] = { > > { 0 otherwise? > > > and then you input > > > x(t) = (A-1)/2 + ((A+1/2)*cos(w0*t) ? > > > let A >= 1. > > > that's what you mean? > > > r b-j > > Yarg... "sinc", not "sync". > > I used the T(a,sample) = cos(a*sample) form. If > you apply V = T(1,V) + T(2,V)... T(n,V) where > V is initially a sine wave, it produces a sinc function. > The complexity of this is O(n*M), (where M is the number > of samples) so it's very inefficient, but it's interesting. Les, do you know what it's like to be a teeny bit rigorous? whatever you wrote above makes no sense at all. do you mean to say Y = T(1,V) + T(2,V)... T(n,V) ? and if V is a sine wave, then T(a,V)=cos(a*V) has nothing to do with Tchebyshev, maybe Bessel, but not Tchebyshev. i dunno how you do it, but i actually can't make *anything* work, unless i'm rigorous. because reality requires it to be right. the DSP or computer that you program doesn't say "oh yeah, i know what you mean...", it only does what you code it to do. so Vlad, maybe you were right. r b-j |