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When a speaker is operating at high volumes, the high frequencies should be frequency modulated by the low frequencies. For the low frequencies the cone moves back and forth slowly but heavily and while doing so it emits the high frequencies. Thus the high frequencies are emitted from a moving sound source and therefor become doppler shifted. The effect should be akin to frequency modulation.

I have little doubt about the above, but my main questions are:

  • Is this effect audible at all ?
  • Does it create an impression of "loud"?
  • Are there any effect devices or plugins which simulate this effect?
  • Has anybody furnished such an effect e.g with the likes of supercollider or ChucK

Edit: After reading the paper referenced by Dave's post on the phyics side I am no longer convinced that "doppler shift" is correct. It may well be a phase-modulation. But in either case, the effect should not just add higher frequencies (as a nonlinear characteristic would), but also lower frequencies.

Edit Just to make it clear: I am not after a regular Guitar distortion effect. I just want to know if it has some audible effect, like e.g. a Compressor which also creates distortion, but it not percieved as such.

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From the point you are standing (usually) and the speed of sound, I would say it's not audible (I'm sure someone will claim that s/he can hear it) Note that Doppler shift happens with uniformly dec/increasing distance. Otherwise it's just garbled. It's the same with a fixed tweeter/subwoofer combo and shaking your head. –  user1306 Apr 11 at 11:41
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Your assumptions are incorrect. The speaker at any given time is moving in response to the sum over all driving frequencies. There is no Doppler Shift taking place. I can see why you might think of the motion that way, but in fact the output pressure vs. time is just a sum of all input waveforms. Think of the higher frequencies as "riding" on the lower ones, perhaps, rather as the signal on an AM radio "rides" on the carrier frequency. The "trick" is that the cone driver "knows" that the low-freq cone motion is there, and the proper adjustment to produce a non-Doppler high freq happens. –  Carl Witthoft Apr 11 at 12:44
    
If I take a 0.01Hz signal which makes the diaphragm move by one meter (for the sake of argument) and I add a 1 kHz signal, which makes the diaphrahm move by 1mm and I sum them up and send them to a speaker, there would certainly be a doppler shift, wouldn't it? –  Martin Drautzburg Apr 11 at 14:42
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The more I think about it, the more I'm confused; so I asked a related question at Physics: physics.stackexchange.com/questions/107965/… –  Dave Apr 11 at 17:12

3 Answers 3

In and of itself, the presence of multiple frequencies does not produce any distortion or frequency modulation. The idea of the high frequency waves "riding on" (or being "pushed by") the low frequency waves can be misleading in this context. Intermodulation effects require some non-linearity in the system, i.e. a distortion model.

Most the most basic distortion models only non-linearly transform the amplitude of the signal; this will not model the effect that you've identified. A model that includes a nonlinear function of both the amplitude and the time-derivative of the amplitude is required to model the effect.

I do not know of any specific models that incorporate this feature, but in terms of the modeling details, the feature that your are looking for is a distortion model that includes the derivative of the signal as one of its inputs.

ROM of the Effect

Let's just assume that we have a tweeter that is affixed to the cone of a low frequency driver as is envisioned in the question (Note: I'm not sure that this is a good model for single-driver speakers...). I'll assume a 1mm (=0.001m) amplitude for the low-frequency driver's amplitude. In order to get a 0.5% (1.005 factor, just less than 10 cents) Doppler shift, the frequency of the "low" frequency driver would need to be about 330Hz. This is in the realm of plausibility and perceptability.

Details

A basic (no-memory) non-linear transformation can be represented as

out(t) = a0 + a1*in(t)+ a2*[in(t)]^2+a3[in(t)]^3...

If we assume that there are just two frequencies, f1 and f2, you can see that

out(t) = a0
       + a1*(f1(t)+f2(t))
       + a2*[f1(t)+f2(t)]^2 = a2*[ f1^2 + f2^2 + 2*f1*f2 ]
       +...

it's in the quadratic, and higher order terms, of the nonlinear transfer function that produce the frequency modulation. This type of non-linear amplitude transformation is what is most commonly used in distortion effects simulations.

One aspect of the speaker problem that is less commonly modeled is an explict dependence of the output on the derivative of the input signal:

out(t) = F( in(t), in'(t) ) 

(in'(t) is the time-derivative of the input signal)

Note that for single-frequency signals the the amplitude of the derivative of the signal is scaled by the frequency itself -- this is an intrinsic high-frequency boost in the derivative signal.

As with the simple model, one can Taylor expand the transfer function to get the intermodulation products; but this will be systematically different from the simple (amplitude only) non-linear model above due to the fact that the time-derivatives are scaled by the frequency.

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I know that adding signals does not add any distortion. My point is that I believe a speaker is a non-linear device. –  Martin Drautzburg Apr 11 at 14:45
    
My point is that typical non-linear distortion models cover (most of) the relevant phenomenology. But now I'll edit to reflect a new (better?) idea of what you are getting at. –  Dave Apr 11 at 14:56
    
I'm leaving in the initial disclaimer (i.e. that "this doesn't happen") because more than one person mis-interpreted your question along these lines. –  Dave Apr 11 at 15:11
    
out(t) = F( in(t), in'(t) ) looks odd to me. I would have assumed something like Out(t) = in (F(in(t)), i.e. the modulator affects the time, not the value. This is not distortion caused by a nonlinear characteristic. If that was the case you would only get harmonics, wheras with fm you get additional frequencies below the fundamental. But I am actually more interested in whether this effect has been observed, what it sounds like and if there are audio effects which mimick it, –  Martin Drautzburg Apr 11 at 20:45

One obvious distortion that the overloaded amplifier produces is the saturation: it must already output all voltage is it capable of for a low frequency component and then cannot give any other more for the higher frequency component that should be added to that it is now on the output. Hence the high frequency component is significantly altered (or even fully suppressed) at the peaks of the low frequency component. This effect can be observed in electronics circuitry and probably also on the speaker membrane.

Another type of distortion is when the speaker membrane already cannot go far enough to match the top and bottom peaks of the sine wave of the single tone, producing a different curve at the output. This adds extra tones (mostly overtones, I guess).

These effects do not involve the Doppler shift. They are audible, and while they are normally associated with the loud sound, some tiny speaker can be overloaded without being actually very load.

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Doppler is very unlikely to be audible. I ran the math for a garden variety 12" guitar driver with the playing the low E at full throttle and playing a high E (12th fret, high E) string at the same time. This basically results in the phase modulation of the high note so the pitch of the high note is modulated at the rate of the low note. The overall modulation is really small though: 3mm excursion (= screamingly loud) the maximum pitch shift is only about 7 cents (7/100 of a semitone). That's less than most people can tune to.

Doppler is indeed a non linearity (as it depends on amplitude) and results mainly in intermodulation distortion (not harmonic distortion). You can model this using the power series as Dave described in his answer but there are actually some closed form solution using Bessel functions that are bit easier.

However intermodulation sounds pretty much always really bad, so it's not a useful effect to model or explore. There are plenty of other non-linearities that can be desirable for guitar but Doppler is not one of them.

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Are you saying: "you cannot hear it, and if you could it would sound bad"? There is no sweet spot in between, where you could tell that it is a loud loudspeaker without noticing a distortion? –  Martin Drautzburg Apr 16 at 16:18
    
Yep, pretty much. This is very non-harmonic whereas "good" distortion such as tube, overdrive, etc. are typically odd order harmonic. –  Hilmar Apr 17 at 2:09

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