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Post by aquin43 on Jun 6, 2020 4:06:17 GMT -5
Causal, I would say. The delay line model is a good description of the basic response of a velocity sensitive pickup. It is purely linear and can't reproduce that detail of the strat waveform even with the addition of any linear filter that I can think of. Add that Hammerstein non-linear filter and now it can.
I am happy to accept the non-linearity of the Hammerstein model as a realistic addition to the delay line model. This could be just a limitation of my imagination; can anyone else suggest a linear filter that can reproduce that behaviour?
So the Hammerstein equation is a "model fitting" technique right? Which is to say, deriving equations that match the observed phenomena, but not inferring what physical phenomena causes that to happen, right? I'm at a loss as to understanding what problem this solves or question this answers, but if the idea is to simply have a mathematical model that approximates measured results, through whatever means, that I can understand. According to the paper, this sort of simplified model is regularly being used in the synthesis of guitar tones and the purpose of the authors was to determine whether one of the more complex non-linear techniques would add much improvement. So, yes, they were treating the string as a black box.
The problem it solves is providing a behavioural model of the non-linearity, rather similar in nature to the commonly used pickup response models that also have no physical basis.
There is a chapter by Zollner in the POTEG collection that deals with pickup non-linearity, though it keeps many of his test results hidden. He investigates intermodulation as well and concludes that there is virtually non between strings.
Arthur
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Post by antigua on Jun 6, 2020 4:49:17 GMT -5
So the Hammerstein equation is a "model fitting" technique right? Which is to say, deriving equations that match the observed phenomena, but not inferring what physical phenomena causes that to happen, right? I'm at a loss as to understanding what problem this solves or question this answers, but if the idea is to simply have a mathematical model that approximates measured results, through whatever means, that I can understand. According to the paper, this sort of simplified model is regularly being used in the synthesis of guitar tones and the purpose of the authors was to determine whether one of the more complex non-linear techniques would add much improvement. So, yes, they were treating the string as a black box.
The problem it solves is providing a behavioural model of the non-linearity, rather similar in nature to the commonly used pickup response models that also have no physical basis.
There is a chapter by Zollner in the POTEG collection that deals with pickup non-linearity, though it keeps many of his test results hidden. He investigates intermodulation as well and concludes that there is virtually non between strings.
Arthur
I suppose if you're building synthesizers and you need a formula for a convincing guitar tone, that makes sense, but personally I am foremost interested in what's happening at a physical level, as I don't develop synth software. We had a similar order of events a couple years ago with modeling eddy currents, John H found an LCR circuit that "fitted" the peculiar transfer function of Filter'trons, but the model didn't seem to describe the physical nature of pickups, it was just being used as a plotting device, more or less. But then one day a member on another forum named TeleTucson was able to curve match a Filter'tron type curve with a three part transformer, and what was missing from the earlier spice modes was a "voltage dependent current source" in the place of the guitar string, and then finally there was a model that was not merely fitted, but which was a "physical model" of the relationship between the tangible parts. I'll be highly interested in learning about what physically phenomena accounts for these distortions that are currently being described in a "black box" manner. Maybe it's hysteresis, but I have this feeling that it's a further consequence of the non-linear relationship between the guitar string and the static magnetic field.
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Post by ms on Jun 6, 2020 5:58:16 GMT -5
Wikipedia: The Hammerstein model consists of a static single valued nonlinear element followed by a linear dynamic element.
That is, the first block is a function that gives an output for each input, but the plot of output vs input does not have to be a straight line. It is up to you to specify what it is. (I suspect that a specific parametrized non-linear static model contained a theMatlab Simulink package is sometimes associated with the term "Hammerstien".)
The following linear block does contain time, and in our case it is a differentiator, converting displacement to velocity.
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Post by aquin43 on Jun 6, 2020 6:36:01 GMT -5
I'll be highly interested in learning about what physically phenomena accounts for these distortions that are currently being described in a "black box" manner. Maybe it's hysteresis, but I have this feeling that it's a further consequence of the non-linear relationship between the guitar string and the static magnetic field. Well, in the simplest model of the strat pickup where the string is magnetised by the pole and the field generated by the magnetised string then couples with the pole and coil there are several static non linearities.
The total flux intercepted by the string varies with the distance d from the pole roughly as 1/(d + k), where k represents a small distance below the pole face. The string magnetisation varies more or less in step with that.
The field radiated by the string is distributed in space in non linear fashion and couples with the coil accordingly.
The precise pattern of the magnetisation of the string varies with distance from the pole.
These are the main sources of non linearity, I would suggest. Hysteresis in the string and pole must also be in the mix somewhere but I would guess its effects are swamped by the simple geometry based magnetic non linearities. The effectiveness of Hammerstein model with its static non linearity would tend to confirm this.
Arthur
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Post by blademaster2 on Jun 6, 2020 8:27:01 GMT -5
I think with the complexities of the motion and interaction of a vibrating string and the nonlinearities of magnetic material behaviour in general, fitting models to observed phenomena might be the pragmatic approach for now.
After all, the root cause of magnetization curve shapes being what they are is not (as I understand it) known at a molecular level, so we characterize materials through test results to obtain a model. If the magnetic material is part of it, then other strings would also intermodulate the signal as they move near the same location as the string in question. It becomes a very. very complicated relationship.
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Post by aquin43 on Jun 6, 2020 11:04:50 GMT -5
I thought it might be worth trying another more intuitive approach. Drive with a displacement wave, differentiate it and then multiply the output by the non linear displacement characteristic. Easily done in Spice. I chose the 1/(1+k*displacement) response from Zollner's article, which by the way is also a good fit for the total flux intercepted by the string. I was curious to know if this would give the same sort of pulse response. The drive is a simple triangular wave which differentiates to a square wave. The result should be obvious, on reflection.
Arthur
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Post by ms on Jun 6, 2020 13:46:09 GMT -5
I thought it might be worth trying another more intuitive approach. Drive with a displacement wave, differentiate it and then multiply the output by the non linear displacement characteristic. Easily done in Spice. I chose the 1/(1+k*displacement) response from Zollner's article, which by the way is also a good fit for the total flux intercepted by the string. I was curious to know if this would give the same sort of pulse response. The drive is a simple triangular wave which differentiates to a square wave. The result should be obvious, on reflection.
Arthur
Yes, it is as simple this: if you reduce the sensitivity as the string goes away from the pickup, you get it back as the string returns towards it.
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Post by ms on Jun 6, 2020 14:24:51 GMT -5
Now consider holding the string in a half sinusoidal shape rather than a triangle and releasing it. This excites fundamental only. Then distort it and take the derivative. The result: This is very different. I think we need to use a realistic waveform (that is, a realistic level of harmonics) and distort it in order to get a better idea of how the distortions appears.
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Post by antigua on Jun 6, 2020 15:05:51 GMT -5
I think with the complexities of the motion and interaction of a vibrating string and the nonlinearities of magnetic material behaviour in general, fitting models to observed phenomena might be the pragmatic approach for now. After all, the root cause of magnetization curve shapes being what they are is not (as I understand it) known at a molecular level, so we characterize materials through test results to obtain a model. If the magnetic material is part of it, then other strings would also intermodulate the signal as they move near the same location as the string in question. It becomes a very. very complicated relationship.Black boxes are certainly pragmatic, I gather that the intention of black box testing is to get around a thing that is not understood in lieu of understanding it. I think the root cause of the magnetization curves are just the net effect of the magnetic domain properties en.wikipedia.org/wiki/Magnetic_domain , the different grades of AlNiCo for example cause the domains to have different properties, is that what you mean or are you referring to something else? As for the strings causing interactions with their neighboring strings, the coupling coefficients are all very low, even between the pickup and the strings, so there could be interactions, but they would be multiplied by that coupling coefficient, which is probably less than 0.01.
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Post by aquin43 on Jun 6, 2020 16:18:59 GMT -5
Here is another iteration of the simple pickup model above. This time I have used the curve of relative total flux intercepted by the string over an Alnico strat pole as calculated using FEMM in the axisymmetric mode. The output scaling is arbitrary. The output load resistor is chosen so that at low levels the current in R3 is equal to that in V2 so that the fundamental of the output can be subtracted out using i(R3)-i(v2). V1 is the static position of the string. The drive is a sine wave at varying levels.
Output Waveform Distortion Components
Notice how the fundamental is not compressed or expanded but harmonics are added. The Hammerstein model behaves in a similar fashion.
Arthur
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Post by blademaster2 on Jun 6, 2020 19:57:23 GMT -5
Predictive models are interesting but I prefer to see what the sound is like for actual pickups and to compare actual physical pickups qualitatively, with the measured and predicted data on hand as a reference.
If it can be done, I would be interested to see test results that can measure distortion due to non-linearity in order to have that comparative quantitative data (perhaps as a THD value, plotted as a function of frequency or amplitude, or any other means of using a measurement that can compare pickups with regard to this attribute).
I can only suggest spectral analysis from a pure sinusoidal stimulus input as a quantitative method, but I am aware of other methods such as spectral analysis of the response to square wave stimulus. Alternatively, possibly a transfer curve plotted for various frequencies, but this would require an accurate measurement of the displacement of the string (or equivalent stimulus source). I am not an expert in the laboratory techniques typically used and what could be done, but these results would be fascinating to see.
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Post by aquin43 on Jun 8, 2020 12:04:43 GMT -5
I can see that I made an error in the Spice schematic for simulating the effect of the string magnetisation. Instead of treating the magnetisation as a multiplier of the differentiated sine output it should have been multiplied by the sine before differentiating. This steepens the slope but leaves the general character the same. With regard to the Hammerstein formula from the paper. The non linear part maps volt seconds to displacement from an origin at 5mm. Volt seconds is, of course, magnetic flux which when differentiated gives the pickup output in volts. The non linear formula, then, seems to be mapping how the flux in the coil varies with displacement of the string from the reference position. There will also be an added fixed amount of flux which is not included since its effect would disappear in the subsequent differentiation.
What I find curious is that the gradient of the curve increases at the extremes for both positive and negative excursions. I can't work out what causes that in such an obviously asymmetrical device as a strat pickup.
Arthur
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Post by antigua on Jun 8, 2020 12:49:01 GMT -5
I can see that I made an error in the Spice schematic for simulating the effect of the string magnetisation. Instead of treating the magnetisation as a multiplier of the differentiated sine output it should have been multiplied by the sine before differentiating. This steepens the slope but leaves the general character the same. With regard to the Hammerstein formula from the paper. The non linear part maps volt seconds to displacement from an origin at 5mm. Volt seconds is, of course, magnetic flux which when differentiated gives the pickup output in volts. The non linear formula, then, seems to be mapping how the flux in the coil varies with displacement of the string from the reference position. There will also be an added fixed amount of flux which is not included since its effect would disappear in the subsequent differentiation.
... What I find curious is that the gradient of the curve increases at the extremes for both positive and negative excursions. I can't work out what causes that in such an obviously asymmetrical device as a strat pickup.
Arthur
Given that the flux through the coil varies with displacement, could the height of the coil have an affect there, like would a very flat coil cause more distortion than a tall coil? Could the extreme at the excursions be due to the way the string has an elliptical movement path, rather than moving perfectly on axis?
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Post by aquin43 on Jun 8, 2020 13:08:42 GMT -5
I can see that I made an error in the Spice schematic for simulating the effect of the string magnetisation. Instead of treating the magnetisation as a multiplier of the differentiated sine output it should have been multiplied by the sine before differentiating. This steepens the slope but leaves the general character the same. With regard to the Hammerstein formula from the paper. The non linear part maps volt seconds to displacement from an origin at 5mm. Volt seconds is, of course, magnetic flux which when differentiated gives the pickup output in volts. The non linear formula, then, seems to be mapping how the flux in the coil varies with displacement of the string from the reference position. There will also be an added fixed amount of flux which is not included since its effect would disappear in the subsequent differentiation.
... What I find curious is that the gradient of the curve increases at the extremes for both positive and negative excursions. I can't work out what causes that in such an obviously asymmetrical device as a strat pickup.
Arthur
Given that the flux through the coil varies with displacement, could the height of the coil have an affect there, like would a very flat coil cause more distortion than a tall coil? Could the extreme at the excursions be due to the way the string has an elliptical movement path, rather than moving perfectly on axis? Do you mean that some of the flux available as the string moves down fails to couple to the coil because the coil is too short?
I would guess that the elliptical movement has little effect because the axial movement of the string is the only one really picked up to any degree - unless the cross mode robs the string of some of its amplitude at large amplitudes.
Whatever the effect is, it is quite large.
Arthur
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Post by ms on Jun 8, 2020 13:19:00 GMT -5
I can see that I made an error in the Spice schematic for simulating the effect of the string magnetisation. Instead of treating the magnetisation as a multiplier of the differentiated sine output it should have been multiplied by the sine before differentiating. This steepens the slope but leaves the general character the same. With regard to the Hammerstein formula from the paper. The non linear part maps volt seconds to displacement from an origin at 5mm. Volt seconds is, of course, magnetic flux which when differentiated gives the pickup output in volts. The non linear formula, then, seems to be mapping how the flux in the coil varies with displacement of the string from the reference position. There will also be an added fixed amount of flux which is not included since its effect would disappear in the subsequent differentiation.
What I find curious is that the gradient of the curve increases at the extremes for both positive and negative excursions. I can't work out what causes that in such an obviously asymmetrical device as a strat pickup.
Arthur That does seem strange; surely the sensitivity must decrease with greater distance from the pickup, leading to asymmetrical distortion.
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