Passive Intermodulation characterization

[blademaster2]

Hello,

I have been impressed by the rigor of the testing and characterization that many of you have performed on pickups. As I understand it these have dealt exclusively with the frequency response, usually looking for a pronounced peak response before rolling off, and I infer from the various posts that the higher the peak the better (or more desirable) the pickup is in many people's view.

Has anyone been performing any testing on the linearity of the response over frequency as a function of stimulus amplitude?

Peak response to a sinusoidal stimulus is one very interesting and important attribute, but the tonal response of a pickup must be at least somewhat influenced by the amplitude/magnetization linearity (more likely the non-linearity, given the metals used for pickup magnets) and hysteresis. These non-linear effects would introduce additional harmonics. Many guitarists will mention the bass and mid response of pickups, and whether pickups are warm or edgy (and I am sure I am missing a large number of the descriptors in this sentence). I expect that these characteristics are strongly influenced by pickup response linearity (probably over frequency if it varies, but I do not know that off-hand), and the resulting tonal characteristics will clearly be a factor of how strong the player picks the string and of the pickup height.

Hysteresis might be harder to test for a continuous-wave vibration source using the method I have seen being used here, but non-linearity and the resulting introduction of harmonics seems plausible especially where the stimulus coil is 'air-core' and therefore linear at the input. 

Just a thought ....


P.S. If this has been done extensively and I missed it ... please accept my apologies (and please also show me where this is located)

[stratotarts]

In the frequency domain:

There are three recognizable regions of the pickup response - the low end which is always extremely "flat", the peak which depends not only on pickup construction but damping, and the rolloff which is a natural result of the RLC response of the pickup. To your question, the low end is both theoretically and practically linear, to the point where it can be taken as a reference level. When I plot various pickups, the expected level between 100Hz and just before the pickup resonance, is less than 0.5 dB or so.

The amplitude of the resonant peak, compared with the rest of the signal, can be tweaked as desired in various circuits and guitars.

In the time domain:

Have any practical experiments been done? I don't know. But consider that the result of non-linearity would definitely be visible in the frequency domain also, because of the presence of harmonic overtones indicating distortion. But those typically haven't been noticed.

The hysteresis part you might explain further, as it is mostly believed that the threshold of any magnetic hysteresis is far greater than any string signal. That is based on existing measurements of magnetic hysteresis of materials that designers use for many purposes besides this one.

It's actually a simple test that you could do - inject a clean fundamental sine wave with an exciter coil. Look for harmonics. Try a few frequencies and plot if desired. But I don't think you'll actually see much there.

Further, there may be a distinction between hysteresis and non-linearity, however in practice hysteresis inevitably introduces non-linearity and therefore harmonics in the frequency domain.

[blademaster2]

Understood.

So if the low end is flat and invariant between pickups, then we conclude that any guitarist who mentions low end response of a particular pickup is only able to hear a different in the mounting stiffness of the pickup into the guitar and nothing about the pickup design itself. I can accept that.

If the peak is only a theoretical RLC resonant peak, then again there are not many differences between pickups except for the amplitude of this peak and its width, which is influenced by the coil wire resistance (damping, as you said). The magnet material and how much of it is present has an effect on eddy currents, and that effects the peak amplitude through damping - but nothing else ... or is that wrong, and if so what other response would be affected?

The roll-off will always be the 40 db/decade of a theoretical RLC circuit.

That means that only difference, and the only figure of merit (if I can call it that) of any pickup, is the amplitude and shape of the peak. We all ignore the effect of the body and mounting in these characterizations (and I will not spark another tonewood debate here, especially since the pickups are tested on the benchtop).

So why are there so many pickups on the market, and so many perceived differences between them?

To me there still seem to be other factors coming into play. If you look at hysteresis plots of magnetic materials there are few regions where it looks even somewhat linear but I would expect any small non-linearity will introduce harmonics that may not be easily detected on a time domain or frequency domain plot. They would be there, but I disagree that we should be able to see this distortion on a time domain plot. Similarly a frequency domain plot (bode plot) only looks at amplitude response to pure swept sinusoidal input and ignores distortion and other harmonics. Human hearing detects attributes in a waveform that human eyes looking at instrument displays cannot. I am not referring to saturation/desaturation of the magnet material, which I agree likely requires a much stronger field waveform than we see in a guitar, however being permanent magnets means that the magnet material has already been driven pretty far up this magnetization curve and remains there, where it is certainly not a linear portion of the curve.

Perhaps your suggestion of introducing a signal of a very pure sinusoid on the exciter coil and looking at the spectral response of the pickup at that one frequency, other frequencies could be seen due to any non-linearities. This would also, I expect, be different depending on the amplitude of the exciter coil input so it would need multiple amplitudes for each frequency tested. If this showed nothing visible on the analyzer it does not convince me that harmonics produced by the material are not audible - non-linearities are producing harmonics whether or not we can easily see it in instrumentation.

[antigua]

The reason people say "this pickup has a lot of bass" or "this pickup sounds compressed" even though a pickup has no compression and they all have the same bass is because the pickup is merely a filter, and those other adjectives are born from describe the end result of the source signal having been filtered.If you take a look at GuitarFreak  guitarnuts2.proboards.com/thread/3627/guitarfreak-guitar-frequency-response-calculator , long time forum member JohnH has created a modeling program through advanced use of Excel, and it attempts to model both factors of the equation, the pickup filtering as well as the source signal from the moving guitar strings. 

The problem is that while the pickup may stay the same, this source signal can vary from guitar to guitar, and the combination of the two is a dynamic sum that changes from one guitar to the next.

Another problem is that when people say they hear certain qualities from a pickup, it might actually be the case that they hear many things, but that they've been conditioned by having read information from the seller, or other people on the internet, to observe specific qualities of the pickup, and the fact that the overall timbre is so rich, and the adjectives so ambiguous, that it makes it possible for many adjectives to simultaneously be true. For example, saying a Little 59 sounds like a full sized '59. On a technically level, they're completely different, but they share enough in common to serve a marketing purposes.

Long story short, I've used people's claims as a basis to investigate pickups, and I've often discovered that what was being claimed had no physical basis in science, despite rigorous attempts to relate the claims with actual science.

[aquin43]

Even if a pickup had a strong non linearity due to some peculiar hysteresis curve in the string (not in the pickup itself since the fractional variation in flux in the pickup structure caused by the moving string is minute), the effect would be confined to a slight alteration in the harmonic structure of the string waveform. Intermodulation between the strings would be negligibly small since the interaction between the strings is small. They behave more or less as individual channels summed in the coil.

The frequency response is a matter of interpretation. A shelving step of a few dB above 2kHz, for example, which is fairly common in pickups with conducting covers, can be interpreted as a rise in the bass.

When you start modelling pickups it becomes apparent that some simple filtering could easily turn one pickup into another. The proliferation of pickup models, most of which from the point of view of string sensing are just minor variants on one or two old designs can be put down to the limitations of amplifier tone controls coupled with the common gut feeling that the only natural and honest way to achieve a particular sound is by tweaking the pickup parameters.

Arthur

[blademaster2]

Okay, I guess I will never obtain consensus from the forum on any non-linearity becoming influential in the tone of a pickup. 


So if the wood makes zero difference (a statement that I do not personally believe), and pickups are such simple gadgets that they essentially only exhibit the frequency response characteristics of a theoretical RLC circuit loaded by a guitar pot and capacitive cable, then any of the differences between all solidbody guitars should be able to be synthesized by adjusting the parameters of an active filter. But unless a magnetization curve can be produced and to show on it where the permanent magnet strength of the pickup sits, and the excursions in the emanating field from that magnet due to string vibration from that highly-biased point in the curve can be shown to be locally very linear in the generation of the electrical signal, I will have difficulty accepting all of that.

Personally, with all of the guitars I own and have played it does not feel nor sound as simple as that to my senses.

I can, however, accept that with the myriad of factors in a guitar, from wood and body construction to scale and strings and pickups, nothing alone can be claimed to dominate the resulting tone.  No instrumentation or simulation will be convincing to me (I could not tell the difference between a celtic harp compared to a guitar if I were to look only at waveforms and spectral measurements).

So, to summarize, I understand that there *have* been experiments performed to detect the production of harmonics due to PIM from pickups and that they indicated that the pickup response is essentially perfectly linear - at least as far as the instrumentation can show. Is that correct?

[ms]

Jun 4, 2020 16:38:28 GMT -5 blademaster2 said:

I could not tell the difference between a celtic harp compared to a guitar if I were to look only at waveforms and spectral measurements.

The altering of the relative strengths of harmonics caused by sampling the string at one or a few narrow regions with one or two coils or multiple pickups  has a definite audible effect and can also be observed in a spectral measurement.  This differentiates an electric guitar from an acoustic stringed instrument.

I agree wood makes a difference.  The vibration of a string, what the pickup senses, is affected by its environment, and also there are various ways to enhance this effect.

There is modeling that shows that non-linear effects can modify the ratios of harmonics.  That must change the sound some, but is it really enough to matter when there are other ways that make larger changes?

Also pickups with strong eddy current effects have an additional degree of freedom in the frequency response compared to those with small or no eddy current effects.  With eddy currents, the damping is a function of something like the square root of frequency, and so the shape of the broad resonant peak is affected.  I do not know if this makes a practical difference.

[antigua]

Jun 4, 2020 16:38:28 GMT -5 blademaster2 said:

Okay, I guess I will never obtain consensus from the forum on any non-linearity becoming influential in the tone of a pickup. 


So if the wood makes zero difference (a statement that I do not personally believe), and pickups are such simple gadgets that they essentially only exhibit the frequency response characteristics of a theoretical RLC circuit loaded by a guitar pot and capacitive cable, then any of the differences between all solidbody guitars should be able to be synthesized by adjusting the parameters of an active filter. But unless a magnetization curve can be produced and to show on it where the permanent magnet strength of the pickup sits, and the excursions in the emanating field from that magnet due to string vibration from that highly-biased point in the curve can be shown to be locally very linear in the generation of the electrical signal, I will have difficulty accepting all of that.

Personally, with all of the guitars I own and have played it does not feel nor sound as simple as that to my senses.

I can, however, accept that with the myriad of factors in a guitar, from wood and body construction to scale and strings and pickups, nothing alone can be claimed to dominate the resulting tone.  No instrumentation or simulation will be convincing to me (I could not tell the difference between a celtic harp compared to a guitar if I were to look only at waveforms and spectral measurements).

So, to summarize, I understand that there *have* been experiments performed to detect the production of harmonics due to PIM from pickups and that they indicated that the pickup response is essentially perfectly linear - at least as far as the instrumentation can show. Is that correct?

The relationship between the guitar string and the magnetic fields are non-linear, and that's not only because the string is moving while the field is static, but also because if the magnet pulls harder on the string, tension upon the string becomes asymmetrical, and that's yet another non linearity, but the point of focusing on RLC characteristics is because, of all the parameters can be defined, it's what most sets different pickups apart in the market place. The fact that one pickup uses AlNiCo 5 and another uses AlNiCo 2 is a triviality if their inductance also varies by more than, say 10%. 

As far as the prominence of wood, let's say it does make a difference, and then let us also acknowledge that few people ever buy a guitar thinking about how the wood will impact the tone, but they do think about how the pickups will impact the tone. Talk about the impact of the wood seems akin to navel gazing. 

[blademaster2]

Hello again,

I will ignore the "naval gazing" comment, but I will note that I have myself heard differences (albeit subtle ones) when experimenting with different body woods in the most controlled manner I could achieve. No one can convince me that I could not hear a difference, and I have no agenda here.

I found and read the following paper: 

NON-LINEAR IDENTIFICATION OF AN ELECTRIC GUITAR PICKUP, by Antonin Novak et al, published in Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016. In the paper some experiments and measurements were made that showed, through simulation that matched experimental results, that the identification of pickup non-linearities was reasonably well modeled using a simple Hammerstein representation. This model uses a non-linear element followed by a linear filter element.

In this paper, the authors state that:

" ...the magneto-electric conversion
done by the pickup is modeled using static non-linearity
followed by a simple derivative. The static non-linearity
represents the non-linear relation between the string displacement
and the magnetic flux which can be evaluated using computer simulations
and implemented as an exponential or N-th order polynomial
."

Regarding their experiment, the results of which the paper shows matched their Hammerstein representation model rather well, the authors noted the following:

"As
expected, the string displacement is distorted just after the excitation.
It becomes less and less distorted as the harmonics of higher
orders fade with time. The output signal of the pickup exhibits the
same kind of behavior with time. One can notice that the output
voltage is more distorted due to pickup non-linearities.
 The displacement signal measured with the vibrometer is then
used as the input of estimated parametric Hammerstein model of
the pickup and both measured and synthesized pickup outputs
are compared on the same graph .... The difference between
estimated and experimental signals is plotted .... showing
that the model succeeds in describing the non linear behavior of
the pickup when used in realistic conditions."


This is the basis of my question originally, and finding that paper seems to answer it - there *has* been work done in this area, but evidently not within this forum. Interestingly the authors used a Seymour Duncan single coil pickup for their experiment, which was what I used in my aforementioned home experiment with body woods.

I can assert that I am not alone in saying that pickups respond non-linearly, and unless the cause of this non-linearity is identical for all pickup designs, which I highly doubt, then it is quite reasonable to expect different pickup designs to introduce harmonics in slightly different ways and therefore to sound different. I am prepared to accept the possibility that these differences are subtle.

[ms]

Jun 4, 2020 18:59:19 GMT -5 blademaster2 said:

it is quite reasonable to expect different pickup designs to introduce harmonics in slightly different ways and therefore to sound different. I am prepared to accept the possibility that these differences are subtle.

You have not shown that the non-linearity has any significant effect on the sound.  Seems premature to suppose that there might be differences between pickups.

[antigua]

Jun 4, 2020 18:59:19 GMT -5 blademaster2 said:

This is the basis of my question originally, and finding that paper seems to answer it - there *has* been work done in this area, but evidently not within this forum. Interestingly the authors used a Seymour Duncan single coil pickup for their experiment, which was what I used in my aforementioned home experiment with body woods.

I can assert that I am not alone in saying that pickups respond non-linearly, and unless the cause of this non-linearity is identical for all pickup designs, which I highly doubt, then it is quite reasonable to expect different pickup designs to introduce harmonics in slightly different ways and therefore to sound different. I am prepared to accept the possibility that these differences are subtle.

I don't mean navel gazing in a bad way, I ponder it often, but being perfectly honest, I think on my part, and others part, it's motivated by a wish for their to be a difference, and not the likelihood of their being a difference. 

I don't know the specifics of their experiment, but there is a known reason that the harmonics dacay more quickly than the fundamental; the rigidity of the guitar string. The higher harmonics are physically more narrow than the lower ones. The string is effectively more stiff from the perspective of those narrow harmonics. Some of the energy gets transferred to the lower harmonics, and some is lost as heat. Maybe their experiment is talking about something else, I don't know.  

Lots of things make a difference, but not everything can be heard. For every ten observed difference, on average, only one of them is genuinely audible. The threshold of human hearing is about 1dB difference in quiet conditions, 2dB in normal conditions, and higher in the midst of cacophony. What makes this sort of investigation difficult, such as determining the significance of pickup microphony, is not discovering whether it happens at all, but discovering if it's audible. That's what makes the wood inquiry so unlikely, being familiar with what sorts of changes result in what sorts of amplitude changes, and realizing that the odds of wood type affecting >1dB changes somewhere along the functional frequency range, all else being equal, is very low.

[blademaster2]

Jun 4, 2020 19:23:47 GMT -5 ms said:

Jun 4, 2020 18:59:19 GMT -5 blademaster2 said:

it is quite reasonable to expect different pickup designs to introduce harmonics in slightly different ways and therefore to sound different. I am prepared to accept the possibility that these differences are subtle.

You have not shown that the non-linearity has any significant effect on the sound.  Seems premature to suppose that there might be differences between pickups.

No - in fact I have not myself shown anything, and I am not making any claim to defend. The point is that harmonics are much smaller in amplitude than the fundamental (this is something I *have* seen myself) and that is where the difference between a guitar and, say, a trumpet exist if playing the same note. So if a difference is observable in time domain as the authors indicated then I would expect the difference in tone to be audible. In fact given the characteristics of sound waves that I have seen on oscilloscopes I would expect to hear differences in tone to be audible even if they were not easily visible on the oscilloscope.

Are you actually asserting that an observable effect of non-linearity in a time domain plot is not likely to be audible in the tone? As I said, I would expect just the opposite.

[antigua]

Jun 4, 2020 22:21:21 GMT -5 blademaster2 said:

Jun 4, 2020 19:23:47 GMT -5 ms said:

You have not shown that the non-linearity has any significant effect on the sound.  Seems premature to suppose that there might be differences between pickups.

No - in fact I have not myself shown anything, and I am not making any claim to defend. The point is that harmonics are much smaller in amplitude than the fundamental (this is something I *have* seen myself) and that is where the difference between a guitar and, say, a trumpet exist if playing the same note. So if a difference is observable in time domain as the authors indicated then I would expect the difference in tone to be audible. In fact given the characteristics of sound waves that I have seen on oscilloscopes I would expect to hear differences in tone to be audible even if they were not easily visible on the oscilloscope.

Are you actually asserting that an observable effect of non-linearity in a time domain plot is not likely to be audible in the tone? As I said, I would expect just the opposite.

I don't understand why you discount magnitudes in this conjecture, asserting that any change, no matter how small, must be audible.

Suppose two pickup makers offer AlNiCo 5 Strat pickups, one with more turns than another, the inductance is different, but that's covered in the RLC distinction that we're discounting from the discussion, what other source of variation exists between these two pickups that you think might impact amplitude with respect to time?

[blademaster2]

Jun 4, 2020 20:59:47 GMT -5 antigua said:

Jun 4, 2020 18:59:19 GMT -5 blademaster2 said:

This is the basis of my question originally, and finding that paper seems to answer it - there *has* been work done in this area, but evidently not within this forum. Interestingly the authors used a Seymour Duncan single coil pickup for their experiment, which was what I used in my aforementioned home experiment with body woods.

I can assert that I am not alone in saying that pickups respond non-linearly, and unless the cause of this non-linearity is identical for all pickup designs, which I highly doubt, then it is quite reasonable to expect different pickup designs to introduce harmonics in slightly different ways and therefore to sound different. I am prepared to accept the possibility that these differences are subtle.

I don't mean navel gazing in a bad way, I ponder it often, but being perfectly honest, I think on my part, and others part, it's motivated by a wish for their to be a difference, and not the likelihood of their being a difference. 

I don't know the specifics of their experiment, but there is a known reason that the harmonics dacay more quickly than the fundamental; the rigidity of the guitar string. The higher harmonics are physically more narrow than the lower ones. The string is effectively more stiff from the perspective of those narrow harmonics. Some of the energy gets transferred to the lower harmonics, and some is lost as heat. Maybe their experiment is talking about something else, I don't know.  

Lots of things make a difference, but not everything can be heard. For every ten observed difference, on average, only one of them is genuinely audible. The threshold of human hearing is about 1dB difference in quiet conditions, 2dB in normal conditions, and higher in the midst of cacophony. What makes this sort of investigation difficult, such as determining the significance of pickup microphony, is not discovering whether it happens at all, but discovering if it's audible. That's what makes the wood inquiry so unlikely, being familiar with what sorts of changes result in what sorts of amplitude changes, and realizing that the odds of wood type affecting >1dB changes somewhere along the functional frequency range, all else being equal, is very low.

I, too, have pondered it a lot. Years ago when I built my first guitar I assumed that the wood made zero difference in the tone. It was only when I experienced surprises (I will not go into them) that I became convinced the other way. I was certainly not the victim of confirmation bias. My more recent experiment, much more controlled, again convinced me of a difference due to the wood but it also showed me that this is rather subtle compared to, say, the location/shape/amplitude of the response peak (however from the plots you have generated I observe that once the pickup is loaded by the cable capacitance these differences in peaks become far less pronounced).

Despite being convinced of the subtle influence of body wood on tone, as I have become over the intervening years, I have no skin in this game. Same goes for pickups and the existence or nonexistence of audible non-linearity on tone. This will never be my career, and I am simply looking for answers to better understand our instrument.

To my ears the timbre of the attack varies for different pickup designs, and it sounds more complex to me than a simple frequency response difference. PIM might explain it but I cannot prove this.

On the other hand it may well be shown that the amplitude response of pickups over frequency ends up remaining the only characterization worth pursuing in this forum, however I would suggest folks track down that paper to see the specifics of their experiment (it was a reasonably controlled lab experiment involving two types of stimulus with metal moving in front of the test pickup). The fact that the authors observe (in time domain) and even mathematically model non-linearities of the pickup response tells me that it is worth looking into it further, if only to be complete in the analysis and modeling of pickups. 

[blademaster2]

Jun 4, 2020 22:27:52 GMT -5 antigua said:

Jun 4, 2020 22:21:21 GMT -5 blademaster2 said:

No - in fact I have not myself shown anything, and I am not making any claim to defend. The point is that harmonics are much smaller in amplitude than the fundamental (this is something I *have* seen myself) and that is where the difference between a guitar and, say, a trumpet exist if playing the same note. So if a difference is observable in time domain as the authors indicated then I would expect the difference in tone to be audible. In fact given the characteristics of sound waves that I have seen on oscilloscopes I would expect to hear differences in tone to be audible even if they were not easily visible on the oscilloscope.

Are you actually asserting that an observable effect of non-linearity in a time domain plot is not likely to be audible in the tone? As I said, I would expect just the opposite.

I don't understand why you discount magnitudes in this conjecture, asserting that any change, no matter how small, must be audible.

Suppose two pickup makers offer AlNiCo 5 Strat pickups, one with more turns than another, the inductance is different, but that's covered in the RLC distinction that we're discounting from the discussion, what other source of variation exists between these two pickups that you think might impact amplitude with respect to time?

I do not discount magnitudes - the RLC response is clearly highly influential. I am only suggesting that other effects are being ignored if bode plots are the only analysis and comparison method.

[antigua]

Jun 4, 2020 22:45:07 GMT -5 blademaster2 said:

Jun 4, 2020 22:27:52 GMT -5 antigua said:

I don't understand why you discount magnitudes in this conjecture, asserting that any change, no matter how small, must be audible.

Suppose two pickup makers offer AlNiCo 5 Strat pickups, one with more turns than another, the inductance is different, but that's covered in the RLC distinction that we're discounting from the discussion, what other source of variation exists between these two pickups that you think might impact amplitude with respect to time?

I do not discount magnitudes - the RLC response is clearly highly influential. I am only suggesting that other effects are being ignored if bode plots are the only analysis and comparison method.

As far as pickup testing in general goes, yes we have experiments with real guitar string and test rigs, this is one I did a couple weeks back guitarnuts2.proboards.com/thread/8983/practical-pickup-sensing-area-test . It's assumed, unless there is cause to believe otherwise, that most all pickups of a given construction will have most of these tested characteristics in common. For example, an SSL-1 isn't different enough from a Fender Fat 50 to think they're physically and spatially distinctive.  

As for cataloging production pickups, of which I've documented over two hundred now, I collect R, L, C, a bode plot that indicated Q factor, and the magnetic strength at the top of the pickup. If there is something else to be collected, I'd need to know what that thing is. 

I know that pickup makers, both large and small, do virtually none of this sort of testing. I suspect they keep track of the magnet type, wire type and turn count, and maybe the tension and turns per layer if they're using a computerized bobbin winder, but the RLC values are almost certainly left up to chance. Their business is making pickups, not necessarily understanding them. When it comes times to make a new pickup design, trial and error alone seems to work well enough, they have no need for deeper analysis. Any unidentified X factor, whatever that might turn out to be, would be just as much left to chance as the RLC values. 

[aquin43]

Jun 4, 2020 18:59:19 GMT -5 blademaster2 said:

Hello again,
...

NON-LINEAR IDENTIFICATION OF AN ELECTRIC GUITAR PICKUP, by Antonin Novak et al, published in Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016. In the paper some experiments and measurements were made that showed, through simulation that matched experimental results, that the identification of pickup non-linearities was reasonably well modeled using a simple Hammerstein representation. This model uses a non-linear element followed by a linear filter element.

In this paper, the authors state that:

" ...the magneto-electric conversion
done by the pickup is modeled using static non-linearity
followed by a simple derivative. The static non-linearity
represents the non-linear relation between the string displacement
and the magnetic flux which can be evaluated using computer simulations
and implemented as an exponential or N-th order polynomial
."
...

I have constructed a quick LTSpice simulation of their Hammerstein model using the quoted coefficients. The input to the non linear section is in millivolt-seconds and the output in millivolts. The output is passed through a differentiator scaled for unity gain at omega = 1000. All of the parts are standard spice devices.

If the model is valid, there is definitely serious non linear distortion and even limiting going on at amplitudes above a millimetre or so.

The listing for the .asc file is given below. Cut and paste it into a file called something.asc and run it.



Version 4
SHEET 1 1248 680
WIRE -80 96 -144 96
WIRE -32 96 -80 96
WIRE 192 96 96 96
WIRE 256 96 192 96
WIRE 384 96 320 96
WIRE 576 96 528 96
WIRE 640 96 576 96
WIRE -144 128 -144 96
WIRE -32 128 -32 96
WIRE 96 128 96 96
WIRE 192 128 192 96
WIRE 384 128 384 96
WIRE 528 128 528 96
WIRE 640 128 640 96
WIRE -144 240 -144 208
WIRE -32 240 -32 208
WIRE -32 240 -144 240
WIRE 96 240 96 208
WIRE 96 240 -32 240
WIRE 192 240 192 208
WIRE 192 240 96 240
WIRE 384 240 384 208
WIRE 384 240 192 240
WIRE 528 240 528 208
WIRE 528 240 384 240
WIRE 640 240 640 208
WIRE 640 240 528 240
WIRE -144 256 -144 240
FLAG -144 256 0
FLAG -80 96 m
FLAG 576 96 out
SYMBOL voltage -144 112 R0
WINDOW 123 1 100 Left 2
WINDOW 39 0 0 Left 0
WINDOW 3 32 154 Left 2
SYMATTR Value SINE(0 {lvl} {f0})
SYMATTR InstName V1
SYMBOL bv 192 112 R0
WINDOW 0 -34 8 Left 2
WINDOW 3 -361 191 Left 2
SYMATTR InstName B1
SYMATTR Value v = v(m)*7.50e-2 + v(m)**2*6.75e-3 + v(m)**3*2.11e-3 + v(m)**4*4.75e-4 + v(m)**5*8.31e-4
SYMBOL voltage 384 112 R0
SYMATTR InstName V2
SYMATTR Value 0
SYMBOL bv 528 112 R0
WINDOW 3 14 103 Left 2
SYMATTR InstName B2
SYMATTR Value v=i(v2)
SYMBOL res -48 112 R0
SYMATTR InstName R1
SYMATTR Value 1G
SYMBOL res 624 112 R0
SYMATTR InstName R3
SYMATTR Value 1G
SYMBOL cap 320 80 R90
WINDOW 0 0 32 VBottom 2
WINDOW 3 32 32 VTop 2
SYMATTR InstName C1
SYMATTR Value 159µ15
SYMBOL res 80 112 R0
SYMATTR InstName R2
SYMATTR Value 1G
TEXT -160 376 Left 2 !.tran 40m
TEXT -160 344 Left 2 !.param lvl = 2
TEXT -32 376 Left 2 !.four 200 v(out)
TEXT -144 40 Left 2 ;.param f0 = 1000/(2*pi)
TEXT 32 344 Left 2 !.param f0 = 200
TEXT -216 304 Left 2 ;B1:


Arthur

[aquin43]

Further to the LTSpice simulation, here is a graph of the variation of distortion with amplitude produced by an auxiliary program from the LTSpice logfile after analysis with the added instruction .step param lvl 0.1 2 0.1

Arthur

[ms]

Jun 4, 2020 22:21:21 GMT -5 blademaster2 said:

Jun 4, 2020 19:23:47 GMT -5 ms said:

You have not shown that the non-linearity has any significant effect on the sound.  Seems premature to suppose that there might be differences between pickups.

The point is that harmonics are much smaller in amplitude than the fundamental (this is something I *have* seen myself) and that is where the difference between a guitar and, say, a trumpet exist if playing the same note. 

If that were true, then an electric guitar should would sound like a sine wave with a bit of distortion.  The relative level of the fundamental varies with conditions and can sometimes be quite small on E6 open since the amp-driver-cabinet can attenuate bass a lot.

[ms]

"As
expected, the string displacement is distorted just after the excitation.
It becomes less and less distorted as the harmonics of higher
orders fade with time. The output signal of the pickup exhibits the
same kind of behavior with time. One can notice that the output
voltage is more distorted due to pickup non-linearities...."

Can anyone explain to me how these non-linearities are observed?  Higher harmonics decay naturally with time.  I think it would be difficult to determine the level of harmonics due to non-linearity in a background of decaying string harmonics.

[blademaster2]

As I understood their statement, the authors noted that there was an initial transient from the stimulus (like a guitar pick creates) and after that had decayed then they observed that the remaining generated pickup signal waveform, now with fewer transients and easier to observe, differed from the measured string movement as a result of non-linearities in the pickup response.

The following are my thoughts, and not explicitly stated in the paper nor defendable with any measurements I have made: Assuming that the observed non-linearities are a property of the pickup response that is also present (but harder to observe visually) during the attack transient created by plucking the string, then it seems to me a reasonable inference that this generates harmonics (as any non-linearity would) during the transient, too. This would, to me, account for some of the tone of the pickup and could well be different between pickup designs and magnet materials used.

[aquin43]

Jun 5, 2020 9:53:57 GMT -5 blademaster2 said:

As I understood their statement, the authors noted that there was an initial transient from the stimulus (like a guitar pick creates) and after that had decayed then they observed that the remaining generated pickup signal waveform, now with fewer transients and easier to observe, differed from the measured string movement as a result of non-linearities in the pickup response.

The following are my thoughts, and not explicitly stated in the paper nor defendable with any measurements I have made: Assuming that the observed non-linearities are a property of the pickup response that is also present (but harder to observe visually) during the attack transient created by plucking the string, then it seems to me a reasonable inference that this generates harmonics (as any non-linearity would) during the transient, too. This would, to me, account for some of the tone of the pickup and could well be different between pickup designs and magnet materials used.

Actually, the non-linearity is so large that it can be observed directly in the pluck response of the pickup and it explains a phenomenon that I found couldn't be modelled by any kind of pluck excitation of my delay line model.

Note how, in the following waveform, the top and bottom of the square pulses both slope downward. I could find no pluck waveform that could reproduce that behaviour. This is the initial response of a strat fingerboard pickup.

.

Now look at the output of the Spice version of the Hammerstein model, driven by a PWL signal shaped to represent the string displaced and released at the centre.

It looks to me as if the non-linearity is an essential ingredient.

Arthur

[antigua]

Another source of harmonics in the transient comes from the fact that the test pickup, an SSL-1 it looks like, has discrete pole pieces rather than a blade, so there are doubled harmonics from "horizontal" movement. Those movements will be strongest when displacement is highest, because the ratio between the pole piece width and the string displacement is initially very low, but by the time the string movement comes to a stop, it's moving around in a small, magnetically uniform region, lessening the doubled harmonic distortion. If a blade style pickup is tested against, not of that distortion occurs since the magnetic field is essentially uniform throughout the string's entire displacement. 

[ms]

Jun 5, 2020 9:53:57 GMT -5 blademaster2 said:

As I understood their statement, the authors noted that there was an initial transient from the stimulus (like a guitar pick creates) and after that had decayed then they observed that the remaining generated pickup signal waveform, now with fewer transients and easier to observe, differed from the measured string movement as a result of non-linearities in the pickup response.

The following are my thoughts, and not explicitly stated in the paper nor defendable with any measurements I have made: Assuming that the observed non-linearities are a property of the pickup response that is also present (but harder to observe visually) during the attack transient created by plucking the string, then it seems to me a reasonable inference that this generates harmonics (as any non-linearity would) during the transient, too. This would, to me, account for some of the tone of the pickup and could well be different between pickup designs and magnet materials used.

OK, I can see you might want to avoid the initial transient.  But it seems that they should compare the pickup output with an independent measure of string velocity (a linear measurement) in order to see the excess harmonics generated by the non-linearity.  They mention a vibrometer (not sure what that is), and it seems to measure displacement.  So in order to make a comparison, they have to differentiate the vibrometer output or integrate the pickup output.  I wonder if this can be done accurately enough to get a good measurement of the harmonics due to the non-linearity. 

[ms]

Jun 5, 2020 10:46:12 GMT -5 aquin43 said:

Actually, the non-linearity is so large that it can be observed directly in the pluck response of the pickup and it explains a phenomenon that I found couldn't be modelled by any kind of pluck excitation of my delay line model.

Note how, in the following waveform, the top and bottom of the square pulses both slope downward. I could find no pluck waveform that could reproduce that behaviour. This is the initial response of a strat fingerboard pickup.

.

Does this feature (both downward sloping pulses) decrease in relative importance when you reduce the intensity of the picking?

[aquin43]

Jun 5, 2020 11:36:25 GMT -5 ms said:

Jun 5, 2020 10:46:12 GMT -5 aquin43 said:

Actually, the non-linearity is so large that it can be observed directly in the pluck response of the pickup and it explains a phenomenon that I found couldn't be modelled by any kind of pluck excitation of my delay line model.

Note how, in the following waveform, the top and bottom of the square pulses both slope downward. I could find no pluck waveform that could reproduce that behaviour. This is the initial response of a strat fingerboard pickup.

.

Does this feature (both downward sloping pulses) decrease in relative importance when you reduce the intensity of the picking?

It seems to, but the whole shape of the pulse becomes more rounded as the amplitude of the pluck is reduced. I find it very difficult to make these measurements and get a clean waveform. The wire breaking method would probably be the best, but the thinnest wire I have is too strong.

[antigua]

Jun 5, 2020 12:10:33 GMT -5 aquin43 said:

I find it very difficult to make these measurements and get a clean waveform. The wire breaking method would probably be the best, but the thinnest wire I have is too strong.

That never worked well for me, I got the best results with a pull pin type string release i.imgur.com/YIq1EhW.jpg . There was still some variation from the act of pulling the pin, but if I did it very fast, the resulting wave form was usually pretty consistent. 

[antigua]

Jun 5, 2020 10:46:12 GMT -5 aquin43 said:

Jun 5, 2020 9:53:57 GMT -5 blademaster2 said:

As I understood their statement, the authors noted that there was an initial transient from the stimulus (like a guitar pick creates) and after that had decayed then they observed that the remaining generated pickup signal waveform, now with fewer transients and easier to observe, differed from the measured string movement as a result of non-linearities in the pickup response.

The following are my thoughts, and not explicitly stated in the paper nor defendable with any measurements I have made: Assuming that the observed non-linearities are a property of the pickup response that is also present (but harder to observe visually) during the attack transient created by plucking the string, then it seems to me a reasonable inference that this generates harmonics (as any non-linearity would) during the transient, too. This would, to me, account for some of the tone of the pickup and could well be different between pickup designs and magnet materials used.

Actually, the non-linearity is so large that it can be observed directly in the pluck response of the pickup and it explains a phenomenon that I found couldn't be modelled by any kind of pluck excitation of my delay line model.

Note how, in the following waveform, the top and bottom of the square pulses both slope downward. I could find no pluck waveform that could reproduce that behaviour. This is the initial response of a strat fingerboard pickup.

....

Is there implied causality in this, or is this just confirmation between measured data and an a fitted model?

[aquin43]

Jun 5, 2020 12:53:35 GMT -5 antigua said:

Jun 5, 2020 10:46:12 GMT -5 aquin43 said:

Actually, the non-linearity is so large that it can be observed directly in the pluck response of the pickup and it explains a phenomenon that I found couldn't be modelled by any kind of pluck excitation of my delay line model.

Note how, in the following waveform, the top and bottom of the square pulses both slope downward. I could find no pluck waveform that could reproduce that behaviour. This is the initial response of a strat fingerboard pickup.

....

Is there implied causality in this, or is this just confirmation between measured data and an a fitted model?

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?

[antigua]

Jun 5, 2020 17:18:53 GMT -5 aquin43 said:

Jun 5, 2020 12:53:35 GMT -5 antigua said:

Is there implied causality in this, or is this just confirmation between measured data and an a fitted model?

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.