Post by ms on Jul 20, 2020 12:35:58 GMT -5
Summary of the process:
Sample and store the signal from a picked string.
Integrate the signal to get the displacement.
Input the signal to a non-linear function and get a modified displacement.
Differentiate the result to get a modified velocity.
Compare the spectra of the velocity signals (before and after the non-linear function is applied).
First consider the non-linear function. In order to simulate the effect of a B field that changes with distance from the pole piece it needs to have a “gain” that falls off with positive string displacement. This could be characterized by a graph of displacement out versus displacement in. I prefer to use the slope of that curve, kind of a gain. The function used is the square root, after adding a constant to the input, and using scale factors before and after as necessary. The first plot shows this function; the gain varies by about a factor of 2.5. This corresponds to variation in B of 2.5 as the string moves from the most negative position (closest to the pole) to the farthest position during vibration. The displacement values are arbitrary, and a constant is added before the square root operation in order to get this curve.
The orange curve on the next plot shows the pickup output early in the excitation just after the pick releases the string. (The pickup output should already have some non-linear effect, but this does not mean we cannot add more.) This is a single coil type bridge pickup, E6 string, picked just on the neck side of the pickup. The pickup has a resonant peak and so its output differs from the velocity at the higher frequencies. (The blue curve is after the distortion is introduced, to be discussed below.) The next plot shows the same curves after about four seconds when the signal has decayed considerably. The orange and blue are almost identical, meaning that there is essentially no distortion (and that the gain in the software has been carefully adjusted).
It is necessary to integrate the measured velocity (pickup output) in order to get the displacement. This integration greatly increases the low frequencies in the beginning of the signal as a result of the string motion induced by the picking. Most of this can be eliminated by starting the analysis when the string is released. However, there still is still the matter of the constant of integration. This does not seem to be a problem because a constant is added to the signal as part of the input to the non-linear function generator. However, my sampling device is not dc coupled, and so the level introduced at the start of the string vibration becomes a decaying exponential as the coupling capacitor charges/discharges. I have subtracted out this exponential after finding its parameters using trial by error and visual inspection. It did not seem necessary to write a fitting program.
The result of integration of the sampled pickup output (velocity) is the orange line in the next plot. (The blue is result of applying the non-linear function to the data shown in the orange curve.) The first msec or so is the result of the combination of the picked wave passing over the pickup and its reflection off the bridge, which is not far away. At about 12 msec we see the combination of that reflected wave and the original wave heading towards the nut after it reflects from the nut and passes over the pickup. And so on.
The distorted displacement is stretched in the negative direction significantly, but there is not so much difference otherwise. The distorted velocity shows that the large peaks have increased in size. The final plot shows the spectrum. (It is in the next post) Power in the lower harmonics relative to the fundamental has increased. Surprisingly, power in some of the higher harmonics has decreased.
I would judge that the nonlinearity has a significant audible effect, but it is not an essential part of the electric guitar sound.
