Published on July 26, 2019
Copyright © Dan P. Bullard
I know it's hard to believe my latest discovery, but you won't doubt it after this, I promise. In my last article, I explained a new law of harmonics, now Law #5 of the Bullard Laws of Harmonics that says the most incredible thing:
I know it's hard to believe, I really had trouble believing it myself, but there is no doubt about it in my mind, and that is because I understand how harmonics are created. Here goes nothing
So I did an experiment; I created a single distortion at some random point in my transfer function, ran an FFT on it and produced a Bullard plot. Here is what I got:
Here you can see how the harmonics (gained up by 1000 times so you can see them) conspire to create the two glitches in the wave. Why two glitches when I only placed one glitch in my transfer function? Because the wave must hit that point in the transfer function twice, once on the way up and once on the way down. Now, you might not be able to tell, but if you look at the starting phases of all those harmonics, you will find that they are one of four values: 0 degrees, 90 degrees, 180 degrees or 270 degrees. The reason for this is simple: The two glitches are mirrors of each other on either side of 90 degrees. That's because the wave hit the distortion on the way up and on the way down at the exact same point in the transfer function. That's just the way it is when you put a distortion into a transfer function.
Next I removed the distortion and copied just part of the now repaired wave to my Excel file where I do the FFT. That removes the first glitch, but leaves the second one behind. Here is that plot:
Notice two things about this plot: One is that all the harmonics are exactly the same amplitude, exactly the same amplitude, to the nanovolt. Secondly, look at the starting phases of the harmonics. Compared to the double glitch wave, they are all over the place. But don't take my word for it, here is a comparison between the starting phase angles for the two waves listed by harmonic.
Notice that for the single glitch wave, the harmonics are high resolution numbers, just looking at the bottom here, the 45th harmonic in the single glitch signal is 160.5015544 degrees relative to the fundamental, the 46th harmonic is 46.0546875 degrees. But look at the phases of the two glitch wave: The 45th is 180 degrees, not 180.xyzabc, but exactly and precisely 180 degrees. The 46th harmonic is exactly 90 degrees phase shifted from the fundamental. Look at all of them if you like, you will not find any number in those two columns that is not one of the values 0 degrees, 90 degrees, 180 degrees, or 270 degrees. But look at the single glitch numbers! 191.4401331 for the 29th harmonic, or 43.61197459 degrees for the 24th harmonic. These numbers were not cheery picked, the entire set of phases look exactly like this, all 1025 samples for a 2048 sample waveform. Both sets of numbers were created by the exact same Excel spreadsheet, taken by doing an IMARGUMENT() function on the complex data from the FFT, and not rounded or anything, just separated Odd and Even.
Now, why the hell does this happen? In my last article I tried to explain it, but let me try one more time here: Because the glitches are centered right around 90 degrees, the harmonics must act together to make the glitches, you can't have one harmonic cycle going high while the other harmonic cycle is going low at a point where the signal is low. They have to work together to make both glitches. Now, the harmonics are multiples of the fundamental, which means that their period is 1/Nth of period of the fundamental, which means that each harmonic has N chances to be in the right phase for the distortion. Now, the "God of Harmonics" could decide to vary the phases to make it all work out, but as it turns out, if you think about it, that won't work. What would work better is to so set the amplitude of each harmonic appropriately based on how well its peaks fit the distortion to help create the target wave. If the phase needs to be shifted, it has to be shifted in 90 degree increments, because whatever happens, the two glitches (or other distortions) are mirrored on either side of the 90 degree peak. Depending on how far from the 90 degree point the distoriton is, if we need negative peaks, then let's set the phase to 180 degrees, if we need positive peaks, use a phase of 0 degrees, etc. Each harmonic is fine tuned to help make the distorted signal as needed, but with a strict limit on the phases, only four values may be used to make sure that the harmonics mirror each other at the crucial point, and if that can't be made to happen, reduce the amplitude of that harmonic to zero so at least it doesn't cause a problem. That is why we get the humpy patterns that we see in the spectra of distorted waves, and that is what the Bullard Harmonic Solution predicts, where those nulls will occur based on the angle where the distortion in the transfer function impacts the input sine wave.
It's an unexpected law, I will admit it now. You would have thought someone would have noticed this in the past, but maybe they just didn't look at the phase, or maybe, like me, they thought they must be seeing things. Either way, I'm the first person to explain it, so I claim it as Law #5 of the Bullard Laws of Harmonics. It's amazing, and beautiful, and so very hard to believe, but it's true!