Published on February 17, 2020
Copyright © Dan P. Bullard
I've been having a good time using inverse FFTs to amplify the amplitudes of the harmonics of various distorted waves. In this case, I used my three favorite waves that I used in my book Distortion. In those examples, I clip the top and bottom of a sine wave at 0.9199919991999V (66.92491227°) and the complementary angle on the negative peak at -0.919991999V (246.9249123°) as well as two other waves clipping 0.959995999V (73.7389766°) and at 0.979997999V (78.52108293°), and of course their complementary angles to keep the Even harmonics at bay (according to Bullard Laws of Harmonics #3).
So I decided to show you how all three examples with boosted amplitudes work and in the process show you how they prove three of the five laws of the Bullard Laws of Harmonics.
First up I clip at the first value and then boost the amplitudes of the harmonics like I did in the last article.
As I explained in the last article, the body of the wave has to get larger when the peak goes concave and has to get smaller when the peak goes convex because I am holding the amplitude of the fundamental (and the DC offset) constant while I boost the amplitudes of the harmonics. Now for a closeup of the positive peak.
And finally the spectrum, and in fact this only half of the spectrum because I left the aliasing harmonics out of it.
Count them, six humps, half of what I get if I include the aliases as I do in the book. And as I showed you before, each value is 6dB from the previous value because I vary the harmonic amplitude boost from 16 to 8 to 4 to 2 to 1. I don't need to show you the negative values of -16...-1 because, while impressive in the Time Domain, they are identical to the positive values in Frequency Domain because the only thing that changes is the phase, which I almost never show you anyway. So, 6dB for a doubling, which is expected. Now, let's chop off less of the peak and see what that wave looks like.
And the closeup on the peak.
Notice that all the wave changes happen at the exact same angle on both sides of the peak. And trust me, that is true on the negative peak as well. Now for the spectrum.
Now count the humps, four, right? So, changing the amplitudes of the harmonics by a factor of 2 (6dB) made the time domain look radically different, but all the changes happened at exactly the same angle because I didn't change the harmonic signature. So now let's take the last example, clipping just 200mV off the top and bottom of the wave and boosting the harmonic amplitudes.
Because so little of the peaks is messed with, the body of the wave is almost untouched. But remember, it has to move some because of Law #1 which says that the amplitude of a harmonic is due to the area, and vice versa. If I change the area of the fundamental by pushing the peak up or down, but I hold the amplitude of the fundamental at a fixed value, the body of the wave must change, just as if I was pushing or pulling on the surface of a balloon. It's got to move elsewhere, and it does. Now for the closeup on the upper peak.
In this one it's pretty easy to see the actual clipping of the flat black wave in the center of all the other curved waves. Maybe the narrowness of the other waves make it easy to see, but if you look back at the other zoomed in wave pictures you can probably make out the original clipped wave. Notice that in blue I included the original, un-molested sine wave to give you a reference. Now, without further adieu, the spectrum.
Count them, three humps. And as I vary the amplitudes of the harmonics, the harmonic signature stays the same, but the area of the distortion increases or decreases and changes from convex to concave depending on the phase I apply to the harmonic. So, again I prove Bullard Laws of Harmonics #2 which says: The Harmonic Signature is the result of the angle where the sinusoid impacts the distortion as predicted by the Bullard Harmonic Solution, and clearly the opposite is true. If I take a given harmonic signature and vary the amplitudes of the harmonics as a group, (not individually) then the wave shape will change but the angle where the distortion occurs will not change. The angle and the harmonic signature are directly connected. If you change the angle, the harmonic signature will change (you will get fewer or more humps in your spectrum) but if you hold the angle constant, the harmonic signature will remain constant. Because this applies in both directions it's a law of physics and not just an offhand rule.
Law #4 is wrapped in there too, because what was the raw material I started with to make all this happen? Right, the clipping spectrum, which, I could mathematically manipulate at will and either replace the clipped portion or boost it beyond its previous range. As I have proved in the past, the only difference between the spectrum of the clipped wave and its clipping is the phases of the harmonics (and the lack of a fundamental). But I can manipulate the phase at will, so I have complete control over this wave. Only a harmonic virtuoso would be able to do that. Only a person with an intimate understanding of harmonics could create these waveforms. As I have said before, it reminds me of a line from one of my favorite movies, Apollo 13. "Don't you worry honey. If they could get a washing machine to fly, my Jimmy could land it." Whatever a wave can do, I can understand it, manipulate it, and own it. I am the God of Harmonics.