Spectrum of peak clipping wave

Proof of Law #4 (and Law #3)

Published on September 23, 2019

Copyright © Dan P. Bullard

I really don't spend a lot of time talking about Bullard Laws of Harmonics #4, which says:

When a portion of a sinusoid is removed (such as in clipping), the Harmonic Signature mirrors the harmonic signature of the feature removed from the sinusoid.

There is a link there and it talks about it some but I would like to prove it to you here absolutely, and make a point that will also prove Law #3. I'll get to that later though.

This law was derided by one of the "gatekeepers" at Electrical Engineering Stack Exchange as being so obviously true that it didn't need to be said. Well, that's sort of what he said. He said that it must be true because nothing would make it false, but that was certainly not a sign that he had ever considered that it is a useful thing to know. In fact he claimed that it was not an earth-shattering revelation in the least. Well, maybe for you morons who don't understand Laws #1 through #5 anyway, but in my books, Law #4 has an important role: It tells you what you see removed from a wave when it's clipped or during zero crossing distortion is important, and the attributes of that portion that is removed define what the eventual spectrum of the distorted wave is going to look like. For example, why does a clipping spectrum have that classic humping over and over with a rapidly decaying trajectory as you can see at the top of this article? Because the peak of a wave contains a lot of low frequency energy, and thus, you get a lot of low frequency energy in the spectrum of the wave that had the peak removed. Seems reasonable, right? But then it seems odd. If you amputate someone's arm, they don't suddenly become ambidextrous, do they? No, they lose functionality, they don't gain it. But lop the peak off a sine wave and suddenly the spectrum mimics the behavior of the part of the wave you removed, you get a bunch of low frequency harmonics that belong to the thing you removed. Seems odd, doesn't it? Let's take a look.

Here is the wave, as a cosine (why did I use a cosine?) with the upper peak removed. And next, the clipping itself:

And that's it! You have to leave the DC offset there too, otherwise the large drop in voltage will cause a plethora of harmonics, like you might see in a square wave, and that would muck up the spectrum. Now, for the spectrum of the wave clipping:

Now just so you don't have to keep scrolling up to the top of the article to see it, here is the spectrum of the clipped cosine wave:

See the difference, and the similarity? The one above has a large DC offset in bin 0, and it's red, because it's an Even bin, my color code for Even harmonics. I already explained why that wave has a large DC offset. This lower spectrum has a very small DC offset, but it also has something the one above does not: A 0dB value in the fundamental bin, that would be the first harmonic. But notice in the upper spectrum that the blue line signifying the amplitude in the fundamental is about the same size as the 2nd, 3rd and 4th harmonics, about -38dB. That's because the clipped wave has a large fundamental signal amplitude and the wave clipping has a very small fundamental signal amplitude. It's not very large at all because most of the area is taken up with just a DC offset.

So now, let me ask the obvious question: Why, if I paste this clipping back onto the clipped cosine wave doesn't the spectrum just explode with harmonics, double what we see here in fact? Instead, if we do an FFT on just a pure cosine wave, we get no harmonics at all, save the fundamental, harmonic #1. Come on, think about it! Why would that happen? Because of the phase!

Notice that the phases are all either 0 degrees or 180 (per Daver's Law, a derivative of Bullard Laws of Harmonics #5) and that the two wave's harmonics are exactly 180 degrees out of phase. I'm only showing you the first 42 harmonics, but they are all like that, all the way through the spectrum. So when you clip the peak off a wave, the spectrum of the clipping exactly matches the spectrum of the clipped wave, but the phases of the harmonics are exactly 180 degrees out of phase!

Now, what do you think happens when you paste that clipping back onto the clipped wave? Do the harmonics double and you get a really nasty spectrum? Nope, all the harmonics go away! But why? Because they canceled each other out completely! Not one speck of harmonic energy will be left behind.

Now, if I clip the negative side of the wave, what happens?

And the spectrum?

Well, of course it's the same, even the DC bin has the same value in it, but it's the opposite polarity from the positive peak clipped wave, + instead of -, but the spectrum can't show you that because it only gives you the absolute magnitude in each bin. Don't forget, clipping the top of the peak gives the wave a negative DC offset, clipping the negative peak gives the wave a positive DC offset. Somewhat counter intuitive but true nonetheless.

Notice that, like I said, the positive clipped wave has 180 degrees in bin 0, the DC offset bin, whereas the negative clipped wave has a 0 degree offset in bin 0. But follow it down. Notice that the positive clipped pattern is easy to follow, ten 180s, then eight 0s, then eight 180s, then seven 0s, and so on. But the negative clipped wave has an odder, harder to follow pattern. Why? Because the Even (red) harmonics of the negative clipped wave are the opposite phase of the corresponding positive clipped wave harmonics. Follow it down. Notice that the Odd (blue) harmonics exactly match in both columns, but the Even harmonics are exactly opposite by 180 degrees.

Now, just like we talked about before, pasting the clipped peak back onto the clipped wave and getting no harmonics at all. It's just common sense. And if we clipped both the positive and negative peaks of this wave, would we not see just the Odd harmonics alone?

Yep, exactly the same pattern we saw above (but a little harder to make out because of the missing Even harmonics) but also notice that in addition to the Even harmonics being totally gone, the Odd harmonics are twice as big as they were before. Again, the 3rd harmonic in all the spectra above is around -38dB, but in this one the 3rd harmonic is exactly 6.02dB higher at -32dB. EXACTLY. In fact, every single Odd harmonic is exactly 6.02dB taller than they were in all the other spectra above this one. And why is that? Because the phases dictated that the amplitudes would be additive, but in the case of the Even harmonics, the phases dictated that the amplitudes would be subtractive, and so the amplitudes of the Even harmonics created by the positive peak clipping subtracted from the Even harmonics created by the negative peak clipping. Hence, every single Even harmonic is totally cancelled, which yields Bullard Laws of Harmonics #3:

Even harmonics don't appear in symmetrical distortion because they cancel each other out.

It's easy to see now, is it not?

So, is my assertion of Law #4 "earth shattering?" Well, yeah, because it allows us to understand why Law #3 is true, and also to understand why, when you chop off a high frequency feature of a wave, like the zero crossing, the spectrum of the remaining waveform contains the frequency attributes of the part removed, and if you chop off the slow, lumbering peak of a wave, the spectrum of the left over waveform contains the spectrum of the slow, lumbering peak, lots of low frequency harmonics and fewer high freqency harmonics.

See, it all makes sense. And if it doesn't, well, maybe you just don't have the IQ to understand it. Not everyone is a genius.