Bullard plot of an impulse

Which came first?

Published on June 30, 2019

Copyright © Dan P. Bullard

When I talk harmonics to my wife, being a farm girl she always brings out the Chicken or the Egg question. Which came first? We know now, thanks to the evolutionary record that the egg came first, all the way back to fish, then amphibians, then dinosaurs, then birds. But still, sometimes you have to ask, which came first, the signal or the harmonics? Most engineers believe that signals make harmonics, which we can see on a spectrum analyzer. But I think that harmonics come first, harmonics make the signal, and without them, there is no signal. Let me prove it to you.

At the top of this article you can see the first 36 or so harmonics of an impulse in a Bullard plot. Fewer probably, because LinkedIn's article editor cropped the picture a little, but you get the idea. You can see that those harmonics all conspire together to make the impulse by pushing up, each one, a little bit, on the otherwise almost dead flat fundamental at the top (in blue) of the plot. Also, to keep those lower frequency signals from causing the signal to spread out and be wider than we want it to be, the higher frequency harmonics cancel out parts of the lower harmonics, that on this plot appear very wide. If you limit the upper harmonics you not only get a smaller impulse, but you also get a wider signal as seen below.

To make this pitiful little wave I started with the impulse above that goes from -1v to +1v, did an FFT on it, then multiplied all the harmonics except the first 100 by zero, then did an inverse FFT. The inverse FFT produced a new time domain waveform from the harmonics I gave it, and since all but the first 100 harmonics were zero, you can see that not only is there not much amplitude, there is a lot of ringing around the impulse. That ringing is there because the higher harmonics were not canceled out by the even higher harmonics that we were relying on that are no longer there. Now let's see what happens if I allow a few more harmonics to come in before doing the inverse FFT, this time 500.

With 500 harmonics (out of a potential of 1024) I have a lot more amplitude, since each additional harmonic adds just a little more amplitude, but also the impulse is narrower and additionally the ringing is much higher in frequency. Those higher frequencies were supposed to be canceled by the higher harmonics, but we prevented them from appearing, so they couldn't stop the ringing. OK, let's try 1000 harmonics this time to see how that impacts our impulse.

Wow, very nice! With 1000 out of a potential 1024 harmonics enabled, we get almost all the amplitude we were getting originally but we still get some high frequency ringing around the base of the impulse. Let's see what happens if we allow just 20 more harmonics into the mix.

Amazingly just those 20 harmonics really cleaned it up, but it still left a bit of fuzziness at the base of the impulse. Perhaps it actually takes all 1024, or just 4 more harmonics to make this a perfect impulse wave.

Yep, that did it! The time domain wave has 2048 samples, and if we make sure it gets all 1024 harmonics, we get a perfect wave!

So why does every semiconductor maker, every circuit board manufacturer and every electronic design engineer strive to increase the bandwidth of their circuits? To make sure that the signals they are trying to create don't lose their integrity by losing harmonics, because it's the harmonics that make the wave, not the other way around. The egg came before the chicken and the harmonics came before the time domain signal, no doubt about it.