Bullard plot of a 4% peak clipped sine wave

How does a spectrum store time?

Published on May 2, 2020

Copyright © Dan P. Bullard

I frequent Quora. I know, it's a sickness. I find it helpful to read questions and try to answer them. It gives me ideas for new experiments and I saw one today that caught my attention.

How is it possible to "remake" a signal by having it's fourier transformation (spectrum of frequency). I mean where is "time stored in that function?"

I know the answer. It's pretty cool, the phase of each harmonic determines where the distortion occurs in a wave. Other answerers had the same general idea, but I can tell that they only know that in a general sense. Like this guy who used this example to show how you can go from time domain to frequency domain and back again:

The problem is that that hand drawn squiggly wave is not the summation of those three harmonics, that's pretty obvious. I don't see what's so hard about using Excel or other math tools to create a wave from harmonics. Obviously they can't see it the way I do, so let me show you how I found a way to prove it.

What I did is I took one of my old Excel files and reworked it and clipped a sine wave at the peak. Here is the Bullard plot:

Notice on the left side I show you a legend of the harmonics that attempts to line up with the harmonics displayed to the right, along with their starting phase angles. The values are color coded too which really helps! Notice that the fundamental is at 0° phase, the 2nd harmonic is at 90° phase, the 3rd harmonic is at 0° phase, the 4th harmonic is at 270° phase, and so on.

So next what I did is I created some code to flip the phase of the even harmonics only. I added 180 degrees in Excel, then I used the MOD() function to reduce the value of any phase that went over 360, so 0°+180° = 270°, so that's alright, and 90°+180° = 270°, which is also OK. But 180°+180° = 360°, so that gets changed to 0°, and 270°+180° = 450°, so that gets changed to 270°. Again, I do this only on the even harmonics, the odd harmonic phases are left alone. So now we get this:

Now you can see that instead of the positive peak being clipped, the negative peak is now clipped. I moved the distortion in time! How did this happen? How was the concept of time transmitted to the waveform? Look at the phases on the far left again. The fundamental is still 0°, but now the 2nd harmonic is 270°. The 3rd is still 0°, but the 4th is now 90°. And look at the harmonics near the positive peak! Now the even harmonics are no longer helping the odd harmonics flatten the wave, they are opposing them! However, because of the phase change the even harmonics are now falling into phase with the odd harmonics over the negative peak, which flattens the negative peak rather than the positive peak. Remember, no other part of the spectrum has changed. The amplitudes were untouched, only the phases were changed just as I described.

After writing this article I played around some more and came up with this animation where I move the distortion forward and backward in time in stages. I start by flipping the phases of the higher harmonics and then flip the phases of the lower harmonics a few at a time until the very lowest even harmonic is flipped.

So what transmits the concept of time to the spectrum? The much overlooked and often forgotten phases of the harmonics. We moved a distortion in time, from the positive peak to the negative peak, just by manipulating the phases of the harmonics. Once you have control of the phase, you can move any feature of a wave to any location in the wave. You just have to remember that the phase is there, working in the background. Even if you can't see it on the spectrum analyzer, it's there. Einstein said it best:

If you can’t explain it simply, you don’t understand it well enough. – Albert Einstein