In search of genius

I have a tee-shirt that a friend bought me, and I wear it proudly. It says JENIUS which is a joke about the time I sent an email to my boss and colleagues when I invented the Pallette AWG for testing ISO14443 Smart Card chips, kind of a eureka moment for me. In the email I misspelled genius, I was missing the "i." I admit it, I make mistakes, but not like the clowns at MIT, or as I have named them, Morons In Training. On their website, they offer their course-ware where I found this howler in the subject Signals and Systems on the topic of Harmonics.

This is supposed to show the harmonic content of a square wave, but it is totally wrong in multiple respects. First off, the phase of the square wave does not match the phase of the fundamental, the first harmonic! If they shifted the phase of the square wave 180 degrees, they would be closer, but still wrong. The second, fourth and sixth harmonics have amplitudes in them, but when a square wave has a 50% duty cycle, like this one appears to have, there are no even harmonics, they cancel each other out due to the symmetry! Morons!

So I thought, wouldn't it be nice to have an Excel spreadsheet that creates diagrams like this automatically from an input wave? In about an hour I had one working, and it is a marvel to behold. What I did is use my standard sine wave distortion template, then I created a new column for each harmonic, adjusting the amplitude from the FFT's amplitude value and adjusting the phase with the phase value from the FFT. The result is simply fabulous! Here is a composite plot with a wave that has 10% chopped off the top and bottom.

At the bottom is the distorted wave, above it in red is the fundamental, which is the predominant wave in this signal. Next is the second harmonic, but it's zero, and hopefully you remember why. Because when the distortion is symmetrical, like this wave, the even harmonics cancel out. Next is the third harmonic, and notice it is in phase with the fundamental, which flattens out the peak of the resulting wave on the top and bottom. Then the fourth, again, canceled out due to the symmetry, then the fifth in green, but notice that this wave is out of phase with the fundamental. I'm not making this up folks, like the "fine people" at MIT, these waves, their amplitudes and phases come directly from the FFT itself. Next is the sixth, zero thanks to the symmetry, then the seventh, in red. Hmmmm, it seems to be zero as well! Maybe it's just too small to see.... But you can clearly see there is something moving in the ninth harmonic at the top, soooo.... When I first saw this I dove back into my code. I must have made a mistake! Should I don my Jenius tee-shirt again? Let's look at the spectrum just to double check.

Aha! There it is! The seventh harmonic (in blue) is down at -70dB. Don't forget to count by twos, this plot is showing only Odd harmonics because the Even (red) harmonics canceled each other out totally. So thanks to the cosine function in the Bullard Harmonic Solution, (read the book, don't wait for the movie) the seventh harmonic is attenuated so low that in time domain, you can't make it out. But it's there, it's got to be there, at that exact level, else this waveform would look a little bit different. Not much maybe, but it doesn't take much to change the appearance of the wave. And now we can see why we see some movement in the ninth harmonic, because that one came back up to -40dB, a whole lot bigger than the seventh at -70dB. Notice that if we kept going, the same thing would happen at the 17th harmonic as well (don't forget to count by twos). That's the cosine function in the Bullard Harmonic Solution making that happen, it's not an artifact, it's real!

Now, let's try this with asymmetrical clipping, removing just the top peak of the wave. What do we get?

OK, only the top peak is clipped, the negative peak is intact. The fundamental didn't change much (it only changes amplitude or phase, because it is a single tone) but suddenly we see the second harmonic and it appears 90 degrees out of phase from the fundamental (1st harmonic). Realize that if we had restored the positive peak and left the negative peak chopped off, the second harmonic would be 270 degrees from with the fundamental, meaning that it's 180 degrees out of phase with this version, so if you chop off both peaks, both versions of the second harmonic cancel each other out. Then there is the third harmonic, then the fourth, in blue, then the 5th, in green, the sixth (in purple), which has some movement on it, but again the seventh (in red) is dead flat! What's going on?

Ah, there it is, the seventh harmonic at (coincidentally) -76dB. Think about that for a minute. I make this point in my books, and it's a point that is well worth the purchase price of the books and here you are getting it for free. When the distortion was symmetrical, the seventh harmonic was hard to see in the time domain plot because, as we saw in the frequency domain plot, it was 70dB below the 1Vpeak reference value. But with only one peak chopped off the seventh harmonic is 76dB below the reference value. What's 76-70? 6dB. So the seventh harmonic is exactly 6.02dB below what it was when the distortion was symmetrical, because one half of the distortion is missing. That's not all. Look at all the other Odd (blue) harmonics, they are ALL exactly 6.02dB lower in the asymmetrical example than they are in the symmetrical example! That is the meaning of Bullard Laws of Harmonics Law #1, "Harmonic amplitudes are proportional to the area of the distortion." If the distortion decreases by half, the harmonic amplitudes go down by 6dB, and not just 6dB, but 6.02dB, the exact number for a factor of two. But wait, what about the Even (red) harmonics? Ah, they obey Law #3. "Even harmonics don't appear in symmetrical distortion because they cancel each other out." So you can't really look at the Even harmonics the same way. They get canceled out with any symmetry, which in some ways once again proves Law #1. Clip the bottom of the wave again and the Even harmonics disappear, because an equal and opposite area was applied to the wave, and while the amplitudes obey Law #1 (the areas were identical) the phases were opposite, so they canceled each other out.

Now, am I being too hard on MIT? Are you kidding me? When a high school graduate and US Navy vet can do stuff like this, and the best they can do is that false and misleading graphic from their course ware, shouldn't I be hard on them? I mean, their graduates will get a job before I will, and I am clearly more qualified for any job as you can plainly see. I think we should just zero out the records, a degree from MIT is worth about as much as a wet paper bag, and as Marvin said in HHGG, "I hate wet paper bags."