In the previous video, I showed how turning the crank on this machine generates 20 different frequencies, also known as harmonics, which are turned into cosines here, multiplied by coefficients, then summed, magnified, and the resulting function plotted here on the front. This process is called synthesis, but this machine is called a harmonic analyzer, which means it can be used to solve the much more difficult inverse problem called analysis. For example, if I want to plot a particular function here, how do I set the amplitude bars to produce it? To illustrate how this machine does analysis, let's start with a square wave. First we use some unique properties of the square wave, since it's periodic and even, all of the information that function carries is contained in half a period. So we'll take that half a period and sample it at 20 equally spaced points. We use the values of these points as the inputs for the machine. When we turn the crank, we produce a new function that reveals the correct coefficients. To see how we get these coefficients, we'll look at the side of the machine. As I turn the crank, the tips of the rocker arms form a sinusoid. The indices on the rocker arms run from high on the left to low on the right. I'll flip the video to make it a little more intuitive. When the rocker arm tips are lined up in a straight line, we'll call that crank number zero. If the horizontal axis runs from zero to pi, and the vertical axis from minus one to plus one, then we can describe the position of the rocker arms with a cosine. Every two turns of the crank increases the frequency of the cosine by one. At these even number cranks, the values of the function on the platen yield the coefficients we're looking for. Let's rewind and watch the plot and the rocker arms simultaneously. If we pause briefly at every second crank, a point is marked on the function. We create a total of 20 points. If we look at the output from the machine, and compare it to a sync calculated by a computer, we see that they are very similar. Now we'll take the data points from the sync, scale them, and use these values on the rocker arms. Remember that our goal is to program the analyzer so that it will plot a square wave. Now, as I turn the crank, the pin writes a horizontal line, then drops and writes another flat section, which amazes me because we're adding only cosines, and then it rises to write another horizontal line. Of course, what we're seeing is a square wave. What I've just shown with the machine is an essential feature of Fourier methods. I can take a function, perform harmonic analysis, extract the coefficients, and then synthesize that function to approximate the original. So now that we see that this machine can do harmonic synthesis and analysis, I'll show you in the next video some details about how to set up the analyzer to perform these calculations. I'm Bill Hammock, the engineer guy. Next up in the series is Operation. If you haven't seen them already, there's also the intro and synthesis videos. You can learn more about the book here, and if you really want to learn more about the book, watch the page by page. If you're a fan of oscillatory motion, you've got to watch the bonus rocker arms video.