Thanks for sticking with me. I'm not the sharpest tool in the shed but I'm willing to work hard.

Whoa there, I wouldn't sell yourself short. This is pretty advanced stuff for mid-leveled high school and you seem to be doing a good job so far. I didn't know how electronics or basic signal processing worked myself until somewhere around sophomore year of college (electrical engineering major).

What is x(t)?

What is X(w)?

x(t) is the signal in the time domain. This is what you would see if you were looking at the signal on an oscilloscope. On the x(t) graph, the x-axis represents a change in time.

X(w) is the SAME signal, but in the frequency domain. This is what you would see if you were looking at an FFT of the signal. On the X(w) graph, the x-axis represents a change in frequency. This explicitly shows which frequency components are present in your signal.

Both graphs provide the same exact information about the signal, but in different ways. For instance, it would be like saying that the US President lives at 1600 Pennsylvania Ave and also saying that he lives at 38.8977° N, 77.0365° W (Lat - Long). Both pieces of information are correct, they just have a different form. Much like x(t) and X(w) both give the same information about the signal, just in different form.

What are the red arrows? Are they trying to keep up with the PWM signal?

The red arrows show the presence of harmonics in the signal. They are called Dirac Delta Impulses, but you don't need to worry about that. The important thing is that they show you which frequencies are most prevalent in your signal. At w=0, we can see there is a red arrow, showing that there is a DC component to the signal. Moving along the x-axis we run into the next red arrow. This occurs at the first positive harmonic (we'll pick w = 10rad/sec since no numbers are given on the graph). We can also see another red arrow at w = -10rad/sec. This pair of red arrows indicates a cosine with a magnitude of twice the height of one the arrows. You will find similar pairs of red arrows at the second (w=20rad/sec), third (w=30rad/sec), and fourth (w=40rad/sec) harmonics, up to infinity. The only difference between all of the pairs are their heights - which shows how prevalent the cosines are compared to others. More height means they play a larger role in shaping the signal.

Once you filter the signal, the x(t) graph will become a constant and the X(w) graph will have only one arrow at w=0rad/sec.

What are the name of these graphs?

You can find similar graphs by searching "Fourier Series".

Is the Fourier Series illustration highlighting the different frequencies that comprise a fundamental frequency at a specific period in time?

Yes and no. It does illustrate the different frequencies of the signal, but is NOT for a specific moment in time. It is for the entire periodic signal from t = -infinity to t = infinity. Also, be careful to note that the fundamental frequency isn't comprised of anything; it just

*is*.

What is the process for selecting the resistor and capacitor? I'm thinking I start with a resistor that doesn't overdrive the PWM pin (at 490Hz the capacitor will have a very low capacitive reactance). I then start 'playing' with different capacitor values that give good responsiveness and nominal ripple?

That question is best answered by research and trial and error. I suggest looking

here first. Good luck!

As for "playing" with different values, I'd suggest doing that with a circuit simulator first such as LTSpice IV.