I think these concepts became clear to me when I learned about the complex plane, Euler's formula, and demodulation of an FM signal. Particularly enlightening

I think these concepts became clear to me when I learned about the complex plane, Euler's formula, and demodulation of an FM signal. Particularly enlightening was the demodulation of a frequency modulated sine wave when the tuner was imperfectly matched to the carrier frequency. Looking at it on an oscilloscope was similar to watching an old TV with the Vertical Hold improperly set. That made me start thinking in terms of a signal (sine wave) that was a cycle rather than a sinusoidal shape. Seeing that you can graph the amplitude and phase of any signal on the complex plane and that the frequency was the change in phase from one moment to the next was the aha! moment. Then if you think about sampling and how if you sample a sinusoid exactly at its peaks how that would graph as a constant point on the complex plane, but if you sampled at any other mismatched frequently, the point would rotate and change amplitude with respect to either axis. The further away from the actual frequency you'd go the the points would look more random, and they'd average out to zero with fewer samples. This would be a fun animation or java applet to make; I'm sure someone has done it.