What are your thoughts on time-frequency duality, in that time as a variable is able to be abstracted away for our analytical purposes (which works!)? And is the frequency domain a more qualified s-plane?
Consider a point on the edge of a rotating circle which is rotating counter-clockwise at a constant frequency. If you plot the point over time in three-dimensional space (starting from 0rad), you get a spiral or helix (like one half of a double-helix DNA strand).
If you view the spiral from the side, you see a sine wave.
If you view the spiral from the top, you see a cosine wave.
If you look down the barrel of the signal, you will see an infinitely bright circle with a radius equal to the amplitude of the signal. The brightness of the circle relates to the energy in the signal, since by looking down the barrel, we have collapsed time and relinquished all phase information.
Pure sinusoids are infinite, so real-world signals will be windowed (note: windowing always introduces artifacts).
Now, what happens if the signal is a composition of rotating circles all spinning at different frequencies and directions? Well, you'll see a bit of a blurry distribution of where the wavefront spends its time. If you think about this a bit, you can ask yourself what you have to do to "see" the frequency components of the signal :).
I have to go now, but will try to post more later.
>the brightness of the circle relates to the energy in the signal
What does this “energy” mechanically correspond to in the complex unit circle and how do these mechanics appear in the time-domain?
The ideal low pass filter’s mathematical application - of the frequency domain effects on natural signals - astonishes me and I can’t quite fathom the transformation; ie is the frequency space physical or a purely mathematical abstraction?
Ultimately: is the complex exponential function a natural algorithm?
