FFT(just about everything)

Listening to Brad Osgood on Fourier Series, and kinda reading at the notes.
In the first lecture, his main point was that periodicity comes from the circle (eg sin, cos) and that frequency and wavelength have reciprocal relationship.  Right away, I'm challenged by the limited formatting that exists (or that I know how to apply) to this blog.  In no time I'm going to need integrals, and Mathcad beckons.  Maybe I can get the ideas down in English!  That's more challenging but might make me understand better.

In the notes, there's a concept of the series of functions periodic on an interval, and finding a coefficient as an integral of the product of exp(n*i*pi) across the interval.  The why is cool: first, the deal is that all periodic functions can be made from a sum of sines & cosines (with integer frequency multiples) and if you multiply that sum by the negative frequency component you're interested in, it isolates one coefficient because for that term, the exponent of e adds to zero and so the e terms cancel and c drops out of the integral. Meanwhile, integrating each of the other terms across the interval yields zero every time. That's the orthogonality of the set of functions I think.  I'm happy with that description! On to the second lecture.  It seems easy so far but I'm getting the value from the notes, and when I dropped in in the middle of the class I was lost, so I'm taking this more ponderous approach, for now.

Second lecture: "periodicity."  It's very tempting to complain about his lecture and chalkboard style!  Maybe, people in glass houses... "Here's a secred of the universe, comin' your way."  Now, that's a joke that deserves a laugh, but he got nothing.

Functions with limited time frame of interest can be matched over that interval only, and you just pretend it was periodic, repeating just that interval.  Sin(2pi*t) has period = 1. That restriction is useful for analysis. Model generic signals by using 2pi*n*t. Over that one period, you can stuff in many frequencies in harmonics of the "base" (longest) signal.  Remember frequency w (omega) = 2pi*n.  Sum a bunch of these and period is still 1 - it's limited by the lowest frequency being represented. Besides different frequencies, we can modify amplitudes and phase of each term, to model different signals.  What about frequency 2pi*1.5? That's not an integral multiple, not periodic over the interval 1, probably not legal to use.

Writing them down, he noted sin(2pi*k*t + p) = sin(2pik)cos(p) + cos(2pik)sin(p).  (p's the phase).  I mention it because in thinking about the angle sum formula, I realized how complex numbers can be used to derive this formula.  Think of p as the rotation angle in a direction cosine matrix. So we have sin(A+p) and the A part is spinning vector and the p is the fixed phase offset of the sin wave out of the real plane. Our question is "what part of this imaginary wave projects on the real axis, and for the answer you just multiply the signal by the DCM. QED. Back to Osgood's point, the fixed values, sin(p) & cos(p) can be thought of as coefficients, and then an arbitrary function is made of sines and cosines with various coefficients and the phase of the net wave is buried in those coefficients, instead of explicitly calling out a phase angle, and using only sine.

Complex notation:  csubk* exp(i*2pi*k*t + p) is the way to write it in complex form. To get coefficients c, isolate them again (as noted from the notes before) using the trick of multiplying the whole sum by the MINUS frequency of the coefficient you want. All the rest of the terms will sum to zero.

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