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.

Boring Archive of Notes on Philosophy Podcasts

This is strictly a memory aid for me.  I'm going to write these up after listening to philosophy podcasts. So, for those few of you paying attention to this blog, it's probably best to go away & wait for the next post. This one's mostly just for me.

What Mary knew is a famous thought experiment.  Australian Frank Jackson invented this to defend against materialists. (...who believe that only things that exist are physical.)  Someone (Mary) who knew everything, but never had an experience of a fat red tomato, or perhaps a black & white limitation in their vision. However smart she is, analytically maybe knowing everything, won't she, upon being cured, suddenly feel she's learned something special and different? The word "qualia" describes the "feely side" of red that she would be now more vividly experiencing.  I felt this was basically empty of meaning, and maybe so does Jackson, because sometime after making this famous argument, he changed his mind and became a materialist after all.

Consequentialism:  This theorizes we should act to produce the best consequences, ie the end justifies the means. Seems like utilitarianism, but Pettit says it's different only in the definition of utility: what yardstick is used to measure the good.  There's a broad and meaningless argument about what "the goods" are.  Non-consequentialism has more inflexible moral absolutes: "no kicking of puppies," for instance.  What if the best outcome requires you to do that? Could you lose your integrity thereby?  Famous example by Williams, sets us up with a scenario about to execute natives: but we could kill one to save 10, should you do it? Consequentialism says of course, shoot the one.  Another objection is that maybe you as an agent should Never be required to treat other people "as means."  Wild west example: sheriff stops the riot by scapegoat someone, hang the innocent, quell the population & save lives.  These objections suggest there is value in living a life of character, even at some cost.  Maybe the greater integrity has value exceeding even the very lives that would be saved by unfeeling consequentialist acts in those scenarios. So say absolutists.  Another approach might ask, would not morality and honesty, ubiquitously applied by all, yield good?  Perhaps, but that tack, if you take it, would be consequentialism! (choosing for the good.) A great example is Kant's case of the wild eyed axeman at your door, asking after your friend Flynn, who's lolling right over there in a hammock. Do you tell the truth because you're fundamentally in favor of doing so? "Flynn's right over there," may well get him killed!  Red lights go on... (that's still consequentialism) Kant would have answered effectively, "Fiat justitia ruat caelum"  This is just a counterexample where consequentialism makes sense. What about the other heartless ones?  He's for them.  How about torturing someone to find where the bomb's buried?  That's ok, but you should have to be tried for it afterwards.  After that he gets fuzzy.

Unity of Value is Ronald Dworkin's thesis that pluralism is wrong. Pluralism posits that different values are necessarily in tension, eg freedom vs respect (Consider, "freedom for the pike is death for the minnows!"). The whole thing seemed half thought. Anybody who says "the way in which" too often  is probably full of baloney.  One phrase I particularly disliked boiled down to, "If you make your argument, somebody who disagrees is of course not going to believe you."  (Isn't the point of engaging in argument to potentially change your mind: his presumption of lack of openness makes  the whole field worthwhile.)