why sine waves? and not another form of periodic motion?
solution to differential eqn, where force towards center proportional to distance
from center
IMPORTANCE OF SINE WAVES... also, the very idea of a sine wave.
the value of sine is proportional to how far it is from the "center". i.e, when the x coord
is far away, large value... when not as far away, smaller value.
double y coord, double sin... ahhh, i see now
double x coord, double cosine.
at zero, no pulling force, so zero.
go through derivation of beats... how modulations works... how the product of a sin and
a cosine can turn into their sum (in the frequency domain)
if equation is linear, then any combination of solutions will work. aha! because
3x = 3 * result of(x) = 3 * 0.
need intuitive reason why integral of cos(mx) and cos(nx) = 0;
sin/cos expansion almost like a taylor series...
continuous, non-differential functions? music: noise, square waves
Fejer: any continuous fn. can be reconstructed from fourier terms.
Gibbs phenomenon: "e" -> continuity not converging. ah... infinity.
amplifier: feed it a square wave, will overshoot and undershoot, trying to reconstruct
discontinuous signal. amplifier has maximum frequency to which it can respond
uniform vs. pointwise convergence.
use complex exponentials because it simplifies things... not because there is any intuitive
reason to.
windowing: actual sounds aren't periodic (periodic fns. go on for infinity).
ahhh... analyze the frequency specturm around a window of time. not at a single point (no
meaning to frequency), and not for the entire duration. time/frequency smearing: heisenberg
uncertainty principle. heisenberg uncertainty principle based on the errors of a fourier
transform!! neat.
energy density at a particular frequency: square of amplitude. integrating over all frequencies
gives energy.
parseval's formula: energy in time domain = energy in freq. domain (makes sense intuitively)
we expect it to happen -- don't get extra energy just by looking at the same signal a
different way.
changing the value at a point does not change value of integral! amazing... eh?
distribution: defined implicity, using multiplication and an integral
delta function is the identity for convolution
(f * delta)t = (delta * f) = f
frequency filter: multiply the frequency domain by a filtering function. in time domain,
a convolution (which is more difficult to understand).
