In
Harmonic tremors and spectrograms I claimed that the Fourier transformation can be used for frequency analysis, which is of course true, but I feel that I should prove the following lemma which proves it.
Lemma. a sin(x) + b cos(x) = (a²+b²)½ sin(x+arccos(a/(a²+b²)½) = (a²+b²)½ sin(x+arcsin(b/(a²+b²)½).
Proof. There is exactly one solution for the following set of equations
- f(0) = b,
- f'(0) = a,
- f''(x) = -f(x),
namely the left hand side of the equation above. But
s sin(x+u) also fulfils the third condition, so in order to fulfil the first two, we must solve
- b = s sin(u) = s (1-cos(u)²)½,
- a = s cos(u) = s (1-sin(u)²)½
and going after
s first we arrive at
- sin(u) = b/s, cos(u) = a/s,
- a = s (1-(b/s)²)½ = (s²-b²)½, b = s (1-(a/s)²)½ = (s²-a²)½
and squaring that we reach both ways
Inserting the (positive) square root of this into the equations under the third last point, we only have to invert
sine and
cosine simultaneously, which can be done, since
( a/(a²+b²)½, b/(a²+b²)½ ) lies on the unit circle, to reach the result.