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11. März 2018

Fourier transformation and amplitude and phase.

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
  •  s² = a²+b².
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.

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