This class guesses harmonic coefficients from a sample.

The algorithm used to guess the coefficients is as follows:

We know \( f(t) \) at some sampling points \( t_i \) and want to find \( a \), \( \omega \) and \( \phi \) such that \( f(t) = a \cos (\omega t + \phi) \).

From the analytical expression, we can compute two primitives : \[ If2(t) = \int f^2 dt = a^2 (t + S(t)) / 2 \] \[ If'2(t) = \int f'^2 dt = a^2 \omega^2 (t - S(t)) / 2 \] where \(S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}\)

We can remove \(S\) between these expressions : \[ If'2(t) = a^2 \omega^2 t - \omega^2 If2(t) \]

The preceding expression shows that \(If'2 (t)\) is a linear combination of both \(t\) and \(If2(t)\): \[ If'2(t) = A t + B If2(t) \]

From the primitive, we can deduce the same form for definite integrals between \(t_1\) and \(t_i\) for each \(t_i\) : \[ If2(t_i) - If2(t_1) = A (t_i - t_1) + B (If2 (t_i) - If2(t_1)) \]

We can find the coefficients \(A\) and \(B\) that best fit the sample to this linear expression by computing the definite integrals for each sample points.

For a bilinear expression \(z(x_i, y_i) = A x_i + B y_i\), the coefficients \(A\) and \(B\) that minimize a least-squares criterion \(\sum (z_i - z(x_i, y_i))^2\) are given by these expressions:

\[ A = \frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i} \] \[ B = \frac{\sum x_i x_i \sum y_i z_i - \sum x_i y_i \sum x_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i} \]

In fact, we can assume that both \(a\) and \(\omega\) are positive and compute them directly, knowing that \(A = a^2 \omega^2\) and that \(B = -\omega^2\). The complete algorithm is therefore:

For each \(t_i\) from \(t_1\) to \(t_{n-1}\), compute: \[ f(t_i) \] \[ f'(t_i) = \frac{f (t_{i+1}) - f(t_{i-1})}{t_{i+1} - t_{i-1}} \] \[ x_i = t_i - t_1 \] \[ y_i = \int_{t_1}^{t_i} f^2(t) dt \] \[ z_i = \int_{t_1}^{t_i} f'^2(t) dt \] and update the sums: \[ \sum x_i x_i, \sum y_i y_i, \sum x_i y_i, \sum x_i z_i, \sum y_i z_i \] Then: \[ a = \sqrt{\frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i } {\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i }} \] \[ \omega = \sqrt{\frac{\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}} \]

Once we know \(\omega\) we can compute: \[ fc = \omega f(t) \cos(\omega t) - f'(t) \sin(\omega t) \] \[ fs = \omega f(t) \sin(\omega t) + f'(t) \cos(\omega t) \]

It appears that \(fc = a \omega \cos(\phi)\) and \(fs = -a \omega \sin(\phi)\), so we can use these expressions to compute \(\phi\). The best estimate over the sample is given by averaging these expressions.

Since integrals and means are involved in the preceding estimations, these operations run in \(O(n)\) time, where \(n\) is the number of measurements.