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 ti and want to find a,
ω and φ such that f (t) = a cos (ω t + φ).
From the analytical expression, we can compute two primitives :
If2 (t) = ∫ f2 = a2 × [t + S (t)] / 2
If'2 (t) = ∫ f'2 = a2 ω2 × [t - S (t)] / 2
where S (t) = sin (2 (ω t + φ)) / (2 ω)
We can remove S between these expressions :
If'2 (t) = a2 ω2 t - ω2 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 t1 and ti for each ti :
If2 (ti) - If2 (t1) = A × (ti - t1) + B × (If2 (ti) - If2 (t1))
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 (xi, yi) = A × xi + B × yi, the
coefficients A and B that minimize a least square criterion
∑ (zi - z (xi, yi))2 are given by these expressions:
∑yiyi ∑xizi - ∑xiyi ∑yizi
A = ------------------------
∑xixi ∑yiyi - ∑xiyi ∑xiyi
∑xixi ∑yizi - ∑xiyi ∑xizi
B = ------------------------
∑xixi ∑yiyi - ∑xiyi ∑xiyi
In fact, we can assume both a and ω are positive and
compute them directly, knowing that A = a2 ω2 and that
B = - ω2. The complete algorithm is therefore:
for each ti from t1 to tn-1, compute:
f (ti)
f' (ti) = (f (ti+1) - f(ti-1)) / (ti+1 - ti-1)
xi = ti - t1
yi = ∫ f2 from t1 to ti
zi = ∫ f'2 from t1 to ti
update the sums ∑xixi, ∑yiyi, ∑xiyi, ∑xizi and ∑yizi
end for
|--------------------------
\ | ∑yiyi ∑xizi - ∑xiyi ∑yizi
a = \ | ------------------------
\| ∑xiyi ∑xizi - ∑xixi ∑yizi
|--------------------------
\ | ∑xiyi ∑xizi - ∑xixi ∑yizi
ω = \ | ------------------------
\| ∑xixi ∑yiyi - ∑xiyi ∑xiyi
Once we know ω, we can compute:
fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
It appears that fc = a ω cos (φ)
and
fs = -a ω sin (φ)
, so we can use these
expressions to compute φ. 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.