Whether an unknown Laplace's Tidal Equation or Mach-Zehnder modulation can be decoded easily is still an open question but here is a trig trick that doesn't require an \\(arcsin \\) function. The puzzle is how to recover \\( g(t) \\) from \\( \sin( k g(t) ) \\), where both \\( k \\) and \\( g(t) \\) are unknown. The hint is that it is easier to recover \\( g'(t) \\) than \\( g(t) \\). I am certain it's in the literature somewhere on inverting MZ modulation but it will take a while to find.

This is a recovery of \\( g'(t) \\) for an LTE/MZ modulation of \\( k=3 \\) for three mixed frequency sine waves (periods 12, 25, 40)


This is a modulation of \\( k=10 \\)


Note that the only problem is that it can't decide whether the recovered signal is \\( g'(t) \\) or \\( -g'(t) \\) and switches between the two whenever a \\( 2 \pi \\) boundary is crossed.

This is a slight modulation of \\( k=0.01 \\), which essentially recovers the derivative of the original mix of sine waves without switching to the opposite sine since no \\( 2 \pi \\) boundary crossings occur.


I think the approach should work fine on an ideal case once an algorithm is applied to prevent jump discontinuities when piecing together the intervals (the zero crossing points are the trickiest to handle). Whether it will work on a non-ideal signal such as ENSO is another question.

This is actually an easy formulation to derive if anyone wants to give it a go.