Why I didn't do this bit of signal processing earlier, I don't know.

Take the Fourier spectrum of the ENSO time-series and shift and reverse the order of frequencies between 0.5 to 1/year and by 0.5/year (i.e. fold it back) to quantify the underlying forcing symmetry.


The correlation coefficient is ~0.6 which is impossible to attain with a stochastic auto-regressive spectrum.

Compare this to the model tidal forcing where the spectrum is not folded as above but the symmetry centered at 0.5 /year (and 1.5 /year) is clearly apparent.


The folded cross-correlation of this ideal model forcing is > 0.9, so likely the LTE modulation is responsible for the multiple splitting of the peaks in the ENSO forcing response, thus degrading the symmetry to some degree.

Analysts may overlook this kind of approach because they will simply take the spectrum, not see any annual spectral peak, and so just assume that an annual signal doesn't exist. Yet, taking an annual impulse and **convolving** that with any kind of sinusoidal waveform with a mean of zero will create **only** satellite peaks in the response, with the annual spectral peak completely missing. Then with any modulation, such as via the LTE tidal response, the spectral response gets reshaped -- much like the Mach-Zehnder modulation creates a "comb" response (from the post from last week).

The insight is that LTE and MZM is essentially the same analysis. After the MZ modulation is applied to the input signal, it combs out as in the rightmost column below. The comb is essentially a large number of precisely spaced spectral lines that are comprised of harmonics of the original signal.

["Characterization of dual-electrode Mach-Zehnder modulator based optical frequency comb generator in two regimes" ](https://www.semanticscholar.org/paper/Characterization-of-dual-electrode-Mach-Zehnder-in-Fontaine-Scott/640e962d202e714aa225887ad33fb45d7f6d77a4)


The reason that the ENSO is difficult to decode is that we are only guessing what the LTE modulation is. In contrast, with Mach-Zehnder, that is part of the device design and one calibrates the modulation beforehand. As I said earlier, this is why MZM is hard to decode when used for crypto applications. With a strong modulation, the time domain trace is shown below, creating erratic harmonics of the fundamental frequency.


Consider that there may be a number of varying input signals and if the comb is in 2D, the output will obviously become quite scrambled, making it impossible to decode without knowing the MTM key.

But here is the next mathematical idea. The MZM modulation is typically expanded as a Taylor's series with the coefficients of the harmonics given by a Bessel function, see this


So with enough data points and enough coefficients in this so-called Jacobi-Anger expansion, we should be able to do a multiple linear regression to optimize for a best fit of the amplitude (z below) and phase (weighting of sin vs cos below) of the modulation (or more than a single modulation if that works)

The reason that this may be possible without iteration and just by applying a regression algorithm is that the input frequencies (inner terms above) will be restricted to the primary tidal periods, each one of these will have an amplitude and phase. Since we will have ~1700 data points over 140 years worth of monthly ENSO data, there will be enough spectral resolution between 0 an 0.5 /year frequency to do a decent regression with a couple dozen parameters -- two modulation and perhaps 10 tidal cycles (amplitude + phase). The tricky part is the Bessel function on z which is non-linear, so perhaps these will need to be expanded and another regression applied.