I found an interesting mathematical simplification relating seasonal aliasing of short-period cycles with a biennial signal.

If we start with a signal of an arbitrary frequency

and then modulate it with a delta function array of one spike per year

This is enough to create a new aliased cycle that is simply the original frequency summed with an infinite series of that frequency shifted by multiples of .

This was derived in a previous post. So as a concrete example, the following figure is a summed series of *f(t)* -- specifically what we would theoretically see for an anomalistic lunar month cycle of 27.5545 days aliased against a yearly delta.

If you count the number of cycles in the span of 100 years, it comes out to a little less than 26 cycles, or an approximately 3.9 year aliased period. If a low-pass filter is applied to this time series, which is likely what would happen in the lagged real world, a sinusoid of period 3.9 years would emerge.

The interesting simplification is that the series above can also be expressed exactly as a *biennial + odd-harmonic* expansion

where T is given by

where the function "int" truncates to the integer part of the period reciprocal. Since is shorter than the yearly period of 1, then the reciprocal is guaranteed to be greater than one.

As a check, if we take only the first biennial term of this representation (ignoring the higher frequency harmonics) we essentially recover the same time series.

What is interesting about this factoring is that a biennial modulation may naturally emerge as a result of yearly aliasing, which is possibly related to what we are seeing with the ENSO model in its biennial mode. Independent of how many lunar gravitational terms are involved, the biennial modulation would remain as an invariant multiplicative factor.

For the ENSO signal, an anomalistic term corresponding to a biennial modulation operating on a 4.085 year sinusoid appears significant in the model

1/4.085 = 1/2 + int(365.242/27.5545) - 365.242/27.5545

which is expanded as this pair of biennially split factors (see ingredient #5 in the ENSO model)

1/4.085 ~ 2/3.91 - 2/1.34