The Fourier spectrum proves to be very complicated, but it can be simplified somewhat by manipulating the data. Linearity is an important factor in spectral analysis; if a nonlinear transformation, such as rectification or hard limiting, is applied to a signal, the spectrum gets messy.
As a philosophical principle, suppose that you subject an observed time series to a nonlinear back-transformation that you guess, and the resulting spectrum becomes simpler than it was before. Does that not imply that the original observations were to some extent contaminated by a previously unsuspected nonlinearity and that removal of the nonlinearity brings the data into a form that is closer to the underlying reality?
In support of this principle, I have derectified the sunspot series and the spectrum immediately simplifies. This move could be important to people looking for a correlation between the solar cycle and a tropospheric phenomenon, such as some parameter of the weather.
There is a clear solar-cycle effect on the density of ionospheric ionization, on auroral frequency, and on terrestrial magnetic variations, but although several meteorological connections have been claimed, none has met with general acceptance. If there was a faint 22-year effect present in the weather, driven by the solar cycle, then it would be better detectable by correlating against the derectified data series. (Bear in mind that the "11-year" cycle varies in length from 8 to 14 years, so correlation is likely to be far more sensitive than looking for a spectral peak.)
A second nonlinearity was guessed by assuming that a sine component from the underlying oscillator appears in the observations as the nth power of the sine. An analytic method for deducing n gives a value 3/2. When the derectified series is raised to the 2/3 power, the spectrum again simplifies.
A third nonlinearity has to do with the statistical result that the ratio of rise time of an 11-year "semicycle" to fall time varies inversely as the peak sunspot number of that cycle. This effect can be partially canceled by a nasty amplitude-dependent shearing distortion.
Plausible physical conjectures can be given for all three of these nonlinearities. The basic technique can be recommended for consideration by anyone that has data series to study.
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