The length of sunspot cycles is usually calculated by measuring peak to peak or trough to trough but this necessarily involves guessing the true min and max, an interpretation, as well as only providing a few spot figures.
An alternative exists in the field of signal processing where a contiguous figure for the period of a varying wave can be computed, called instantaneous frequency (in this case the reciprocal is more useful, the period).
I was going to explain in detail but Rog thought what I had written looked too complicated, fair comment. This is a cut down version, better than nothing.
To do this we need to produce the analytic signal from the sunspot data. This is simple in concept, a pair of signals where one is 90 degrees phase shifted relative to the other, the twist here, is at all frequencies. One way of visualising this is by delaying the signal by an amount directly proportional to each frequency component.
If that sounds hairy, it is yet in many ways it is trivially simple. A very specific kind of filter can do this and the general name for the procedure is the Hilbert Transform.
I can explain some more if asked.
The sunspot data is “difficult” so simply passing that wave through such a filter produces a mess where we can’t get what we want.
- bandpass filter the sunspot data keeping say 7 year to 19 year information, no hard and fast rules here. Lower traces of figure 1 show ssn and the bandpass filtered version.
- Apply the filter to the data
- Do various computations on the resultant pair of signals.
Ever seen Lissajous figures, XY plots of A against B?
In phase signals give a 45 degree slope straight line, antiphase -45 degree straight line and between the two it opens out. At 90 degree shift you get a circle.
Like this bandpass filtered ssn data.
A party trick is that sin and cos are complementary where for a pair at 90 degrees they sum to unity. If you take the RMS value sqrt(x^2+y^2) you get the envelope (amplitude) of the original signal, shown in figure 1, simple as that point by point.
It is also possible to compute the instantaneous phase. At this point I am going to point you at Liverpool John Moore university where I hope some insight can be gleaned, pdf and with simple plots. This is an extremely important topic in for example medical imaging yet there is no exact solution, only heuristics.
If you have the phase you can also compute the frequency. The meaning of that is vague, hence bandpass filtering has restricted the window onto what is of interest.
As an aside I’ve experimentally used a completely different method as part of a software development, it gives roughly the same result.
This is noisier but is less fooled by the 1790 event, moreover the end data shown is trustworthy. (keep in mind the early ssn data is very poor goodness)
Commentary on result
Figure 1, there is a spike in period around 1800, probably quite wrong, clip the plot, figure is >30 years. However, there is good if disputed data which suggests there was a sunspot cycle which is missing because the data back then is very poor. If true there is a short cycle thereabouts.
In broad terms this agrees with the piecewise estimations. It also tends to show amplitude and period are complementary, nothing new.
Simple XLS spreadsheet of raw data and results, here. (appears empty, is a wide sheet, go right)
Post by Tim Channon, co-moderator