# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 4 Spectral Analysis of the Sunspot Series  
The spectral analysis of the sunspot time series incorporated five different preprocessing steps in order to achieve an expressive spectral representation. Accordingly:1. The whole time series was filtered with an 11-point median filter so that outliers were rejected and weak harmonics were easily revealed (Figure 2). 2. Taking into consideration that the period of the 11-year circle is approximately 4000 days, we partitioned the time series into sections of 8000 days so that each section contained two 11-year circles. 3. Using an overlapping of 4000 samples, we obtained a total of 17 sections of 8000-day samples. 4. Each segment of 8000 samples was extended to 32,000 samples by adding 12,000 zero samples to each edge. This made the final spectrum finer with a distance between harmonics of \(-\)1/32,000 causing the first harmonic after the dc to represent a period of 32,000 days and, accordingly, the 11-year circle to be represented by the 9 th harmonic. At the 16,001 position of the spectrum, we obtained the highest frequency, which represented a 2-day period. 5. The periodogram of the abovementioned 17 time-series pieces of 32,000-length samples was calculated in order to obtain the final spectrum. Accordingly, the total spectrum was calculated as an average of 17 spectra. Any frequency that would have come from an artifact would have vanished by the averaging process. In Figures 6-8, three parts of the evaluated spectrum are provided. In Figure 6, the low-frequency part of the sunspot series spectrum, which involves frequencies from dc to 100, which corresponds to a 320-day period, is presented. As depicted in Figure 6, a strong peak is present at spectral position 8 with almost equivalent spectral energy values at position 7 and 9, which correspond to the orbital period of Jupiter and its synodic periods with Saturn and Uranus. The 11-year cycle is also represented by the 8 th spectral component, which lies at position 9. Figure 6: The first 100 harmonics of the sunspot time series from a total of 16,001. Since at position 16,001 we have the harmonic that corresponds to 2 days, the harmonic at position 101 corresponds to 32,000/100 = 320 days. Six is the 6 th periodic planetary movement presented in Table 1, which corresponds to Jupiter’s rotation. It lies at the 8 th harmonic position, i.e., it corresponds to a 4000-day period (11-year cycle). Positions 4 and 5 are the synodic periods of Saturn and Uranus with Jupiter, respectively. Their strength is greater than the period of Jupiter. Figure 8: Spectrum of the sunspot time series with harmonics from 500 to 1300 from a total of 16,001. Since at position 16,001 we have the harmonic that corresponds to 2 days, the harmonic at position 1301 corresponds to 32,000/1300 = 24.6 days. Harmonics 950 to 1250, in the circle indicated as 30 from the corresponding information in Table 1, represent the rotational behavior of the body of the Sun, as it rotates at different periods in its equator (25.6 days) from its mass near the poles (33.5 days). Figure 7: Spectrum of the sunspot time series with harmonics from 20 to 500 from a total of 16,001. Since at position 16,001 we have the harmonic that corresponds to 2 days, the harmonic at position 501 corresponds to 32,000/500 = 64 days. The numbers above the red circles correspond to the serial numbers in Table 1. Indicatively, number 12 is the orbital period of Mars and corresponds to the 48 th harmonic. Eighteen corresponds to Earth’s orbital period, which is the 89 th harmonic, while numbers 28 and 29 represent the positions of the 361 and 365 harmonics, which represent the synodic periods of Mercury and Jupiter, and the orbital period of Mercury correspondingly. All the relative information is presented in Table 1.  
Figure 7 presents the harmonics up to a value of 400. The reader can find the peaks corresponding to various orbital planetary and synodic periods.