# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 3 Basics of the Bispectrum  
The bispectrum, obtained by applying a two-dimensional Fourier Transform to the third-order autocorrelation function, \(R_{xxx}(\tau_{1},\ \tau_{2})\), proved to be effective in identifying and measuring the Quadratic Phase Coupling (QPC) [21; 25; 26]. Concurrently, the bispectrum serves to elucidate the statistical associations among various frequency components within a signal. Its formal definition is as follows:  
\[B(f_{1},f_{2})=E\Big{\{}X(f_{1})X(f_{2})X^{*}(f_{1}+f_{2})\Big{\}} \tag{2}\]  
For the bispectrum to manifest non-zero values at frequencies \((f_{1},\ f_{2})\), it necessitated that the Fourier Transforms at the individual frequency components of \(f_{1},f_{2}\), and \(f_{1},\ +\ f_{2}\) were non-zero. Additionally, these three spectral components must exhibit correlations. It is essential to highlight that, following the expectation process, the bispectrum becomes zero due to a phase randomization. However, when the phases are coupled, this does not hold true. In contrast to the power spectrum, the bispectrum is a complex component, even when the signal is real-valued. The symmetric regions of the bispectrum are illustrated in Figure 3[21; 25; 26]. Consequently, the analysis typically focused on a single, non-redundant region. In this paper, the notation \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\#} & **Planets’ Periods** & **Orbital or Synodic** & **Position in the 32,000 Harmonics** \\  & **Planets’ Synodic Periods** & **Periods: P** & **Spectrum, Including Position 1** \\ \hline
4 & Jupiter–Uranus & 5045 & 7.34 \\ \hline
5 & Jupiter–Neptune & 4669 & 7.85 \\ \hline
6 & Jupiter & 4331 & 8.4 \\ \hline
7 & Jupiter–Mars & 816.52 & 41.2 \\ \hline
8 & Mars–Earth & 780 & 42.0 \\ \hline
9 & Mars–Saturn & 733.92 & 44.6 \\ \hline
10 & Mars–Uranus & 702.78 & 46.5 \\ \hline
11 & Mars–Neptune & 694.98 & 47.0 \\ \hline
12 & Mars & 687 & 47.6 \\ \hline
13 & Earth–Venus & 584 & 55.8 \\ \hline
14 & Earth–Jupiter & 399 & 81.2 \\ \hline
15 & Earth–Saturn & 378 & 85.6 \\ \hline
16 & Earth–Uranus & 370 & 87.4 \\ \hline
17 & Earth–Neptune & 367 & 88.2 \\ \hline
18 & Earth & 365.25 & 88.6 \\ \hline
19 & Venus–Mars & 333.91 & 96.8 \\ \hline
20 & Venus–Jupiter & 237 & 136 \\ \hline
21 & Venus–Saturn & 229.5 & 140.4 \\ \hline
22 & Venus–Uranus & 226.36 & 142.4 \\ \hline
23 & Venus–Neptune & 225.55 & 142.8 \\ \hline
24 & Venus & 224.7 & 143.4 \\ \hline
25 & Mercury–Venus & 141.46 & 227.2 \\ \hline
26 & Mercury–Earth & 116 & 276.9 \\ \hline
27 & Mercury–Mars & 100.89 & 318.2 \\ \hline
28 & Mercury–Jupiter & 88.95 & 360.7 \\ \hline
29 & Mercury & 87.9691 & 364.8 \\ \hline
30 & Sun rotation period & 25.6–33.5 & 956.2–1251 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. \(B(f_{1},\ f_{2})\) was employed to represent the bispectrum in the non-redundant triangular region shown in blue in Figure 3. It is defined in \(\{(f_{1},\ f_{2}):\ 0\ \leq f_{2}\leq\ f_{1}\leq\ f_{e}/2;\ f_{1}+f_{2}\leq\ f_{e}/2\}\), where \(f_{e}\) is the sampling frequency. Further details about the computational regions can be found in [21,25]. The bispectrum proves to be a valuable tool for effectively addressing practical problems, as exemplified by [21,27,28,29]:  
* In the case of a stationary zero-mean Gaussian process, \(x(n)\) its bispectrum consistently remains zero. * Unlike the power spectrum, which discards phase information, the bispectrum retains it. The bispectrum, serving as a quantified expression of Higher-Order Statistics (HOSs), represents the Fourier Transform of the third-order cumulant or moment. Nonlinearity influences these cumulants, and the bispectrum captures such effects. The estimated bispectrum \(\hat{B}\ (f_{1},\ f_{2})\) is defined as:  
\[\hat{B}(f_{1},f_{2})=\frac{1}{M}\sum_{k=1}^{M}X_{k}(f_{1})X_{k}(f_{2})X_{k}^{* }(f_{1}+f_{2})\approx E\Big{\{}X(f_{1})X(f_{2})X^{*}(f_{1}+f_{2})\Big{\}} \tag{3}\]  
The expectation operation holds significant importance in this context, particularly for the QPC detection.