# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 3 Basics of the Bispectrum
It implies \"ensemble averaging\" for an estimate: when phases are random, the bispectrum tends towards zero, whereas if phases are coupled, it does not. Case of frequency and phase summation:  
Consider three signals, \(x_{1}=cos(\lambda_{1}x+\varphi_{1})\), \(x_{2}=cos(\lambda_{2}x+\varphi_{2})\), and \(x_{3}=cos((\lambda_{1}+\lambda_{2})x+(\varphi_{1}+\varphi_{2}))\), where \(\lambda_{1}<\lambda_{2}\). The phase of signal \(x_{3}\) arises from the sum of the phases of the signals \(x_{1}\) and \(x_{2}\), indicating an expected coupling edge at the coordinates \((\lambda_{1},\lambda_{2})\) of the bispectrum. For instance, if \(\lambda_{1}=4\) and \(\lambda_{2}=16\), coupled components can  
Figure 3: The symmetrical structure of the bispectrum. The blue region is the space that we have to investigate in order to recognize the frequency coordinates that contribute to the formation of coupled components. Frequencies in both axes are normalized in the sampling frequency \(F_{3}\) or \(2\pi\). appear at \(\lambda_{3}=20\) (see Figure 4a). The coupling edge is anticipated at coordinates \((4,16)\) in the bispectrum, as is demonstrated in Figure 4b in the red region. Case of frequency and phase subtraction:  
Consider three signals, \(x_{1}=cos(\lambda_{1}x+\varphi_{1})\), \(x_{2}=cos(\lambda_{2}x+\varphi_{2})\), and \(x_{3}=cos((\lambda_{2}-\lambda_{1})x+(\varphi_{2}-\varphi_{1}))\), where \(\lambda_{1}<\lambda_{2}\). The phase of the signal \(x_{3}\) results from  
Figure 4: (**a**). The three harmonics that appear at positions \(4\), \(16\), and \(20\) (dc is at position \(1\)). From the power spectrum, it is not possible to distinguish that the \(20\)th harmonic is the result of coupling the other two. The \(64\)th harmonic is at position \(65\), which coincides with half of the sampling frequency, \(F_{s}/2\) or \(\pi\). (**b**). Since (1,1) is the position of \(\mathrm{dc}\), the coupling edge is present at coordinates \((5,17)\), i.e., for coupled frequencies \(4\) and \(16\). The created \(20\)th harmonic does not present any footprint in the region of the bispectrum, which is enclosed in the red line. On both axes, the \(64\)th harmonic is at position \(65\), which coincides with half of the sampling frequency, \(F_{s}/2\) or \(\pi\). the subtraction of the phases of the signals \(x_{1}\) and \(x_{2}\), suggesting that a coupling edge is anticipated at the coordinates \((\lambda_{1},\lambda_{2}-\lambda_{1})\). For example, if we have \(\lambda_{1}=4\) and \(\lambda_{2}=16\), a coupling edge is expected at coordinates \((4,12)\). See Figure 5.  
Figure 5: (**a**). The three harmonics that appear at positions \(4\), \(12\), and \(16\) (dc is at position \(1\)). From the power spectrum, it is not possible to distinguish the coupled harmonic. The \(64\)th harmonic is at position \(65\), which coincides with half of the sampling frequency, \(F_{s}/2\) or \(\pi\). (**b**). Since (1,1) is the position of dc, the coupling edge is present at coordinates \((5,13)\), i.e., for coupled frequencies \(4\) and \(12\). The original \(16\)th harmonic does not present any footprint in the region of the bispectrum, which is enclosed in the red line. On both axes, the \(64\)th harmonic is at position \(65\), which coincides with half of the sampling frequency, \(F_{s}/2\) or \(\pi\). It is worth noting that, in the case of a subtraction, the result aligns with the case of adding signals \(x_{1}\) and \(x_{3}\). In other words, if initially we had the signals \(x_{1}=cos(\lambda_{1}x+\varphi_{1})\), and \(x_{3}=cos((\lambda_{1}+\lambda_{2})x+(\varphi_{1}+\varphi_{2}))\), where \(\lambda_{1}<\lambda_{2}\), the phase of the signal \(x_{2}=cos(\lambda_{2}x+\varphi_{2})\) can be derived by adding the phases of the signals \(x_{1}\) and \(x_{3}\); then, a coupling edge is expected at the coordinates \((\lambda_{1},\lambda_{2}-\lambda_{1})\). Quadratic phase coupling is the most well-understood [21; 25], non-linear procedure that creates a periodicity as the summation of two other harmonics.