# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 3 Basics of the Bispectrum
Although all three harmonic components appear in the power spectrum, in the bispectrum, there is only a peak at the coordinates determined by the two initial harmonics. This is explained in the subsequent sections in the paper:  
_There must exist a significant quadratic coupling nonlinearity, which applies to almost all periodicities that are found in the sunspot time series in such a way that intensive coupled harmonics appear only for specific summations, namely \(f_{1}+f_{2}=f_{140}\), \(f_{1}+f_{2}=f_{340}\) and \(f_{1}+f_{2}=f_{1200}\)._  
These couplings become obvious from the intense spikes that appear in the diagonal ellipses in the Figures in the Section 5. In summary, for three signals, where one has a phase resulting from the addition or subtraction of the phases of the other two, coupling edges appear at the coordinates corresponding to the two lowest frequencies of these signals. In the field of signal processing, the phenomenon of coupling edges, as described in the context of signal addition and subtraction, highlights the complex relationships between signal phases. When the signals \(x_{1}\) and \(x_{2}\) are combined to yield \(x_{3}\), the coupling edge appears at specific coordinates, revealing a harmonic interaction between the frequencies \(\lambda_{1}\) and \(\lambda_{2}\). This coupling effect, visualized in Figure 4b, provides information on the synchronization of the signal phases. Conversely, in the signal subtraction scenario, where its phase, \(\varphi_{3}\), results from the differentiated interaction between \(\varphi_{1}\) and \(\varphi_{2}\), the resulting coupling edge is realized in coordinates determined by \((\lambda_{1},\lambda_{2}-\lambda_{1})\). This complex dance of frequencies, exemplified in Figure 5, shows the delicate balance required to de-phase the signal to yield a coherent result. Moreover, the interesting surprise is that, in the case of signal subtractions, the results mirror those of signal additions, which highlights the symmetric nature of these signal functions. This symmetric behavior, represented by coupling spikes at fixed coordinates, speaks to the underlying mathematical elegance that governs signal interactions. It is important to note that these observations extend to three-signal scenarios, where the phase relationships are more complex. In cases where the phase of one signal is obtained by adding or subtracting the phases of two other signals, the coupling spikes appear fixed at coordinates corresponding to the two smaller phases. This generalization adds a level of universality to the coupling phenomenon, demonstrating its applicability to various signal processing scenarios. In conclusion, exploring the coupling edges in signal processing reveals a rich fabric of relationships where the addition or subtraction of signal phases orchestrates complex patterns at specific frequency coordinates. These insights not only deepen our understanding of signal behavior, but also open avenues for further explorations of the colorful field of signal processing. It is worth mentioning that the strength of the coupling is expressed by the function of bicoherence [5; 21], which is actually the bispectrum normalized by the amplitude, i.e., the strength of the spectral lines that contribute to the value of the bispectrum at the specific position. Most of the time, the value of bicoherence is large, which means that the coupling is strong, regardless of the strength of the bispectrum.