## 2 Theory and Models
### Cyclostationary Description  
The further analysis of cyclostationary characteristics based on second order two-dimensional products can be performed more effectively if the pair of independent variables forming axes spanning the time domain (\(t_{1}\), \(t_{2}\)) or frequency domain (\(f_{1},f_{2}\)) are changed for other pairs of variables. This comes after the concept of symmetric transformation, exploited by [PERSON] [38] and illustrated in [46] by example of a second order CS process. The substitution  
\[\left\{ \begin{array}{l}t_{1}=t+\tau/2,\\ t_{2}=t-\tau/2,\end{array} \right. \tag{17}\]  
is aimed at transforming the DCCF into the symmetric form of the cross correlation function (CCF)  
\[\mathcal{R}_{vi}(t,\tau)=v(t+\tau/2)i^{*}(t-\tau/2), \tag{18}\]  
which is a function of two variables: the current time, \(t\), and the lag, or time shift, \(\tau\). Similarly, after applying the change in the frequency variables  
\[\left\{ \begin{array}{l}f_{1}=f+\alpha/2,\\ f_{2}=f-\alpha/2,\end{array} \right. \tag{19}\]  
the bifrequency spectrum (14) will be transformed into the _cross spectral correlation density_ (CSCD)  
\[\mathcal{S}_{vi}(\alpha,f)=V(f+\alpha/2)I^{*}(f-\alpha/2), \tag{20}\]  
where \(\alpha\) is traditionally called a cyclic frequency, whereas \(f\) is considered as a frequency exploited in the integral Fourier transform (6). After substituting (19) into (14) and formally rearranging the summands, the analytical expression of voltage-current CSCD can be expressed in the form of a double summation, involving weighted and shifted two-dimensional Dirac delta functions \(\delta(\alpha,f)\):  
\[\mathcal{S}_{vi}(\alpha,f)=\sum_{m=-\infty}^{+\infty}\sum_{k=-\infty}^{+\infty }\mathcal{S}_{vi}^{(mk)}\delta\bigg{(}\alpha-mF,f-\bigg{(}k+\frac{((m))_{2}} {2}\bigg{)}F\bigg{)}, \tag{21}\]  
where \(\big{(}(m)\big{)}_{2}\) denotes the modulo operation, or the remainder after \(m\) is divided by 2, and complex scalars, \(\mathcal{S}_{vi}^{(mk)}\), being indexed by the pair of integers \((m,k)\), are defined via the coefficients put in the complex form of the Fourier series for the waveforms of the voltage (3) and current (4):  
\[\mathcal{S}_{vi}^{(mk)}=M_{v}^{([k+\frac{m+((m))_{2}}{2}]F)}\text{conj}\bigg{(} M_{i}^{([k+\frac{m+((m))_{2}}{2}]F)}\bigg{)}, \tag{22}\]where conj(0) marks the complex conjugation. Practically, the introduction of the modulo operation in (21) and (22) has been performed to formally reflect the fact that the Dirac delta functions at \(\alpha=mF\) turn out to be shifted along the \(f\)-axis by the step \(F/2\) for any odd integer, \(m\). Since CSCD is a generalized function whose components are present at cyclic frequencies in, at most, a countable set, A\({}_{2}=\{\alpha\mid\alpha=mF\}\), the application of the decomposition (16) to the term \(\delta(\alpha,f)\) will allow rewriting Expression (21) in the form:  
\[\mathcal{S}_{vi}(\alpha,f)=\sum_{m=-\infty}^{+\infty}\mathcal{S}_{vi}^{mF}(f) \delta(\alpha-mF), \tag{23}\]  
where \(\mathcal{S}_{vi}^{mF}(f)\) represents _spectral correlation function_ (SCF) at the cyclic frequency \(\alpha=mF\):  
\[\mathcal{S}_{vi}^{mF}(f)=\sum_{k=-\infty}^{+\infty}\mathcal{S}_{vi}^{(m,k)} \delta\bigg{(}f-\bigg{(}k+\frac{\big{(}(m)\big{)}_{2}}{2}\bigg{)}F\bigg{)}. \tag{24}\]  
_Cyclic cross-correlation function_ (CCCF), which is a function of lag parameter \(\tau\), taken at cyclic frequency \(mF\) is introduced as  
\[\mathcal{R}_{vi}^{mF}(\tau)=\frac{1}{T}\int\limits_{T}\mathcal{R}_{vi}(t,\tau )\exp\Big{(}-j2\pi\frac{m}{T}t\Big{)}dt, \tag{25}\]  
where the integration is carried out over any interval of a one period length, e.g., [0, \(T\)) or [\(-T/2\), \(T/2\)).