## 2 Theory  
### Cyclic Spectral Analysis  
The concept of cyclostationary signals has been around for almost 40 years [13; 14]. In recent years, the theory of cyclostationary signals has been increasingly used in the field of mechanical fault detection. Vibration signals extracted from bearings and rotating machines are non-stationary signals containing some hidden periodicities in their background. To extract the periodicity many signal processing tools are available. The cyclostationary signal analysis technique is one of the most powerful techniques from those currently available [15]. The statistics of cyclostationary signals have periodicity with respect to time [10]. Currently, second-order cyclostationarity which is known as cyclic correlation [10] has an important role in practical application. Despite the strengths of this technique and its popularity in signal processing of rotating machinery, it has not attracted much attention in the field of wind turbine condition monitoring and signal processing. To describe this technique we start with a random process.  
In general, a random process will have a time-varying autocorrelation:  
\[R_{x}(t,\tau)=E\left\{x\left(t+\frac{\tau}{2}\right)x^{*}\left(t-\frac{\tau}{ 2}\right)\right\} \tag{1}\]  
where \(E\{\dots\}\) denotes the mathematical expectation operator, and \(\tau\) is the time delay. If the autocorrelation is considered periodic with a period of \(T_{0}\), the ensemble average can be estimated with a time average as below:  
\[R_{x}(t,\tau)=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N}x\left(t+nT_{0}+\frac{ \tau}{2}\right)x^{*}\left(t+nT_{0}-\frac{\tau}{2}\right) \tag{2}\]  
The autocorrelation function in Equation (1) can be written in form of Fourier series due to its periodicity  
\[R_{x}(t,\tau)=\sum_{n}R_{x}^{n}(\tau)e^{2\tau\alpha t} \tag{3}\]where \(\alpha=m/T_{0}\) and \(m\in\mathbb{Z}\). Combining the above equation with the Equation (2), the Fourier coefficients can be written as below  
\[\begin{split} R_{x}^{\alpha}(\tau)&=\lim_{T\to \infty}\frac{1}{T}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}x\left(t+\frac{\tau}{ 2}\right)x^{*}\left(t-\frac{\tau}{2}\right)e^{j2\pi at}\ dt\\ &=\left\langle x\left(t+\frac{\tau}{2}\right)x^{*}\left(t-\frac{ \tau}{2}\right)e^{j2\pi at}\right\rangle_{t}\end{split} \tag{4}\]  
where \(R_{x}^{\alpha}(\tau)\) is known as cyclic autocorrelation function with \(\alpha\) being the cyclic frequency. The \(\left\langle\dots\right\rangle\) symbol represents the time averaging operator. Next, to obtain the cyclic spectrum, the Fourier transform can be applied to the cyclic autocorrelation function with respect to the time delay \(\tau\), yielding  
\[S_{x}^{\alpha}(f)=\int_{-\infty}^{\infty}R_{x}^{\alpha}(\tau)e^{-j2\pi/\tau} \ d\tau \tag{5}\]  
that is known as cyclic spectrum or the spectral correlation function.  
In this work the [PERSON]'s averaged periodogram technique has been used, that due to its high computational efficiency is one of the most common estimators for cyclic spectrum [16].  
The cyclic coherence that measures the strength of the correlation between spectral components distanced by cyclic frequencies can be calculated using [16]  
\[C_{x}^{\alpha}(f)=\frac{S_{x}^{\alpha}(f)}{\sqrt{S_{x}^{0}(f+\alpha/2)S_{x}^{0 }(f-\alpha/2)}} \tag{6}\]