## 2 System Model
### OFDM Signal Cyclic Correlation Analysis  
The cyclic prefix in an OFDM signal is intricately designed to reduce the impact of multipath fading. This prefix is a repetition of a segment of the signal within the OFDM symbol, resulting in a correlation between the cyclic prefix and the informative part of the signal. Furthermore, in the context of single-carrier signals, the transmission symbols are independently and identically distributed, such that correlation exists solely when there is no delay. Utilizing this property, the modulation type of an OFDM signal can be identified.  
In the \(k\)-th OFDM symbol, the auto-correlation of the time-domain data \(c_{k,l}\) at the delayed sample point \(n_{\tau}(0\leq n_{\tau}\leq N)\) equals  
\[\begin{array}{c}E(c_{k,l}c_{k,l-n_{\tau}}^{*})=1/NE[\sum_{n=0}^{N-1}a_{k,n}e ^{(j2\pi(1-D)n/N)}\cdot\sum_{n=0}^{N-1}a_{k,n}^{*}e^{(-j2\pi(1-n_{\tau}-D)n/N)} ]=\\ \frac{1}{N}\sum_{n=0}^{N-1}E(a_{k,n}a_{k,n}^{*})\exp[j2\pi(l-D)n/N-\\ j2\pi(l-n_{\tau}-D)n/N]=\frac{\sigma_{2}^{2}}{N}\sum_{n=0}^{N-1}\exp(j2\pi n_{ \tau}n/N).\end{array} \tag{7}\]  
In practice, this function is approximated as  
\[E(c_{k,l}c_{k,l-n}^{*})=\begin{cases}\sigma_{a}^{2},&n_{\tau}=0,N,\\ 0,&others.\end{cases} \tag{8}\]  
The correlation properties introduced by the cyclic prefix, disregarding the initial delay, frequency offset, and noise, result in the auto-correlation function of the received signal being  
\[\begin{array}{c}R_{r}(t,\tau)=E[r(t)\cdot r^{*}(t-\tau)]=\\ E\bigg{[}\sum_{k}\sum_{l=0}^{G-1}c_{k,l}g(t-lT_{c}-kT_{s})\cdot\sum_{k}\sum_{l =0}^{G-1}c_{k,l-n}^{*}g^{*}(t-lT_{c}-kT_{s}-\tau)]=\\ \sum_{k}E\bigg{[}\sum_{l=0}^{G-1}c_{k,l}g(t-lT_{c}-kT_{s})\cdot\sum_{l =0}^{G-1}c_{k,l-n}^{*}r^{g^{*}}(t-lT_{c}-kT_{s}-\tau)],\end{array} \tag{9}\]  
wherein the delay \(\tau=n_{\tau}T_{c}\).  
As indicated by Equation (9), the cyclic period of the OFDM signal has two instances, which can be expressed as  
\[R_{r}(t,\tau)=\begin{cases}\sigma_{a}^{2}\sum_{k}\sum_{l=0}^{G-1}g(t-lT_{c}- kT_{s})\cdot g^{*}(t-lT_{c}-kT_{s}-\tau),&|\tau|<T_{c},\\ \sigma_{a}^{2}\sum_{k}\sum_{l=N}^{G-1}g(t-lT_{c}-kT_{s})\cdot g^{*}(t-lT_{c}- kT_{s}-\tau),&|\tau_{N}|<T_{c},\\ 0,&\tau=others,\end{cases} \tag{10}\]  
wherein \(\tau_{N}=|-NT_{c}|\).  
From the above proof, it can be determined that the OFDM signal possesses second-order cyclostationary characteristics, with the OFDM signal having two cyclic periods, \(T_{c}\) and \(T_{s}\). The autocorrelation function \(R_{r}(t,\tau)\) exhibits periodicity. Based on the properties of the Fourier transform, the Fourier transform with respect to \(t\) of the autocorrelation function \(R_{r}(t,\tau)\) will result in discrete spectral lines. The positions of these spectral lines are at \(\alpha=\frac{m}{T_{c}}\) or \(\alpha=\frac{m}{T_{s}}\), where \(m\in\mathbb{Z}\). Here, \(\alpha\) is referred to as the cyclic frequency. Performing the Fourier transform of \(R_{r}(t,\tau)\) with respect to \(t\) yields:  
\[|R_{r}^{\alpha}(\tau)|=\begin{cases}\frac{\sigma_{k}^{2}}{T_{s}}|\frac{\sin[ \pi\alpha(T_{c}-|\tau|)]}{\pi\alpha}|,&\alpha=m/T_{c},|\tau|\leq T_{c},m\in Z, \\ \frac{\sigma_{k}^{2}}{T_{s}}|\frac{\sin(\pi\alpha T_{c})}{\sin[\pi\alpha T_{c} )}|\cdot|\frac{\sin[\pi\alpha(T_{c}-|\tau_{N}|)]}{\pi\alpha}|&,\alpha=m/T_{c},| \tau_{N}|\leq T_{c},m\in Z,\\ 0,&others.\end{cases} \tag{11}\]