## 3 Cyclostationarity  
### LCL-PTV Optimal FRESH Filtering  
A complex-valued zero-mean signal is said to exhibit wide-sense (second-order) cyclostationarity (WSCS) with _cycle frequency_ (CF) \(\alpha\
eq 0\) if and only if the cyclic autocorrelation function (CAF) denoted by the following mathematical equation [14; 15; 22]  
\[R_{\mathrm{xx}}^{\alpha}=<E\Big{[}x\Big{(}t+\frac{\tau}{2}\Big{)}x\cdot\Big{(} t-\frac{\tau}{2}\Big{)}\Big{]}e^{-\mathrm{j}2\pi\alpha t}=<R_{\mathrm{xx}}(t,\tau)e^{- \mathrm{j}2\pi\alpha t}> \tag{14}\]  
exists and is not zero for some values of the lag parameter \(\tau\). In (14), \(<\)-\(>\) and \(E\) [.] denote infinite-time and ensemble average, respectively, and \(R_{\mathrm{xx}}\)(t,\(\tau\)) is the statistical autocorrelation function of x(t). The discrete time domain CAF can be expressed by the following mathematical equation  
\[R_{\mathrm{xx}}^{\alpha_{\mathrm{k}}}(1)=\left(\frac{1}{\mathrm{N}_{0}}\right) \sum_{n=0}^{\mathrm{N}_{0}-1}R_{\mathrm{xx}}(n,1)e^{-\mathrm{j}2\pi\alpha_{ \mathrm{k}}n} \tag{15}\]  
where \(\alpha_{\mathrm{k}}=\mathrm{k}/\mathrm{N}_{0}\), \(\mathrm{k}=0\), \(1\), \(\ldots\), \(\mathrm{N}_{0}-1\), are referred to as the cyclic frequencies and \(\mathrm{N}_{0}\) is the cyclostationary period. The spectral correlation function (SCF), \(S_{\mathrm{xx}}^{\alpha_{\mathrm{k}}}\)(f), can be computed by the Fourier Transform of \(R_{\mathrm{xx}}^{\alpha_{\mathrm{k}}}(n)\) and can be expressed mathematically by  
\[S_{\mathrm{xx}}^{\alpha_{\mathrm{k}}}(f)=\sum_{n=0}^{\mathrm{N}_{0}-1}R_{ \mathrm{xx}}^{\alpha_{\mathrm{k}}}(n)e^{\mathrm{j}2\pi\mathrm{fn}} \tag{16}\]  
A FRESH filter is an optimal widely linear filter which exploits the spectral redundancy in the input signal to extract the desired cyclostationary signal. LCL-PTV filters belong to the class of FRESH filters, which exploit both the spectral redundancy and conjugate spectral redundancy to derive an estimate of the desired signal. The output signal \(\hat{y}(n)\) for the input mixed signal \(r(n)\) (comprising desired signal \(y(n)\) and some unknown interference signal) is given by [19; 20; 21; 22; 28]  
\[\hat{y}(n)=\sum_{1=-\infty}^{\infty}h(n,1)\cdot r(l)+\sum_{m=-\infty}^{\infty} h^{c}(n,m)\cdot r^{*}(m) \tag{17}\]  
The impulse responses of the FRESH filter \(h(n,1)\) and \(h^{c}(n,l)\) can be expressed as  
\[ \begin{array}{l}h(n,l)=\sum_{p}h_{p}(n-1)\cdot e^{\mathrm{j}2\pi\alpha_{p}l} \\ h^{c}(n,l)=\sum_{q}h_{q}(n-1)\cdot e^{\mathrm{j}2\pi\alpha_{q}^{c}l}\end{array}  \tag{18}\]Relationship between the input \(\tau(n)\) and estimated output \(\hat{y}(n)\) of the FRESH filter is given by  
\[\hat{y}(n)=\sum_{l=-\infty}^{\infty}(\sum_{p}h_{p}(n-l)\cdot r_{p}(l)+\sum_{q}h_{ q}(n-l)\cdot r_{q}^{c}(l)) \tag{19}\]  
where \(r_{p}(n)=r(n)\cdot e^{-|2\pi\alpha_{p}n}\), and \(r_{q}^{c}(n)=r^{*}(n)\cdot e^{-|2\pi\alpha_{q}^{c}n}\). Based on (17)-(19), the system operation of LCL-FRESH filter can be represented by  
\[\hat{y}(n)=\sum_{p}h_{p}(n)\otimes\left[r(n)\cdot e^{|2\pi\alpha_{p}n}\right]+ \sum_{q}h_{q}(n)\otimes\left[r^{*}(n)\cdot e^{-|2\pi\alpha_{q}^{c}n}\right] \tag{20}\]  
From Equation (22), it is evident that the LCL-FRESH filter can be regarded as two parallel FIR filter-bank systems. The 1 st filter-bank and 2 nd filter-bank are driven by the received signal and its conjugated version respectively. The outputs of the 2 filter-banks are added together to form the estimated version, \(\hat{y}(n)\) of the desired SOI. The frequency shifts represented by \(e^{|2\pi\alpha_{p}n}\) are used for aligning the spectral redundancies within \(r_{p}(n)\), which are combined by the 1 st filter-bank, \(\{h_{p}(n)\}\). Similarly, the frequency shifts represented by \(e^{-|2\pi\alpha_{q}^{c}n}\) are used for aligning the conjugate spectral redundancies within \(r_{q}^{c}(n)\), which are combined by the 2 nd filter-bank, \(\{h_{q}(n)\}\). The optimal FRESH filter weights can be derived by minimizing the mean squared error (MSE) between the desired cyclostationary signal and its estimate.