## 3 Proposed FRESH-Filter Receivers
### Cyclic Frequency Offset Estimation and Compensation  
Ideally, the FIR filter at each branch of the FRESH filter behaves as an LPTV filter, which can be expressed as  
\[\left(\mathbf{w}_{k}[n]\right)_{i}=e^{j2\pi\lambda_{k}n}\left(\mathbf{w}_{o,k }[n]\right)_{i} \tag{13}\]  
Figure 4: Details in the proposed adaptive FRESH filter with cyclic frequency offset estimation and compensation. The thin and thick lines with arrows respectively represent the passing-through signals as real and complex.  
where \(\left(\mathbf{w}_{k}[n]\right)_{i}\) denotes the \(i\)-th coefficient of the \(k\)-th branch of the FRESH filter and \(\mathbf{w}_{o,k}[n]\) represents the optimal coefficients of the \(k\)-th branch of the FRESH filter when there is no cyclic frequency offset. Note that the cyclic frequency offset will induce the term \(e^{j2\pi\Delta_{k}n}\) with the result that \(\left(\mathbf{w}_{k}[n]\right)_{i}\) varies periodically. We can extract offset information by calculating the following ratio:  
\[\begin{array}{ll}r_{k,i}&=\frac{\left(\mathbf{w}_{k}[n_{2}]\right)_{i}}{ \left(\mathbf{w}_{k}[n_{1}]\right)_{i}}\\ &=e^{j2\pi\Delta_{k}N_{\tau}}\\ &=\cos\left(2\pi\Delta_{k}N_{\tau}\right)+j\sin\left(2\pi\Delta_{k}N_{\tau} \right)\end{array} \tag{14}\]  
where \(N_{\tau}=n_{2}-n_{1}>0\) is the interval during which the phase variation of the coefficient is observed. To obtain a better estimation of \(\Delta_{k}\), we can select the branch that exhibits the largest phase variation as follows:  
\[\widetilde{R}_{k}=\max_{k,i}\left|R_{k,i}\right| \tag{15}\]  
where \(R_{k,i}=\mathbb{I}\mathrm{m}\{r_{k,i}\}/\ \mathbb{R}\mathrm{e}\{r_{k,i}\}= \tan\left(2\pi\Delta_{k}N_{\tau}\right)\). Thus, the estimate of cyclic frequency offset at the \(k\)-th branch of the FRESH filter can be expressed as  
\[\widehat{\Delta}_{k}=\frac{1}{2\pi N_{\tau}}\tan^{-1}\left(\widetilde{R}_{k}\right) \tag{16}\]  
As shown in Figure 4, estimation \(\widehat{\Delta}_{k}\) can be used to compensate for the unknown cyclic frequency offset. This relaxes the required tracking ability of the adaptive FRESH filter. To quantify the effectiveness of the estimation method, we propose a performance metric called \"cyclic frequency relative error\" (CFRE), which is defined as follows:  
\[\begin{array}{ll}\mathrm{CFRE}_{\text{CSIN}}&=\frac{\alpha_{1}^{\prime}- \widehat{\Delta}_{1}}{\alpha_{1}^{\prime}}\times 100\%\\ &=\frac{\left(\alpha_{1}+\Delta_{1}\right)-\left(\alpha_{1}+\widehat{\Delta}_ {1}\right)}{\alpha_{1}+\Delta_{1}}\times 100\%\\ &=\frac{\Delta_{1}-\widehat{\Delta}_{1}}{\alpha_{1}+\Delta_{1}}\times 100\% \end{array} \tag{17}\]  
where \(\alpha_{1}^{\prime}\) is the fundamental cyclic frequency susceptible to cyclic frequency offset, and \(\widehat{\alpha}_{1}\) is the estimate of \(\alpha_{1}^{\prime}\). For the mains frequency of 60 Hz and frequency offset of 4%, the corresponding fundamental cyclic frequency \(\alpha_{1}^{\prime}=120\times 1.04=124.8\) Hz, i.e., \(\Delta_{1}=4.8\) Hz.  
Note that observation interval \(N_{\tau}\) should be selected carefully to avoid phase ambiguity caused by the arctangent function \(\tan^{-1}\left(\cdot\right)\). Assuming that the variation in the frequency of the mains voltage is less than \(\epsilon\), the maximum cyclic frequency of the \(k\)-th branch is \(2k(1+\epsilon)/T_{AC}\). Thus, the corresponding maximum cyclic frequency offset can be expressed as follows:  
\[\left|\Delta_{k,\max}\right|=2k\epsilon/T_{AC} \tag{18}\]  
Combining (16) with (18), we find that the maximum observation period \(N_{\tau}\) must satisfy the following equation:  
\[N_{\tau}\leq\left|\frac{1}{2\pi\Delta_{k,\max}}\tan^{-1}\left(\widetilde{R}_{ k}\right)\right|\leq\frac{T_{AC}}{8\epsilon k} \tag{19}\]