## 3 Matched Filter Parallel Interference Cancellation  
In this section, we present a MF-PIC approach to mitigate the ICI in OFDM systems. The banded structure is employed to reduce the complexity of MF-PIC in the matched filtering stage and the cancellation stage. This is because the elements of \(\mathbf{H}_{D}\) outside the shaded area are omitted for complexity reduction. The LLR calculation of MF-PIC is also discussed as follows. The ICI terms are mostly contributed by \(2Q\) adjacent subcarriers, as illustrated in Figure 2 and reported in [4]. Hence, the residual ICI outside the band is considered as noise. The matched filtered signals are expressed as follows:  
\[\begin{split}\mathbf{\hat{y}}&=\mathbf{H}_{D}^{H} \mathbf{y}\\ &=\mathbf{H}_{D}^{H}\mathbf{H}_{df}\mathbf{s}+\mathbf{H}_{D} \mathbf{v}\\ &=\mathbf{R}_{Df}\mathbf{s}+\mathbf{\tilde{v}},\end{split} \tag{6}\]  
where the vector \(\mathbf{\hat{y}}=[\hat{y}_{0},\ldots,\hat{y}_{k},\ldots,\hat{y}_{N_{s}-1}]^{T}\) represents the MF outputs. We consider the quadrature phase shift keying (QPSK) here to simplify the exposition, even though we remark that it is straightforward to generalize the LLR processing to other constellations. Hence, the LLR values of \(s_{k}(i),i=1,2\) for the channel decoder and the soft symbol estimates for iterative interference cancellation can be computed as [16]:  
\[\mathbf{L}(s_{k}(1))=\ln\frac{\text{Pr}(s_{k}(1)=+1|\hat{y}_{k})}{\text{Pr}(s _{k}(1)=-1|\hat{y}_{k})}=\frac{4\Re(\hat{y}_{k})}{\sigma_{\hat{v}}^{2}} \tag{7}\]  
Figure 2: The matrix representations of the banded channel matrix \(\mathbf{H}_{D}\), the \(k\)-th truncated channel matrix \(\mathbf{H}_{k}\) and the reduced channel matrix in the blue square.  
Accordingly, \(\text{L}(s_{k}(1)=\frac{4\Im(\hat{y}_{k})}{\sigma_{\bar{v}}^{2}}\), where the quantity \(s_{k}(i)\) denotes the \(i\)-th bit of the symbol \(s_{k}\) at the \(k\)-th subcarrier, and \(i=1,2\), because QPSK symbols carry two information bits. Subsequently, \(\text{L}(s_{k}(i))\) from Equation (7) is deinterleaved and then fed to the channel decoder as the _a priori_ LLRs. The extrinsic LLRs of \(\text{L}(s_{k}(i))\) can be obtained from the channel decoder, and then, the soft symbol estimates \(\hat{s}_{k}\) are fed back for soft interference cancellation after the interleaver, as given in Equation (8). The received data vector after cancellation is given by:  
\[\mathbf{\bar{y}}=\mathbf{\hat{y}}-\mathbf{\tilde{R}}_{Df}\mathbf{\hat{s}}, \tag{8}\]  
where \(\mathbf{\bar{y}}=[\bar{y}_{0},\ldots,\bar{y}_{k},\ldots,\bar{y}_{N_{s}-1}]^{T },\ \mathbf{\hat{s}}=[\hat{s}_{0},\ldots,\hat{s}_{k},\ldots,\hat{s}_{N_{s}-1}]^{T}\), \(\mathbf{\tilde{R}}_{Df}\) is the matrix with zero diagonal elements, and the new LLR value \(\text{L}(s_{k}(i))\) can be re-computed in Equation (7) to replace \(\hat{y}_{k}\) by \(\bar{y}_{k}\). According to [PERSON]'s theorem [26] and Equation (7), the soft symbol estimate of the \(k\)-th subcarrier for interference cancellation is given by:  
\[\hat{s}_{k}=\tanh(\text{L}(s_{k}(1))/2)+\tanh(\text{L}(s_{k}(2))/2)i, \tag{9}\]  
where \(\sqrt{-1}=i\). Therefore, the MF-PIC can cancel the ICI in one shot once the _a priori_ LLR from the channel decoder is known and then fed the new LLR after the ICI cancellation to the channel decoder.