## 4 Proposed Multi-Feedback Interference Cancellation
### Problem Formulation and Solution  
The received signal in Equation (5) after subtracting the soft symbol estimates is given by:  
\[\mathbf{\tilde{y}}_{k}=\mathbf{h}_{k}s_{k}+\underbrace{\sum_{\bar{k}=k-Q, \bar{k}\
eq k}^{k+Q}\mathbf{h}_{\bar{k}}(s_{\bar{k}}-\hat{s}_{q})}_{\text{ soft residual interference inside band}}+\underbrace{\sum_{\bar{k}\
eq[k-Q,k+Q]}\mathbf{h}_{\bar{k}}s_{\bar{k}}+ \mathbf{v}_{k}}_{\mathbf{\hat{v}}}, \tag{10}\]  
where \(\mathbf{\hat{s}}_{\bar{k}}=[s_{k-Q},\ldots,\hat{s}_{k-1},\hat{s}_{k+1},\ldots, s_{k+Q}]^{T},\hat{s}_{q}\in\mathbf{\hat{s}}_{\bar{k}}\) denotes the soft symbol vector omitting the desired symbol at the \(k\)-th subcarrier \(\hat{s}_{k}\), which are inside the band obtained by previously-detected subcarriers or the LLR from the channel decoder. The vector \(\mathbf{\tilde{v}}\) denotes the interference outside the band plus noise. The residual interference inside the band can be approximated by the summation of received signals \(\mathbf{\tilde{y}}^{(b)},\ b=1,2,\ldots,B\), after subtracting multiple interference symbols \(\mathbf{s}_{k}^{(b)}\) as follows:  
\[\sum_{\bar{k}=k-Q,\bar{k}\
eq k}^{k+Q}\mathbf{h}_{\bar{k}}(s_{\bar{k}}-\hat{s} _{\bar{k}})\approx\sum_{b=1}^{B}(\mathbf{\tilde{y}}_{k}^{(b)}-\mathbf{h}_{k}s _{k}-\mathbf{\tilde{n}})\text{Pr}(\mathbf{\tilde{y}}_{\bar{k}}^{(b)}), \tag{11}\]where:  
\[\mathbf{\tilde{y}}_{k}^{(b)}=\mathbf{h}_{k}s_{k}+\underbrace{\sum_{\bar{k}=k-Q, \bar{k}\
eq k}^{k+Q}\mathbf{h}_{\bar{k}}(s_{\bar{k}}-s_{\bar{k}}^{(b)})}_{\text {hard residual interference inside band}}+\mathbf{\tilde{v}}, \tag{12}\]  
and \(\mathbf{s}_{k}^{(b)}=[s_{k-Q}^{(b)},\ldots,s_{k-1}^{(b)},s_{k+1}^{(b)},\ldots,s _{k+Q}^{(b)}]^{T}\), \(s_{k}^{(b)}\in\mathbf{s}_{k}^{(b)}\). The quantity \(B\) denotes the number of feedback candidates. Substituting Equation (11) into Equation (10), we arrive at:  
\[\mathbf{y}_{k}^{*}\approx\mathbf{\tilde{y}}_{k}-\sum_{b=1}^{B}(\mathbf{\tilde{ y}}_{k}^{(b)}-\mathbf{h}_{k}s_{k}-\mathbf{\tilde{v}})\text{Pr}(\mathbf{ \tilde{y}}_{\bar{k}}^{(b)})\approx\mathbf{h}_{k}s_{k}+\mathbf{\tilde{v}}, \tag{13}\]  
where \(\mathbf{y}_{k}^{*}\) denotes the received signal if the residual interference inside the band is perfectly removed by Equation (11), and the probability \(\text{Pr}(\mathbf{y}_{k}^{(b)})\) denotes the _a priori_ probability of received signals after interference cancellation, which are mainly determined by \(\text{Pr}(\mathbf{s}_{k}^{(b)})\). Hence, the terms \(\text{Pr}(\mathbf{y}_{\bar{k}}^{(b)})\) and \(\text{Pr}(\mathbf{s}_{\bar{k}}^{(b)})\) are interchangeable (\(\text{Pr}(\mathbf{\tilde{y}}_{k}^{(b)})\approx\text{Pr}(\mathbf{s}_{\bar{k} }^{(b)})\)). If we assume no residual interference inside the band in Equation (13) after a rearrangement, it becomes:  
\[\mathbf{\tilde{y}}_{k}-\sum_{b=1}^{B}\mathbf{\tilde{y}}_{k}^{(b)}\text{Pr}( \mathbf{\tilde{y}}_{\bar{k}}^{(b)})=\mathbf{h}_{k}s_{k}+\mathbf{\tilde{v}}- \mathbf{h}_{k}s_{k}\sum_{b=1}^{B}\text{Pr}(\mathbf{\tilde{y}}_{\bar{k}}^{(b )})-\mathbf{\tilde{v}}\sum_{b=1}^{B}\text{Pr}(\mathbf{\tilde{y}}_{\bar{k}}^{( b)}), \tag{14}\]  
where \(\sum_{b=1}^{B}\text{Pr}(\mathbf{\tilde{y}}_{\bar{k}}^{(b)})=\sum_{b=1}^{B} \text{Pr}(\mathbf{s}_{\bar{k}}^{(b)})\approx 1\), if all possible combinations of symbol vectors \(\mathbf{s}_{k}^{(b)}\) are considered in the summation. Therefore, \(\mathbf{\tilde{y}}_{k}-\sum_{b=1}^{B}\mathbf{\tilde{y}}_{k}^{(b)}\text{Pr}( \mathbf{\tilde{y}}_{\bar{k}}^{(b)})\approx 0\). Note that the conditional probability of \(\mathbf{\tilde{y}}_{k}^{(b)}\) given \(\mathbf{s}_{\bar{k}}^{(b)}\) and \(\mathbf{\tilde{y}}_{k}\) given \(\mathbf{\hat{s}}_{\bar{k}}\) in Equation (14) will be mutually exclusive and conditionally independent, if the residual interference term dominates.