## 5 Multi-Feedback Generation Mechanism
### Reduced Channel Matrix for Multi-Feedback Interference Cancellation  
Conventionally, the more interference is mitigated, the better the SINR performance achieved. However, the output SINR of a matched filter with a reduced size \(Q=D\) channel matrix may not suffer significant performance loss compared to that with a normal-sized \(Q=2D\) channel matrix. According to the statements in [19], the truncated channel matrix can also be reduced to the matrix in blue, as shown in Figure 2. The representation of the reduced channel matrix is illustrated in the blue square. It can be found that the elements in the upper-left corner and lower-right corner have been omitted compared to the matrix with \(Q=2D\)  
To show the pre-processing SINR difference between the reduced size channel matrix \(Q=D\) and the normal-sized channel matrix \(Q=2D\), the pre-processing SINR obtained by observing \(\tilde{\mathbf{y}}_{k}^{(b)}\) in Equation (12) is:  
\[ \begin{split}\text{SINR}_{Q}^{(b)}&\approx\frac{E_{ s}\mathbb{E}\{\mathbf{h}_{k}^{H}\mathbf{h}_{k}\}}{\sum_{q=k-Q,q\
eq k}^{k+Q} \mathbb{E}\{\mathbf{h}_{k}^{H}\mathbf{h}_{\bar{k}}\}|s_{\bar{k}}-s_{\bar{k}}^ {(b)}|^{2}\bar{P}(s_{\bar{k}}^{(b)})+\mathbb{E}\{\mathbf{\tilde{n}}_{k}^{H} \mathbf{\tilde{n}}_{k}\}}\\ &\approx\frac{\tilde{E}_{s}}{\sum_{\bar{k}=k-Q,\bar{k}\
eq k}^{k+ Q}\mathbb{E}\{\mathbf{h}_{\bar{k}}^{H}\mathbf{h}_{\bar{k}}\}|s_{\bar{k}}-s_{\bar{k}}^ {(b)}|^{2}\bar{P}(s_{\bar{k}}^{(b)})+\mathbb{E}\{\mathbf{\tilde{n}}_{k}^{H} \mathbf{\tilde{n}}_{k}\}},\end{split}  \tag{27}\]  
where \(\bar{P}(s_{q}^{(b)})=\text{Pr}(s_{q}\
eq s_{q}^{(b)})\), and the quantity \(E_{s}\) denotes the average transmit signal power. Note that the exact symbol error probability \(\bar{P}(s_{q}^{(b)})\) in a coded system cannot be easily obtained. The residual interference term inside the band is omitted for simplicity. Because the first term in the denominator of Equation (27) will be significantly suppressed at high SNR by the MBMF-SIC, the entire ICI power \(\sigma_{c0}^{2}\) and ICI power \(\sigma_{cQ}^{2}\) outside \(Q\) central terms can be evaluated by an upper bound [19,28] as:  
\[ \begin{split}\sigma_{c0}^{2}&\approx\frac{E_{s}}{12}(2 \pi T_{sa}N_{s})^{2}(\sum_{l=0}^{L-1}\sigma_{l}^{2}\sigma_{Dl}^{2})-\frac{E_{s }}{240}(2\pi T_{sa}N_{s})^{4}(\sum_{l=0}^{L-1}\sigma_{l}^{2}\sigma_{Dl}^{4})\\ \sigma_{cQ}^{2}&\approx\sigma_{c0}^{2}(1-\frac{6}{ \pi^{2}}\sum_{q=1}^{2D}\frac{1}{q^{2}})\end{split}  \tag{28}\]  
where \(\sigma_{l}^{2}\) denotes the variance of the \(l\)-th channel tap. \(\sigma_{Dl}^{2}=\int_{-f_{d}}^{f_{d}}P_{l}(f)f^{2}df\), and \(P_{l}(f)\) is the Doppler power spectral density (PSD) of the \(l\)-th path. \(\sigma_{Dl}^{2}=f_{d}^{2}/2\), and \(\tilde{E}_{s}=E_{s}-\sigma_{c2D}^{2}\) denotes the transmit signal power inside the band with a normal channel matrix \(Q=2D\). The quantity \(T_{sa}=\frac{T_{\text{OFDM}}}{N_{s}}\). For \(Q=2D\), Equation (27) can be rewritten as:  
\[\text{SINR}_{Q}^{(b)}\approx\frac{Q\tilde{E}_{s}}{\mathbb{E}\{\mathbf{\hat{n} }_{k}^{H}\mathbf{\hat{n}}_{k}\}} \tag{29}\]  
Without the residual interference inside the band, the SINR for the \(b\)-th feedback is mainly determined by the noise plus the ICI outside the band. Furthermore, the power of the signal \(\tilde{E}_{s}\) for \(Q=D\) will be identical to that for \(Q=2D\), because the power of the desired signal is derived from the same channel coefficients \(\mathbf{h}_{k}\) for \(Q=D\) or for \(Q=2D\). Similar to [19], \(\text{SINR}_{2D}^{(b)}>\text{SINR}_{D}^{(b)}\) if no pre-whitener is employed.