## 5 Multi-Feedback Generation Mechanism
### Reduced Channel Matrix for Multi-Feedback Interference Cancellation
However, the post-processing SINR of the matched filter \(\mathbf{h}_{k}^{H}\) will partially contradict the above discussion, which can be evaluated by:  
\[\text{PSINR}_{Q}^{(b)}\approx\frac{\tilde{E}_{s}^{2}/E_{s}}{\text{tr}(\mathbb{ E}\{\mathbf{h}_{k}\mathbf{h}_{k}^{H}\}\mathbb{E}\{\mathbf{c}_{k}\mathbf{c}_{k}^{H} \})+\sigma_{\bar{v}}^{2}} \tag{30}\]  
where the vector \(\mathbf{c}_{k}=[c_{k-D,k}\ c_{k-D+1,k}\ldots c_{k+D,k}]^{T}\) denotes the residual ICI outside the band for the \(k\)-th subcarrier and \(c_{k-i,k}=\frac{1}{\sqrt{E_{s}}}\sum_{k\
ot\in[k-Q-i,k+Q-i]}H_{\bar{k},k-i}s_{ \bar{k}}\), which implies that the ICI outside the band interfere with the desired signals at the neighbor subcarriers. The autocorrelation function of the channel coefficients \(\mathbf{h}_{k}\) can be derived by [19]:  
\[ \begin{split}\mathbb{E}\{h_{df}(d,k)h_{df}^{*}(d+r,k)\}& \approx 4\pi^{2}T_{sa}^{2}\left(\sum_{l=0}^{N_{h}-1}\sigma_{l}^{2} \sigma_{Dl}^{2}\right)\\ &\cdot\frac{1}{(1-e^{-j2\pi k/N_{s}})(1-e^{j2\pi(k+r)/N_{s}})} \end{split}  \tag{31}\]  
where \(d\in[-D,D]\). Note that there are singularity points when \(d=0\) or \(d=-r\) in Equation (31), and the autocorrelation function become unavailable. For the first case (\(d=-r\) and \(d\
eq 0\)), let \(d=-d\) and \(r=-r\) to avoid the singularity point. For the second case (\(d=0\)), thanks to the Hermitian symmetric property of the autocorrelation function, the value of the singularity point can be obtained by setting \(r=0\) and \(d=r\). For the third case (\(d=0\) and \(r=0\)), \(\mathbb{E}\{h_{f}(0,k)h_{f}^{*}(0,k)\}=\frac{E_{s}-\sigma_{c0}^{2}}{E_{s}}\). Additionally, the autocorrelation function of ICI can be obtained by [19]:  
\[\mathbb{E}\{c_{k-q,k}c_{k-q+r,k}^{*}\}\approx 4\pi^{2}T_{sa}^{2}\gamma(Q,r,N_{s}) \sum_{l=0}^{N_{h}-1}\sigma_{l}^{2}\sigma_{Dl}^{2} \tag{32}\]where:  
\[\gamma(D,r,N_{s})=\sum_{ \begin{subarray}{c}m\
ot\in[-D+q,D+q]\\ \cup[-D-r+q,D-r+q]\end{subarray} }\frac{1}{(1-e^{-j2\pi(k-q)/N_{s}})(1-e^{j2\pi(k +r-q)/N_{s}})}, \tag{33}\]  
and the quantity \(q=d-k\) denotes the subcarrier offset index. Further simplification of Equation (33) can be obtained with some modifications as in [19]. It can be expressed as:  
\[\gamma(D,r,N_{s})=\gamma_{0}(D,r,N_{s})-\gamma_{1}(D,r,N_{s}), \tag{34}\]  
where:  
\[\gamma_{0}(D,r,N_{s})= \begin{cases}\dfrac{N_{s}^{2}}{12},&r-q=0,\\ \dfrac{N_{s}^{2}}{2\pi^{2}(r-q)^{2}},&r-q\
eq=0,\end{cases}  \tag{35}\]  
and:  
\[\gamma_{1}(D,r,N_{s})=\sum_{ \begin{subarray}{c}k\in[-D+q,D+q]\\ \cup[-D-r+q,D-r+q]\\ \setminus\{0,q-r\}\end{subarray} }\frac{N_{s}^{2}}{4\pi^{2}k(k+r-q)} \tag{36}\]  
Hence, the autocorrelation matrix \(\mathbb{E}\{\mathbf{h}_{k}\mathbf{h}_{k}^{H}\}\) and \(\mathbb{E}\{\mathbf{c}_{k}\mathbf{c}_{k}^{H}\}\) can be computed with the aid of Equations (31) and (32), respectively. The analytical pre-processing SINR (\(\text{SINR}_{Q}^{(b)}\)) and post-processing SINR after the matched filter (\(\text{PSINR}_{Q}^{b}\)) can be calculated by Equations (29) and (30). In the following, we present some simple numerical results and analytical results of pre-processing SINR (\(\text{SINR}_{Q}^{(b)}\)) and post-processing SINR (\(\text{PSINR}_{Q}^{(b)}\)) in Figure 3, and \(f_{d}T_{\text{OFDM}}=0.65\). We can observe that the post-processing SINR (\(\text{PSINR}_{Q}^{(b)}\)), with the same size \(Q\), can be improved by the matched filter compared to pre-processing SINR (\(\text{SINR}_{Q}^{(b)}\)) at low SNR values. Furthermore, the post-processing SINR  
Figure 3: Pre-processing and post-processing signal-to-interference-plus-noise-ratio (SINR) comparison for the \(b\)-th feedback with different \(Q\). \(N_{s}=128,f_{d}T_{\text{OFDM}}=0.65\). performance gap between \(Q=D=1\) and \(Q=2D=2\) is small (around \(1\) dB at \(12\) dB SNR). We can conjecture that the error probability of \(\text{Pr}(s_{k}^{(b)}\
eq s_{k})\), with \(Q=D\) in Equation (10), will not be significantly degraded compared to that with \(Q=2D\).