## 5 Sum-Rate Performance with SQ-Based CSI Feedback and RVQ-Based DI Feedback  
In Sections 3 and 5, the analyses are based on the fact that DI is perfectly known to the receivers. In this section, the quantization error resulting from the decoders' feedback is taken into consideration. Since the decoder's size is much lower than its channel matrix, we quantize the CSI and DI via SQ and RVQ, respectively, to achieve a trade-off between complexity and feedback overhead.  
After the receivers quantize and feed back the channel matrices via SQ, the transmitters determine precoders \(\mathbf{V}_{k}\) and decoders \(\mathbf{U}_{k}\) according to quantized channel matrices \(\mathbf{\hat{H}}_{ki}\). According to the IA principle, \(\mathbf{u}_{k}^{mH}\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l}=0\) holds for \((k,m)\
eq(i,l)\). Then, transmitter \(T_{k}\) continues quantizing decoding vectors \(\mathbf{u}_{k}^{m},1\leq m\leq d\) and feeds them back to receiver \(R_{k}\). In this section, we continue to adopt RVQ to quantize each decoding vector with \(B_{u}\) bits; thus, \(\mathbf{u}_{k}^{m}\) is quantized to be \(\mathbf{\hat{u}}_{k}^{m}\).  
According to the distribution of the RVQ error, when an \(N\times 1\) complex unit vector \(\mathbf{u}_{k}^{m}\) is quantized with \(B_{u}\) bits, the mean quantization error is \(E(z)=2^{B_{u}}Beta\left(2^{B_{u}},\frac{N}{N-1}\right)\). Based on the RVQ principle, \(\mathbf{\hat{u}}_{k}^{m}\) can be rewritten as  
\[\mathbf{\hat{u}}_{k}^{m}=\cos\theta\mathbf{u}_{k}^{m}+\sin\theta\mathbf{\hat{u }}_{k}^{m}, \tag{24}\]  
where \(\theta\) is the angle, \(\mathbf{\hat{u}}_{k}^{m}\) is independent of \(\theta\), and \(\mathbf{\hat{u}}_{k}^{m}\) is isotropically distributed in the \(N-1\)-dimensional subspace which is orthogonal to \(\mathbf{u}_{k}^{m}\).  
Therefore, for receiver \(R_{k}\), its inter-stream or inter-user interference can be denoted by  
\[\mathbf{\hat{u}}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{i}^{l}=\sin\theta\mathbf{ \hat{u}}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{i}^{l}+\cos\theta\mathbf{u}_{k}^{mH }\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l},\ \forall(k,m)\
eq(i,l). \tag{25}\]  
The derivation of (25) is shown as (A1) in Appendix A.  
Notice that \(\mathbf{\hat{u}}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{i}^{l}\) is a complex Gaussian random variable with zero mean and unit variance, and \(\mathbf{u}_{k}^{mH}\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l}\) can be approximated as a complex Gaussian random variable with zero mean and \(\sigma_{R}^{2}=\frac{3}{2^{B}}\) variance. Thus, we have  
\[E\left(\left|\mathbf{\hat{u}}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{i}^{l}\right|^ {2}\right)\approx 2^{B_{u}}Beta\left(2^{B_{u}},\frac{N}{N-1}\right)+\left(1-2^{B_{u} }Beta\left(2^{B_{u}},\frac{N}{N-1}\right)\right)\frac{3}{2^{2B}}. \tag{26}\]  
The derivation of (26) is shown as (A2) in Appendix A.  
Therefore, the average achievable sum-rate for IA with SQ-based CSI feedback and RVQ-based DI feedback can be approximated by  
\[R_{IA} \approx E\left(\sum_{k}\sum_{m}\log_{2}\left(1+\frac{\frac{p}{d}| \hat{\textbf{u}}_{k}^{mH}\textbf{H}_{kk}\textbf{v}_{k}^{m}|^{2}}{(Kd-1)\frac{p}{ d}\sigma_{FB}^{2}+\sigma^{2}}\right)\right)=Kd\log_{2}(e)e^{1/\rho\
eq E}E_{1}(\frac{1}{ \rho_{\textit{off}}}),\]  
where  
\[\rho_{\textit{off}}=\frac{\rho}{\rho(Kd-1)\sigma_{FB}^{2}+1}, \tag{28}\]  
and  
\[\sigma_{FB}^{2}=2^{B_{u}}Beta\left(2^{B_{u}},\frac{N}{N-1}\right)+\left(1-2^{B _{u}}Beta\left(2^{B_{u}},\frac{N}{N-1}\right)\right)\frac{3}{2^{2B}}. \tag{29}\]  
\(\rho_{\textit{off}}\) can be treated as the equivalent SNR after the quantization errors of SQ and RVQ are both introduced.