## 4 Sum-Rate Performance with SQ-Based CSI Feedback
### Achievable Sum-Rate with SQ-Based CSI Feedback  
It is assumed that the entries in \(\mathbf{H}\) are all i.i.d. according to \(\mathcal{CN}(0,1)\); therefore, the real and imaginary parts of entries in \(\mathbf{H}\) are all i.i.d. real Gaussian variables with zero mean and \(1/2\) variance.  
For a real Gaussian variable obeying \(\mathcal{N}(0,\sigma^{2})\), since the probability that its value falls out of interval \([-3\sigma,3\sigma]\) is less than \(0.3\%\), we quantize this real Gaussian variable within \([-3\sigma,3\sigma]\). For simplicity, the uniform quantization method is adopted; accordingly, interval \((-3\sigma,3\sigma)\) is divided into \(M_{q}=2^{B}\) equal quantized intervals with \(B\) quantization bits, where the length of each quantized interval is \(\Delta=\frac{6\sigma}{M_{q}}=\frac{3\sigma}{2^{B-1}}\). The first quantized interval is from \(m_{0}=-3\sigma\), and the right-end point of the \(i\)-th quantized interval is \(m_{i}=m_{i-1}+\Delta,1\leq i\leq M_{q}\). The codeword for the \(i\)-th quantized interval is defined by  
\[q_{i}=\frac{m_{i-1}+m_{i}}{2},1\leq i\leq M_{q}. \tag{19}\]  
Therefore, the codebook for uniform scalar quantization in interval \([-3\sigma,3\sigma]\) with \(B\) quantization bits is \(C=\{q_{i}=(i-0.5)3\sigma/2^{B-1},1\leq i\leq 2^{B}\}\). By applying uniform quantization with \(B\) bits, the sample of Gaussian random variable \(x\) obeying \(\mathcal{N}(0,1/2)\) is quantized as a new random variable, \(\hat{x}\), and the variance of the quantization error is defined and approximated as [5]  
\[E(x-\hat{x})^{2}\approx\frac{\Delta^{2}}{12}=\left(\frac{3\sigma}{2^{B-1}} \right)^{2}/12=\frac{3}{2^{2B+1}}. \tag{20}\]  
Therefore, if the entries in \(\mathbf{H}\in\mathbb{C}^{N\times M}\) are all i.i.d. according to \(\mathcal{CN}(0,1)\), \(\mathbf{H}\) is quantized as \(\mathbf{\hat{H}}\) with \(2B\) quantization bits for each entry, and we have \(\mathbf{H}=\mathbf{\hat{H}}+\mathbf{\hat{H}}\), where \(\mathbf{\hat{H}}\) is the error matrix with each entry i.i.d. with zero mean and variance of \(\sigma_{R}^{2}=\frac{3}{2^{2B}}\).  
With SQ, the transmitters compute precoders \(\mathbf{v}_{k}^{m}\) and decoders \(\mathbf{u}_{k}^{m}\) according to \(\mathbf{\hat{H}}_{ki}\), where \(\mathbf{u}_{k}^{mH}\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l}=0\) holds for \((k,m)\
eq(i,l)\). Thus, the interference term can be represented by  
\[\mathbf{u}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{i}^{l}=\mathbf{u}_{k}^{mH}( \mathbf{\hat{H}}_{ki}+\mathbf{\hat{H}}_{ki})\mathbf{v}_{i}^{l}=\mathbf{u}_{k}^ {mH}\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l}. \tag{21}\]  
Since \(\mathbf{v}_{k}^{m}\) and \(\mathbf{u}_{k}^{m}\) are unit vectors, and the entries of \(\mathbf{\hat{H}}_{ki}\) all are all i.i.d. with zero mean and variance of \(\frac{3}{2^{2B}}\), the entries of \(\mathbf{u}_{k}^{mH}\mathbf{\hat{H}}_{ki}\mathbf{v}_{i}^{l}\) are also i.i.d. with the same distribution as the entries of \(\mathbf{\hat{H}}_{ki}\). Therefore, the achievable sum-rate of IA with uniform SQ-based CSI feedback can be approximated by  
\[R_{SQ}\approx E\left(\sum_{k,m}\log_{2}\left(1+\frac{\frac{P}{d}\left|\mathbf{ u}_{k}^{mH}\mathbf{H}_{ki}\mathbf{v}_{k}^{m}\right|^{2}}{(Kd-1)\frac{P_{R}^{2}}{d}+ \sigma^{2}}\right)\right)\approx Kd\log_{2}(e)e^{1/\rho_{\textit{eff}}}E_{1}( \frac{1}{\rho_{\textit{eff}}}), \tag{22}\]  
where  
\[\rho_{\textit{eff}}=\frac{\rho}{\rho(Kd-1)\sigma_{R}^{2}+1}=\frac{\rho}{\rho( Kd-1)\frac{3}{2^{2B}}+1} \tag{23}\]  
denotes the effective SNR with the introduced quantization error from uniform SQ. The accuracy of (22) is also demonstrated in Section 6.