## 3 Sum-Rate Performance with RVQ-Based CSI Feedback
### Similarity between RVQ Error and Gaussian Channel Error  
It is complicated to derive the achievable rate by directly using the RVQ error. However, we can approximate the RVQ error as the equivalent Gaussian channel error on the basis of the similarity between the expressions of these two kinds of errors and then cross-verify the validness of this approximation operation by simulation. Notice that the relationship between original vector \(\mathbf{h}\) and quantized vector \(\mathbf{\hat{h}}\) in VQ can be rewritten as  
\[\mathbf{h}=\cos\theta\mathbf{\hat{h}}+\sin\theta\mathbf{\hat{h}}, \tag{11}\]  
where \(\theta\) is the angle between \(\mathbf{h}\) and \(\mathbf{\hat{h}}\), \(\mathbf{\hat{h}}\) is independent of \(\theta\), and \(\mathbf{\hat{h}}\) is isotropically distributed in the \(M_{h}-1\)-dimensional subspace orthogonal to \(\mathbf{\hat{h}}\)[11]. As for the Gaussian channel error model, the CSI imperfection can be described by the Gauss-Markov model [4] as  
\[\mathbf{H}=\sqrt{1-\beta^{2}}\mathbf{\hat{H}}+\beta\mathbf{E}, \tag{12}\]  
where the entries of \(\mathbf{H}\), \(\mathbf{\hat{H}}\) and \(\mathbf{E}\) are all independent identically distributed (i.i.d.) according to \(\mathcal{CN}(0,1)\). \(\mathbf{H}\) is the actual channel matrix, \(\mathbf{\hat{H}}\) is the imperfect observation of \(\mathbf{H}\) due to the additive noise term \(\mathbf{E}\), and \(\beta\) represents the variance of the Gaussian channel error. If \(\beta=0\), no Gaussian channel error exists. According to the similarity between (11) and (12), we made two hypotheses whereby we can adopt the Gaussian channel error model to approximate the RVQ error when the following two conditions are both satisfied: (1) the value of \(\beta\) is small enough; (2) two kinds of errors, i.e., the equivalent VQ error of the Gaussian channel error and the original RVQ error, have similar probability distribution functions (PDFs). The equivalent VQ error of the Gaussian channel error is defined as the sin square value of the angle between vectorized and normalized \(\mathbf{H}\) and \(\mathbf{\hat{H}}\), denoted by  
\[\tilde{z}=\sin^{2}\angle(vec(\mathbf{H}),vec(\mathbf{\hat{H}})). \tag{13}\]  
If the two conditions are satisfied simultaneously, the RVQ error could be transformed into the equivalent Gaussian channel error, and therefore the existing closed-form sum-rate expression based on the Gaussian channel error model in [4] can be used. The validness of this hypothesis can be confirmed by numerical results. In Figure 2, we transformed Gaussian channel error models with different \(\beta\) values into the equivalent VQ error models and compared the expectations of equivalent VQ errors \(\hat{z}\) and \(\beta^{2}\). In Figure 2, the expectation of \(\hat{z}=\sin^{2}\!\theta\) and \(\beta^{2}\) are almost identical for \(\beta\leq 0.4\). Next, we set \(E(z)=\beta^{2}\) and compared the PDFs of RVQ error \(z=\sin^{2}\!\theta\) and equivalent VQ error \(\hat{z}=\sin^{2}\!\angle(vec(\mathbf{H}),vec(\hat{\mathbf{H}}))\) separately from the RVQ error model (11) and the Gaussian channel error model (12). In quantizing a \(2\times 2\) complex matrix, if \(B=20\), the mean of the RVQ error is \(E(z)=0.0088\) and corresponds to \(\beta=0.0938\) in (12). The PDFs of RVQ error \(z\) and equivalent VQ error \(\hat{z}\) with \(\beta=0.0938\) are illustrated in Figure 3, where they are quite similar, with limited drift. As shown in Figures 2 and 3, we can conclude that the hypothetical conditions are feasible, and then the sum-rate based on RVQ can be transformed into the sum-rate analysis based on Gaussian channel errors. In RVQ, the quantization error mean is \(E(z)=2^{B}Beta(2^{B},\frac{NM}{NM-1})\) using \(B\) bits to quantize an \(N\times M\) complex matrix. Then, \(E(z)\) can be treated as the Gaussian error variance \(\beta^{2}\), i.e., \(\beta=\sqrt{E(z)}\).