## 3 Sum-Rate Performance with RVQ-Based CSI Feedback  
### Vector Quantization of Channel Matrix  
With VQ-based CSI feedback, channel matrices \(\mathbf{H}\in\mathbb{C}^{N\times M}\) are vectorized first and then quantized. The normalized \(M_{li}\)-dimensional unit complex vector from \(\mathbf{H}\) with \(M_{h}=N\times M\) is denoted by  
\[\mathbf{h}=\frac{vec(\mathbf{H})}{\|vec(\mathbf{H})\|}. \tag{5}\]  
Before VQ, the codebook is already known to transmitters and receivers. With \(B\) bits to quantize \(\mathbf{h}\), the codebook has \(N_{B}=2^{B}\) codewords, and each codeword is an \(M_{h}\)-dimensional unit complex vector. The codebook can be denoted as \(C=\big{\{}\mathbf{w}_{1},\mathbf{w}_{2},...,\mathbf{w}_{N_{B}}\big{\}}\). The quantization of \(\mathbf{h}\) is to find the codeword nearest to \(\mathbf{h}\) in codebook \(C\), i.e., to search for the index of the vector achieving the minimum inner product with \(\mathbf{h}\), which can be denoted by [9]  
\[I_{h}=\arg\min_{j=1,2,...,2^{B}}\sin^{2}\angle\big{(}\mathbf{h},\mathbf{w}_{j} \big{)}. \tag{6}\]  
Here \(\angle\big{(}\mathbf{h},\mathbf{w}_{j}\big{)}\) is the angle of two unit complex vectors. Index \(I_{h}\) is fed back to the corresponding transmitter, and the transmitter recovers the quantized channel according to vector \(\mathbf{\hat{h}}\), which is exactly the \(I_{h}\)-th codeword. \(z=\sin^{2}\angle\big{(}\mathbf{h},\mathbf{\hat{h}}\big{)}\) is defined as the quantization error in VQ, whose distribution depends on the codebook design criterion.  
For simplicity, the RVQ codebook is used for performance analysis in this paper. Though the quantization error of the RVQ codebook is higher than that of the Grassmannian codebook, the sum-rate derivation framework is applicable to different codebooks. It has been proved that with the increase in \(B\), the quantization errors of the RVQ and Grassmannian codebooks converge [9].  
In RVQ, define \(y=1-z=\cos^{2}\angle\Big{(}\mathbf{h},\mathbf{\hat{h}}\big{)}\) as the complementary quantization error. Then, the distribution properties of quantization error \(z\) can be obtained by analyzing the distribution of \(y\). In \(M_{li}\)-dimensional complex space, the cumulative distribution function (CDF) of the cosine square value of the angle between two random independent isotropic unit complex vectors is [11]  
Figure 1: A cellular multi-cell downlink scenario and transmission procedures of IA.  
\[F_{\rm cos}(x)=1-(1-x)^{M_{h}-1}. \tag{7}\]  
Then, the CDF of \(y=\cos^{2}\angle\left(\mathbf{h},\mathbf{\hat{h}}\right)\) is  
\[F_{Y}(y)=\sum_{i=0}^{2^{B}}C_{2^{B}}^{i}(-1)^{i}(1-y)^{i(M_{h}-1)}. \tag{8}\]  
Its expectation is  
\[E(y)=\int_{0}^{1}yf(y)dy=1-2^{B}Beta\left(2^{B},\frac{M_{h}}{M_{h}-1}\right), \tag{9}\]  
where \(Beta(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\) is the Beta function, and \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt\) is the Gamma function [9].  
Therefore, the CDF of the quantization error in RVQ is  
\[F_{z}(z)=1-F_{X}(x)|_{x=1-z}=1-\left(1-z^{M_{h}-1}\right)^{2^{B}}, \tag{10}\]  
and its expectation is \(E(z)=2^{B}Beta\left(2^{B},\frac{M_{h}}{M_{h}-1}\right)\), with upper bound \(E(z)\leq 2^{-\frac{B}{M_{h}-1}}\).