# A Novel Multi-User Codebook Design for 5G in 3D-MIMO Heterogeneous Networks
## 3 System Model
Then the \(U_{0}^{(k)}\) received site gnal can be expressed as:  
\[y_{0k}=\sqrt{\rho}\;d_{k}^{-\frac{\alpha}{2}}\mathbf{h}_{0k}^{H}\mathbf{x}_{0}+\sqrt{ \rho^{\prime}}\;\sum_{m\in\Phi(\lambda_{M})}\;l_{m}^{-\frac{\alpha}{2}}\mathbf{h}_ {mk}^{H}\mathbf{x}_{m}+\sqrt{\rho}\;\sum_{i\in\Phi(\lambda)/T_{0}}\;D_{i}^{-\frac{ \alpha}{2}}\mathbf{h}_{ik}^{H}\mathbf{x}_{i}+\mathbf{n}_{0k} \tag{1}\]  
Figure 2: 3D-MIMO antenna beamforming. where, \(\rho=P/K,\rho^{\prime}=\frac{nP}{K}\), equivalent power allocation is used for analysis, \(l_{m}\) is the distance between the user and the \(m\)th macro base station in micro base station \(T_{0}\), \(D_{i}\) is the distance between the user and the \(i\)th micro base station in the micro base station \(T_{0}\), \(n_{0k}\) is the additive white Gaussian noise with variance \(\sigma^{2}\). In this paper, a limited feedback mechanism is adopted, and the base station can only acquire part of the channel state information (CSI) [21]. The micro base station \(T_{0}\) and the user \(U_{0}^{(k)}\) keep the same quantization codebook \(C_{k}=\left\{c_{k1},\ c_{k2},\ \dots\ c_{kN}\right\}\), which contains \(N=2^{B}\) unit norm vector codewords. The \(k\)th user selects the quantization vector (codeword) \(\hat{\mathbf{h}}_{k}\) closest to the channel direction \(\overline{\mathbf{h}}_{k}=\mathbf{h}_{k}/\mathbf{h}_{k}\) from the codebook with \(B=\ln(N)\) bit, and the user selects the optimal codeword from the codeword by receiving the SNR maximization criterion:  
\[\hat{\mathbf{h}}_{k}=\arg\ \max_{c_{ki}\in C_{k}}\ \left|\overline{\mathbf{h}}_{k}^{H}C_{ki} \right|^{2} \tag{2}\]  
After searching for the best-quantized codeword, the user feedbacks the index of the codeword to the base station using \(B\) bit. After the base station receives the index, the codeword corresponding to the index is found as a beamforming vector in the codebook, and then the related signal processing is made [22]. The received signal interference-noise ratio (SINR) of user \(U_{0}^{(k)}\) is given by:  
\[SINR=\frac{\rho S_{0k}d_{k}^{-\alpha}}{I_{r}+I_{p}+I_{q}+\sigma^{2}} \tag{3}\]  
\[I_{r}=\sum_{m\in\Phi(\lambda_{M})}\rho^{\prime}\|\mathbf{h}_{mk}^{H}\mathbf{W}_{m}\|l_ {m}^{-\alpha} \tag{4}\]  
\[I_{p}=\sum_{i\in\Phi(\lambda)/T_{0}}\rho\|\mathbf{h}_{ik}^{H}\mathbf{W}_{i}\|D_{i}^{- \alpha} \tag{5}\]  
\[I_{q}=\sum_{j\in K,\ j\
eq k}\rho\Big{|}\mathbf{h}_{k}^{H}\mathbf{W}_{j}\Big{|}^{2}d_{ k}^{-\alpha} \tag{6}\]  
Among them, \(\mathbf{W}=[\mathbf{w}_{1},\mathbf{w}_{2},\ \dots,\ \mathbf{w}_{K}]\), \(I_{r}\) is the inter-cell interference from the macro base station Poisson domain \(\Phi(\lambda_{M})\), \(I_{p}\) is the inter-cell interference from the micro base station Poisson domain \(\Phi(\lambda)/T_{0}\), \(I_{q}\) is the inter-user interference in the cell estimated from the quantized CSI. The system throughput, that is, the regional spectrum efficiency, is defined as the product of the absolute coverage probability and the maximum sum rate of transmission information per unit area. Let \(T\) be the \(SINR\) threshold, then the system throughput can be expressed as:  
\[T=\lambda\sum_{k=1}^{K}P(T,\alpha)\ln(1+T) \tag{7}\]  
The flowchart in Figure 3 explains the system procedure.