# A Novel Multi-User Codebook Design for 5G in 3D-MIMO Heterogeneous Networks
## 4 System Performance Analysis Based on Random Geometric Finite Feedback  
According to the statistical characteristics of the random distribution of channel gain and Laplace transform, the system outage probability can be obtained [21; 23]:  
\[P(T,\alpha)=P(SNR>T)=L_{r}(n\xi_{k})L_{p}(\xi_{k})L_{q}(\xi_{k})e^{-\frac{\xi_{k }e^{2}}{P}} \tag{8}\]  
where, \(\xi_{k}=Td_{k}^{\alpha}\). According to the probability density function (PDF) and cumulative distribution function (CDF) of the radius' \(r\) in the 3D Poisson process, the inter-cell normalization from the Poisson domain \(\Phi(\lambda)\)  
Figure 3: Flowchart of the proposed 3D-MIMO codebook. of the microcell can be obtained through the probability generation functional (PGFL). The Laplace transform \(L_{p}\) of the interference \(\overline{I}_{p}\) can be expressed as:  
\[L_{p}(\xi_{k})=\exp\left\{ \begin{array}{l}-4\pi\lambda\\ \int_{0}^{\infty}\Big{[}\int_{d_{k}}^{\infty}\Big{(}1-e^{-Td_{k}^{n}Sr^{-n}} \Big{)}r^{2}dr\Big{]}f(s)\mathrm{d}s\end{array} \right\} \tag{9}\]  
By expanding and integration process we get:  
\[=\exp\left\{ \begin{array}{l}-4\pi\lambda\\ \int_{0}^{\infty}\Big{[}\int_{d_{k}}^{\infty}r^{2}dr-\int_{d_{k}}^{\infty}e^{- Td_{k}^{n}Sr^{-n}}r^{2}dr\Big{]}f(s)\mathrm{d}s\end{array} \right\} \tag{10}\]  
\[=\exp\left\{ \begin{array}{l}-4\pi\lambda\\ \int_{0}^{\infty}\Big{[}\int_{d_{k}}^{\infty}r^{2}dr-e^{-STd_{k}^{n}}\int_{d_{ k}}^{\infty}e^{-r^{-n}}r^{2}dr\Big{]}f(s)\mathrm{d}s\end{array} \right\} \tag{11}\]  
Putting limits and rearranging to PGFL form we get:  
\[=\exp\left\{ \begin{array}{l}\frac{4\pi\lambda d_{k}^{3}}{3}-\frac{4\pi \lambda T^{3/a}d_{k}^{3}}{\alpha}\\ E_{S}\Big{[}(S)^{3/\alpha}\big{[}\Gamma\big{(}-\frac{3}{\alpha},\ TS\big{)}- \Gamma\big{(}-\frac{3}{\alpha}\big{)}\big{]}\Big{]}\end{array} \right\} \tag{12}\]  
where \(\Gamma(\cdot)\) is the PGFL. The Laplace transform \(L_{r}\) of the inter-cell normalized interference \(\overline{I}_{r}\) from the macro-cell Poisson domain \(\Phi(\lambda_{M})\) can be expressed as:  
\[L_{r}(n\xi_{k})=\exp\left\{ \begin{array}{l}\frac{4\pi\lambda_{M}d_{k}^{3}}{ 3}-\frac{4\pi\lambda_{M}(nT)^{3/a}d_{k}^{3}}{\alpha}\\ E_{S}\Big{[}(S)^{3/\alpha}\big{[}\Gamma\big{(}-\frac{3}{\alpha},\ nTS\big{)}- \Gamma\big{(}-\frac{3}{\alpha}\big{)}\big{]}\Big{]}\end{array} \right\} \tag{13}\]  
For the interference term \(\left|\mathbf{h}_{k}^{H}\mathbf{w}_{j}\right|^{2}=\left\|\mathbf{h}_{k}\right\|^{2}\left| \overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{j}\right|^{2}\). \(\varphi_{k}\) represents the quantized distortion angle between \(\mathbf{h}_{k}\) and \(\hat{\mathbf{h}}_{k}\), decomposing the normalized channel vector \(\overline{\mathbf{h}}_{k}=(\cos\varphi_{k})\hat{\mathbf{h}}_{k}+(\sin\varphi_{k})\mathbf{v }_{k}\), where \(\mathbf{v}_{k}\) is an isotropic unit norm vector that is distributed in the null space of \(\hat{\mathbf{h}}_{k}\) and is independent of \(\sin\varphi_{k}\). Since \(\mathbf{w}_{j}\) and \(\mathbf{v}_{k}\) are independent, they are \(\left|\mathbf{v}_{k}^{H}\mathbf{w}_{j}\right|^{2}\sim B(1,\ N_{S}-2)\)[23] and are independent of the quantization error \(\sin\varphi_{k}\). To normalize interference between users, we can write:  
\[\overline{I}_{q}=d_{k}^{-\alpha}\|\mathbf{h}_{k}\|^{2}\sin^{2}\varphi_{k}\sum_{j \in K,j\
eq k}B(1,N_{s}-2) \tag{14}\]  
Among them, \(X=\|\mathbf{h}_{k}\|^{2}\sin^{2}\varphi_{k}\sim\Gamma(N_{s}-1,\ \delta)\), \(\|\mathbf{h}_{k}\|^{2}\), \(\varphi_{k}\) and \(B(1,N_{s}-2)\) are independent of each other,\[\beta_{2}(\lambda,\ T,\ \alpha)=\frac{4(nT)^{3/\alpha}}{\alpha(1+T\delta)^{K-1}} \times E_{5}\bigg{[}(S)^{3/\alpha}\bigg{[}\Gamma\bigg{(}-\frac{3}{\alpha},\ nTS \bigg{)}-\Gamma\bigg{(}-\frac{3}{\alpha}\bigg{)}\bigg{]}\bigg{]} \tag{18}\]  
Among them, \(\Gamma(x)=\int_{0}^{\infty}\epsilon^{x-1}e^{-t}dt\) is a gamma function. From Equations (16)-(18), the throughput is related to the SNR at the base station density and the feedback overhead. The quantization error is reflected by the feedback overhead.