# Hybrid Precoding Design for Subarray-Structure-Enabled mmWave URLLC System
## 3 Problem Solution
### Optimal Digital Pre-Encoder Design  
From Equation (4), we find that the achievable rate \(R\) contains two parts; the first part is the spectral efficiency of the Shannon theorem, and the second part is the channel dispersion function. Both of them are related to \(\gamma\), which cannot be solved directly [17]. For ease of solving, we split \(R\) into two parts and analyze them separately, Let \(\mathbf{w}=\mathbf{A}\mathbf{f}\), and we have [18; 19]  
\[\max_{\mathbf{w}} R\] (8) such that \[\|\mathbf{w}\|^{2}\leq P_{\text{tot}}. \tag{9}\]  
Here, Equation (4) can be rewritten as  
\[R=f(\mathbf{w})-a^{\prime}g(\mathbf{w}), \tag{10}\]  
where \(f(\mathbf{w})=\log_{2}(1+\gamma)\), \(a^{\prime}=Q^{-1}(\varepsilon)/\ln 2\sqrt{M}\), and \(g(\mathbf{w})=\sqrt{1-(1+\gamma)^{-2}}\). To ensure that \(R\) is a convex function for easy solving, we analyze the convexity of the two expressions on the right side of (8) separately. (1) Convex analysis for \(f(\mathbf{w})\)  
Expanding expression \(f(\mathbf{w})\), we have \(f(\mathbf{w})=\log_{2}(1+|\mathbf{h}^{H}\mathbf{w}|^{2}/\sigma^{2})\). According to the principle of convex optimization, \(|\mathbf{h}^{H}\mathbf{w}|^{2}\) can be judged to be a convex expression; namely, \((1+|\mathbf{h}^{H}\mathbf{w}|^{2}/\sigma^{2})\) is convex. As a composition function of logarithms and powers, \(f(\mathbf{w})\) meets the characteristics of a logarithmic concave function. (2) Convex approximation for \(g(\mathbf{w})\)  
Expanding expression \(g(\mathbf{w})\), we have \(g(\mathbf{w})=\sqrt{(1-(1+|\mathbf{h}^{H}\mathbf{w}|^{2}/\sigma^{2})^{-2})}\). It can be found that \((1-(1+|\mathbf{h}^{H}\mathbf{w}|^{2}/\sigma^{2})^{-2})\) is a concave function, whose value is within interval \((0,1)\), which means \(g(\mathbf{w})\) is supposed to be concave. However, there exists a minus sign before \(g(\mathbf{w})\), which means Equation (8) can not be solved by the optimization algorithm. The function \(g\) at \(\gamma^{k}\) can be expanded with the first-order Taylor expansion as  
\[ \begin{split} g(\gamma)=&\sqrt{1-\frac{1}{(1+ \gamma)^{2}}}\\ \approx&\sqrt{1-\frac{1}{(1+\gamma^{k})^{2}}}+\frac {\gamma-\gamma^{k}}{(1+\gamma^{k})^{3}\sqrt{1-\frac{1}{(1+\gamma^{k})^{2}}}} \\ \triangleq&\hat{g}(\gamma).\end{split}  \tag{11}\]  
The achievable rate maximization optimization problem can be written as  
\[R=\psi(\gamma)-a^{\prime}\hat{g}(\gamma). \tag{12}\]  
In traditional communication scenarios, the optimal value of the digital precoding vector can be obtained by using the MRT solution when the transmit power is maximized. However, in the context of the URLLC, the achievable rate function \(R\) of the system is not a jointly concave function, and the maximum spectral efficiency may not be achieved based on the MRT solution. Therefore, we use an MRT solution that contains an unknown variable \(P_{x}\), and substitute it into the original optimization problem. Before further analysis, we first present the following theorem [20]. Theorem 1: _The digital precoding vector of problem (8) is in the form of a Maximum-Ratio Transmission (MRT)_  
\[\mathbf{w}^{*}=\sqrt{P_{0}}\frac{\mathbf{h}^{H}}{\|\mathbf{h}\|}, \tag{13}\]_where \(P_{0}\) represents the actual transmit power at the base station._  
Proof. : We assume there exists an arbitrary solution \(\mathbf{w}_{1}=P_{1}\bar{\mathbf{a}}\) that satisfies \(P_{1}\leq P_{\text{tot}}\), where \(|\bar{\mathbf{a}}|=1\), and \(P_{1}\) represents the power of the digital precoding vector \(\mathbf{w}_{1}\). In this case, we can always find another feasible solution \(\mathbf{w}_{2}=\sqrt{P_{2}}\bar{\mathbf{h}}^{H}/|\bar{\mathbf{h}}|\) with power \(P_{2}=P_{1}\bar{\mathbf{h}}\bar{\mathbf{a}}/|\bar{\mathbf{h}}|\), where \(P_{2}\) represents the power of the beamforming vector \(\mathbf{w}_{2}\). Since \(|\bar{\mathbf{h}}\bar{\mathbf{a}}|^{2}\leq|\bar{\mathbf{h}}|^{2}\), we have \(P_{1}\leq P_{2}\). Since \(P_{1}\leq P_{\text{tot}}\), there exists \(P_{2}\leq P_{\text{tot}}\), which means that we can always obtain an MRT solution for the precoding vector.