# Intelligent Transmission Control Scheme for 5G mmWave Networks Employing Hybrid Beamforming
## 4 Proposed Work
### Optimized Beam Training for Adaptive mmWave Communication
#### 4.4.1 Digital and Analog Hybrid Beam Training (DAH-BT) Approach  
We suggest a quick beam training approach by incorporating the suggested DA-hybrid design, which we dub the new beam training method DAH-BT. In this research, we consider the possibility of MS being devices like vehicles with large antenna arrays by assuming \(N_{t}\gg L\) and \(N_{t}\gg L^{5}\). Our approach is based on the sparse nature of the angular equivalent channel matrix \(H_{a}\) and the direct correspondence between AoAs/AoDs and entries in \(H_{a}\). \(S_{d}\) is described as the collection of predominant values in \(H_{a}\), where \(S_{d}=\{(ik):|H_{a}(i,k)|>\epsilon\}\) with the threshold \(\epsilon\) being suitably chosen. If the cardinality of the set \(S_{d}\) is greater than or equal to \(d=\)\(|S_{d}|\ll G_{r}G_{t}\) we say that \(H_{a}\) is efficiently containing few non-zero elements. This study derived the formula for \(H_{a}\) using a \"uniform linear array (ULA)\" positioned at both the \"base station (BS)\" and \"mobile station (MS)\". It can be concluded that the \((i,\ k)\)-th element of \(H_{a}\) for a \"uniform planar array (UPA)\" located at both BS and MS is defined as follows:  
\[H_{a}(i.k)\approx\sum_{t=1}^{L}\alpha_{f}f_{N_{t}^{\pi^{2}}}\bigg{(}\frac{m(i)} {N_{\pi^{2}}^{\alpha z}}-\theta_{l}^{\alpha z}\bigg{)}f_{N_{t}^{pl}}\bigg{(} \frac{n(i)}{N_{t}^{pl}}-\theta_{l}^{\alpha z}\bigg{)}\times f^{*}{}_{N_{t}^{ \pi^{2}}}\bigg{(}\frac{p(k)}{N_{t}^{\alpha z}}-\mathcal{L}_{l}^{\alpha z} \bigg{)}f^{*}{}_{N_{t}^{\pi^{2}}}\bigg{(}\frac{q(k)}{N_{t}^{pl}}-\mathcal{L}_{ l}^{\alpha l}\bigg{)} \tag{29}\]  
\[\text{where }m(i)\triangleq\lfloor\frac{i}{G_{r}^{pl}}\rfloor+1n(i) \triangleq mod\{\frac{i}{G_{r}^{pl}}\}p(k)\triangleq\lfloor\frac{k}{G_{r}^{ pl}}\rfloor\text{, }q(k)\triangleq mod\Big{\{}\frac{i}{G_{r}^{pl}}\Big{\}}\text{; }f_{N_{t}^{\pi^{2}}}(.)\text{,}f_{N_{t}^{pl}}(.)\text{,}\]  
\(f_{N_{t}^{\pi^{2}}}(. )\), and \(f_{N_{t}^{pl}}(. )\) are the Dirichlet kernels, according to the following outlined method of ordered arrangements:  
\[f_{N}(\theta)=\frac{1}{N}{\sum_{i=1}^{N}}\,e^{-j\pi i\theta}\text{.} \tag{30}\]  
\[f_{N}(\theta)=\frac{1}{N}e^{-j\pi(N+1)\theta}\frac{sin\,sin\,\left(N\pi\theta \right)}{sin\,sin\,\left(\pi\theta\right)} \tag{31}\]  
The sinc function's property suggests that the dominating values in \(H_{a}\) correspond to AoA/AoD pairs, showing that \(H_{a}\) is effectively L-sparse with \(L\) paths. As a result, we can locate the matching beams by using standard CS techniques to approximately recover \(H_{a}\). Algorithm 3 provides a summary of the beam training phase 1 procedure, which is optimized to effectively estimate the angles of arrival (AoA) in mmWave communication systems with reduced training overhead. In beam training, the angles of arrival (AoA) are estimated. The following actions should be taken: Broad beams are sent in all directions by BS. A basic Digital Beamforming (DBF) circuit is used by MS to receive. To extract dominant AoAs, orthogonal matching pursuit (OMP) should be used. The best AoAs should be saved for the following stage. This is beneficial because it narrows the search space in dynamic situations and lowers the overhead of beam training. This phase starts with the hybrid beamforming module sending signals in every direction from the base station (BS) and with the mobile station (MS) receiving them omnidirectionally with a low-cost digital beamforming (DBF) circuit. After gathering the received signals, the OMP algorithm is utilized to estimate the leading AoAs by extracting the highest signal components. This is conducted through iterative picking of the maximum-energy beam directions from the channel response matrix and the subsequent adaptation of the estimation process via an adaptive thresholding strategy. The selected top AoA values are stored for refinement in the subsequent stage of beam training.