# Intelligent Transmission Control Scheme for 5G mmWave Networks Employing Hybrid Beamforming
## 4 Proposed Work
### Accurate Channel Estimation for mmWave Systems
#### 4.2.1 Sparse Signal Recovery Using LAMP Network  
The TRICE framework, in contrast to its counterpart, requires AMP as an iterative thresholding algorithm that directly restores the sparse angular channel by estimating the true spatial frequencies with the soft-thresholding function. Therefore, we can reconstruct every trajectory of the linked channel by utilizing grid points that are near the actual spatial frequencies. While the AMP algorithm works well for sparse recovery in general, its channel estimation performance is constrained by using identical linear transform matrices and shrinkage parameters throughout all iterations. To enhance AMP's efficiency for the issue of cascaded channel estimation, it is possible to unfold the AMP algorithm iterations into trainable layers using deep unfolding, as follows:  
\[\widetilde{H}_{e,k}^{d}=\mathcal{T}(\widetilde{H}_{e,l-1}^{d}+X_{l}Y_{l-1}); \frac{\alpha_{l}}{\sqrt{AH}}\|Y_{l-1}\| \tag{10}\]  
\[Y_{l}=Z-Q_{l}\widetilde{H}_{e,k}^{d}+\frac{1}{AH}\|\widetilde{H}_{e,k}^{d}\|_ {0}Y_{l-1} \tag{11}\]  
where \(\widetilde{H}_{e,k}^{d}\) and \(Y_{l}\), the calculated angles, are, respectively, the series of communication channels and the remaining interference set as the initial values \(\widetilde{H}_{e,k}^{d}=0\) and \(Y_{0}=Y\). \(X_{l}\) is the consistent CS literature notation. The function \(\mathcal{T}\), known as a soft-thresholding operator, has additional features that offer noise reduction functionality as well as enhancing clarity \(\widetilde{H}_{e,k}^{d}\), taking into account the lack of density. The definition of the function pertaining to the soft-thresholding approach is shown as follows:  
\[\left[\mathcal{T}(X;\;\lambda)\right]_{k}=\left(X_{k}-\lambda\frac{X_{k}}{|X_ {k}|}\right)\,I(|X_{k}>\lambda|) \tag{12}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline \(H\) & Channel matrix in mmWave communication & \(RSSI\) & Received signal strength indicator \\ \hline \(A\) & Measurement matrix used in LAMP network & \(P_{signal}\) & Received signal power \\ \hline \(\lambda\) & Soft-thresholding parameter & \(P_{noise}\) & Noise power in mmWave communication \\ \hline \(b\) & Congestion window growth rate (cwnd) & \(T_{delay}\) & End-to-end delay (ms) \\ \hline \(\theta_{A_{0}A}\) & Angle of arrival & LAMP &
\begin{tabular}{c} Learned approximate message \\ passing network \\ \end{tabular} \\ \hline \(\theta_{A_{0}D}\) & Angle of departure & \(RTT\) & Round trip time \\ \hline \(MSE\) & Mean squared error & \(PL\) & Packet loss percentage \\ \hline \(SNR\) & Signal-to-noise Ratio & \(T_{throughput}\) & Throughput (Mbps) \\ \hline \(N_{u}\) & Number of users & \(LSTM\) & Long short-term memory network \\ \hline _cwnd_ & Congestion window size (TCP) & \(LLCCS\) & Low-latency congestion control scheme \\ \hline \(DAH-BT\) & Digital and analog hybrid beam training & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations of symbols. At index negative lambda, the absolute value is represented by a larger than symbol, which may reduce the elements of the set to zero if their amplitude is lower than the function that acts as an indicator. Additionally, the term \(\frac{1}{AH}\left.\left\|\widetilde{H}_{e,k}^{d}\right\|_{0}Y_{l-1}\right.\) is introduced for the calculation of \(Y_{l}\) to accelerate convergence. To boost the cascaded channel approximation demonstration, we can optimize \(\{Q_{l},\,X_{l},\alpha_{l}\}_{l=1}^{L}\) as the tuneable parameters of the LAMP network through DL, where the linear transform matrices \(\{\,Q_{l}\}_{l=1}^{L}\) and \(\{X_{l}\}_{l=1}^{L}\), respectively, replace the measurement matrix \(A\) and the matched filter \(A^{H}\) in the original AMP, and \(\{\,\alpha_{l}\}_{l=1}^{L}\) denotes the shrinkage parameters for the soft-thresholding function.