# 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
Lastly, the cascaded channel is improved by altering the valued angular cascaded channel \(\widetilde{H}_{e,L}^{d}\) into the time domain channel via the following:  
\[H_{e,L}^{d}=\left(A_{S}^{*}\bigoplus A_{B}\right)\widetilde{H}_{e,L}^{d} \tag{13}\]  
Nevertheless, the LAMP system solves the combined recovery issue to directly restore the cascaded channel. \[Z=vec(Y)\approx\Big{(}\Theta^{T}\otimes Q^{H}\Big{)}vec\bigg{(}A_{B} \widetilde{H_{c}}A_{S}^{H}\bigg{)}+n=\Big{(}\Theta^{T}\otimes Q^{H}\Big{)}(A_ {S}^{*}\otimes A_{B})\widetilde{H}_{e}+n=A\widetilde{H}_{e}+n \tag{14}\]  
\[\text{Somewhere, }\widetilde{H_{e}}=vec\left(\widetilde{H_{c}}\right) \in C^{K_{b}K_{a}\times 1},n=vec\left(N\right)\in C^{AT\times 1},\text{ and }A=\left(\Theta^{T}\otimes Q^{H}\right) \left(A_{S}^{*}\otimes A_{B}\right).\]  
The angular cascaded channel \(\widetilde{H}_{e}\) is recovered directly in the joint CS method using the 3D measurement matrix A made up of \(K_{b}K_{a}\) atoms. Each particle characterizes an AoA at the BS and a pair of active and advancement AoDs at the IRS. The computational difficulty of the LAMP network is \(O(LT\ AK_{b}BK_{s})\), and even with lower grid resolutions, the computational cost is too high for practical systems. To solve this problem, our suggestion is a \"two-phase LAMP network that incorporates row compression (RCTS-LAMP)\" for sequential channel estimation. This will raid an improved trade-off between estimation enactment and computational intricacy. Enhanced Two-Stage LAMP Network with Row Compression  
Instead of the joint CS method, which directly retrieves \(\widetilde{H_{c}}\) utilizing a single LAMP network and the 3D measurement matrix \(A\), the proposed \"two-stage LAMP (TS-LAMP)\" network sequentially retrieves \(\widetilde{H_{c}}\) with two LAMP networks. Therefore, the TS-LAMP network can take advantage of this approach by utilizing an efficient LAMP network, and it can benefit from reduced complexity by breaking down the process of channel recovery. Furthermore, it also utilizes the angular cascade's sparsity in rows. In channel \(\widetilde{H_{c}}\), a row compression (COMP) mechanism is introduced to detect the row support \(\widehat{\Omega}\). Hence, the suggested RTCS-LAMP network is capable of further decreasing the computational burden streamlining by just recovering the rows of matrix \(\widetilde{H_{c}}\) that are on the row back \(\widehat{\Omega}\). The LAMP network utilizes the structure that resides in the overall measurement matrix \(Y\), as follows:  
\[Y\approx B_{1}\ \widetilde{H_{c}}\ B_{2}^{T}+N \tag{15}\]  
where \(B_{1}=Q^{H}A_{B}\ \epsilon\ C^{A\times K_{b}}\) and \(B_{2}=\theta^{T}A_{S}^{*}\epsilon\ C^{T\times K_{c}}\) are the matching extent conditions for \(\widetilde{H_{c}}\) at the BS and IRS. According to the row organization of \(\widetilde{H_{c}}\), the suggested TS-LAMP is designed for the network to break down the series channel estimation into two parts, as outlined in Algorithm 1, to develop a more simplified estimation of the mmWave cascaded channel. Step 1: Several rows should be parceled off with a LAMP network to recover a row-sparse matrix. During this stage, only the most significant rows are selected, and this stage compresses the channel. Step 2: Once the row support is computed in step 1, a second LAMP network will estimate the angular cascaded channel using only the support of the compressed row. The output is now the full cascaded channel estimate after transforming from angular domain back into time domain. The benefit appears to be in cutting the computational time down whilst maintaining a good level of accuracy by taking something challenging and breaking it down into two less challenging transformations. One should begin by laying the foundation and setting the groundwork to address the row-sparse matrix recovery issue, as follows:  
\[Y=B_{1}H_{row}+N \tag{16}\]  
where \(H_{row}=\overset{\sim}{H_{c}}B_{2}^{T}\) is an \(L_{G}\) row-sparse matrix since is \(\overset{\sim}{H_{c}}\) has \(L_{G}\) non-zero rows.