# Beamforming Based on a SSS Angle Estimation Algorithm for 5G NR Networks
## 3 Proposed Beamforming Based on SSS Angle Estimation Algorithm  
An illustration of the proposed BSAE algorithm's concept is presented in Figure 3.  
To achieve better system performance levels in the effective decoding of the transmitted information, the UE can additionally use the SSS in combination with DMRSs to determine the impact of the propagation channel and then apply the inverse to the PBCH payload before the decoding attempt, since these two signals experience similar propagation-channel effects. However, the UE processing is carried out after analog-to-digital (ADC) conversion, and thus, to efficiently perform better UE beam angle adjustments, equivalent channel estimation of the analog part is required. When considering a DL system with a BS equipped with \(N_{tx}\) transmit antennas transmitting SSBs over \(B\) beams, and a single user equipped with a single radio frequency (RF) chain, \(N_{rx}\) receive antennas, and \(S\) possible beams at the  
Figure 3: BSAE algorithm concept. IBE stage, the received SSS signal \(\widetilde{d}_{k,s}\in\mathbb{C}\) at the \(k\)th subcarrier for the \(s\)th analog beam, \(s\in\{0,\ldots,S-1\}\), at the UE, is given by  
\[\widetilde{d}_{k,s}=\mathbf{w}_{a,s}^{H}\mathbf{H}_{k}\mathbf{f}_{a,b}d_{k}+ \mathbf{w}_{a,s}^{H}\mathbf{n}_{k,s}, \tag{1}\]  
where the \(d_{k}\in\mathbb{C}\) is the transmitted SSS at the subcarrier \(k\), and the vectors \(\mathbf{f}_{a,b}\in\mathbb{C}^{N_{tx}}\), \(b\in\{0,\ldots,B-1\}\), and \(\mathbf{w}_{a,b}\in\mathbb{C}^{N_{rx}}\) model the analog beam at the BS and the UE, respectively. Additionally, \(\mathbf{H}_{k}\in\mathbb{C}^{N_{rx}\times N_{tx}}\) is the frequency-domain channel at subcarrier \(k\), and the parameter \(\mathbf{n}_{k,s}\in\mathbb{C}^{N_{rx}}\) denotes the zero-mean Gaussian noise, with variance \(\sigma_{n}^{2}\). Having the transmitted and received SSSs, the equivalent channel \(h_{eq,k,s}=\mathbf{w}_{a,s}^{H}\mathbf{H}_{k}\mathbf{f}_{a,b}\in\mathbb{C}\) is estimated by  
\[\widetilde{h}_{eq,k,s}=\widetilde{d}_{k,s}d_{k}^{-1}. \tag{2}\]  
At this point, the equivalent channel estimated in Equation (2) contains the needed analog counterpart estimation. Considering \(\mathbf{W}_{a}=\left[\mathbf{w}_{a,1},\ldots,\mathbf{w}_{a,S}\right]\in \mathbb{C}^{N_{rx}\times S}\), where \(S\) is the number of beams in the beam-sweeping procedure, \(\mathbf{N}=\left[\mathbf{n}_{1}d_{k}^{-1},\ldots,\mathbf{n}_{S}d_{k}^{-1} \right]\in\mathbb{C}^{N_{rx}\times S}\), and \(\widetilde{\mathbf{h}}_{eq,k}=\left[\widetilde{h}_{eq,k,1},\ldots,\widetilde {h}_{eq,k,S}\right]^{T}\in\mathbb{C}^{S}\). Then, from Equation (2), we have,  
\[\widetilde{\mathbf{h}}_{eq,k}=\mathbf{W}_{a}^{H}\mathbf{H}_{k}\mathbf{f}_{a,b} +\mathbf{W}_{a}^{H}\mathbf{N}. \tag{3}\]  
Therefore, by applying the pseudo-inverse of the \(\mathbf{W}_{a}\) to Equation (3), we can estimate the channel in the analog part, \(\mathbf{h}_{a,k}=\mathbf{H}_{k}\mathbf{f}_{a,b}\in\mathbb{C}^{N_{rx}}\), by  
\[\widetilde{\mathbf{h}}_{a,k}=\left(\mathbf{W}_{a}\mathbf{W}_{a}^{H}\right)^{ -1}\mathbf{W}_{a}\widetilde{\mathbf{h}}_{eq,k}. \tag{4}\]  
Hence, the receive beamformer, \(\widetilde{\mathbf{w}}_{a}\in\mathbb{C}^{N_{rx}}\), can be found by solving the following optimization problem, through the interactive greedy method:  
\[ \begin{array}{c}\overline{w}_{a}^{H}=\arg_{\theta}\max\Bigl{|}w_{a}^{H}( \theta)\widetilde{h}_{a,k}\Bigr{|},\\ \overline{w}_{a}^{H}\in\Omega_{a}\end{array}  \tag{5}\]  
where \(\Omega_{a}\) is the set of feasible vectors and \(\mathbf{w}_{a}^{H}(\theta)\) is the array response vector for angle \(\theta\in\{0,\ldots,N_{\theta}-1\}\) such that \(N_{\theta}\) is the dictionary size  
\[\mathbf{w}_{a}(\theta)=\frac{1}{\sqrt{N_{rx}}}\Bigl{[}1,e^{j\mathrm{I}kd\sin( \theta)},e^{j2 kd\sin(\theta)},\ldots,e^{j(N_{rx}-1)kd\sin(\theta)}\Bigr{]}^{T}, \tag{6}\]  
where \(k=2\pi/\lambda\), \(\lambda\) is the wavelength and \(d=\lambda/2\) is the inter-element spacing. The proposed BSAE algorithm is summarized in the following algorithm (Algorithm 1).