# Instantaneous Beam Prediction Scheme against Link Blockage in mmWave Communications
## 3 System Model
### Signal and Channel Models  
Consider a time division duplexing (TDD) transmission and reception mode in the mmWave channel to leverage reciprocity, where the channel state is determined beforehand and exchanged between the MS and MS. The downlink signal model at the MS is then formulated as,  
\[y_{\text{MS}}{=}\ P_{\text{tr}}C_{\text{MS}}^{\text{H}}P_{\text{BS}}\text{H}\ \text{z}+C_{\text{MS}}^{\text{H}}w \tag{1}\]  
where the variables \(P_{\text{tr}}\), \(C_{\text{MS}}\), \(P_{\text{BS}}\), H, \(z\) and \(w\) represent the transmitted power level, the MS combining matrix composed of \(c_{i}^{t}\) combining vectors, BS beamforming matrix composed of \(p_{j}^{t}\) beamforming vectors, mmWave channel block, control or data signal, and the additive white Gaussian noise (AWGN), modeled as \(w\ \sim N(0,\ \sigma_{w}^{2})\), where \(\sigma_{w}^{2}\) is the variance, respectively.  
The channel here follows the geometric model due to the sparse structure, where the geometry of objects in the channel highly impact the number and distribution of rays in the received signal profile. In addition, the wavelengths are much smaller than the geometries of objects in the channel. The geometric channel is formulated as [12]  
\[\text{H}=\sqrt{\frac{N_{\text{BS}}N_{\text{MS}}}{\Gamma_{bl}}}\sum_{k=1}^{K} \sum_{l=1}^{L}h_{l}P_{\text{BS}}C_{\text{MS}}^{\text{H}}, \tag{2}\]  
where \(\Gamma_{bl}\) denotes the blockage path loss and \(h_{l}\) is the channel gain from path \(l\in L\) total paths (rays) that follow an exponential arrival rate in few \(K\) clusters that follow a Poisson rate. Note that the path gains follow a Rician distribution to account for the transition between Line-of-sight (LoS) and non-line-of-sight (NLOS), \(h_{l}\sim R(0,\ \zeta)\). Here the variable \(\zeta\) denotes the power ratio between the first arriving LoS ray and subsequent rays. Furthermore, the beamforming \(P_{\text{BS}}\) and combining \(C_{\text{MS}}\) matrices in Equation (1) determine the array response at the BS and MS, respectively. Here, the array factor (AF) is used to gauge the angular directions (azimuth \(\theta_{\text{MS}}\) and elevation \(\phi_{\text{MS}}\)) along the array. Hence the AF for the MS, \(A_{\text{MS}}\), is expressed as (similar approach is adopted at the BS) [29; 30],  
\[A_{\text{MS}}=\sum_{n=1}^{N_{\text{MS}}}I_{n}\exp(\tilde{j}va\sin\theta_{ \text{MS}}\cos(\phi_{\text{MS}}-\phi_{n})+\varphi_{i}^{t}), \tag{3}\]  
where \(I_{n}\) is the \(n\)-th antenna amplitude at the MS, \(\tilde{j}\) is the imaginary number, \(v\) symbolizes the wave-number, i.e., \(v=2\pi/\lambda\), where \(\lambda\) is the wavelength, \(\lambda=\alpha/f_{\alpha}\), where \(\alpha\) is the speed of light and \(f_{\alpha}\) is the carrier frequency. The variable \(\phi_{n}\) in Equation (3) denotes the \(n\)-th antenna angular position, expressed by \(\phi_{n}{=}\ 2\pi N_{\text{MS}}/n_{\text{MS}}\). Finally, the variable \(\varphi_{i}^{t}\) is the principal radiation at the MS affiliated with \(c_{i}^{t}\) vector at (\(t\))th time step, expressed as,  
\[\varphi_{i}^{t}=-va\sin\theta_{\text{MS}}\cos(\phi_{\text{MS}}-\phi_{n}). \tag{4}\]