# Multi-Backup Beams for Instantaneous Link Recovery in mmWave Communications
## 3 System Model  
### Beamforming Design at the MS  
Consider a MS equipped with an analog beamformer that is composed of parallel uniform linear arrays (ULA), where each ULA radiates a single beam, i.e., forming simultaneous multiple beams radiation in different directions. Each antenna is connected to a single analog phase shifter to provide continuous scanning capabilities (as opposed to step scanning in digital phase shifters). The ULAs are then connected to a single RF chain, as shown in Figure 1. Consider the design details. In this paper, a novel multi-beam parallel array model is proposed at the MS. Hence, consider a MS equipped with a group of \(r=1\), \(2\),..., \(R\) parallel arrays, each composed of \(n=1\), \(2\),..., \(\mathbf{N}_{r}\) co-polarized antenna elements arranged in a linear geometric setting, i.e., forming one-dimensional radiation (1D). The elements are uniformly oriented with \(d_{r}\) equi-spacing, i.e., \(d~{}=~{}\lambda/2\), where \(\lambda\) represents the mmWave wavelength, \(\lambda~{}=~{}c/f_{c}\), \(c\) is the speed of light, and \(f_{c}\) is the carrier frequency. This spacing value is chosen so that grating lobes and pattern blindness are avoided, as well as to ensure there are minimal mutual coupling effects. Thus, it satisfies the formula \(d<\left(1+\left|cos\theta_{0}^{r}\right|\right)\), where the variable \(\theta_{0}^{r}\) is the observation angle from array \(r\) at the MS in azimuth direction. The maximum radiation pattern for the array points along \(\theta_{0}^{r}\) scanning directions, for an even number of elements at any spatial direction is expressed by the closed-form normalized array response vector at each array \(r\) at the MS; which, in turn, is represented by the periodic array factor (AF), \(A_{MS}^{r}\). This is expressed as  
\[A_{MS}^{r}=\frac{1}{N_{\text{MS}}^{r}}\sum_{n=1}^{N_{\text{MS}}^{r}}a_{n}exp(j(N _{\text{MS}}^{r}-1)(k_{0}d_{r}\cos\theta_{0}^{r}+\beta_{\text{MS}}^{r})=A_{ \text{MS}}^{r}=\frac{1}{N_{\text{MS}}^{r}}\sum_{n=1}^{N_{\text{MS}}^{r}}a_{n} exp(j(N_{\text{MS}}^{r}-1)(\varphi_{MS}^{r}) \tag{1}\]  
where the variable \(a_{n}\) denotes the amplitude excitation for the \(n\)-th antenna element, \(k_{v}=2\pi/\lambda\) represents the wave number, and \(\beta_{MS}^{r}\) symbolizes the relative progressive phase shift between the interconnected antenna elements at array \(r\) at the MS. Note that \(\varphi_{\text{MS}}=~{}kd\cos\theta_{0}^{\text{MS}}+\beta_{\text{MS}}\) is a compact form that represents the array phase function at the MS with a visible region that varies between \(-kd\leq\varphi_{\text{MS}}\leq kd\). Moreover, the half-power beamwidth (HPBW) at the broadside and scanning directions, i.e., \(\forall\phi_{0}^{\text{MS}}\Big{(}\theta_{0}^{\text{MS}}:~{}0<~{}\theta_{0}^{ \text{MS}}~{}\leq~{}\pi\Big{)}\), is expressed as [11]  
\[\phi_{bnd}^{\text{MS}}=\cos^{-1}\Bigg{(}\frac{\lambda}{2\pi d}k_{w}d\cos\theta_ {0}^{\text{MS}}\pm\frac{2.782}{N_{\text{MS}}^{y}}\Bigg{)}\,0<\theta_{0}^{\text {MS}}\leq\pi, \tag{2}\]  
whereas the HPBW at the endfire direction is computed as [12],  
\[\phi_{endfire}^{\text{MS}}=2\cos^{-1}(1-1.391\lambda/N_{\text{MS}}^{y}d),~{}for~ {}\theta_{0}^{\text{MS}}=0,\pi. \tag{3}\]  
One important remark here, is that the spatial footprint of the array increases proportionally to a broadening factor of \(b,b=1/\cos\theta_{0}^{\text{MS}}\) for directions scanned off the broadside. Moreover, the array gain is gauged by \(G_{\text{MS}}\)\(=~{}A_{\text{MS}}G_{a}\), where \(G_{a}\) is the gain for a single antenna element. For example, microstrip rectangular patch antennas are widely chosen for mmWave transceivers, and they provide a gain range of 5-7 dBi [13; 14]. Each antenna array is fed in parallel by an array of \(P\) phase shifters, in particular, quadrature varactor-loaded transmission-line phase shifters. Note that the total number of phase shifters is equal to the number of antennas.