## 2 Initial Cell Search Procedure
### Received Signal  
Given a large array, analog beamformers are easier to implement and integrate with the array antennas compared with their digital counterparts since digital beamformers for large arrays require a high data rate through the analog to digital converter (ADC). Furthermore, digital beamformers usually require higher power consumption for the ADC and processors that perform multiple functions; such as combining signals from antenna elements, signal processing or data packaging in the system, etc. Thus, we consider analog beamforming in this paper, where the BS antenna system generates one beam at a time. The complex received signal at the receiver after the beamforming can then be written as:  
\[z(t)=\sum_{l=1}^{N_{slot}}G_{tx,l}G_{rx,l}PL_{linear}p_{l}(t-\tau)+d(t) \tag{1}\]  
Figure 2: Periodical Primary Synchronization Signal (PSS) signal transmission.  
where \(l\) is the time slot index, and \(G_{tx,l}\), \(G_{rx,l}\) are the gain values of the transmitter (TX) and receiver (RX), respectively. The path loss coefficient of the coverage area is denoted by \(PL_{linear}\), \(p_{l}(t)\) is the reference PSS signal waveform at time slot \(l\), \(\tau\) is the propagation delay of the PSS signal, and \(d(t)=N\big{(}0,{\sigma_{d}}^{2}\big{)}\) is the AWGN with \({\sigma_{d}}^{2}\) representing the PSD of the noise, which is assumed to be constant in time. Thus, if the TX antenna is omnidirectional, the value of \(G_{tx,l}\) will be constant at each time slot \(l\). In this paper, the antenna array is assumed to have isotropic elements, and the array factor is used as the antenna radiation pattern [8]. As a result of this assumption, i.e., that the array factor is the antenna pattern, the mutual coupling between antenna elements, and individual element pattern of array elements, which would not be isotropic in real applications, are neglected in our analysis. The radiation pattern for an N \(\times\) M element can then be calculated from its array factor formulation as in (2).  
\[AF(\theta,\phi)=\sum_{n=1}^{N}\sum_{m=1}^{M}I_{nm}e^{j(m-1)(kd_{x}sin\theta cos \phi+\beta_{x})}e^{j(n-1)(kd_{y}sin\theta sin\phi+\beta_{y})} \tag{2}\]  
where \(\beta_{x}=-kd_{x}sin(\theta_{0})cos(\phi_{0})\), \(\beta_{y}=-kd_{y}sin(\theta_{0})sin(\phi_{0})\) with \(k=\frac{2\pi}{\lambda}\), \(\lambda\) is the operating wavelength, \(d_{x}\), \(d_{y}\) are the spacing between adjacent elements along x and y axes, \((\theta_{0},\phi_{0})\) denote the scanned position of the main beam, and \((\theta,\phi)\) are the observation angles on the conventional spherical co-ordinate. The gain values \(G_{tx,l}\), \(G_{rx,l}\) of the antennas in (1) when scanned to position \((\theta_{0},\phi_{0})\) are then calculated as  
\[G(\theta_{0},\phi_{0})=\frac{4\pi[AF(\theta_{0},\phi_{0})][AF(\theta_{0},\phi_ {0})]^{*}}{\int_{0}^{2\pi}\int_{0}^{\pi}[AF(\theta,\phi)][AF(\theta,\phi)]^{*} sin\theta\ d\theta d\phi} \tag{3}\]  
The value of \(PL_{linear}\) in (1) depends on the channel characteristics. For line-of-sight (LOS) channels, the omnidirectional path loss coefficient is calculated based on the Friis' transmission principles. For non-line-of-sight channels (NLOS), the \"Empirical NYC\" model [4; 27], which is based on an extensive field measurement procedure in downtown New York city at both 28 GHz and 73 GHz is used in our analysis. Besides the \"Empirical NYC\" model for NLOS channels, various empirical/theoretical models can be used such as the Walfisch and Bertoni model [28], which includes the effect of buildings on the propagation path, or the prediction model of [29], which obtains NLOS formulas for low-rise or high-rise environments.