## 2 Related Work  
There are many publications regarding CB, yet it remains far from ready for practical applications. A detailed description of CB in the context of signal processing and beam pattern, with regard to phase synchronization being found in [4], mainly related to problems such as phase jitters and localization errors. Nevertheless, this paper concentrates mainly on the theory of CB, However, it does not include practical considerations. In particular, the authors focus on the nodes' spatial distribution and its effect on the beam. Following the same idea, in [8], the authors also consider the advantages of some nodes' spatial distribution, concluding that when using a _Gaussian_ distribution, by increasing the number of notes, we can reduce the mean beam pattern of the side lobe region as well as the mainlobe decays exponentially with rate proportional to the spread of the notes (or variance). The mainlobe in spatial _Gaussian_ distribution is wider than in _uniform_ distribution, thus the beam directivity using _uniform_ distribution is higher. A key point to perform CB, is the absolute synchronization between all motes. These motes are required to adjust their individual carrier signals in frequency and phase in such a way that the individual transmissions combine coherently at the BS [4; 7]. The motes will need to perform a distributed and collaborative transmission for a CB in a synchronized manner and it is not valid just a relative or event synchronization, taking into account that we do not have any wired network as in a traditional antenna array. In the literature, there are many papers that focus on time synchronization for WSN such as Timing-Sync Protocol for Sensor Networks (TPSN) [9], Reference Broadcast Synchronization (RBS) [10], Flooding Time-Synch Protocol (FTSP) [11], Rate Adaptive Time Synchronization (RATS) [12], Adaptive Time Window [13], etc. With accurate time synchronization, we can introduce a calculated phase offset to the carrier signal at each mote for CB. Nonetheless, none of these time synchronization protocols are suitable for CB that requires high scale coherent transmission [7]. The precision required is on the order of hundreds of nanoseconds (ns), in particular when using the frequency bands of standard IEEE 802.15.4 [1] (with frequency bands: 868/915 MHz and 2450 MHz) or the amendment IEEE 802.15.4n [14] (with frequency bands: 174-216 MHz, 407-425 MHz, 608-630 MHz). A further discussion is done in Section 4. It is worth mentioning that, in [7], a master-slave synchronization protocol is proposed to adjust phase and carrier frequency at each mote in a distributed manner. Moreover, CB performance is examined using theory and simulations, taking into account the effects of imperfect synchronization. The main conclusion is that it is theoretically possible to achieve CB with imperfect synchronization. A phase skew of \(30^{\circ}\) (\(\frac{\pi}{6}\)) and \(60^{\circ}\) (\(\frac{\pi}{3}\)) will result in 96% and 70%, respectively, of the maximum Signal-to-Noise Ratio (SNR) that results from a perfect phase synchronization. Hence, these results could be achieved when considering commercial motes. Other studies analyzing CB in WSN have assumed perfect synchronization as a fact, such as [4; 15]. Other attempts were made to account for synchronization errors, such as [16], where phase errors and other technical factors (due to the noise from internal carrier oscillator, node position errors, and other timing synchronization errors in terms of Bit Error Rate) were analyzed. Their authors conclude that most technical factors that affect the performance of CB are affiliated to synchronization. In [5], authors present a survey of different approaches focusing on synchronizing the phase between modes in a WSN in order to be able to perform CB. These approaches are based on an iterative process to find out the phase for the coherent transmission at each mode, using both open and closed loop phase synchronization.