## 2 Related Work
### Beamforming
Another metric used in WSN is the total energy consumed in a CB-DCB transmission, so one way to minimize it would be to minimize the number of internal transmissions within WSN and also in a CB-DCB, [37]. The fundamental objective of CB-DCB is to reach large distances in the transmission, to transmit to a BS far away from the WSN area. Therefore, a very suitable metric to evaluate the range is the signal-to-noise ratio (SNR) in the receiver. The SNR is related to the power emitted in CB-DCB mode towards the receiver, and the received signal strength (RSS) in the receiver, this is in turn related to the capacity of the transmission channel C = \(\log_{2}(1+\text{SNR})\). In these cases, the energy consumption in the WSN is not taken into account, being the objective to reach enough energy to send the signal with sufficient quality to the receiver. But the main problem presented by CB-DCB is synchronization. Most of the work published to date that presents CB-DCB algorithms and that intend to optimize the targeting, radiation pattern, and energy, they consider that a perfect synchronization of the motes is possible. However, without a perfect synchronization of the motes, all of these algorithms cannot be effective. Usually, the motes are not synchronized, there are offsets of carrier and time, and this offset depends on the battery level in each node, as well as environmental parameters such as temperature, [14], without a perfect synchronization the CB-DCB is impossible. \[G_{N}=\sum_{k=1}^{N}a_{k}e^{i(2\pi(f_{x}+\Delta f_{k})\cdot(t+\Delta t_{k})-x_ {k}\cos\varphi\sin\theta+y_{k}\sin\varphi\sin\theta-a_{k})},|a_{k}|\leq 1. \tag{1}\]  
The above expression is the CB-DCB gain, which is the metric we use to evaluate the power in the direction of the receiver, which would be in the direction \(\theta\), \(\varphi\), and which depends on the frequency and time offsets \(\Delta f_{k}\) and \(\Delta t_{k}\). The WSN would be deployed on the ground (\(z_{k}=0\)), in positions \(x_{k}\), \(y_{k}\), with carrier \(f_{c}\), and with emission powers related to \(a_{k}\), and phases \(\alpha_{k}\). The \(\Delta f_{k}\) and \(\Delta t_{k}\) offsets are unknown due to the lack of synchronism. Without an adequate correction of these phase shifts it is impossible to apply all the previous CB-DCB procedures. This metric is linearly related to the received signal strength (RSS) and the normalized received signal strength (NRSS) which in turn is related to the synchronization of the motes through \(\Delta f_{k}\) and \(\Delta t_{k}\). This is the metric that will be used in this work. The previous expression contemplates the possibility that the motes emit with different power, and the normalized CB-DCB gain would be G/N, being N the total number of motes. A great deal of effort has been put into proposing synchronization procedures. Synchronization procedures can be divided into two groups: closed loop and open loop. In closed-loop procedures there is an exchange of information between WSN and BS. BS evaluates the RSS of the signal received from WSN and according to a certain criterion responds to WSN so that the motes correct the phases [19]. Depending on the bits used in the feedback and the phase selection procedure, these procedures can take time to converge, and can be costly in terms of energy consumed, thus the energy necessary to know the channel state is avoided in [39]. Although 1-bit closed-loop convergence is demonstrated in [40], it consumes energy and is susceptible to improvement as demonstrated in [41; 42]. For that reason, techniques are introduced that in closed loop send more information (in addition to RSS) to make convergence faster [43; 44], introducing phase predictors such as the Kalman filter [45], which show significant computing power needs, in some cases requiring the use of Digital Signal Processors (DSP), [46].