# Optimized Combination of Local Beams for Wireless Sensor Networks
## 1 Introduction  
Wireless sensor networks (WSNs) have been widely studied and applied for activity-monitoring applications in military [1; 2], weather [3; 4], and commercial [5; 6] areas. To share monitored data among nodes in WSN, recent WSNs generally utilize the communication protocol designed for industrial, scientific, and medical (ISM) or lower frequency bands. However, as the demand of high-capacity and high-rate data exchange grows, the millimeter wave (mmWave) bands have also been considered as the promising spectrum resources [7; 8; 9; 10]. Although these frequency bands suffer from the severe path loss caused by the atmospheric absorption and scattering [11], they are still attractive because their small wavelengths allow to use phased array architectures such as uniform linear arrays (ULAs) [12] and coprime arrays [13; 14], which can be implemented in the WSN nodes [15; 16; 17]. Usually, the nodes are clustered within a small region and surrounded by far-away access points (APs) [18]. When one of them, (i.e., a sink node \(\mathfrak{S}\)), needs to establish a direct uplink with one of the APs, it is possible to solely use its own beam. However, because of the two reasons that the longer transmission may cause a more serious path loss [19; 20], and the small physical size and low power supply of the nodes [21] limit the number of the radiating elements in ULAs, it is more efficient to combine multiple beams from the neighbor nodes by utilizing the concept of a collaborative or cooperative beamforming (CB). This approach was introduced in [22] to analyze the possibility of extracting the directive beam from uniformly distributed nodes, and the cases of Gaussian andarbitrary distributions are also analyzed in [23] and [24], respectively. In addition, [PERSON] et al. proposed the novel methods to efficiently obtain the beamforming weight of the nodes [25; 26; 27], and [PERSON] et al. deeply surveyed the fundamentals and applications of the CB. These studies give us the verified result that the CB can provide a well-defined and deterministic mainlobe in the direction of a desired AP. Unfortunately, unpredictable sidelobes caused by the random positions of the nodes also affect unwanted results over the whole angular domain, which would result in the unacceptable levels of interference in the directions of undesired APs. To minimize these malignancies, some types of techniques have been suggested and examined in the various literatures. [PERSON] et al. formulated the combinatorial optimization problem with the node-selection scheme to prune the sidelobes [28], [PERSON] et al. utilized the decentralized cross-entropy optimization (CEO) having the significantly reduced complexity compared to [29], [PERSON] et al. adjusted the excitation amplitude and phase of the nodes by the firefly algorithm [30], and [PERSON] et al. proposed the nondominated sorting genetic algorithm II (NSGA-II) to solve the multi-objective amplitude and phase optimization having the goal of minimizing the peak sidelobe level minimization and maximizing the directivity simultaneously [31]. Although these schemes showed the prominent effects on reducing the sidelobe levels in the undesired directions, they are confined to the case that the nodes are equipped with omni-directional antennas and operate in the conventional frequency bands, which provides the motivation of our paper. We assume the scenario that the nodes of the WSNs, operating in the mmWave band, are installed on the irregular surface of the practical sensing area and equipped with ULAs steering beams toward the desired direction, as shown in Figure 1. From simulation results, it is verified that the CB under the given scenario not only provides the power improvement in the desired direction, but also causes the considerable sidelobes. Thus, we design the combinatorial optimization for minimizing the average interference-to-noise ratio (INR).