## 1 Introduction  
Wireless Sensor Networks (WSNs) are composed of spatially distributed autonomous sensor devices, named motes. These motes have their own power supply, processing unit, memory, sensors and wireless communications. However, they are limited by strong constraints in terms of energy, bandwidth, memory size and computational capabilities. Most of these WSNs are based on standard IEEE 802.15.4 [1]. Usually, the motes are placed in the near field and exchange packets using multi-hop routing protocols. In these networks, usually at least one mote called a sink (or Base Station (BS)), acts as a gateway to connect with other networks. In the last decade, WSNs have been used in many different applications, such as environment, habitat, precision horticulture, seismic, volcano hazard monitoring, coil mine monitoring, etc. Most of these applications are analyzed in [2], a recent survey based on real world deployments. The main function of these motes is to monitor the surrounding environment (e.g., humidity, temperature, motion, etc. ), collect raw data and transmit it to the BS. However, when this BS is located in a far field (for example, more than 5 km [2]), to perform these long range communications in a WSN, different techniques have been proposed. These techniques are classified as Collaborative Beamforming (CB), also known as Distributed or Cooperative Beamforming, virtual antenna array and cooperative Multiple Input Multiple Output (MIMO) transmission [3; 4]. The benefits and advantages of CB for distributed WSN are defined in [4; 5]. The purpose of CB is to achieve long distance communication using constructive interference between signals transmitted from several motes to achieve power gain in the resulting signal to be effectively received in the BS, located in the far field. The constructive interference between these signals that provides gain in one direction will result in destructive interference in most other directions. To do beamforming, all motes will transmit the same signal (including the modulated data), synchronizing their individual carrier signals in frequency and adjusting the carrier phase of each transmitter, in such a way that the individual transmissions combine coherently at the BS [4]. The resulting signal would be transmitted as a unidirectional beam, given certain conditions as described in this paper. Thus, the formation of such beams requires synchronous cooperation between the motes. Figure 1 shows an example of CB where the different motes placed on a surface transmit the same signal with scheduled delay, creating a constructive interference illustrated as a beam pointing to a BS, in this case placed on a satellite. Further details are given in the following sections. CB is already a mature technology that is implemented in daily life appliances to overcome the restrictions imposed by a single transmitter in terms of transmission power as well as the environment such as path loss, shadowing, and multipath fading [6]. However, CB in the context of WSN is relatively new. In a traditional (centralized) multi-antenna transmitter, one way to perform beamforming is by exploiting reciprocity to estimate the complex channel gains to each antenna element [7]. These channel gains are computed in a centralized manner with reference to a carrier signal supplied by a local oscillator. However, in a distributed setting such as a WSN, each mote has separate carrier signals supplied by separate local oscillator circuits. These carrier signals are not synchronized a priori. This research considers the non-availability of a central clock that synchronizes all motes in the WSN. Due to the lack of carrier synchronization, it is not possible to estimate and precompensate the channel phase responses at the BS. The research on CB for WSN has been done mainly based on theoretical aspects.