# Position Accuracy and Distributed Beamforming Performance in WSNs: A Simulation Study
## 2 Distributed Beamforming  
Before starting to investigate distributed beamforming, it is essential to understand the distinction between near-field and far-field radiation in the context of single antenna and large antenna aggregates, including distributed configurations such as our case study of WSNs, where sensors are assumed to communicate cooperatively using distributed beamforming techniques. This distinction depends on frequency, the aperture size of individual antennas, and, in the case of antenna arrays, the overall array dimensions,as illustrated in Figure 1. In the far-field, the distance between the antenna array and the receiver is sufficiently large that the differences in path lengths from each array element to the receiver can be considered negligible, and the wavefront can be approximated as planar. As outlined in [10], the minimum distance \(d_{F}\) to reach the far-field region is typically defined by the Fraunhofer distance, which for an antenna aggregate of size \(N\times N\) is given by  
\[d_{F}\geq\frac{2D^{2}}{\lambda} \tag{1}\]  
where \(D=N\times L\), \(L\) is the physical size of a single antenna element, and \(\lambda\) is the wavelength. On the other hand, the near-field region can be sub-divided into two areas: the reactive near-field, dominated by evanescent waves and magnetic induction, which occurs at distances below  
\[d_{R}\leq 0.62\sqrt{\frac{D^{3}}{\lambda}} \tag{2}\]  
and the radiating near-field (also known as the Fresnel region), which lies between \(d_{R}\) and \(d_{F}\), as illustrated in Figure 1.  
For distances greater than or equal to the Fraunhofer distance \(d_{F}\), the differences in path lengths from each array element (or node) to the receiver are negligible, making phase adjustments for beamforming in the far-field unnecessary. However, due to the typically large distances required to satisfy the Fraunhofer condition, especially at higher frequencies and for large antenna aggregates, this study focuses on the radiating near-field region. In the context of WSNs, the receiver is generally in the far-field of each individual sensor but in the near-field of the entire sensor network. Therefore, our analysis concentrates on the radiating near-field, where the distance to the receiver is short enough to cause significant phase and/or amplitude variations across the array, yet sufficiently large to neglect direct hardware interactions between transmitter and receiver [12]. To establish effective distributed beamforming, precise synchronisation among the nodes of the system is essential. This synchronisation process involves carefully adjusting the timing of each node's transmission, either by delaying or advancing its output signal, to ensure that the signal components from the desired direction arrive at the receiver in phase. This temporal alignment enables constructive interference among the signals, result  
Figure 1: Near- and far-field regions [13]: (**left**) single antenna (**right**) antenna array. ing in a significant enhancement of the received signal strength [14]. Figure 2 represents a possible sensor distribution. Let \(s_{n}(t)\) be the transmitted signal from the \(n^{th}\) sensor being obtained by applying a phase shift \(\theta_{n}\) to the measured data, \(x(t)\) according to Equation (3):  
\[s_{n}(t)=\Re\left\{x(t)e^{i\omega_{n}t}\right\}=\Re\{x(t)e^{i\omega_{n}(t_{0}-t _{\alpha_{n}})}\} \tag{3}\]  
Since we assume that all nodes are perfectly synchronized transmitting in \(t_{0}=0\), we can simplify this equation, obtaining Equation (4):  
\[s_{n}(t)=\Re\{x(t)e^{-i\omega_{n}t_{\alpha}}\}=\Re\{x(t)e^{-i\beta_{n}}\} \tag{4}\]  
This phase adjustment, \(\theta_{n}\), is crucial for achieving in-phase addition at the receiver and maximizing the received signal strength.