# Position Accuracy and Distributed Beamforming Performance in WSNs: A Simulation Study
## 1 Introduction  
WSNs consist of numerous low-cost sensor nodes that communicate wirelessly. These miniature devices typically incorporate sensing, data processing, and communication capabilities within a compact form factor, ideal for various applications, including environmental monitoring, smart agriculture, and healthcare [1, 2, 3]. A key strength of WSNs resides in their collaborative nature, which is necessary to combat energy limitations, where a network of nodes collectively performs sensing tasks, offering significant advantages over traditional, isolated sensor deployments. This collaborative behaviour can be broadly categorised into two primary approaches: centralised and distributed [2, 3]. The difference between these two approaches is that in a centralised model, all the measurements are collected and processed at the central base station, like a Central Process Unit (CPU) or an anchor node, while in a distributed model, computation occurs at each node. Due to sensor hardware restrictions, distributed scheme solutions are considered cost-effective [2]. In recent years, research on antenna arrays and their applications has increased, leading to the development of novel technologies such as Large Intelligent Surfaces (LISs) [4] and radio stripes [5]. These advancements often rely on uniformly spaced Wireless Sensor Networks (WSNs) [6], which can be regarded as power-constrained distributed MIMO networks [7, 8]. One key area of focus within this research domain has been signal quality optimisation, with beamforming emerging as a prominent solution. The uniform spacingof antennas simplifies localisation, as determining the position of a single antenna within the array is sufficient to deduce the positions of all other antennas. Due to this, the challenges associated with beamforming become more notorious in non-uniform WSNs. Inherent position inaccuracies in the nodes can lead to significant phase errors, increasing the likelihood of destructive interference and degrading beamforming performance. To address this issue, numerous localisation techniques have been investigated to assess the feasibility of implementing effective beamforming in such scenarios, as demonstrated in [9]. As mentioned before, beamforming became a solution to improve signal quality at the receiver. This technique involves coherently combining individual signals by adjusting the phase of each one. This technique relies on precise phase adjustments, so accurate node position estimation is crucial for effective beamforming [10]. Recognising the inherent challenges of achieving perfect localisation in practical scenarios [9], this article presents a study that evaluates the system performance in the presence of random position errors in each node. This study focuses on distributed beamforming, where all nodes must communicate the same message synchronously at the same time [11]. To evaluate the impact of node position errors, we first determine the ideal signal amplitude under error-free conditions by applying the phase adjustment formulas for a given Signal-to-Noise Ratio (SNR). Subsequently, we introduce random position errors to the sensor nodes and re-calculate the SNR, allowing us to quantify the degradation in system performance due to these position inaccuracies, comparing the BER graphs. Furthermore, to understand the relationship between network size and system robustness, we employ the Cramer-Rao Lower Bound (CRB). The CRB provides a theoretical lower bound on the variance of any unbiased estimator, offering valuable insights into the fundamental limitations of position estimation accuracy as the number of sensors in the network increases. A practical scenario was simulated to validate these concepts. Signal degradation is primarily attributed to path loss, modelled using the Friis free-space path loss equation. A simulation testbed was developed to investigate the impact of position errors and topology configurations on the received WSN signal, incorporating the CRB and beamforming techniques.