# A Simple and Efficient Method for RSS-AOA-Based Localization with Heterogeneous Anchor Nodes
## 1 Introduction  
With the rapid development and widespread utilization of wireless sensor networks (WSNs) [1, 2, 3, 4, 5] and mobile devices, the demand for accurate and reliable localization has increased significantly. This is particularly evident in indoor positioning [6], autonomous navigation [7], and intelligent transportation systems [8], where the ability to locate objects, vehicles, and people with high accuracy is essential. Additionally, in the context of the Internet of Things (IoT) [9], localizing devices and objects has become increasingly vital for optimizing IoT system performance, reducing energy consumption, and improving user experiences. Consequently, the creation of localization techniques that are accurate, reliable, and efficient has emerged as an essential research focus in recent years. One of the most promising localization approaches involves the use of hybrid received signal strength (RSS) and angle of arrival (AOA) measurements. By merging these two signals, it is possible to achieve greater accuracy and robustness than by relying on either signal independently. Hybrid RSS-AOA measurements have been widely used for localization. The measurement model utilizes RSS measurements to represent distance (where the RSS measurement value decreases as the distance increases), and AOA measurements to represent angle. This approach allows three-dimensional localization to be achieved with only one anchornode whose location is known, while using a single measurement model requires three or more anchor nodes. Additionally, the RSS-AOA measurement model offers high robustness against shadowing and multipath effects [10], which makes it a more resilient option compared to other localization methods. Numerous localization methods based on hybrid RSS-AOA measurements have been proposed. Among them, linearly weighted least squares (LWLS) methods are promising and most widely used, offering both high accuracy and low computational complexity. The main procedure of this approach involves converting the distance between anchor nodes and the target node into a linear expression of the target node's location using spherical coordinate transformation. The LWLS method is then utilized to construct a closed-form solution for the target node's location. The accuracy of these methods depends on the choice of weight matrices, and various weight matrix computation methods have been proposed in the literature to achieve high localization accuracy. Many commonly used weight matrix computation methods have been proposed in the literature. The least squares (LS) method in [11], which did not use any weight matrix (i.e., the weight matrix was a unit matrix), was often used as a benchmark for comparison in simulations. The distance-related WLS method, as proposed by [PERSON] et al. in [12], has been shown to improve localization accuracy without adding complexity. However, using weights that only change based on the distance between the target node and anchor nodes may not be optimal. [PERSON] et al. [13] developed the error covariance-weighted least squares (ECWLS) method, which computes weights based on an approximate error covariance matrix calculated from the LS method estimation of the target location. This approach led to more accurate weights, but estimating the variance of measurement noise was challenging, and directly multiplying the estimated variance into the weight could increase error. [PERSON] presented the two-step error variance-weighted least squares (TSLS) method based on AOA measurement [14], which estimated the target location using the LS method and then calculated the variance as the weight. However, this method only considered the noise variance of the evaluation function items and not the noise value of the measurements. To address this, [PERSON] et al. [15] proposed an error variance and noise value WLS (ENWLS) that could adjust the weight to account for measurement noise values, giving more weight to measurements with smaller noise values.