# A Simple and Efficient Method for RSS-AOA-Based Localization with Heterogeneous Anchor Nodes
## 4 Simulation Results  
In this section, simulations are conducted to demonstrate the performance of the proposed localization method. The simulations can be considered as locating a smartphone (target node) using routers (anchor nodes), where the AOA can be calculated using the MUSIC algorithm with the router's antennas, and the RSS can be obtained directly from the communication protocol, for example, Wi-Fi (IEEE 802.11) or Bluetooth (IEEE 802.15.1). The performance of the proposed method is compared with existing state-of-the-art methods, including LS [11], WLS-d [12], ECWLS [13], TSLS [14], ENWLS [15], SDP-SOCP [18], and GTRS [19]. These methods do not use time series measurements, so the average values of time series measurements over time are taken as the input data for the compared methods. It is worth noting that the average measurement values over time are more accurate than the measurement values at any given time, and the variance of the measurement noise becomes \(\frac{1}{T}\) of the measurement noise variance at any given time. The CRLB is also included in each simulation to provide a theoretical benchmark for comparison with the actual performance of the considered localization methods.  
In all simulations, the anchor and target nodes were randomly distributed within a cubic space with a side length of \(40\text{ m}\). \(M\) independent Monte Carlo (Mc) runs were performed, where the locations of both the anchor and target nodes were re-randomized in each run. The measurement noise for each anchor node followed a zero-mean Gaussian distribution. Note that the variance of each measurement noise is not identical. The standard deviation of the measurement noise for RSS, azimuth angle, and elevation angle followed an exponential distribution with mean values of \(\mu_{n}\), \(\mu_{m}\), and \(\mu_{v}\), respectively. The performance is assessed through the calculation of root mean square error (RMSE), which is computed by  
\[\text{RMSE}=\sqrt{\frac{1}{M}\sum_{i=1}^{M}\lVert\mathbf{x}_{i}-\hat{\mathbf{x}}_{i} \rVert^{2}}, \tag{33}\]  
where \(\mathbf{x}_{i}\) and \(\hat{\mathbf{x}}_{i}\) are the true location and the estimated location, respectively, of the target node in the \(i\)-th Mc run. In this paper, we set \(M=3000\) for all simulations. Based on common wireless communication environments and empirical experience, the reference received signal power, \(P_{0}\), is set to \(-10\,\text{dBm}\), and the reference distance, \(d_{0}\), is set to \(1\,\text{m}\). The path-loss exponent, \(\gamma\), represents the rate at which the signal attenuates with increasing distance. In open spaces, \(\gamma\) is typically 2. However, in complex environments, \(\gamma\) can be larger, usually ranging between 2 and 6. Without loss of generality, this paper selects \(\gamma=2.7\). In the comparison, we vary the number of anchor nodes \(N\), the length of the measurement sequence \(T\), and the distribution variables \(\mu_{n}\), \(\mu_{m}\), and \(\mu_{v}\). These values will be introduced in each scenario. The MATLAB toolbox CVX [24] is utilized to solve the SDP and SOCP, with SeDuMi [25] as the selected solver.  
The computational complexity of the proposed method and the compared methods are first presented, followed by the comparison between the proposed method and the state-of-the-art methods.