## 4 Impact of the Constellation Mesh Area on Localization Accuracy
### Simulation Results  
In this work, the accuracy of the position estimator is firstly evaluated by a set of simulations. This permits to generate a test bench by which evaluate the results, and in prospective provide a model for the localization estimator in a given environment. This eventually permits to plan an effective anchor network topology in an optimal and reliable way, avoiding a lengthy on-site verification. Given the fact that the selected localization algorithm relies on RSSI measurements as defined for IEEE 802.11x and IEEE802.15.4 wireless network standards, the synthetic localization error based on CRB evaluation provide a first order approximation of the obtainable results, not always reliable yet necessary the initial stage to define the real distributed positioning anchors network. This is due to the inherent formulation of the CRB, cf. Section 3.2, that provides the analytical characterization of the localization error assuming the steering vectors composed of the received signals at the antenna section. In [12], it is discussed how the RSSI can be considered as an output variable of a complex digital correlation algorithm with a high process gain. Given this, the effective system performance should be analyzed through an algorithm capable of encompassing every data processing layer, [26]. In particular, the RSSI noise model should be described using its own detection error parameters rather \begin{table}
\begin{tabular}{c c c} \hline \hline
**Configuration** & **Anchor Positions** & **Mesh Area** \\ \hline \multirow{2}{*}{(A) Large} & \#1 (1.50, 5.00) m; \#2 (1.50, 1.00) m & 17 m\({}^{2}\) \\  & \#3 (6.00, 5.00) m; \#4 (6.00, 1.00) m & 40\% of room area \\ \hline \multirow{2}{*}{(B) Medium} & \#1 (2.20, 3.50) m; \#2 (2.20, 1.20) m & 7.6 m\({}^{2}\) \\  & \#3 (5.50, 3.50) m; \#4 (5.50, 1.20) m & 17\% of room area \\ \hline \multirow{2}{*}{(C) Tiny} & \#1 (2.70, 2.65) m; \#2 (2.70, 1.70) m & 1.19 m\({}^{2}\) \\  & \#3 (3.90, 2.65) m; \#4 (3.90, 1.70) m & 2.6\% of room area \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Anchor–mesh configurations. Figure 9: **(a)** Bidimensional map of the experimental test site, **(b)** with the superimpose of the three constellations adopted in this work, and **(c)** the corresponding picture. than the effective physical signal-to-noise ratios: such detection error parameters are defined within protocol standards, and they depict a less noisy measurement scenario than the straight physical one [27]. The RSSI noise model can be brought back to an Additive White Gaussian Noise (AWGN) model, as described in [18]. The simulation process applies the localization estimator described in Equation (3), by considering the following steering vector in the (x,y) domain:  
\[\hat{S}=\langle s_{11}\lor s_{11}\lor\ldots\lor s_{MN}\rangle^{H}=\text{M}(x,y) +\hat{T}+\hat{N}, \tag{8}\]  
where the steering vector \(\hat{S}\) length is \(\sum\limits_{i=1}^{N}M_{i}=M\!N\), while \(\hat{T}\) contains the radiation pattern of the mobile tag node observed by the N anchors:  
\[\hat{T}=\left( \begin{array}{c}\begin{array}{c}\mathcal{T}_{1}\\ \mathcal{T}_{2}^{\prime}\\ \vdots\\ \mathcal{T}_{N}^{\prime}\end{array}\end{array} \right)=\left( \begin{array}{c} \begin{array}{c}\text{G}_{\text{TAG}}(\mathfrak{s}_{1},\varphi_{1})\cdot I_{ M_{1},1}\\ \text{G}_{\text{TAG}}(\mathfrak{s}_{2},\varphi_{2})\cdot I_{M_{2},1}\\ \vdots\\ \text{G}_{\text{TAG}}(\mathfrak{s}_{N},\varphi_{N})\cdot I_{M_{N},1}\end{array} \end{array} \right). \tag{9}\]  
The term \(\text{G}_{\text{TAG}}(\mathfrak{s}_{1},\varphi_{1})\) is the mobile tag antenna gain seen by each anchor antenna element. The anchor and the mobile tag are considered as points in space. The unknown mobile tag spherical coordinate \((\varphi_{i},\mathfrak{s}_{i})\), as observed by each anchor node installed on the ceiling at \((x_{i},y_{i})\), are:  
\[\left\{