## 2 Proposed TDoA-Based Method  
### Application Scenario and Analytical Model  
The key point of the method proposed in this work is the capability of extracting coherent phase-displacement information given an array system that is intrinsically phase-impaired in the working frequency band. For a better understanding of phase-impairing problem, consider an ideal Uniform Circular Array (UCA) in a tridimensional scenario [21] as in Figure 1.  
In Figure 1, an RF-continuous wave is impinging towards the UCA, propagating on a trajectory defined by angle \((\theta,\varphi)\) with respect to Considering a simple wave propagation model, the propagating plain wave at frequency \(f_{0}\) impinging on each i-th antenna can be written as  
\[s(t,\mathrm{i})=\Re\Big{\{}A_{0}\,\mathrm{e}^{\mathrm{i}\left(2\pi f_{0}t- \overline{\mathbf{k}}(\vartheta,\varphi)\cdot\overline{\mathbf{r}}_{\mathrm{i}} \right)}\Big{\}}=A_{0}\cos\Bigl{(}2\pi f_{0}t-\overline{\mathbf{k}}(\vartheta,\varphi)\cdot\overline{\mathbf{r}}_{\mathrm{i}}\Bigr{)} \tag{1}\]  
with  
\[\overline{\mathbf{r}}_{\mathrm{i}}=\left( \begin{array}{c}x_{\mathrm{i}}\\ y_{\mathrm{i}}\\ z_{\mathrm{i}}\end{array} \right),\quad\overline{\mathbf{k}}(\vartheta,\varphi) =\left(\frac{2\pi}{\lambda_{0}}\right)\mathbf{\hat{k}},\quad\mathbf{\hat{k}}= \left( \begin{array}{c}\sin\vartheta\cos\varphi\\ \sin\vartheta\sin\varphi\\ \cos\vartheta\end{array} \right) \tag{2}\]  
where \(x_{\mathrm{i}},y_{\mathrm{i}},z_{\mathrm{i}}\) are equal to i-th antenna center of phase coordinates respect reference system \(\overline{\mathbf{O}}\) as indicated in Figure 1, and \(\overline{\mathbf{k}}\) is the wave vector which portrays the wave propagation direction and speed. Applying (1), the phase difference between signals received at two different i-j antennas results  
\[\Delta\alpha_{ji}=\angle s(t,\mathrm{j})-\angle s(t,\mathrm{i})=\overline{ \mathbf{k}}(\vartheta,\varphi)\cdot\overline{\mathbf{r}}_{\mathrm{j}}- \overline{\mathbf{k}}(\vartheta,\varphi)\cdot\overline{\mathbf{r}}_{\mathrm{i }}=\overline{\mathbf{k}}(\vartheta,\varphi)\cdot(\overline{\mathbf{r}}_{ \mathrm{j}}-\overline{\mathbf{r}}_{\mathrm{i}}) \tag{3}\]  
and it is time unvarying but dependent on physical antenna positions and signal trajectory. The phase difference is directly proportional to the projection of i-j antennas distance on wave propagation direction (Figure 1). As described in [16; 22], the tridimensional DoA estimation can be placed as finding the solution of the following problem:  
\[\left(\hat{\vartheta},\hat{\varphi}\right)=\operatorname{argmin}_{\left( \vartheta,\varphi\right)}\mathrm{C}\left(\vartheta,\varphi\right)\quad\text{ with }\mathrm{C}\left(\vartheta,\varphi\right)=\mathrm{F}\bigl{(} \overline{\mathbf{S}},\mathbf{M}(\vartheta,\varphi)\bigr{)} \tag{4}\]  
where \(\mathrm{C}\left(\vartheta,\varphi\right)\) is a ML estimator (i.e., MUSIC [21; 23; 24], ESPRIT [25], minVAR [16; 18]), \(\overline{\mathbf{S}}\) is the steering vector (or rather the vector comprising amplitude and phase signal readings for each antenna), and \(\mathbf{M}(\vartheta,\varphi)\) is the reference map as shown in [16]. The reference map contains the ideal steering vectors given by model [26; 27] for every \(\left(\vartheta,\varphi\right)\) DoA, and the ideal goal for the ML estimator is to get its minimum value only when \(\left(\hat{\vartheta},\hat{\varphi}\right)\) exactly corresponds to the real DoA. Different ML estimators can overcome different kinds of inequalities between measured and ideal steering vectors, and as shown in [16; 22; 28], the minVAR estimator resulted to be the most feasible one in case of phase-less scalar power antenna or received signal strength indicator (RSSI) readings. Considering an array with \(N\) antenna elements, the generic steering vector \(\overline{\mathbf{S}}_{0}\) is defined as  
\[\overline{\mathbf{S}}_{0}=\left( \begin{array}{c}s_{01}\\ s_{02}\\ \vdots\\ s_{0N}\end{array} \right)\quad\text{with }