## 2 Proposed TDoA-Based Method
### Phase-Wrapping Issues  
Now, let us consider to implement a coherent sensing chain able to get complete amplitude-phase measurements for each term \(s_{i}\) of (5). Dealing with a simple scenario as the one in Figure 1, it is trivial that it must result  
\[\alpha_{5}\leq\alpha_{4}\leq\alpha_{3}\leq\alpha_{1}\leq\alpha_{2} \tag{6}\]  
where the sorting order is strictly dependent on signal direction of arrival \((\vartheta,\varphi)\). Note that such a sequence is preserved also applying a linear transform on each measured phase, so normalizing the steering vector respect to a generic R-th reference antenna as in (5) it results  
\[\Delta\alpha_{5R}\leq\Delta\alpha_{4R}\leq\Delta\alpha_{3R}\leq\Delta\alpha_{ 1R}\leq\Delta\alpha_{2R}\quad\text{with}\;\;\Delta\alpha_{iR}=\left(\alpha_{ i}-\alpha_{R}\right) \tag{7}\]  
and the \(\Delta\alpha_{iR}\) terms are the ones depicted in (3).  
Considering an ideal sensing chain the phase difference terms as defined in (3) is always monotone with respect to antenna distance. Unfortunately, a real phase detection is always impaired by phase wrapping issue, so measured \(\Delta\alpha_{iR}\) terms must be placed as  
\[\Delta\hat{\alpha}_{iR}=\Delta\alpha_{iR}\;\text{mod}\;2\pi=\left(\alpha_{i} -\alpha_{R}\right)\;\text{mod}\;2\pi \tag{8}\]  
where the introduction of modulus operator (mod) has dramatic effects on phase information coherency, disrupting the phase terms sorting order as given in (7). To avoid such phenomena, a condition can be placed imposing that the maximum \(\Delta\alpha_{iR}\) term should be lower than \(2\pi\). Because phase displacement terms are dependent on signal DoA, this would require to dynamically change the reference antenna considering an a priori knowledge of expected direction of arrival, which is nonsensical.  
A more affordable condition could be the bounding of the maximum distance between center of phases of different antennas, so that given any reference antenna all terms of (7) will be surely lower than \(2\pi\).  
Considering (3), such condition can be written as  
\[\max\left\{\overline{\mathbf{k}}(\vartheta,\varphi)\cdot\left(\overline{\mathbf{ r}}_{\mathrm{j}}-\overline{\mathbf{r}}_{\mathrm{i}}\right)\right\}\leq 2\pi\qquad \forall\;\mathrm{i}\in[1,N],\mathrm{j}\in[1,N],(\vartheta,\varphi) \tag{9}\]  
and given an array structure it allows to identify a sub-domain for signal DoAs where the phase ambiguity is avoided (or rather, a \"Field of View\" similar to one defined by Nyquist sampling conditions for MIMO arrays [12; 33; 34]).  
By array geometry, surely it can be put that  
\[\overline{\mathbf{k}}(\vartheta,\varphi)\cdot\left(\overline{\mathbf{r}}_{ \mathrm{j}}-\overline{\mathbf{r}}_{\mathrm{i}}\right)\leq\left|\frac{2\pi}{ \lambda_{0}}\right|\cdot\max_{\mathrm{ij}}\left|\Delta\overline{\mathbf{r}}_{ \mathrm{j}}\right|\leq 2\pi\qquad\forall\;\mathrm{i}\in[1,N],\mathrm{j}\in[1,N],( \vartheta,\varphi) \tag{10}\]  
so it results  
\[\max_{\mathrm{i},\mathrm{j}}\left|\Delta\overline{\mathbf{r}}_{\mathrm{j}} \right|\leq\lambda_{0}\qquad\forall\;\mathrm{i}\in[1,N],\mathrm{j}\in[1,N] \tag{11}\]  
and by (11) it is clear how much the dimensions of the entire array structure must be constrained in order to achieve an effective signal DoA estimation on the overall \((\vartheta,\varphi)\) domain.  
As an example an array working in \(f_{0}=2.45\) GHz, the IEEE 802.11 bg WiFi frequency band should have the maximum distance between antennas center of phases equal to \(\lambda_{0}=12.24\) cm, while shifting to the upper \(f_{0}=5.2\) GHz band for 802.11n such distance lowers to \(\lambda_{0}=5.7\) cm. Dealing with a SBA system, as, for instance, the one described in Section 3, each antenna of the array will have physical dimensions comparable to \(\lambda_{0}\), thus fulfilling such dimensional constraints will lead to serious device engineering issues.