## 2 Proposed TDoA-Based Method
### Phase Reconstruction Method
Consider applying the condition on monochromatic signal fit as proposed in Figure 2, imposing equivalence between equivalent and received signals peaks, given that all the different antenna readings share the same time reference as specified in Figure 4.  
Assuming a monochromatic signal propagation the i-th steering vector term is  
\[s_{\mathrm{i}}=A_{\mathrm{FIT}}\,\mathrm{G}_{\mathrm{i}}(\theta,\varphi)H_{0} \,\mathrm{e}^{-\beta_{\mathrm{FIT}}}\quad\text{ with }\ \beta_{\mathrm{FIT}}=\overline{\mathbf{k}}_{\mathrm{FIT}}(\theta,\varphi) \cdot\overline{\mathbf{r}}_{\mathrm{i}} \tag{32}\]  
and taking in account that the equivalent signal must have the same peak as the real one it must be  
\[A_{\mathrm{FIT}}=\int_{-\frac{\mathrm{B}}{2}}^{\frac{\mathrm{B}}{2}}A_{ \mathrm{TX}}\,\mathrm{d}f=A_{\mathrm{TX}}\cdot\mathrm{B}. \tag{33}\]  
The normalized term respect the R-th reference antenna results in  
\[s_{\mathrm{iR}}=\frac{\mathrm{G}_{\mathrm{i}}(\theta,\varphi)}{\mathrm{G}_{ \mathrm{R}}(\theta,\varphi)}\,\mathrm{e}^{-j\overline{\mathbf{k}}_{\mathrm{ FIT}}(\theta,\varphi)\cdot(\overline{\mathbf{r}}_{\mathrm{i}}-\overline{\mathbf{r}}_{ \mathrm{R}})}\,. \tag{34}\]  
Because the steering vector term is the complex coefficient which models phase shift and amplitude gain given by the i-th antenna reception on signal, considering the baseband signal observations it results that  
\[s_{\mathrm{FIT}}(t,\mathrm{i})=s_{\mathrm{i}}\cdot\mathrm{e}^{\mathrm{i}(2\pi f _{\mathrm{FIT}}t)}\quad\text{ with }\ f_{\mathrm{FIT}}=\frac{\mathrm{B}}{M}=\frac{1}{\mathrm{T}_{\mathrm{ period}}} \tag{35}\]  
thus the imposition of peaks equivalence is placed as  
\[S_{\text{BBRX}}(f)=S_{\text{RX}}(f+f_{0}) \tag{28}\]0  
where \(\tau_{\mathrm{PKi}}\) is known by signal observations. Merging (35) with (36) it results that  
\[S_{\text{BBRX}}(f)=S_{\text{RX}}(f+f_{0}) \tag{28}\]1  
so the \(s_{i}\) terms can be extracted by features measurements of received \(s_{\mathrm{BBRX}}(t,\mathrm{i})\) signals, each one evaluated in the peak instant time \(\tau_{\mathrm{PKi}}\). As indicated in Section 2.1, all steering vector terms can be normalized respect the same coefficient. Considering a generic R-th reference antenna, each term \(s_{i}\) can be rewritten as  
\[S_{\text{BBRX}}(f)=S_{\text{RX}}(f+f_{0}) \tag{28}\]2  
and the latter shows the measurement formula for each antenna of the array. Note that the phase term could lead to phase-wrapping issues and its presence is related to ambiguity given by signal periodicity (Figure 3). To avoid such drawback a further condition can be put on exponential argument in (38). In fact, if it is placed  
\[S_{\text{BBRX}}(f)=S_{\text{RX}}(f+f_{0}) \tag{28}\]3  
the exponential term can be neglected avoiding any phase-wrapping issue. In addition to this it must be assured that the sample time \(\mathrm{T}_{\mathrm{sample}}\) is lower than minimum delay between different antennas to be sure that every antenna will measure a different peak arrival time. Considering the maximum signal propagation speed of \(c_{0}\), the equation above can be placed with respect to distance between antenna center of phases  
\[\left\{ \begin{array}{l}\mathrm{T}_{\mathrm{period}}=\frac{M}{B}\geq\max_{ \mathrm{i}\mathrm{j}\in[1,N]}\frac{|\Delta\tilde{\mathbf{r}}_{\mathrm{j}}|}{c_{ 0}}\\ \mathrm{T}_{\mathrm{sample}}=\frac{1}{B}\leq\min_{\mathrm{i}\mathrm{j}\in[1,N] }\frac{|\Delta\tilde{\mathbf{r}}_{\mathrm{j}}|}{c_{0}}\end{array} \right. \tag{40}\]  
and when the latter equations are satisfied the steering vector obtained as in (38) will present consistent phase information. Note that while the conditions above enable to get a phase consistent steering vector, the analytical form for \(\overline{\mathbf{k}_{\mathrm{FIT}}}(\vartheta,\varphi)\) as in (19) is still undetermined due to lack of knowledge about the \(\mathrm{v}_{\mathrm{group}}\) term.