## 2 Proposed TDoA-Based Method
### Phase Reconstruction Method
Relaxing condition (22) straightly enables to recover phase-information on phase-impaired scenario as in \[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]3 A pictorial proof of such concept is shown in \[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]4  
\[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]5 Example of a set of impinging multi-tone signals on an array. In Figure 3, three hypothetical impinging signal traces, with transmitted signal defined as in (1), are depicted. As in Figure 2b, the wavelength for \(f_{0}\) frequency is too small respect to the signal period; in fact, it is not possible to get a meaningful phase difference value considering the RF signal carrier. Conversely, the maximum distance between signal peaks is large, making possible to define a unambiguous phase delay \(\Delta\alpha_{21},\Delta\alpha_{31}\) also for higher time delays, or higher distances between different antennas. As a drawback dealing with periodic signals still leaves a risk of ambiguity, and in next Section such point will be discussed. By a practical standpoint it means that the receiver chain should sample the received signal envelope depicted in Figure 2, and a postprocessing unit has to extract signal features in order to get peak amplitude \(A_{i}\), proportional to each antenna gain \(\mathrm{G_{i}}(\theta,\varphi)\), and relative phase \(\beta_{\mathrm{FIT}}\) parameters related in some way to received signal relative time shiftings (Figure 3). The right time shifting information is preserved only if the antenna switching mechanism is synchronized to the transmitted signal period, to ensure the coherency of time domains between different acquisitions. Considering Figure 3, it is conceptually clear that doing feature estimations on different antenna signals and embedding these information in a steering vector, the latter shall contain an overall phase displacement information not impaired by wrapping. By this, a generic DoA estimation algorithm can be involved without any kind of issue. Note that in (22) the \(\mathrm{v_{group}}\) is still undetermined, and despite a generic definition of such term is given in (13), achieving an analytical formula for \(\lambda_{\mathrm{FIT}}\) could be a challenging task. In the following, a computational way to get the \(\lambda_{\mathrm{FIT}}\) and the \(\overline{\mathbf{k}}_{\mathrm{FIT}}\) terms is given.