## 2 Proposed TDoA-Based Method
### Phase Reconstruction Method
Consider that to have an ideal coherent IQ receiver connected through a switch to each different antenna of the array, as in \[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]6 For either I and Q demodulator output channels, the signal is sampled fulfilling the Nyquist sampling condition for signal given in (14) providing \(f_{\mathrm{SAMPLE}}\geq\mathrm{B}\), with bandwidth \(\mathrm{B}\) defined as in \[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]7 The antenna switching mechanism is synchronized to signal period \(\mathrm{T_{period}}\) to ensure time domain coherency between different antenna readings. Given a transmitted signal \(s_{\mathrm{TX}}(t)=s(t)\), an ideal radio-link of distance \(R\) would provide a received signal equal to  
\[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]8  
where \(\tau\) corresponds to the propagation delay, \(\mathrm{G_{R}}\) the receiver antenna gain, and \(H_{0}\) the coefficient modeling transmission gain and radio link path-loss, as the effects of the radio channel. Applying concepts of dynamic system modeling, it can also be written that  
\[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]9  
\[s_{\text{TX}}(t)=\left(\sum_{n=-\frac{M}{2}}^{\frac{M}{2}}A_{n}\,\text{e}^{i/2 \pi\cdot n\Delta f\cdot t}\right)\text{e}^{i/2\pi\cdot f_{0}\cdot t}\quad\text{ with }\Delta f=\frac{\text{B}}{M} \tag{14}\]0 Scheme of an IQ demodulator switched beam array-based receiver chain. where \(H(f)\) is the radio channel transfer function with related impulsive response \(h(t)\). Considering a wide-band signal, the path-loss cannot be defined by a single value, so the channel transfer function must be written as  
\[H(f)=|H(f)|\cdot\mathrm{e}^{i\angle H(f)}\quad\text{ with }\left\{ \begin{array}{l}|H(f)|= \mathrm{G}_{R}\,\mathrm{G}_{T}\Big{(}\frac{\varsigma_{0}}{f}\frac{1}{4\pi R} \Big{)}^{2}\\ \angle H(f)=-2\pi\cdot f\cdot\tau+\gamma_{0}\end{array} \right. \tag{25}\]  
in the case of broadband transmitting and receiving antennas with constant gains \(\mathrm{G}_{T}\) and \(\mathrm{G}_{R}\) along the entire signal bandwidth, and a generic phase misalignment \(\gamma_{0}\) between \(f_{0}\) receiver local oscillator and transmitter one [36]. Factorizing the variation term on signal components frequencies with respect to the transform function reference value in the carrier frequency \(f_{0}\), the transfer function can be rewritten as  
\[H(f)=\mathrm{G}_{R}\,H_{0}\cdot H_{\Delta}(f)\quad\text{with }\left\{ \begin{array}{l}H_{0}=\mathrm{G}_{T}\Big{(}\frac{\varsigma_{0}}{f_{0}}\frac{ 1}{4\pi R}\Big{)}^{2}\,\mathrm{e}^{j\gamma_{0}}\\ H_{\Delta}(f)=\Big{(}\frac{f_{0}}{f}\Big{)}^{2}\,\mathrm{e}^{-j/2\pi\cdot f \cdot\tau}\end{array} \right. \tag{26}\]  
The postprocessing unit in Figure 4 is able to get a complete time observation of complex envelope signal as in Figure 2, because IQ demodulation provides the frequency shifting of bandpass signal centered in \(f_{0}\) to baseband keeping spectrum asymmetries and relative tone phases.