## 2 Proposed TDoA-Based Method
### Phase Reconstruction Method  
The constraint on array structure dimensions is given by condition (11), which is a direct consequence of (3). A straight way to relax such condition is to increase the signal wavelength \(\lambda_{0}\), but this would require to shift working frequency band. This work aims to introduce a method able to unwrap actual phase detections by synthetically lowering the equivalent signal wavelength as placed in (3). For propagating single tone signals, the wavelength is given by [35]  
\[\lambda_{0}=\frac{\
u_{\mathrm{phase}}}{f_{0}}\qquad\mathrm{with}\;\;\
u_{ \mathrm{phase}}=2\pi\left|\frac{\overline{\mathbf{k}}}{f_{0}}\right|^{-1}, \quad\left|\overline{\mathbf{k}}\right|=\frac{2\pi}{\lambda_{0}}=2\pi\left( \frac{f_{0}}{c_{0}}\right)\sqrt{\varepsilon_{R}\mu_{R}} \tag{12}\]  
where \(\
u_{\mathrm{phase}}\) is the phase speed related to single tone propagation on propagation direction, \(c_{0}\) is the speed of light, and \(\varepsilon_{R},\mu_{R}\) are the relative dielectric and diamagnetic constants for propagation medium. Checking (12), it is clear that there is not any degree of freedom left for varying the equivalent \(\lambda_{0}\) parameter. Dealing with non-monochromatic signals, or rather modulated RF signals, the propagating speed of the entire signal envelope on propagating direction is no more equal to phase velocity but it equals the group velocity (13) [35]. \[\
u_{\mathrm{group}}=2\pi\left(\left.\frac{\vartheta\left|\overline{\mathbf{k }}\right|}{\partial f}\right|_{f=f_{0}}\right)^{-1}\quad\mathrm{with}\;\; \overline{\mathbf{k}}=\mathrm{f}\left(A_{-\frac{M}{2}},f_{-\frac{M}{2}},\ldots,A_{\frac{M}{2}},f_{\frac{M}{2}}\right)\
eq\frac{2\pi}{\lambda_{0}} \tag{13}\]  
The group speed definition as given in (13) is too complex to be useful for inferring relation between modulating signal parameters and equivalent \(\lambda_{G}\) obtained. Nevertheless,by (13) is clear how further degrees of freedom can be added by introducing a modulated signal in place of a single tone. In place of a single-tone signal, consider using a multi-tone composed signal such as  
\[s_{\text{TX}}(t)=\left(\sum_{n=-\frac{M}{2}}^{\frac{M}{2}}A_{n}\,\text{e}^{i/2 \pi\cdot n\Delta f\cdot t}\right)\text{e}^{i/2\pi\cdot f_{0}\cdot t}\quad\text{ with }\Delta f=\frac{\text{B}}{M} \tag{14}\]  
where B is the pass-band signal bandwidth, \(M\) is the number of tones (even), and \(A_{i}\) is the signal amplitude for the i-th tone. To simplify the scenario, (14) can be simplified considering a constant signal amplitude \(A_{i}=A_{0}\) leading to signal in Figure 1(a). Figure 2 shows frequency and time domain plots for signal as defined by (14). Because the signal is composed by a discrete number of harmonics, its envelope is periodic with period and pulse width equal to  
\[\text{T}_{\text{period}}=\frac{1}{\Delta f}=\frac{M}{\text{B}},\quad\text{T} _{\text{pulse}}\approx\frac{1}{B}\quad\text{with }\ M<\text{B} \tag{15}\]  
It is possible to exploit signal periodicity trying to fit a simpler signal model on transmitted signal. For instance in Figure 1(b) a monochromatic signal coarsely fits on actual signal, preserving the matching simply the signal maximums to envelope maximums. The fit signal model can be written as  
\[s_{\text{FIT}}(t,\beta_{\text{FIT}})=A_{0}\,\text{e}^{i\,(2\pi f_{\text{FIT}} t-\beta_{\text{FIT}})} \tag{16}\]  
where \(f_{\text{FIT}}\) and \(\beta_{\text{FIT}}\) are chosen to achieve the hard matching between real signal and fit signal maximums. Note that \(f_{\text{FIT}}\) is known by reference signal periodicity and it results  
\[f_{\text{FIT}}=\frac{1}{\text{T}_{\text{period}}}=\frac{\text{B}}{M} \tag{17}\]  
Figure 2: Example of a generic multi-tone signal with constant tones amplitude and Sine Fit. while \(\beta_{\text{FIT}}\) provides the exact alignment of maximums by shifting the fit signal. Until now, the proposed signal model has been fit considering a specific starting time for the signal envelope, thus providing a fixed value for \(\beta_{\text{FIT}}\).