## 2 Proposed TDoA-Based Method
### Definition of a Valid Observation Model  
As summarized in Section 2.1, a valid observation model is needed to calculate the ideal expected steering vectors belonging to the reference map [16; 22; 24]. Such a model must be entirely defined, as in (4) the reference map \(\mathbf{M}(\vartheta,\varphi)\) shall be defined as a function of DoA domain only. The easiest way to define the latter is to implement the definition for the \(s_{i}\) terms as given in (32). Let consider the expected physical steering-vector related to RF signal propagation. Considering the relationship between the baseband signal and the RF signal it results that  
\[s_{i\mathrm{R}}=\frac{s_{\mathrm{RX}}(\tau_{\mathrm{PKi}}^{\tau_{\mathrm{Ki}} },\mathrm{i})\,\mathrm{e}^{-j\,2\pi f_{0}\tau_{\mathrm{Ki}}}}{s_{\mathrm{RX}}( \tau_{\mathrm{P}\mathrm{K}}\mathrm{R},\mathrm{R})\,\mathrm{e}^{-j\,2\pi f_{0 }\tau_{\mathrm{KR}}}} \tag{41}\]  
where \(\tau_{\mathrm{PKi}}^{\tau_{\mathrm{Ki}}}\) is the real peak delay respect the beginning of signal transmission. Until now, the \(\tau_{\mathrm{PKi}}\) terms have been defined as the time instant associated to the i-th received signal peak within the baseband observation window. The real received peak time \(\tau_{\mathrm{PKi}}^{\tau_{\mathrm{Ki}}}\) is related to measured \(\tau_{\mathrm{PKi}}\) as  
\[\tau_{\mathrm{PKi}}=\tau_{\mathrm{PKi}}-\mathrm{t}_{0\mathrm{i}}\quad\forall \mathrm{i}\in[1,N] \tag{42}\]  
where \(\mathrm{t}_{0\mathrm{i}}\) is the time offset between the transmission and the beginning of signal period observation. Despite such offset is unknown if the period synchronization between different antenna time windows is ensured (Figure 4) we can suppose that \(\mathrm{t}_{0\mathrm{i}}=\mathrm{t}_{0}\) is constant for the entire set of readings. By this the difference between measured time peaks result  
\[\tau_{\mathrm{PKi}}-\tau_{\mathrm{PKj}}=\tau_{\mathrm{P}\mathrm{Ki}}-\tau_{ \mathrm{P}\mathrm{Kj}}\quad\forall\mathrm{i},\mathrm{j}\in[1,N]. \tag{43}\]  
Considering the transmitted signal spectrum as in (29) and the transfer function as in (26) it results  
\[s_{\mathrm{RX}}(\tau_{\mathrm{P}\mathrm{Ki}},\mathrm{i})\, \mathrm{e}^{-j2\pi f_{0}\tau_{\mathrm{Ki}}} =\int_{f_{0}-\frac{B}{2}}^{f_{0}+\frac{B}{2}}S_{\mathrm{TX}}(f)H( f)\,\mathrm{e}^{i2\pi(f-f_{0})\tau_{\mathrm{Ki}}}\,\mathrm{d}f \tag{44}\] \[=A_{\mathrm{TX}}\,\mathrm{G}_{i}(\vartheta,\varphi)H_{0}\int_{f_{ 0}-\frac{B}{2}}^{f_{0}+\frac{B}{2}}\mathrm{e}^{-j\,2\pi f\cdot\tau_{\mathrm{i} }}\mathrm{e}^{j2\pi(f-f_{0})\tau_{\mathrm{Ki}}}\,\mathrm{d}f\] \[=A_{\mathrm{TX}}\,\mathrm{G}_{i}(\vartheta,\varphi)H_{0}\, \mathrm{e}^{-j2\pi f_{0}\tau_{\mathrm{Ki}}}\int_{f_{0}-\frac{B}{2}}^{f_{0}+ \frac{B}{2}}\mathrm{e}^{j2\pi f(\tau_{\mathrm{Ki}}-\tau_{\mathrm{i}})}\, \mathrm{d}f\] \[=\left(A_{\mathrm{TX}}\mathrm{B}H_{0}\right)\mathrm{G}_{i}( \vartheta,\varphi)\,\mathrm{sinc}\left(\mathrm{B}\cdot(\tau_{\mathrm{PKi}}^{ \tau_{\mathrm{PKi}}}-\tau_{\mathrm{i}})\right)\mathrm{e}^{-j2\pi f_{0}\tau_{ \mathrm{Ki}}}\]so (41) becomes  
\[s_{i\text{R}}=\frac{\text{G}_{i}(\vartheta,\varphi)}{\text{G}_{R}(\vartheta, \varphi)}\cdot\frac{\text{sinc}\left(\text{B}\cdot\left(\tau_{\text{F}\text{i} }-\tau_{\text{j}}\right)\right)}{\text{sinc}\left(\text{B}\cdot\left(\tau_{ \text{F}\text{R}\text{R}}-\tau_{\text{R}}\right)\right)}\cdot\text{e}^{-2\pi f \left(\tau_{\text{F}\text{i}}-\tau_{\text{F}\text{i}}\right)}\,. \tag{45}\]  
Considering that for each antenna the shape of transmitted pulse is the same it results that  
\[\tau_{\text{F}\text{K}\text{i}}-\tau_{\text{i}}=\tau_{\text{F}\text{K}\text{j }}-\tau_{\text{j}}\quad\forall\text{i},\text{j} \tag{46}\]  
so finally we get  
\[s_{i\text{R}}=\frac{\text{G}_{i}(\vartheta,\varphi)}{\text{G}_{R}(\vartheta, \varphi)}\cdot\text{e}^{-2\pi f_{0}\left(\tau_{\text{F}\text{i}}-\tau_{\text{ F}\text{R}\text{R}}\right)} \tag{47}\]  
and the steering vector term defined as this has exactly the same form as given in (5) for the propagation of a monochromatic signal.