## 5 RF Switch Model
As a result, the IQ components can be expressed in Equation (28),  
\[s(t) =Ae^{j(2\pi(f_{\Delta}+f_{o})t+\psi)}\] \[=\underbrace{A\cos(2\pi(f_{\Delta}+f_{o})t+\psi)}_{I(t)}+j \underbrace{A\sin(2\pi(f_{\Delta}+f_{o})t+\psi)}_{Q(t)}, \tag{28}\]  
where \(A\) is the amplitude, \(\psi\) is the initial phase, \(f_{\Delta}=250\) kHz is the frequency deviation considering LE 1M physical layer [4], and \(f_{o}\) is the carrier frequency offset (CFO) which is in order of 10 kHz [29]. Without loss of generality, let us consider that IQ samples from the reference period correspond to the first antenna of the L-shaped array. From Equation (28),  
Figure 4: Depiction of the transmitter and receiver operations. the phase shift between two consecutive samples of the reference period without considering AWGN is evidenced by Equation (29),  
\[h_{1}(t+\Delta t)=e^{j2\pi\Delta f/\Delta t}h_{1}(t), \tag{29}\]  
where \(\Delta\) t = 1 \(\upmu\)s and \(\Delta f=f_{\Delta}+f_{o}\). In other words, it is possible to estimate the phase shift over a 1 \(\upmu\)s time period by using the samples of the reference period. However, we are interested in \(\Delta f\) since we can use it to estimate the phase shift over other time periods. Assuming the carrier frequency offset is constant during the CTE, we can estimate \(\Delta f\) by calculating the average of the phase difference between two consecutive IQ samples of the reference period, as shown in Equation (30),  
\[\widehat{\Delta f}=\frac{1}{7(2\pi\Delta t)}\sum_{i=1}^{7}\arg\Bigl{(}\frac{h _{1}(t+i\Delta t)}{h_{1}(t+(i-1)\Delta t)}\Bigr{)} \tag{30}\]  
By adopting the Round Robin switch pattern, each antenna from the L-shaped array carries out IQ sampling sequentially, as shown in Figure 5. Note that the switch pattern begins in the last reference period slot. As a result, the L-shaped array samples 75 IQ samples in total, 74 samples from the switch sample period, and 1 sample from the last reference period slot. Moreover, the array observation, in this case, is defined as one single sequence of the Round Robin pattern, that is, when all antennas in the array complete the IQ sampling. Observe that antenna \(k\) performs an IQ sampling \(2(k-1)\upmu\)s after the array observation starts. It means that the phase shift is \(e^{j2\pi\Delta f/\Delta t_{k}}\) without considering AGWN as indicated by Equation (31),  
\[h_{k}(t+\Delta T_{k})=e^{j2\pi\Delta f/\Delta t_{k}}h_{k}(t), \tag{31}\]  
where \(\Delta T_{k}=2(k-1)\mu s\). We can generalize the observation of Equation (31). As a result, let \(\mathbf{h}_{ss}(t)\) be an array observation of a Round Robin sequence that begins at timestamp \(t\), the array \(\mathbf{h}_{ss}(t)\) relates to \(\mathbf{h}(t)\) by the phase shift matrix due to the RF switch labeled as \(\mathbf{O}\) in accordance with Equation (32),  
\[\mathbf{h}_{ss}(t)= \begin{bmatrix}h_{1}(t)\\ h_{2}(t+\Delta T_{2})\\ \vdots\\ h_{2M-1}(t+\Delta T_{2M-1})\end{bmatrix}\begin{bmatrix}h_{1}(t)\\ h_{2}(t)e^{j2\pi\Delta f/\Delta T_{2}}\\ \vdots\\ h_{2M-1}(t)e^{j2\pi\Delta f/\Delta T_{2M-1}}\end{bmatrix} =\mathbf{O} \begin{bmatrix} h_{1}(t)\\ h_{2}(t)\\ \vdots\\ h_{2M-1}(t)\end{bmatrix} =\mathbf{O} \begin{bmatrix}h_{1}(t)\\ h_{2}(t)\\ \vdots\\ h_{2M-1}(t)\end{bmatrix} =\mathbf{O} \begin{bmatrix}h_{2}(t)\\ \vdots\\ h_{2M-1}(t)\end{bmatrix} =\mathbf{O} \begin{bmatrix}h_{1}(t)\\ h_{2}(t)\\ \vdots\\ h_{2M-1}(t)\end{bmatrix} =\mathbf{O}\begin{bmatrix}h  
For example, a receiver could not be able to decode both transmitter IDs of the transmitters reliably as each transmission would interfere with the other. Therefore, in this scenario, DOA methods can only estimate a single DOA only per execution. As result, the L-shaped array observation model of one sequence of the Round Robin pattern is defined in Equation (34),  
\[\mathbf{h}_{\mathrm{ss}}(t)=\mathbf{O}\mathbf{h}(t)=\mathbf{O}\mathbf{a}(\alpha,\beta)s(t)+\mathbf{n}_{h}(t). \tag{34}\]  
Note, \(s(t)\) is a scalar that represents one single signal in opposition to the vector \(\mathbf{s}(t)\) found in Equations (2) and (3).