# Performance Aspects of Retrodirective RFID Tags +
## 3 Results  
In this section, the performance of the retrodirective double-antenna tag is verified. Taking advantage of the presented channel model, we examine the case of a two-element tag equipped with a retrodirective signaling matrix. The reader uses one omni antenna, both for transmitting and for receiving. The tag uses two dipole antennas spaced at \(\lambda/2\). We assume it is a flat-fading, time-invariant, and narrowband channel (neglecting delays). We refer to Figure 3[12]. Figure 3: The 1x NTAG x1 backscatter channel is examined in the current work. Note that this is the well-known pin- (key-)hole two-way channel model, with the added information that the signal’s direction of departure is the same as the signal’s arrival due to the use of the retrodirective concept, embodied as the scattering matrix of the array antenna’s passive (self-)feeding network. The performance of the retrodirective array is to be compared with a broadside array, thus we will use the retrodirective and the identity signaling matrix. For the system in Figure 3, the channel matrix for the forward link reads:  
\[\mathbf{H}_{f}= \begin{bmatrix}1\\ e^{jrsin\theta_{f}}\end{bmatrix} h_{f}[1]^{T}, \tag{4}\]  
while the channel matrix for the backscattering link obeys:  
\[\mathbf{H}_{b}=[1]h_{b} \begin{bmatrix}1\\ e^{jrsin\theta_{b}}\end{bmatrix} ^{T}. \tag{5}\]  
The total channel matrix and receive signal follows:  
\[\mathbf{H}_{b}\cdot\mathbf{S}\cdot\mathbf{H}_{f}\Rightarrow z=\frac{1}{2}\mathbf{H}_{b}\cdot \mathbf{S}\cdot\mathbf{H}_{f}x+\mathbf{n}. \tag{6}\]  
We can use as a scattering matrix the identity one, Equation (2), or the retrodirective one [15] that obeys:  
\[S_{r}=j \begin{bmatrix}0&\Gamma\\ \Gamma&0\end{bmatrix} . \tag{7}\]  
For the two signaling matrices the total channel matrix reads:  
\[ \begin{split}\mathbf{H}_{i}&=h_{b}h_{f}\mathbf{\Gamma}\Big{(}1+e^{ jkd(\sin(\theta_{f})+\sin(\theta_{b}))}\Big{)}\\ \mathbf{H}_{r}&=jh_{b}h_{f}\mathbf{\Gamma}\Big{(}e^{j(kd\sin(\theta_{f}))}+e^{j(kd \sin(\theta_{b}))}\Big{)}\end{split} \.\end{split} \tag{8}\]  
\(\mathbf{H}_{i}\) is formed using the identity signaling matrix to model the broadside array while \(\mathbf{H}_{r}\) is formed using the retrodirective signaling matrix to model the retrodirective tag. Note that in the signaling matrix no coupling is taken into account for both cases. Since Tx antenna array works also as the Rx antenna array, then the \(\theta_{b}=\theta_{f}=\theta\) and the forward and backward channels are fully correlated, (\(h_{b}=h_{f}=h\)):  
\[ \begin{split}\mathbf{H}_{i}&=h^{2}\cdot\Gamma\cdot \Big{(}1+e^{jkdsin(\theta)}\Big{)};\\ \mathbf{H}_{r}&=j2h^{2}\cdot\Gamma\cdot e^{jkdsin( \theta)}.\end{split}  \tag{9}\]  
Note that the respective case when the two tag antennas are uncorrelated reads:  
\[\mathbf{H}_{diversity}=\Gamma\cdot\Big{(}h_{1}^{2}+h_{2}^{2}\Big{)}. \tag{10}\]  
When there is only one antenna in (10) only the \(h_{1}\) coefficient exist. Since termination \(\Gamma\) is between two values \(\Gamma^{\pm}\) then the received baseband complex envelope for four interesting cases reads:  
\[2\cdot|V|=\big{|}H^{+}-H^{-}\big{|}= \begin{cases}|\Delta\Gamma|h^{2}&\text{ One Antenna}\\ |\Delta\Gamma|\big{(}h_{1}^{2}+h_{2}^{2}\big{)}&\text{Spatial diversity}\\ |\Delta\Gamma|2h^{2}&\text{Retrodirective}\\ |\Delta\Gamma|h^{2}\Big{(}1+e^{j2 kdsin(\theta)}\Big{)}&\text{Broadside Array}\end{cases} . \tag{11}\]  
In Equation (11), apart from the broadside and retrodirective array, the one antenna and the two antenna spatial diversity cases were included. In the previous the average DC component of the envelope is assumed zero since it can be easily eliminated, via a DC blocking capacitor, [16], from the total received signal. Equation (11) indicates that the results are scaled with the amplitude of \(\mathbf{\Delta\Gamma}\) and thus this cannot differentiate the Tag's response between the available study cases.