## 2 System Model
### Bit Error Rate (BER) Analysis  
For the BER evaluation of radio frequency (RF) backscattering channels, we use the MGF as it transforms a more complex mathematical structure into a more tractable expression. Once the number of tag antennas is \(L=2\) and taking two time slots, the backscatter coding matrix is given by \(\mathbf{G}=\left(\begin{array}{cc}s_{0}&s_{1}^{*}\\ s_{1}&-s_{0}^{*}\end{array}\right)\). If \(\mathbf{G}\) is transmitted, using binary phase shift keying (BPSK) modulation there are three possible codewords as follows:  
\[\left(\begin{array}{cc}s_{0}&s_{0}^{*}\\ s_{1}&-s_{1}^{*}\end{array}\right)\,,\,\left(\begin{array}{cc}s_{1}&s_{0}^{*} \\ s_{1}&-s_{0}^{*}\end{array}\right)\,,\,\left(\begin{array}{cc}s_{0}&s_{1}^{*} \\ s_{0}&-s_{1}^{*}\end{array}\right) \tag{3}\]  
The Hamming distance is given by the codeword difference from the received symbol (one of the above) and the true transmitted symbol. Normalizing the Hamming distance for the transmitted codeword \(\mathbf{G}^{u}\) and received \(\mathbf{G}^{v}\) we have \(||\mathbf{g}^{u}-\mathbf{g}^{v}||=1\). For the quadrature phase shift keying (QPSK), the BER is defined as:  
\[P_{e}=Q\bigg{(}\sqrt{g\big{\|}|Y|_{F}^{2}E_{b}/N_{0}}\bigg{)} \tag{4}\]  
Figure 1: The backscatter system configuration.  
where \(Q(\cdot)\) is the \(Q\) function, \(E_{b}\) is the bit energy, and \(g\) is the constant gain that depends on the modulation scheme. For the QPSK modulation, \(g\) = 2 and we express \(\gamma=E_{b}/N_{0}\). The BER of our backscatter system, averaged over the channel \(Y=\mathbf{h}^{b}\mathbf{D}^{f}\) is given by:  
\[\begin{array}{l}p_{e}(\mathbf{G}^{u}\rightarrow\mathbf{G}^{v}|Y)\leq\frac{1 }{2}\exp\Bigl{(}\Bigl{|}\Bigl{|}Y\Bigr{|}\Bigr{|}\Bigr{|}_{F}^{2}\gamma\\ p_{e}(\mathbf{G}^{u}\rightarrow\mathbf{G}^{v}|Y)\leq\frac{1}{2}\exp\Bigl{(}- \sum\limits_{i=1}^{L}\gamma\delta_{i}\rho_{ii}\Bigr{)}\end{array} \tag{5}\]  
Because the two random variables are independent, we can write:  
\[\begin{array}{l}\mathbf{E}_{\delta,\rho}\Bigl{[}\frac{1}{2}\exp\Bigl{(}- \sum\limits_{i=1}^{L}\gamma\delta_{i}\rho_{ii}\Bigr{)}\Bigr{]}\\ =\frac{1}{2}\prod\limits_{i=1}^{L}\mathbf{E}_{\delta_{i}\rho_{ii}}\biggl{[} \exp\Bigl{(}-\sum\limits_{i=1}^{L}\gamma\delta_{i}\rho_{ii}\Bigr{)}\biggr{]} \end{array} \tag{6}\]  
In a real propagation environment, the line-of-sight (LoS) component is present and more predominant. Therefore, the Rician channel model has been chosen to represent forward and backward channels.  
We consider two antennas attached to a tag with vertical and horizontal polarizations, therefore, the channel vectors \(\mathbf{h}^{b}\) and \(\mathbf{D}^{f}\) become uncorrelated Rician random variables as in [8].  
Then, the final expression is given by:  
\[p_{e}(\overline{\gamma},L)=C_{L}\Bigl{(}C_{1}^{f}C_{2}^{b}\Bigr{)}^{L}\Bigl{(} \Bigl{(}\ln(\overline{\gamma})+C_{3}^{fb}\Bigr{)}\overline{\gamma}^{-1} \Bigr{)}^{L} \tag{7}\]  
where \(\overline{\overline{\gamma}}\) is the MGF of \(\Bigl{|}\bigl{|}Y\bigr{|}\bigr{|}_{F}^{2}\), \(C_{L}=(\Gamma((1/2)+Lb)/2\sqrt{\pi}\Gamma(1+Lb))\), \(C_{1}^{f}=K_{2}e^{-K_{f}}\), \(C_{2}^{b}=K_{1}e^{-K_{b}}\), \(C_{3}^{fb}=e^{K_{f}}-1-\ln(K_{1}K_{2})\), \(K_{1}=K_{b}+1\), and \(K_{2}=K_{f}+1\). The parameters \(K_{f}\) and \(K_{b}\) characterize the Rician channel and are for forward and backward channels, respectively. Optionally, and for a more tractable system behavior analysis, the Nakagami-m expression is an alternative. Similarly, for the Nakagami-m, the parameters \(m_{f}\) and \(m_{b}\) characterize the forward and backward Nakagami-m channels, respectively.  
The diversity gain of our system is given as in [9] and is \(L\) = 2:  
\[d=LN-L(N-1)=L=2 \tag{8}\]  
where \(d\) is the diversity gain and \(N\) = 1 is the number of the reader receiving antennas.