## 3 Proposed Wi-Fi Backscatter System Using Multiple Antennas
When the tag sensor is off, there is only the direct channel environment from the helper to the reader without the tag's reflection. Diversity order of the proposed system is expressed according to the number of channel paths. If one antenna is used in reader, the channel environment between tag sensor and reader in the proposed system can be expressed as follows:  
\[\mathbb{H}=\left[\widetilde{h}_{11}*\hat{h}_{11}+\widetilde{h}_{21}*\hat{h}_ {12}\right]. \tag{3}\]  
Figure 2: Frame structure in the proposed system. In this case, diversity order of proposed system is 2 because there are two channel paths. If \(N\) antennas are used in reader, the channel path between tag sensor and reader is expressed as follows:  
\[\mathbb{H}=\left[ \begin{array}{c}\dfrac{\widetilde{h}_{11}*\hat{h}_{11}+ \widetilde{h}_{21}*\hat{h}_{12}}{\widetilde{h}_{11}*\hat{h}_{21}+\widetilde{h}_ {21}*\hat{h}_{22}}\\ \vdots\\ \widetilde{h}_{11}*\hat{h}_{N1}+\widetilde{h}_{21}*\hat{h}_{N2}\end{array} \right] \tag{4}\]  
There are \(2N\) channel paths, when the number of antennas used in reader is \(N\). Therefore, diversity order of the proposed system is \(2N\). The method of increasing the data rate determines how the antennas of the tag sensor are reflecting depending on the symbol (\(0\sim 3\)) to be sent from the tag sensor. The symbol \(0\) means turning off reflecting property of all antennas of the tag sensor. For transmitting symbol \(1\), the first antenna of the tag sensor is ON (fully reflecting an RF wave), and the second antenna is OFF (absorbing an RF wave). For symbol \(2\), the first antenna is OFF, and the second antenna is ON. Finally, the tag sensor turns on reflecting property of both antennas for the symbol \(3\). The channel environment, \(\mathbb{H}\), with the switch on and off of each antenna is expressed as the following equations. \[\mathbb{H}_{TA_{0}}=\left[ \begin{array}{c}h_{11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right] \tag{5}\]  
\[\mathbb{H}_{TA_{1}}=\left[ \begin{array}{c}\hat{h}_{11}&0\\ \hat{h}_{21}&0\\ \vdots\\ \hat{h}_{N1}&0\end{array} \right]\left[ \begin{array}{c}\widetilde{h}_{11}\\ 0\end{array} \right]+\left[ \begin{array}{c}h_{11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right]=\left[ \begin{array}{c}\widetilde{h}_{11}*\hat{h} _{11}\\ \widetilde{h}_{11}*\hat{h}_{21}\\ \vdots\\ \widetilde{h}_{11}*\hat{h}_{N1}\end{array} \right]+\left[ \begin{array}{c}h_{ 11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right] \tag{6}\]  
\[\mathbb{H}_{TA_{2}}=\left[ \begin{array}{c}0&\hat{h}_{12}\\ 0&\hat{h}_{22}\\ \vdots\\ 0&\hat{h}_{N2}\end{array} \right]\left[ \begin{array}{c}0\\ \widetilde{h}_{21}\end{array} \right]+\left[ \begin{array}{c}h_{11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right]=\left[ \begin{array}{c}\widetilde{h}_{21}*\hat{h} _{12}\\ \widetilde{h}_{21}*\hat{h}_{22}\\ \vdots\\ \widetilde{h}_{21}*\hat{h}_{N2}\end{array} \right]+\left[ \begin{array}{c}h_{ 11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right] \tag{7}\]  
Figure 3: Modeling of channel environments in the proposed system. \[\mathbb{H}_{TA_{3}}=\left[\begin{array}{ccc} \begin{array}{c}\hat{h}_{11}&\hat{h}_{ 12}\\ \hat{h}_{21}&\hat{h}_{22}\\ \vdots\\ \hat{h}_{N1}&\hat{h}_{N2}\end{array} \right]\left[ \begin{array}{c}\overline{h }_{11}\\ \overline{h}_{21}\end{array} \right]+\left[ \begin{array}{c}h_{11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right]=\left[ \begin{array}{c}\overline{h}_{11}\ast\hat {h}_{11}+\overline{h}_{21}\ast\hat{h}_{12}\\ \overline{h}_{11}\ast\hat{h}_{21}+\overline{h}_{21}\ast\hat{h}_{22}\\ \vdots\\ \overline{h}_{11}\ast\hat{h}_{N1}+\overline{h}_{21}\ast\hat{h}_{N2}\end{array}  \right]+\left[ \begin{array}{c}h_{11_{HR}}\\ h_{21_{HR}}\\ \vdots\\ h_{N1_{HR}}\end{array} \right] \tag{8}\]  
where \(\mathbb{H}_{TA_{0}}\), \(\mathbb{H}_{TA_{1}}\), \(\mathbb{H}_{TA_{2}}\), and \(\mathbb{H}_{TA_{3}}\) are channels when sending symbols 0, 1, 2, and 3, respectively. The reader evaluates the variation of a channel from CSI of the received Wi-Fi packet.