# WIPP: Wi-Fi Compass for Indoor Passive Positioning with Decimeter Accuracy
## 3 System Description
### Angle of Arrial (AOA) Measurement Estimation
\begin{array}{cccccccc}csi_{1,1}&csi_{1,2}&\cdots&csi_{1, 11}&csi_{2,1}&csi_{2,2}&\cdots&csi_{2,10}&csi_{2,11}\\ csi_{1,2}&csi_{1,3}&\cdots&csi_{1,12}&csi_{2,2}&csi_{2,3}&\cdots&csi_{2,11}&csi_{2,12 }\\ csi_{1,3}&csi_{1,4}&\cdots&csi_{1,13}&csi_{2,3}&csi_{2,4}&\cdots&csi_{2,12}&csi_{2, 13}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\ csi_{1,20}&csi_{1,21}&\cdots&csi_{1,30}&csi_{2,20}&csi_{2,21}&\cdots&csi_{2, 29}&csi_{2,30}\\ csi_{2,1}&csi_{2,2}&\cdots&csi_{2,11}&csi_{3,1}&csi_{3,2}&\cdots&csi_{3,10}&csi_{ 3,11}\\ csi_{2,2}&csi_{2,3}&\cdots&csi_{2,12}&csi_{3,2}&csi_{3,3}&\cdots&csi_{3,11}&csi_{ 3,12}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\ csi_{2,20}&csi_{2,21}&\cdots&csi_{2,30}&csi_{3,20}&csi_{3,21}&\cdots&csi_{3,29}& csi_{3,30}\end{array} \right]. \tag{11}\]  
In Equation (11), each of column denotes a subset of CSI corresponding to 20 subcarriers at two neighboring antennas. Then, based on Equation (11), the direction vector, \(\mathbf{a}\left(\theta_{k},\tau_{k}\right)\), in Equation (5) can be rewritten as  
\[\mathbf{a}\left(\theta_{k},\tau_{k}\right)=\left[ \begin{array}{cccc}\mathbf{ a}_{1}\left(\theta_{k},\tau_{k}\right)&\mathbf{a}_{2}\left(\theta_{k},\tau_{k} \right)\end{array} \right]^{\mathsf{T}}, \tag{12}\]  
where \(\mathbf{a}_{m}\left(\theta_{k},\tau_{k}\right)=\left[ \begin{array}{cccc}a_{ m,1}\left(\theta_{k},\tau_{k}\right)&a_{m,1}\left(\theta_{k},\tau_{k}\right)&...&a_{m, 20}\left(\theta_{k},\tau_{k}\right)\end{array} \right]^{\mathsf{T}}\!\!\left(m= 1,2\right)\). At the same time, the amplitude attenuation vector, \(\mathbf{\beta}\), can be rewritten as  
\[\mathbf{\beta}=\left[ \begin{array}{cccccccc}\beta_{1}a_{1,1}\left(\theta_{1 },\tau_{1}\right)&\cdots&\beta_{1}a_{11,1}\left(\theta_{1},\tau_{1}\right)& \beta_{1}a_{12}\left(\theta_{1},\tau_{1}\right)&\cdots&\beta_{1}a_{112}\left( \theta_{1},\tau_{1}\right)\\ \beta_{2}a_{1,1}\left(\theta_{2},\tau_{2}\right)&\cdots&\beta_{2}a_{11,1}\left( \theta_{2},\tau_{2}\right)&\beta_{2}a_{12}\left(\theta_{2},\tau_{2}\right)& \cdots&\beta_{2}a_{112}\left(\theta_{2},\tau_{2}\right)\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \beta_{K-1}a_{1,1}\left(\theta_{K-1},\tau_{K-1}\right)&\cdots&\beta_{K-1}a_{11,1 }\left(\theta_{K-1},\tau_{K-1}\right)&\beta_{K-1}a_{12}\left(\theta_{K-1},\tau _{K-1}\right)&\cdots&\beta_{K-1}a_{112}\left(\theta_{K-1},\tau_{K-1}\right)\\ \beta_{K}a_{1,1}\left(\theta_{K},\tau_{k}\right)&\cdots&\beta_{K}a_{11,1}\left( \theta_{K},\tau_{k}\right)&\beta_{K}a_{12}\left(\theta_{K},\tau_{k}\right)& \cdots&\beta_{K}a_{112}\left(\theta_{K},\tau_{k}\right)\end{array} \right] \tag{13}\]  
Finally, we rely on Equations (6), (7), (10), and (11) to achieve the joint estimation of AOA and TOA measurements. In addition, we conduct some simulations to examine the performance of the joint estimation of AOA and TOA measurements. In the simulations, the number of antennas for each AP is set as three. The 30 signal frequencies occupy the range from 5.19 to 5.22 GHz with the same interval of 1 MHz. By assuming that the signal from different paths is incoherent, Figure 5 shows the result of joint estimation of AOA and TOA measurements with the signal having the AOA measurement 25.5\({}^{\circ}\), 15\({}^{\circ}\), 9\({}^{\circ}\), \(-\)13.5\({}^{\circ}\), \(-\)42\({}^{\circ}\), or \(-\)59.5\({}^{\circ}\) and TOA measurement 43 ns, 10 ns, 25 ns, 37 ns, 8 ns, or 76 ns. From  
Figure 4: Diagram of Channel State Information (CSI) separation. this figure, we can find that both the AOA and TOA measurements are estimated accurately by using the proposed algorithm. We continue to conduct some other simulations to investigate the performance of the proposed 2D spatial smoothing algorithm used for signal decorrelation before the joint estimation of AOA and TOA measurements.