# WIPP: Wi-Fi Compass for Indoor Passive Positioning with Decimeter Accuracy
## 3 System Description
### Angle of Arrial (AOA) Measurement Estimation
\(\tau_{k}\) and \(\theta_{k}\) are the TOA and AOA measurements of the signal from the \(k\)-th signal path. Since the CSI contains the information of amplitude attenuation and phase shift of each subcarrier with respect to each antenna, we conduct the Fourier transformation of Equation (4) to jointly estimate the TOA and AOA measurements of the signal, which is described as  
\[\mathbf{CSI}=\mathbf{A}\mathbf{\beta}+\mathbf{n}, \tag{7}\]  
where \(\mathbf{CSI}=\left[ \begin{array}{cccccccc}csi_{1,1}\left(t\right)&csi_{1,2 }\left(t\right)&\cdots&csi_{1,30}\left(t\right)&csi_{2,1}\left(t\right)&csi_{2,2}\left(t\right)&\cdots&csi_{2,30}\left(t\right)&csi_{3,1}\left(t\right)\\ csi_{3,2}\left(t\right)&\cdots&csi_{3,30}\left(t\right)\end{array} \right]^{ \mathrm{T}}\) is a \(1\times 90\) vector of CSI collected at \(3\) antennas. \(\mathbf{\beta}=\left[ \begin{array}{cccccccccccc}\beta_{1}&\beta_{2}&\cdots& \beta_{K}\end{array} \right]^{\mathrm{T}}\) is a \(K\times 1\) amplitude attenuation vector. \(\mathbf{\beta}_{k}\) is the amplitude attenuation of the signal from the \(k\)-th signal path. We continue to define the covariance matrix of CSI as  
\[\mathbf{R}=\mathrm{E}\left[\mathbf{CSI}\times\mathbf{CSH}^{\mathrm{H}}\right], \tag{8}\]  
where the notation \"H\" denotes the complex conjugate transpose operation. The previous \(K\) eigenvectors of \(\mathbf{R}\) are corresponding to the signal, while the latter \(90-K\) eigenvectors are corresponding to the noise. Thus, the noise space is constructed as  
\[\mathbf{E}_{N}=\left[\mathbf{v}_{K+1},\mathbf{v}_{K+2},...,\mathbf{v}_{90} \right], \tag{9}\]  
where \(\mathbf{v}_{k}(k=K+1,K+2,\cdots,90)\) is the \(k\)-th eigenvector corresponding to the noise. Since the eigenvectors corresponding to the noise are orthogonal to the direction vectors in \(\mathbf{A}\), the pseudo-spectrum which is also used in [34] can be calculated by  
\[P_{\mathrm{music}}\left(\theta,\tau\right)=\frac{1}{\mathbf{a}^{\mathrm{H}} \left(\theta,\tau\right)\mathbf{E}_{N}\mathbf{E}_{N}^{\mathrm{H}}\mathbf{a}^{ \mathrm{H}}\left(\theta,\tau\right)}. \tag{10}\]  
Since the signal from different paths received at the same antenna is generally coherent, the covariance matrix of signal is probably singular, which will result in the unavailability of the conventional MUSIC algorithm [35]. The existing approaches of de-correlating the coherence of signal normally follow the process of dividing the antennas into several sub-antenna arrays, and then estimating the AOA measurement by averaging the output of the covariance matrix of sub-antenna arrays [34,35]. However, these approaches fail to utilize the information of subcarriers,and consequently cannot guarantee the effectiveness of the estimation of AOA measurement. To de-correlate the coherence of signal, we introduce a new enhanced 2D spatial smoothing algorithm. To address this problem, the WIPP separates the antennas into several sub-antenna arrays, as well as the subcarriers into several subsets. This process is shown in Figure 4. As the rectangular window slides, the CSI can be separated into 22 different subsets, and each of them contains 20 subcarriers with two neighboring antennas. The purpose of selecting 20 subcarriers is to guarantee that the number of direction measurement units is larger than the number of signal paths. Based on the result of CSI separation, we reconstruct the matrix of CSI collected at three antennas as  
\[\mathbf{CSI}=\left[