# Multi-Information Fusion Indoor Localization Using Smartphones
## 4 Fusion Localization Architecture
### Location Initialization  
The transmission and reception of acoustic signals is achieved by chirp modulation. The chirp signal is a pulse compression signal, which has good autocorrelation characteristics and can be extracted from severe signal fading. The chirp signal is characterized as follows:  
\[m(t)=e^{\beta 2\pi(f_{0}t+\frac{1}{2}k_{0}t^{2})},te[0,T] \tag{3}\]  
where \(f_{0}\) is the initial frequency, \(k_{0}\) is the modulation rate, and \(T\) is the duration time. In this paper, the frequency range of the chirp signal is between 17.5 and 19.5 kHz, and 40 ms per frame is shown in Figure 3. We installed the beacons with microphones and speakers at the trial scenes. The target passively listens to the beacons and saves the messages. The server terminal calculates the target location with the CHAN algorithm. The initial location of the target is achieved by ultrasonic-base estimation. The CHAN algorithm is a non-recursive hyperbolic solution that has high localization accuracy and low computational complexity. This algor  
Figure 3: Time domain, frequency domain, and time\(-\)frequency characteristic of chirp signal with time duration of 40 ms between 17.5 and 19.5 kHz. limit of Calimero in the line-of-sight environment and has been widely used in practical engineering. Assuming a target location \(T(x,\,y)\), and beacon locations \(A_{i}(x_{i},y_{i}),\,\,i=1,\,2,\,3,\ldots,\) the spatial position with three beacons and the target \(T\) are shown in Figure 4.  
The distance difference \(d_{i1}\) between the \(i\)-th beacon and the target can be calculated as follows:  
\[\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}}-\sqrt{\left(x_{1}-x \right)^{2}+\left(y_{1}-y\right)^{2}}=d_{i1}, \tag{4}\]  
Expanding (4), we can obtain:  
\[d_{i1}{}^{2}+2d_{i1}d_{1}=x_{i}{}^{2}+y_{i}{}^{2}-x_{1}{}^{2}-y_{1}{}^{2}-2x(x_ {i}-x_{1})-2y(y_{i}-y_{1}) \tag{5}\]  
Assuming that \(r_{i}=x_{i}{}^{2}+y_{i}{}^{2}\), \(x_{i1}=x_{i}-x_{1},\,\,y_{i1}=y_{i}-y_{1}\), Equation (5) can be simplified as follows:  
\[x_{i1}x+y_{i1}y+d_{i1}d_{1}=\frac{1}{2}\Big{(}r_{i}-r_{1}-d_{i1}{}^{2}\Big{)} \tag{6}\]  
Equation (6) can be expressed using a matrix as follows:  
\[h=g_{a}F \tag{7}\]  
\[\text{where }h=\frac{1}{2} \begin{bmatrix}r_{2}-r_{1}-d_{21}{}^{2}\\ \vdots\\ r_{n}-r_{1}-d_{n1}{}^{2}\end{bmatrix} ,\,g_{a}= \begin{bmatrix}x_{21}&y_{21}&d_{ 21}\\ \vdots&\vdots&\vdots\\ x_{n1}&y_{n1}&d_{n1}\end{bmatrix} ,\,F=[x\,y\,d_{1}]^{T}.\]  
Due to observation noise, the error vector can be expressed as:  
\[e=h-g_{a}F \tag{8}\]  
The covariance matrix of the error vector e is  
\[\Sigma=Cov(\text{e},\text{e})=E\Big{(}\text{e}\text{e}^{T}\Big{)}=c\,\,^{2}RqR \tag{9}\]  
where \(R=diag(d_{2},\,d_{3},\ldots,\,d_{n})\) and \(q\) is the covariance matrix. The location estimation can be obtained through weighted least squares:  
\[\hat{F}=\Big{(}g_{a}{}^{T}pg_{a}\Big{)}{}^{-1}g_{a}{}^{T}ph \tag{10}\]  
Figure 4: Spatial geometric location map with three beacons \((A_{1},\,\,A_{2},\,A_{3})\) and the target \(T\). where \(p\) is the inverse of the covariance matrix \(\Sigma\). After obtaining the localization for the first time, we can obtain the equation:  
\[e^{\prime}=h^{\prime}-g_{a}{}^{\prime}F^{\prime} \tag{11}\]  
with the constraint:  
\[\left\{ \begin{array}{l}F_{1}=x^{0}+\partial_{1};\\ F_{2}=y^{0}+\partial_{2};\\ F_{3}=\partial_{1}{}^{0}+\partial_{3}.\end{array} \right. \tag{12}\]  
where \(\partial_{1},\ \partial_{2},\ \partial_{3}\) are the estimation \(\hat{F}\). \(F^{\prime}= \begin{bmatrix}(x-x_{1})^{2}\\ (y-y_{1})^{2}\end{bmatrix} \), \(h^{\prime}= \begin{bmatrix}(F_{1}-x_{1})^{2}\\ (F_{2}-y_{1})^{2}\\ F_{3}{}^{2}\end{bmatrix} \), \(g_{a}{}^{\prime}= \begin{bmatrix}1&0\\ 0&1\\ 1&1\end{bmatrix} \). We obtain:  
\[\hat{F}^{\prime}=\left(g_{a}{}^{\prime}T^{\prime}g_{a}{}^{\prime}\right)^{-1} g_{a}{}^{\prime}T^{\prime}p^{\prime}h^{\prime} \tag{13}\]  
where  
\[\left\{