# An Overview of Indoor Localization System for Human Activity Recognition (HAR) in Healthcare
## 4 Localization Methods
### Pseudo-Range Multilateration  
Pseudo-range multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately calculating the difference in arrival time (TDOA) of a signal emitted by the object to three or more receivers [35]. In this case, the time of emission of the traveling signal is unknown. This technique bases the estimation of the position of a node on the minimization of the difference between the estimated distance via TDOA and the actual distance obtained from the known coordinates. Considering that the time of arrival is proportional to the space traveled by the signal itself, it is possible to identify the position of a target. For example, consider a target that emits a signal in an unknown position with \(x\), \(y\), and \(z\) coordinates in an area equipped with a multilateration system  
Figure 8: Trilateration.  
Figure 9: Triangulation.  
with \(n\) receivers (\(P_{1}\), \(P_{2}\),..., \(P_{n}\)). The time the signal takes to reach each receiver by the emitter is given by dividing the space by the signal speed assumed as the speed of light \(c\).  
\[\begin{array}{l}T_{1}=\frac{1}{c}(\sqrt{\left(x-x_{1}\right)^{2}+\left(y-y_{1} \right)^{2}+\left(z-z_{1}\right)^{2}})\\ T_{2}=\frac{1}{c}(\sqrt{\left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}+ \left(z-z_{2}\right)^{2}})\\ T_{3}=\frac{1}{c}(\sqrt{\left(x-x_{3}\right)^{2}+\left(y-y_{3}\right)^{2}+ \left(z-z_{3}\right)^{2}})\\.........................................\\ T_{n}=\frac{1}{c}(\sqrt{\left(x-x_{n}\right)^{2}+\left(y-y_{n}\right)^{2}+ \left(z-z_{n}\right)^{2}})\end{array} \tag{10}\]  
Assuming that the point \(P_{n}\) in 3D coincides with the origin of the system, we have that \(T_{n}\) is expressed as  
\[T_{n}=\frac{1}{c}(\sqrt{x^{2}+y^{2}+z^{2}}). \tag{11}\]  
The differences in the time of arrival concerning the reference site are  
\[\begin{array}{l}\tau_{1}=T_{1}-T_{n}=\frac{1}{c}(\sqrt{\left(x-x_{1}\right)^ {2}+\left(y-y_{1}\right)^{2}+\left(z-z_{1}\right)^{2}}-\sqrt{x^{2}+y^{2}+z^{2 }})\\ \tau_{2}=T_{2}-T_{n}=\frac{1}{c}(\sqrt{\left(x-x_{2}\right)^{2}+\left(y-y_{2} \right)^{2}+\left(z-z_{2}\right)^{2}}-\sqrt{x^{2}+y^{2}+z^{2}})\\ \tau_{3}=T_{3}-T_{n}=\frac{1}{c}(\sqrt{\left(x-x_{3}\right)^{2}+\left(y-y_{3} \right)^{2}+\left(z-z_{3}\right)^{2}}-\sqrt{x^{2}+y^{2}+z^{2}})\\.............................\\ \tau_{n-1}=T_{n-1}-T_{n}=\frac{1}{c}(\sqrt{\left(x-x_{n-1}\right)^{2}+\left( y-y_{n-1}\right)^{2}+\left(z-z_{n-1}\right)^{2}}-\sqrt{x^{2}+y^{2}+z^{2}})\end{array}, \tag{12}\]  
where (\(x_{i}\), \(y_{i}\), \(z_{i}\)) with variables from 1 to \(n\) are the known locations of the various receivers. Each of the above equations represents a hyperbola; from the system's resolution, the emitter coordinates are obtained.  
From a graphical point of view, the coordinates of the source can be determined by the intersections of the above hyperbolas, as shown in Figure 10. A hyperbola is defined as a geometric place of the plane having as a constant the difference in distances from the foci that, in our case, are the reference sensor that is the first to receive the signal from the target and the unprecenth sensor that receives the same signal.  
In addition, to avoid conflict between the emitters, a minimum separation distance between the various nodes must be ensured by defining a protection zone around each node so that any other node does not infiltrate it.