## 3 Basics of Acoustic Target Tracking
### Acoustic Target Tracking Model  
We assume that all motes (sensor nodes) have microphones for sensing sound waves. Localization and time synchronization of all motes are done with high accuracy. A target in an unknown location \((x_{o},y_{o})\) produces a specific detectable sound wave at time \(t_{o}\). This sound is broadcasted and reaches the motes after some time delay. The delay depends on the distance of motes from the target object. Figure 1 shows the basic schema of acoustic target tracking in 2-dimensional space by sensing sound waves of a moving target [27]. When a mote \(i\) senses a sound wave and detects that this sound belongs to a target object of interest, it generates a record consisting of three fields in the form of \((x_{i},y_{i},t_{i})\). The first two fields represent the 2-dimensional coordinates of the sensing mote and the third field represents the time at which this mote has sensed the sound of the target object. Each mote broadcasts this record to its neighboring motes in its communication range. The goal is to cooperatively compute the three unknown variables \((x_{o},y_{o},t_{o})\) of a target object in a distributed way. This record represents the spatio-temporal information of the concerned target object, implying that the target object has generated a sound at time \(t_{o}\) in position \((x_{o},y_{o})\) that has been detected by motes after some delay. To compute these three unknown variables, we need the sensing information of at least three sensor nodes to create their simultaneous equations.  
Based on the \(\Delta x=v.\Delta t\) relation, the distance \(\Delta x\) of a mote from a target object is equal to sound propagation speed (v) times by the time delay from the sensing sound of target object by a mote from the sound generation time of the target object (\(\Delta t\)). The sound propagation speed v is considered 344.0 m/s in this paper. To track a target object, the three unknown variables \((x_{o},y_{o},t_{o})\) of the target object must be calculated. To calculate these three unknown variables, at least three distinct independent equations are required. In other words, at least three different motes must have detected the sound of a target object at a given position in order to be able to compute the spatio-temporal information of a moving target object. The sensing information of three sensor nodes gives us the simultaneous equations of target tracking in Equation (1):  
Figure 1: Basic schema of target tracking in 2-dimensional space [27].  
\[\begin{cases}\sqrt{(x_{o}-x_{1})^{2}+(y_{o}-y_{1})^{2}}=(t_{1}-t_{o})*v=r_{1}\\ \sqrt{(x_{o}-x_{2})^{2}+(y_{o}-y_{2})^{2}}=(t_{2}-t_{o})*v=r_{2}\\ \sqrt{(x_{o}-x_{3})^{2}+(y_{o}-y_{3})^{2}}=(t_{3}-t_{o})*v=r_{3}\end{cases} \tag{1}\]  
We rewrite simultaneous equations of target tracking and get Equation (2):  
\[\begin{cases}(x-x_{1})^{2}+(y-y_{1})^{2}=(t_{1}-t)^{2}v^{2}\\ (x-x_{2})^{2}+(y-y_{2})^{2}=(t_{2}-t)^{2}v^{2}\\ (x-x_{3})^{2}+(y-y_{3})^{2}=(t_{3}-t)^{2}v^{2}\end{cases} \tag{2}\]  
Let \(C\) be a region in 2-dimensional filed. It is said to be \(k\)-covered if each point in \(C\) belongs to the sensing range of at least \(k\) sensors. From this point onward in this paper, coverage is taken synonymous to sensing coverage rather than connectivity.