# FPGA Applied to Latency Reduction for the Tactile Internet
## 3 Discrete Model of the Tactile Internet
Essentially, the majority of the computational effort of MD is in this subsystem. At each \(n\)-th instant \(t_{s}\) the HMD processes the array \(\mathbf{b}(n)\) generating an information array \(\mathbf{c}(n)\) expressed by  
\[\mathbf{c}(n)=\left[c_{1}(n),\ldots,c_{i}(n),\ldots,c_{N_{HMD}^{f}}(n)\right], \tag{3}\]  
where \(c_{i}(n)\) is the \(i\)-th signal generated by HMD towards the subsystem NW at the \(n\)-th instant \(t_{s}\) and \(N_{HMD}^{f}\) is the numbers of signals. \(N_{HMD}^{f}<N_{MD}\) is expected to minimize latency during the transmission in the NW subsystem. The NW subsystem, as shown in Figure 1, characterizes the communication medium that links OP to ENV. In this model, the data propagates through two different channels called the forward channel, that transmits the OP data towards the ENV, and the backwards channel, that transmits the ENV signals towards the OP. The signal transmitted by the forward and backwards channels may be disturbed and delayed. In the case of the forward channel, the received signal, \(\mathbf{v}(n)\), may be expressed as  
\[\mathbf{v}(n)=\left[v_{1}(n),\ldots,v_{i}(n),\ldots,v_{N_{HND}^{f}}(n)\right], \tag{4}\]  
where  
\[v_{i}(n)=c_{i}\Big{(}n-d_{i}^{f}(n)\Big{)}+r_{i}^{f}(n) \tag{5}\]  
in which, \(r_{i}^{f}(n)\) represents the added noise and \(d_{i}^{f}(n)\) represents a delays associated with the \(i\)-th information sent in \(\mathbf{c}(n)\). In this model, the noise can be characterized as a random Gaussian variable of zero mean and \(\sigma_{rf}^{2}\) variance, and the delays are characterized as integers, that is, they occur at a granularity of \(t_{s}\). It is important to note that the NW subsystem can take the shape of the Internet, a metropolitan network (MAN), a local area network (LAN), or even a direct connection between an MD and a workstation or computer. As shown in Figure 1, the HSD receives the \(\mathbf{v}(n)\) signal through the forward channel and has the role of generating control signals to the SD through the signal  
\[\mathbf{u}(n)=\left[u_{1}(n),\ldots,u_{i}(n),\ldots,u_{N_{HSD}^{f}}(n)\right], \tag{6}\]  
where \(N_{HSD}^{f}\) is the number of control signals and \(u_{i}(n)\) is \(i\)-th control signal at the \(n\)-th instant \(t_{s}\) associated with the array \(\mathbf{u}(n)\). It is important to note that there may be various types of SD: from real robotic handlers to virtual tools in computational environments. Thus, it can be stated without loss of generality that HSD can perform an inverse processing to HMD in addition to specific algorithms associated with the type of SD. For example, if the SD is a robotic handler, HSD must additionally implement closed loop control algorithms, whereas if SD is a virtual arm HSD must implement positioning algorithms for a given virtual reality platform. SD does not have to correspond directly with MD, e.g., MD can be a glove while SD is a drone. However, it is desirable that the stimulus generated by the SD is a copy of the stimulus generated by the OP, that is, within the model presented in Figure 1, it can be understood that SD generate a signal expressed as  
\[\mathbf{\hat{a}}(n)=\big{[}\hat{a}_{1}(n),\ldots,\hat{a}_{i}(n),\ldots,\hat{a }_{N_{OP}}(n)\big{]}, \tag{7}\]  
where \(\hat{a}_{i}(n)\) is an estimate of the \(i\)-th stimulus \(a_{i}(n)\) generated by the OP. Thus, the estimate of the stimulus generated by OP, \(\hat{a}_{i}(n)\), is applied to the ENV subsystem representing a given real or virtual environment in which OP is interacting. In the backwards direction, the stimulus actions generated by OP, \(\mathbf{a}(n)\), and represented by \(\mathbf{\hat{a}}(n)\), receives a group of reactions from the ENV subsystem that can be characterized in the model by the set of signals expressed by  
\[\mathbf{o}(n)=\big{[}o_{1}(n),\ldots,o_{i}(n),\ldots,o_{N_{ENV}}(n)\big{]}, \tag{8}\]  
where \(N_{ENV}\) is the number of stimulus signals and \(o_{i}(n)\) is \(i\)-th stimulus signal at the \(n\)-th instant \(t_{s}\).