# Device Free Detection in Impulse Radio Ultrawide Bandwidth Systems
## 6 Performance Analysis and Discussion  
In this section, the performance of the receiver is discussed by using the probability of false alarm and detection as calculated in the previous section. We will first look at the affect of the pulse duration on the probability of false alarm followed by the effects of interference noise ratio (INR) and transmitted energy of the interference on the UWB system. Usually, the change in environment due to the presence of the device is modelled as a sinusoidal [47; 50; 51; 59; 60]. The two unknown parameters, frequency and time shift, help in determining the presence of a device. Furthermore, the device will be moving indoors and the speed will be less than 5 km/h, which will result in a normalized doppler frequency of \(1\times 10^{-7}\), having a very slow effect on the channel coefficients. The simulations are performed based on the IEEE 802.15.4a channel model final report [54]. However, the SV parameters were simulated using an indoor industrial environment [53], as it is corresponds to the most challenging environment with a large number of reflections. Note that, our simulations were carried out based on the Saleh-Valenzuela (S-V) channel model, which is characterized by the parameters \(1/\Lambda=14.11\) ns, cluster decay rate \(=2.63\) ns and ray power decay = 4.58 ns. The term INR is defined as \(INR=\sigma_{i}^{2}/\sigma^{2}\). The simulation parameters are shown in Table 1.  
Let us study the effects of threshold over the frame duration. Figure 1 shows the false alarm probability as a function of the threshold \(\gamma\) for different values of frame duration. These curves are plotted with the help of (23), where \(\mu_{a}\) and \(\sigma_{a}\) are calculated using (29) and (34), respectively. In this simulation, the INR was fixed to 5 dB. It can be observed that as the frame duration increases, the value of threshold decreases. For a false alarm probability equal to \(10^{-6}\), it can be observed that a threshold of 2.26 will be required to detect a frame length of 300 ns. However, for a frame length of 100 ns, the threshold value should be around 3.25. This satisfies the result as calculated in (A7). From (A7), it can be analysed that as the frame duration \(T_{f}\) increases, \(\sigma_{a}^{2}\) decreases, resulting in a lower value of \(\gamma\), as shown in (24). It can be also observed from the figure that for \(\gamma=\mu_{a}=1\), the \(P_{FA}=0.5\). \begin{table}
\begin{tabular}{c c c} \hline Path Loss exponent & n & 1.2 \\ \hline Shadowing Standard Deviation & \(\sigma_{s}\) & 6 dB \\ \hline Path Loss at 1 m distance & \(PL_{0}\) & 56.7 dB \\ \hline Antenna Loss & \(A_{ant}\) & 3 dB \\ \hline Frequency dependence of Path Loss & \(\kappa\) & \(-1.103\) \\ \hline Nakagami-\(m\) factor mean & \(m_{0}\) & 0.36 dB \\ \hline Nakagami-\(m\) factor variance & \(\hat{m}_{0}\) & 1.13 \\ \hline Nakagami-\(m\) for strong components & \(\hat{m}_{0}\) & 12.99 dB \\ \hline \end{tabular}
\end{table} 
Table 1: The path loss and small-scale fading parameters for the simulated line of sight (LoS) Industrial ultra-wide bandwidth (UWB) channel, as mentioned in [53; 54]. Figure 1: Probability of false alarm versus threshold when using different frame duration at interference noise ratio (INR) = 5 dB. In Figure 2, NP criteria is adopted to calculate the detection probability (\(P_{D}\)) and false alarm probability (\(P_{FA}\)) in STH-UWB systems. Figure 2 was simulated using (23) and (27). The value for \(\mu_{a}\), \(\mu_{p}\), \(\sigma_{a}\) and \(\sigma_{p}\) are calculated using (29), (36), (34) and (41), respectively. Figure 2 illustrates the effect of interference to noise ratio (INR). From this figure, it is obvious that as the INR increases the \(P_{D}\) improves and \(P_{FA}\) decreases, which results in easier detection of the device. It can be observed from Figure 2 that, for \(P_{FA}=0.2\), the device is detected with probability 0.98, when an INR of \(-5\) dB.