## 4 A Transmission Strategy for Minimizing Energy Consumption
### Sensitivity Analysis
For example, for 40-bytes packets, if the transmission power is underestimated in only 1.4 dB, the number of received packets will be halved with respect to the objective \(n_{obj}\), and if the transmission power is underestimated in more than 4 dB, the number of received packets will be practically zero. Figure 4: Effect of the error in the transmission power on the energy consumption and on the number of received packets. The behavior shown in these figures is valid for both TOA-based and RSS-based localization schemes, but as the probability distribution of the distance estimation is different for the two localization schemes (see Appendix A and B), the probability distribution of \(\tilde{P}_{TX}-P_{TX}^{*}\) will also be different. Therefore, the frequency of underestimations and overestimations of the transmission power will depend on the distance estimation method, and this will affect the probability distribution of the energy consumption and the number of received packets, as we will see next. This error in the transmission power can be calculated as:  
\[P_{TX}^{\prime}-P_{TX}^{*}(dB)=10\eta\log\frac{\tilde{d}}{d} \tag{17}\]  
where \(\tilde{d}\) is the estimated distance. Appendix A and B derive the expressions of \(\tilde{d}\) for TOA-based and RSS-based localization respectively. In a TOA-based localization scheme the distance estimation is given by Equation (24), therefore the difference between calculated and optimum transmission power will be given by:  
\[P_{TX}^{\prime}-P_{TX}^{*}=10\eta\log\frac{d+\mathcal{N}(0,\sigma)}{d}=10\eta \log\mathcal{N}(1,\sigma/d) \tag{18}\]  
where \(\sigma=c\cdot\sigma_{t}/\sqrt{n}\) or \(\sigma=c\cdot\sigma_{t}/2\sqrt{n}\), as explained in Appendix A. On the other hand, in a RSS-based localization scheme the distance estimation is given by Equation (28), thus the difference between calculated and optimum transmission power will be given by:  
\[P_{TX}^{\prime}-P_{TX}^{*}=\mathcal{N}(0,\sigma) \tag{19}\]  
with \(\sigma=\sigma_{RSS}/\sqrt{n}\). Figure 5 shows the probability density function of the relative energy consumption (\(E^{\prime}/E_{min}\)) for the cases of TOA-based and RSS-based localization. The different curves correspond to different values of \(\sigma/d\) and \(\sigma\) respectively. Notice that in the TOA case the individual values of \(\sigma\) and \(d\) do not have any influence, as long as the relation \(\sigma/d\) remains the same. This can be easily understand from Equation (18). In both cases, the probability of consuming more energy increases with \(\sigma\). Furthermore, the mean value of \(E^{\prime}/E_{min}\) also increases when the standard deviation increases. Figure 6 shows the probability density function of the relative number of received packets (\(n/n_{obj}\)) for TOA-based and RSS-based localization. The different curves correspond to different values of \(\sigma/d\) and \(\sigma\) respectively, and were calculated for a packet length of \(l=5\) bytes (different packet lengths produce similar results, but a little more sharper as the packet length increases). In both cases, when the standard deviation is bigger, the probability of not receiving the desired number of packets is higher. It can be also seen that if the standard deviation is over a certain threshold, the pdf begins to be bimodal. This is because the margin of \(SNR\) for which the \(PRP\) takes intermediate values is small, so if the standard deviation is big enough, the probability of receiving packets with a very low or very high \(SNR\) (enough to make \(PRP=0\) or \(PRP=1\) respectively) is significant. In conclusion, for both TOA and RSS-based localization methods, a greater variance in the measurements will produce a greater variance in the optimum transmission power estimation and thus, more energy consumption and a higher probability of losing packets. Figure 5: Probability density function of \(E^{\prime}/E_{min}\) for TOA-based and RSS-based localization schemes. (**a**) TOA; (**b**) RSS.