## 4 A Transmission Strategy for Minimizing Energy Consumption
### Sensitivity Analysis  
It is interesting to analyze how much energy will be consumed if the best transmission strategy is not perfectly achieved, due to an error in the estimated transmission power. As it has been mentioned above, the error in the estimated transmission power is due to an error in the distance estimation, which depends itself on the RSS or TOA measurement errors and on the number of packets that are averaged at the receiver. In this section, we first evaluate how the energy consumption is affected by the errors in the estimated transmission power and then we study how the distance estimation errors propagate to errors in the estimated transmission power, and therefore affect the energy consumption. Let us suppose a transmitter node X that has to send packets to another node Y, so that Y receives in average \(n_{obj}\) packets during the period \(T_{0}\). As the optimum \(snr\) at receiver node Y is known, node X can calculate the optimum value of \(T\) (\(T^{*}\)) that satisfies the desired accuracy using Equation (11). On the other hand, the optimum transmission power \(P_{TX}^{*}\) can be obtained from Equation (3), assuming that the distance between transmitter node X and receiver node Y is known. An error in the distance estimation will produce an error in the estimated optimum transmission power, which will lead to the use of a transmission power \(P_{TX}^{\prime}\) different from the optimum \(P_{TX}^{*}\). If the estimated transmission power is greater than \(P_{TX}^{*}\), the energy consumption will increase with respect to the minimum \(E_{min}\) and the  
Figure 3: Ratio between the desired number of received packets and the mean value of \(n\). number of received packets will increase as well with respect to \(n_{obj}\), although it is not necessary. If the estimated transmission power is lower than \(P_{TX}^{*}\), the energy consumption will diminish, but the number of received packets will not be sufficient. Figure 4 shows how the errors in the transmission power affect the consumed energy and the number of received packets. In particular, Figure 4 represents the ratio between the energy consumption \(E^{\prime}\) corresponding to the estimated transmission power \(P_{TX}^{\prime}\) and \(E_{min}\) (corresponding to the optimum transmission power \(P_{TX}^{*}\)) as a function of the error in the transmission power (\(P_{TX}^{\prime}\)-\(P_{TX}^{*}\)). This relation is valid for any value of \(n_{obj}\), for any \(W/v_{b}\) ratio for any packet length \(l\) and for any distance between transmitter and receiver. On the other hand, Figure 4 shows the relation between the number of received packets \(n\) corresponding to the estimated transmission power \(P_{TX}^{\prime}\) and the desired number of received packets \(n_{obj}\) (corresponding to the optimum transmission power \(P_{TX}^{*}\)) as a function of the error in the transmission power. This relation is valid for any value of \(n_{obj}\), for any \(W/v_{b}\) ratio and for any distance between transmitter and receiver, but depends on the packet length \(l\). As shown in Figure 4, the consumed energy \(E^{\prime}\) will be greater than the minimum value (\(E_{min}\)) if the transmission power is overestimated. Furthermore, the relation between the error in the transmission power (\(\tilde{P}_{TX}-P_{TX}^{*}\)) and the corresponding energy consumption is exponential. For example, if the transmission power is overestimated in 1 dB the energy consumption will be a 26% higher than \(E_{min}\), if it is overestimated in 2 dB the energy consumption will be a 59% higher than \(E_{min}\), and if it is overestimated in 3 dB the energy consumption will be doubled with respect to \(E_{min}\). On the other hand, as shown in Figure 4, the number of received packets \(n\) will be lower than the desired \(n_{obj}\) if the transmission power is underestimated. Notice that the behavior is quite sharp, especially for longer packets.