## 4 A Transmission Strategy for Minimizing Energy Consumption
### Energy Consumption Analysis
Finally, note that \(E_{TX}\) is a random variable composed of a constant multiplied by a log-normal random variable, therefore, it is itself another log-normal random variable, with parameters \(\mu_{E}=\ln\left(\frac{n\ T_{p}\ v_{b}\ d^{\eta}\ n_{rx}}{2\ WA}\frac{x^{2}}{(1-Q( x))^{8l}}\right)\), \(\sigma_{E}=\sigma_{\mathcal{X}}\) and mean value given by:  
\[\overline{E}_{TX}=\frac{n\ T_{p}\ v_{b}\ d^{\eta}\ n_{rx}}{2\ WA}\cdot\frac{x ^{2}}{(1-Q(x))^{8l}}\cdot e^{\frac{1}{2}\left(\frac{x\ln 10}{10}\right)^{2}} \tag{9}\]  
Deriving Equation (9) with respect to the \(snr\) at the receiver, it can be obtained that the value of \(x\) (or \(snr\)) for which the average energy consumption reaches a minimum (\(x^{*}\)) satisfies the following expression:  
\[\frac{x^{*}e^{-x^{*2}/2}}{(1-Q(x^{*}))}=\frac{\sqrt{2\pi}}{4l} \tag{10}\]In conclusion, if we manage to achieve the \(snr^{*}\) corresponding to this \(x^{*}\), the energy consumption will be minimized. To provide some illustrative results, Figure 1 shows the energy consumption as a function of the \(SNR\) for the parameters of our sensor network, which performs a localization once a second, that is, \(T_{0}\) = 1 s, and uses 8-bytes packets, with \(T_{p}\) = 256 \(\mu\)s. The estimated values of the channel parameters are \(\eta\) = 2.5 and \(A\) = -80 dBm and the noise level at the receiver is \(N_{RX}\) = -87 dBm. The energy consumption was calculated for \(n\) = 5 and \(d\) = 2 m. We can see in Figure 1 that if the optimum \(SNR\) value is not achievable, it would be better to use a higher transmission power, as the increase of power consumption with the \(SNR\) is slower for higher values of \(SNR\). On the other hand, note that the desired accuracy imposed a relationship between \(snr\) and \(T\), thus, there is an optimum value of \(T\) corresponding to \(snr^{*}\), which can be calculated from Equations (1) and (4):  
\[T^{*}=\frac{T_{0}}{n}\cdot\left(1-Q\left(\sqrt{2 snr^{*}\cdot\frac{W}{v_{b}}} \right)\right)^{sl} \tag{11}\]  
In conclusion, among the pairs \(snr\)-\(T\) that satisfy a desired accuracy, there is a pair \(snr^{*}\)-\(T^{*}\) for which the energy consumption is minimum. For that reason, in order to measure the RSS or the TOA with certain accuracy, the strategy that we propose consists in transmitting packets with a packet rate of \(1/T^{*}\) and such a transmission power \(P_{TX}\) that the \(snr\) at the receiver is equal to \(snr^{*}\). As the \(snr\) at the receiver is proportional to the transmission power for a given distance \(d\) between transmitter and receiver (see Section 3.2), in order to calculate the optimum value of \(P_{TX}\), an estimate of the distance between transmitter and receiver is needed. In practice, this distance estimation can be obtained from the RSS or TOA measurements of the previous localization period. Figure 1: Energy consumption (in joules) as a function of the \(SNR\) at the receiver. Note (from Equation (10)) that the optimum value \(x^{*}\) only depends on the packet length, therefore, the optimum value of \(snr\) at the receiver to achieve the lowest energy consumption does not depend on the accuracy we want to obtain or on the propagation characteristics, it only depends on the packet length and on the \(W/v_{b}\) ratio, _i.e._, its value is constant for a given network. This is an interesting result from a practical point of view, as it implies that the value of \(snr^{*}\) does not need to be calculated in real time, which would be costly from a computational point of view, as it involves solving expression (10) iteratively. Therefore, the proposed strategy only involves very simple operations to calculate the optimum values of \(T^{*}\) and \(P^{*}_{TX}\) from the known value of \(snr^{*}\) and, yet, guarantees the desired accuracy in the measurements while achieving the minimum energy consumption. This results in a very useful and attractive technique for battery-powered wireless devices that are deployed to perform the localization of people or objects.