## 4 A Transmission Strategy for Minimizing Energy Consumption
### Practical Considerations
In other words, this margin must be calculated as the difference between the mean value of the \(SNR\) of the successfully received packets (which can be calculated numerically using Equation (14)) and \(SNR^{*}\). Figure 2 shows the value of this margin as a function of the standard deviation of the shadowing for different packet lengths. This margin was seen to be independent of the value of \(W/v_{b}\), and nearly independent of the packet length, as it can be seen in the figure. On the other hand, even when the optimum transmission power is used, the final \(snr\) at the receiver may not be the desired one, as the propagation is again affected by random shadowing. In fact, the \(snr\) at the receiver will be a log-normal random variable and, therefore, the \(PRP\) and \(n\) will also be random variables and their probability density functions can be calculated easily according to Equations (1) and (4). In particular, the probability density function of the \(PRP\) can be calculated as:  
\[f_{PRP}=\frac{f_{snr}}{\frac{\partial PRP}{\partial snr}}=\frac{\sqrt{2\pi}v_{ b}}{8 lW}\cdot\frac{x\left(1-Q(x)\right)}{PRP\cdot e^{x^{2}/2}}\cdot\frac{1}{snr \cdot\sigma_{snr}\sqrt{2\pi}}\cdot e^{-\frac{\left(\ln snr-\mu_{snr}\right)^ {2}}{2\sigma_{snr}^{2}}} \tag{15}\]  
where \(\mu_{snr}=\ln 10^{\overline{SNR}/10}=\ln snr^{*}\), \(\sigma_{snr}=\sigma\cdot\ln 10/10\), \(x=\sqrt{2 snr\cdot W/v_{b}}\) and \(snr=v_{b}/2W\cdot\left(Q^{-1}\left(1-PRP^{1/8l}\right)\right)^{2}\). Thus, knowing that \(n=k\cdot PRP\), the probability density function of \(n\) can be calculated as:  
\[f_{n}(n)=k\cdot f_{PRP}(n)=\frac{n_{obj}}{PRP^{*}}\cdot f_{PRP}(n) \tag{16}\]  
where \(n_{obj}\) is the desired number of received packets and \(PRP^{*}\) is the value of the packet reception probability corresponding to the optimum \(snr\): \(PRP^{*}=\left(1-Q\left(\sqrt{2 snr^{*}\cdot\frac{W}{v_{b}}}\right)\right)^{8l}\). The mean value of \(n\) can be then calculated numerically, resulting that this value is lower than the desired one \(n_{obj}\). The difference between the two depends on \(l\) and \(W/v_{b}\): the mean value of \(n\) is lower when the packet is longer or when \(W/v_{b}\) is higher. Figure 3 shows the relation between these two values for different values of \(l\) and \(W/v_{b}\) for the case of RSS-based localization as a function of the standard deviation of the \(SNR\) distribution. This factor must be used to correct the number of transmitted packets, in order to counteract the effects of the propagation in real deployments. Thus, the estimated value of \(T^{*}\) calculated from Equation (11) must be divided by this factor. Figure 2: Correction margin for the transmission power calculated from estimated distances in RSS-based localization. Although the energy consumption will increase when these two corrections are included in the transmission strategy, they are necessary to achieve the desired number of received packets in real environments and, therefore, to achieve the desired accuracy; otherwise, the accuracy would not be guaranteed. Nevertheless, the final energy consumption is very small and still entails significant savings with respect to other transmission strategies.