## 4 A Transmission Strategy for Minimizing Energy Consumption
### Application to Real Hardware  
The proposed strategy has been derived from the assumption that the power consumption of the transmitter is equal to the transmitted power. However, for real devices, this assumption may not be true due to the inefficiencies of the hardware. In this case, the theoretical strategy previously proposed has to be slightly modified, by substituting the transmission power \(p_{tx}\) in Equation (5) by the power consumption at transmission \(p_{T}\), which in general may depend on the transmission power in a non-linear way. Therefore, the real energy consumption is related to the ideal one, given by Equation (5), by the following expression:  
\[E_{real}=E_{TX}\cdot\frac{p_{T}}{p_{tx}}=E_{TX}\cdot g(p_{tx}) \tag{20}\]  
where \(g(p_{tx})=p_{T}/p_{tx}\) is a nonlinear function of \(p_{tx}\) that will be different for each type of hardware. Similarly to the ideal case, we can minimize this energy consumption with respect to the \(snr\) by setting the derivative equal to zero (and working with the mean values of the random variables). After some algebraic calculations, we finally obtain the following expression:  
\[\left.\frac{\partial E_{real}}{\partial snr}\right|_{x=x^{*}}= 0\quad\Rightarrow\quad g(p_{tx})\cdot\left(2-\frac{8 lx\cdot e^{-\frac{ \kappa^{2}}{2}}}{(1-Q(x))\cdot\sqrt{2\pi}}\right)+\frac{v_{b}}{W}\cdot x^{2} \cdot\frac{\partial g(p_{tx})}{\partial p_{tx}}\cdot\frac{d^{\eta}\;n_{rx}}{a }\cdot e^{\frac{1}{2}\left(\frac{\sigma\cdot 1n10}{10}\right)^{2}}=0 \tag{21}\]  
The value of \(x\) (and therefore, the value of \(snr\)) for which this expression is equal to 0 gives the minimum average energy consumption. Clearly, the result will depend on the hardware characteristics. As an illustration, we next apply this method for two common radio chips used in wireless sensor networks devices: the ChipCon chips CC1000 and CC2420. Clearly, real devices also include other components apart from the radio chip, which will also affect the energy consumption. However, this example illustrates how to proceed in a general case where there is a nonlinear relationship between transmission power and consumption. Several values of \(p_{T}\) as a function of \(p_{tx}\) can be obtained from the datasheets of the two chips [30; 31]. In order to have an approximate expression of \(g(p_{tx})\) and its derivative, we approximated these sets of points by an analytical expression of the form \(g(p_{tx})=U\cdot p_{tx}^{V}\). Table 1 shows the values of \(U\) and \(V\) for the two chips and the correlation coefficient of the fitting. As it can be seen, there is a high correlation in both cases, which means that the considered expression fits well to the data. \begin{table}
\begin{tabular}{l c c} \hline  & **CC1000** & **CC2420** \\ \hline \(U\) & 36.8596 & 48.8470 \\ \(V\) & –0.8256 & –0.8728 \\ \(R^{2}\) & 0.9945 & 0.9986 \\ \hline \end{tabular}
\end{table} 
Table 1: Fitting of \(U\cdot p_{tx}^{V}\) to the real values of \(g(p_{tx})=p_{T}/p_{tx}\) for the CC1000 and CC2420 radio chips. Introducing this expression and its derivative in Equation (21), and after some algebraic manipulation, we finally obtain that the value of \(x\) for which the average energy consumption reaches a minimum (\(x^{*}\)) satisfies the following expression:  
\[\frac{x^{*}e^{-x^{*2}/2}}{(1-Q(x^{*}))}=\frac{\sqrt{2\pi}}{4l}\cdot(1+V) \tag{22}\]  
Thus, if we manage to achieve at the receiver the corresponding value of signal to noise ratio \(snr^{*}\) (using Equation (8)), the real energy consumption will be minimized. Similarly to the ideal case, the optimum value of the \(snr\) 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 \(l\) and on the \(W/v_{b}\) ratio, _i.e._, its value is constant for a given network and does not need to be calculated in real time. Again, as in the ideal case, there is an optimum value of \(T\) corresponding to \(snr^{*}\), which can be calculated from Equation (11).