# Flexible, Brain-Inspired Communication in Massive Wireless Networks
## 4 Analytical Power Consumption Model  
In this section, the analytical power consumption model of the analyzed analog and digital transmission, based on the simulation results, have been proposed. Thus, let us approximate the required SNR for FM-based system vs. the bandwidth extension curves by the exponential function:  
\[\mathrm{SNR}_{\mathrm{req}}^{\mathrm{FM}}\left[\mathrm{dB}\right]=a^{\mathrm{ FM}}\exp\left(\frac{b^{\mathrm{FM}}B_{\mathrm{sig}}}{f_{\mathrm{m}}}\right)+c^{ \mathrm{FM}}\exp\left(\frac{d^{\mathrm{FM}}B_{\mathrm{sig}}}{f_{\mathrm{m}}} \right)\,, \tag{11}\]  
where \(a_{\mathrm{FM}}\), \(b_{\mathrm{FM}}\), \(c_{\mathrm{FM}}\) and \(d_{\mathrm{FM}}\) are the fitting parameters for a given MSE. The coefficients, obtained by least squares fitting to results visible in Figure 6, are shown in Table 2. Based on Equation (11), the transmission power for a given channel attenuation and the noise power can be calculated by:  
\[P_{\mathrm{TX}}^{\mathrm{FM}}\left[\mathrm{dB}\right]=\mathrm{SNR}_{\mathrm{ req}}^{\mathrm{FM}}\left[\mathrm{dB}\right]-10\log_{10}\left(\left|h\right|^{2} \right)+N\left[\mathrm{dB}\right]. \tag{12}\]  
Substituting Equation (12) to Equation (5), the analytical power consumption model which depends on the distance between transmitter and receiver and bandwidth extension can be determined:  
\[P_{\mathrm{totA}}=\frac{4}{\pi}\sqrt{10^{\frac{\mathrm{FM}}{10}\frac{\mathrm{ FM}\left[\mathrm{dB}\right]}{10}}\cdot P_{\mathrm{T}}\left(t\right)}+P_{\mathrm{CA}}. \tag{13}\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Analog Scheme** & \(\mathbf{a^{\mathrm{FM}}}\) & \(\mathbf{b^{\mathrm{FM}}}\) & \(\mathbf{c^{\mathrm{FM}}}\) & \(\mathbf{d^{\mathrm{FM}}}\) \\ \hline \(\mathrm{MSE}=10^{-4}\) & 52.18 & \(-\)0.4481 & 11.20 & \(-\)0.001007 \\ \(\mathrm{MSE}=10^{-5}\) & 43.78 & \(-\)0.2342 & 13.32 & 0.000255 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The fitting parameters for the analog transmission scheme. Figure 9: The power consumption for digital transmission with QAM and ASK modulation as a function of the distance between transmitter and receiver for \(\mathrm{MSE}=10^{-5}\) for the bandwidth extension which minimize the power consumption of the digital transmission with QAM modulation (top figure) and the digital transmission with ASK modulation (bottom figure)In the same way, the analytical power consumption model can be determined for the digital schemes. Due to the finite set of parameters, we determine the lower and upper bound for the required SNR in a function of bandwidth extension. The approximation of the required SNR for coded digital scheme can be defined by the exponential function:  
\[\mathrm{SNR_{req}^{QAM}\left[dB\right]}=a_{\mathrm{coded}}^{\mathrm{QAM}}\exp \left(\frac{b_{\mathrm{coded}}^{\mathrm{QAM}}B_{\mathrm{sig}}}{f_{\mathrm{m}} }\right)+c_{\mathrm{coded}}^{\mathrm{QAM}}\exp\left(\frac{d_{\mathrm{coded}}^{ \mathrm{QAM}}B_{\mathrm{sig}}}{f_{\mathrm{m}}}\right), \tag{14}\]  
while for the uncoded QAM scheme by the function:  
\[\mathrm{SNR_{req}^{QAM}\left[dB\right]}=a_{\mathrm{uncoded}}^{\mathrm{QAM}} \left(\frac{B_{\mathrm{sig}}}{f_{\mathrm{m}}}\right)^{b_{\mathrm{uncoded}}^{ \mathrm{QAM}}}+c_{\mathrm{uncoded}}^{\mathrm{QAM}}, \tag{15}\]  
where the fitting parameters are \(a_{\mathrm{coded}}^{\mathrm{QAM}}\), \(b_{\mathrm{coded}}^{\mathrm{QAM}}\), \(c_{\mathrm{coded}}^{\mathrm{QAM}}\), \(d_{\mathrm{coded}}^{\mathrm{QAM}}\), \(a_{\mathrm{uncoded}}^{\mathrm{QAM}}\), \(b_{\mathrm{uncoded}}^{\mathrm{QAM}}\) and \(c_{\mathrm{uncoded}}^{\mathrm{QAM}}\). These coefficient values, obtained based on least squares fitting to data from Figure 6, are given in Tables 3 and 4 for coded and uncoded QAM system, respectively.