## 3 Feedback Power Control
### Practical Feedback Power Control Strategies
\begin{table}
\begin{tabular}{|c|c|c|} \hline \(\gamma_{k}\) & \(\Delta E_{\rm c,k}\) & Binary Feedback \\ [dB] & [dB] & Command \\ \hline \hline \((\gamma_{k}^{\rm(tgt)}+\Delta E_{\rm max}-\Delta E_{\rm step},+\infty]\) & \(-\Delta E_{\rm max}\) & \(\frac{-1-1\cdots-1}{\Delta E_{\rm max}/\Delta E_{\rm step}}\) \\  &... &... \\ \((\gamma_{k}^{\rm(tgt)}+\Delta E_{\rm step},\gamma_{k}^{\rm(tgt)}+2\Delta E_{ \rm step}]\) & \(-2\Delta E_{\rm step}\) & -1-1 \\ \((\gamma_{k}^{\rm(tgt)},\gamma_{k}^{\rm(tgt)}+\Delta E_{\rm step}]\) & \(-\Delta E_{\rm step}\) & -1 \\ \((\gamma_{k}^{\rm(tgt)}-\Delta E_{\rm step},\gamma_{k}^{\rm(tgt)}]\) & \(\Delta E_{\rm step}\) & +1 \\ \((\gamma_{k}^{\rm(tgt)}-2\Delta E_{\rm step},\gamma_{k}^{\rm(tgt)}-1]\) & \(+2\Delta E_{\rm step}\) & +1+1 \\... &... &... \\ \((-\infty,\gamma_{k}^{\rm(tgt)}-\Delta E_{\rm max}+\Delta E_{\rm step}]\) & \(+\Delta E_{\rm max}\) & \(\frac{+1+1\cdots+1}{\Delta E_{\rm max}/\Delta E_{\rm step}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Practical feedback power control commands and energy corrections. The maximum energy correction is denoted as \(\Delta E_{\rm max}\) (dimension: [dB]) and the minimum energy correction step is \(\Delta E_{\rm step}\) (dimension: [dB]). a step of \(\Delta E_{\rm step}\) (dimension: [dB]). The power control command transmitted back to the sources corresponds to a sequence of bits where each bit is \"\(+1\)\" for an \"up\" correction (\(+\Delta E_{\rm step}\)) and \"-1\" for a \"down\" correction (\(-\Delta E_{\rm step}\)). For instance, if the power control command is equal to +5 dB and \(\Delta E_{\rm step}=1\) dB, then the transmitted binary sequence is \"\(+1\)\(+1\)\(+1\)\(+1\).\" The structure of the binary power control commands is shown in the last column of Table 1. In the following sections, the presented numerical results will be obtained considering \(\Delta E_{\rm max}=20\) dB and \(\Delta E_{\rm step}=1\) dB. Obviously, should \(\Delta E_{\rm step}\to 0\) and \(\Delta E_{\rm max}\rightarrow\infty\), the proposed practical power control strategies would reduce to the corresponding power control strategies with unlimited power control range. We assume that the end of the binary sequence operating the command is indicated by a proper end-of-command \"flag.\" We remark that, since we consider block faded channels, the power control command can be sent only once before each packet transmission. It is worth noting that the proposed power control scheme compares favorably with classical power control schemes (see, for example, [27]), where the transmit power is continuously adjusted at fixed rate. By denoting as \(T_{\rm TPC}\) the fixed transmit power control interval, classical power control schemes allow to compensate for a time varying fading provided that \(T_{\rm TPC}f_{\rm d}\ll 1\), \(f_{\rm d}\) being the fading Doppler frequency. Hence, the power control scheme proposed in this paper may be extended to time-varying channels provided that \(T_{\rm TPC}f_{\rm d}\ll 1\) and the power control command is transmitted continuously during the packet. Obviously, in this case the overloading effect of feedback communications might be relevant and could affect the overall system performance. The extension of the proposed power control scheme to time-varying channels, and the corresponding impact of feedback overloading, is an interesting research direction and will be the subject of future work. As for the reliability of the feedback power control command, in the presence of quasi-static channels it is reasonable to assume error-free feedback channels. More precisely, this can be obtained by protecting the power control commands through the use of low-rate channel codes, e.g., repetition codes. On the basis of these considerations, in Section 4. we will consider error-free (ideal) feedback channels. However, the impact of noisy feedback channels will be investigated in Subsection 5.2..