## 3 Feedback Power Control
### Practical Feedback Power Control Strategies  
The proposed feedback power control strategies require that a source might need to increase, in principle, its transmit energy without limit-for instance, this might be the case over a link characterized by an extremely small fading coefficient. Moreover, the proposed power control schemes assume the presence of an ideal communication (transmission and reception) scheme which achieves the system capacity bounds. Therefore, the power allocation strategies proposed so far will lead to reference performance results. In the remainder of this subsection, practical versions of the balanced SNR and unbalanced SNR feedback power control strategies are proposed, such that (i) the sources can adapt their transmit energies within a _limited_ range \(\pm\Delta E_{\rm max}\) at quantized steps of width \(\Delta E_{\rm step}\) and (ii) a proper energy gap, with respect to the ideal operational points, is considered. For both practical versions of the proposed power control schemes, we assume that each node sends a pilot signal to the AP at a fixed initial SNR defined as follows:  
\[\frac{E_{\rm c-start}}{N_{0}}\triangleq\delta\,\overline{\gamma}^{(\rm tgt)- bal} \tag{20}\]  
where \(\overline{\gamma}^{(\rm tgt)-bal}\) is the minimum target SNR given by the solution of the optimization problem (14) according to the balanced SNR power control strategy and \(\delta>1\) is a coefficient which takes into account the non-idealities of the receiver (e.g., the suboptimal iterative decoding scheme). In other words, the starting target common SNR at the sensors would guarantee that the network operational point lies in the feasible SNR region in the absence of fading over the communication links. Note that \(\overline{\gamma}^{\rm(tgt)-bal}\) does not depend on the channel gains (as already observed) and, hence, may be assumed to be known at the transmitters--we are also implicitly assuming that the nodes know the variance of the AWGN at the AP. Upon receiving the pilot signals from all nodes, the AP is assumed to perform a perfect estimation of the fading coefficients and, therefore, of the received SNRs over all links:  
\[\gamma_{k}=\delta\,\overline{\gamma}^{\rm(tgt)-bal}|\alpha^{(k)}|^{2}\qquad k =1,\ldots,n\]  
At this point, the AP compares, over each link, the received SNR with the corresponding target SNR, which depends on the chosen power control strategy. In the case of the balanced SNR power control strategy, the target SNR for the \(k\)-th source will be \(\delta\,\overline{\gamma}^{\rm(tgt)-bal}\) (i.e., the same for all sources); in the case of unbalanced SNR power control strategy, the target SNR will be \(\delta\,\gamma_{k}^{\rm(tgt)-unbal}\). On the basis of the outcome of this comparison, the AP sends to the \(k\)-th source a power control command, in terms of required per-symbol energy variation \(\Delta E_{c,k}\) (e.g., see note 2), according to the rule in Table 1, where the generic target SNR is denoted as \(\gamma_{k}^{\rm(tgt)}\) and depends on the chosen power control strategy, i.e.,  
\[\gamma_{k}^{\rm(tgt)}=\left\{ \begin{array}{ll}\delta\,\overline{\gamma}^{ \rm(tgt)-bal}&\mbox{balanced SNR}\\ \delta\,\overline{\gamma}_{k}^{\rm(tgt)-unbal}&\mbox{unbalanced SNR}\\ \end{array} \right.\]  
In particular, we assume that the maximum per-bit energy variation which can be carried out by a source is \(\Delta E_{\rm max}\) (dimension: [dB]) and that the power control command is quantized with