## 3 Feedback Power Control  
### Feedback Power Control Strategies with Unlimited Transmit Power  
As discussed in Subsection 2.1., according to the considered block-faded sensor-AP channel model, the instantaneous per-link SNR at the AP is subject to per-packet fading fluctuations due to the time-varying nature of the channel. A feedback power control strategy for a multiple access scheme consists of a rule, depending on the (ideally perfectly estimated) links' statuses, according to which a power control command is sent, by the AP, to each sensor. Equivalently, the power control strategy is based on the determination of proper target SNRs at the AP, denoted as \(\{\gamma_{k}^{\rm(tgt)}\}_{k=1}^{n}\), for all sensors. On the basis of these target SNRs, the AP will send (without errors) the corresponding power control commands to the sources. Therefore, the \(k\)-th source will ideally set its transmit power in order to reach the target SNR at the AP. Under the assumption of unlimited transmit power at the sources, the transmit energy at the \(k\)-th node (assuming fixed bit duration) will be set as follows:  
\[E_{\rm c}^{(k)}=\frac{N_{0}\gamma_{k}^{\rm(tgt)}}{|\alpha^{(k)}|^{2}}=\frac{N _{0}\left(2^{2\lambda_{k}^{\rm(tgt)}}-1\right)}{|\alpha^{(k)}|^{2}} \tag{13}\]  
where \(\lambda_{k}^{\rm(tgt)}=\log_{2}(1+\gamma_{k}^{\rm(tgt)})/2\) is the target capacity for \(k\)-th link. In this setting, a power control strategy consists in allocating the target SNRs \(\{\gamma_{k}^{\rm(tgt)}\}_{k=1}^{n}\) and, thus, the corresponding transmit energies \(\{E_{\rm c}^{(k)}\}_{k=1}^{n}\), so that the constraints (12) are satisfied for all users. Since the constraints (12) can be satisfied in infinite ways, we now propose two possible approaches: classical (in the sense of target SNR equalization over all links) and optimized (in the sense of overall transmit energy minimization). The classical approach for power control in multiple access systems tries to \"balance\" the SNRs at the AP over all possible links. More precisely, the AP fixes a _common_ target SNR, denoted as \(\overline{\gamma}^{\rm(tgt)}\), for all sources. Obviously, the common target SNR will have to be higher than the minimum required common SNR to guarantee that the operational point lies within the feasible SNR region introduced in Subsection 2.2.. The minimum common target SNR, denoted as \(\overline{\gamma}^{\rm(tgt)-bal}\), can be obtained by solving the following optimization problem with constraints given by (11):  
\[ \begin{array}{ll}\mbox{minimize}&\overline{\gamma}^{\rm(tgt)}\\ \mbox{subject to}&p\,\frac{\log_{2}(1+\overline{\gamma}^{\rm(tgt)})}{2}\geq r \left[H\left(n\right)-H\left(n-p\right)\right]\hskip 28.452756 ptp=1,\ldots,n \end{array}  \tag{14}\]  
where we have used the fact that, under the considered common target SNR, it holds that  
\[\lambda_{k_{m}}=\frac{\log_{2}(1+\overline{\gamma}^{\rm(tgt)})}{2}\hskip 56.905512 pt \forall k_{m}\in\{1,\ldots,n\}\]  
From (14) one obtains, after a few mathematical passages, the following expression for the minimum common target SNR:  
\[\overline{\gamma}^{\rm(tgt)-bal}=2^{\chi}-1 \tag{15}\]  
where  
\[\chi\triangleq\max_{p=1,\ldots,n}\left\{\frac{2r}{p}\left[H(n)-H(n-p)\right]\right\}\]If multiple access schemes with uncorrelated sensors are considered, i.e., \(\rho=0.5\), it holds that \(H(n)=n\) and, therefore, \(H(n)-H(n-p)=p\). In this case, the minimum target SNR in (15) reduces to  
\[\overline{\gamma}^{(\rm tgt)-bal}=2^{2r}-1 \tag{16}\]  
In other words, the target SNR for each sensor is the minimum SNR which fulfills the Shannon capacity bound for a given rate \(r\). Therefore, one can conclude that balancing the SNRs at the AP is optimal. On the other hand, it is straightforward to observe that for \(\rho>0.5\) it follows that \(H(n)-H(n-p)<p\), i.e., \(\overline{\gamma}^{(\rm tgt)-bal}\) reduces with respect to the value for \(\rho=0.5\), thus allowing the system to reach a feasible point at lower energy consumption.