## 3 Feedback Power Control
### Feedback Power Control Strategies with Unlimited Transmit Power
However, in the presence of correlated sources, the above solution might no longer be optimal. This motivates one to investigate another power control strategy, as described in the following paragraph. We now derive an optimized a transmit power allocation strategy which allows to achieve a feasible operational point at the lowest overall energy cost. In this case, the power control strategy can be cast into the following optimization problem with respect to the unknown vector of link capacities \(\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{n})\):  
\[ \begin{array}{ll}\text{minimize}_{\boldsymbol{\lambda}}&f(\boldsymbol{ \lambda})=\sum\limits_{k=1}^{n}\frac{\left(2^{2\lambda_{k}}-1\right)}{|\alpha ^{(k)}|^{2}}\\ \text{subject to}&\sum\limits_{m=1}^{p}\lambda_{k_{m}}\geq r\left[H\left(n \right)-H\left(n-p\right)\right]\qquad p=1,\ldots,n&\{k_{1},\ldots,k_{p}\} \subseteq\{1,\ldots,n\}\end{array}  \tag{17}\]  
Once the solution \(\boldsymbol{\lambda}^{(\rm tgt)-unbal}\) of the problem (17) is computed and recalling that \(\lambda_{k}^{(\rm tgt)-unbal}=\log_{2}(1+\gamma_{k}^{(\rm tgt)-unbal})\), the transmit energy at the \(k\)-th sensor is allocated as follows:  
\[E_{\rm c-unbal}^{(k)}\triangleq\frac{N_{0}\left(2^{2\lambda_{k}^{(\rm tgt)- unbal}}-1\right)}{|\alpha^{(k)}|^{2}}=\frac{N_{0}\gamma_{k}^{(\rm tgt)-unbal}}{| \alpha^{(k)}|^{2}} \tag{18}\]  
The problem (17) is a convex optimization problem which may be solved using standard convex optimization solvers [26]. It can be shown that in the case with \(\rho=0.5\) the optimal power control strategy derived in (17) returns the same target SNR shown in (16) for all users. Moreover, it is straightforward to observe that in the case of similar links, i.e., when the fading coefficients are the same in all links, the optimized power control strategy in (17) returns the same solution of the power control strategy (14). Hence, in this case as well the same target SNR is set for all sensors. In general, however, the optimized power control strategy leads to different target SNRs for the sources. We now make a comment on the power control strategies described so far. The AP carries out its optimization strategy determining, after solving of (14) or (17), the target SNRs _at the receiver_ (i.e., at the AP), which correspond to \(\overline{\gamma}^{(\rm tgt)-bal}\) or \(\{\gamma_{k}^{(\rm tgt)-unbal}\}_{k=1}^{n}\), respectively. In the following sections, where the performance will be analyzed, we will consider the average SNR _at the transmitters_, which is defined as the arithmetic average of the actual SNRs (after power control) at the transmitters \(\{E_{\rm c}^{(k)}/N_{0}\}_{k=1}^{n}\), i.e.,  
\[\frac{\overline{E}_{\rm c}}{N_{0}}\triangleq\frac{\sum_{k=1}^{n}E_{\rm c}^{(k )}}{n\,N_{0}} \tag{19}\]  
Analyzing the performance as a function of the average SNR at the transmitters will be representative, for a given performance level, of the energy savings brought by each power control strategy. Before evaluating extensively, through simulations, the performance of the proposed feedback power control strategies in the following section, in Figure 3 we show an illustrative comparison, in terms of target SNRs at the AP and at the transmitters, between (a) (ideal) unbalanced SNR and (b) (ideal) balanced SNR feedback power control strategies in a scenario with \(n=4\) sources and \(\rho=0.95\). This is done in order to highlight the difference between the two strategies.