## 2 System Model in the Absence of Non-Idealities
### Feasible SNR Region of a Multiple Access Scheme  
It is well known that distributed source coding allows to reduce the amount of data to be transmitted to the AP without needing extra inter-sensor communications. In particular, the performance achievable by distributed source coding is identical to that which could be achieved if the sources were encoded jointly. The SW theorem allows to determine the achievable rate region for the case of separate lossless encoding of correlated sources. Denoting by \(r_{\mathrm{s},k}\) the achievable compression rate for \(k\)-th transmitter, one obtains the following bounds:  
\[\sum_{m=1}^{p}r_{\mathrm{s},k_{m}}\geq H\left(x_{i}^{(k_{1})},x_{i}^{(k_{2})}, \ldots,x_{i}^{(k_{p})}|x_{i}^{(j_{1})},x_{i}^{(j_{2})},\ldots,x_{i}^{(j_{n-p})}\right) \tag{4}\]  
where \(p\in\{1,\ldots,n\}\), \(k_{i}\
eq j_{f}\left(i\in\{1,\ldots,p\},f\in\{1,\ldots,n-p\}\right)\), and \(\{k_{1},\ldots,k_{p}\}\cup\{j_{1},\ldots,j_{n-p}\}=\{1,\ldots,n\}\). In other words, \(H(x_{i}^{(k_{1})},x_{i}^{(k_{2})},\ldots,x_{i}^{(k_{p})}|x_{i}^{(j_{1})},x_{i }^{(j_{2})},\ldots,x_{i}^{(j_{n-p})})\) is the conditional joint entropy of the group of \(p\) sources indexed by \(k_{1},\ldots,k_{p}\), conditioned on the remaining \(n-p\) sources. By exploiting the well known relation between joint and conditional entropies [24], one gets:  
\[H\left(x_{i}^{(k_{1})},x_{i}^{(k_{2})},\ldots,x_{i}^{(k_{p})}|x_{i}^{(j_{1})},x_{i}^{(j_{2})},\ldots,x_{i}^{(j_{n-p})}\right)=H\left(x_{i}^{(1)},\ldots,x_ {i}^{(n)}\right)-H\left(x_{i}^{(j_{1})},x_{i}^{(j_{2})},\ldots,x_{i}^{(j_{n-p} )}\right) \tag{5}\]  
The considered correlation model between the sources is such that the joint entropy depends only on the number of considered sources, as shown in (2). Therefore, the family of inequalities in (4) can be equivalently rewritten as follows:  
\[\sum_{m=1}^{p}r_{\mathrm{s},k_{m}}\geq H\left(n\right)-H\left(n-p\right) \tag{6}\]  
By assuming that source coding (compression) is followed by channel coding, the actual channel code rates \(\{r_{\mathrm{c},k}\}_{k=1}^{n}\) may be expressed as  
\[r_{\mathrm{c},k}=r_{\mathrm{s},k}\times r \tag{7}\]  
where \(r\) is the (already introduced) transmission rate equal to \(L/N\). The channel code rates must satisfy the following Shannon bounds:  
\[r_{\mathrm{c},k}\leq\lambda_{k}\qquad k=1,\ldots,n \tag{8}\]  
where \(\lambda_{k}\) is the capacity of the \(k\)-th link, i.e.,  
\[\lambda_{k}\triangleq\frac{1}{2}\log_{2}\left(1+\gamma_{k}\right) \tag{9}\]and \(\gamma_{k}\) is the SNR, at the AP, relative to the \(k\)-th link, i.e.,  
\[\gamma_{k}=\frac{|\alpha^{(k)}|^{2}E_{\mathrm{c}}^{(k)}}{N_{0}} \tag{10}\]  
As discussed in Section 1., compressing each source up to the SW limit and then utilizing independent capacity-achieving channel codes allows to reach the ultimate performance limits. Therefore, combining (6), (9), and (10), the link capacities \(\{\lambda_{k}\}_{k=1}^{n}\) have to satisfy the following inequalities:  
\[\sum\limits_{m=1}^{p}\lambda_{k_{m}}\geq r\;[H\left(n\right)-H\left(n-p\right) ]\;\;\;\;\;\;\;\;p\in\{1,\ldots,n\}\;\;\;\;\text{and}\;\;\;\;\{k_{1},\ldots,k_{ p}\}\subseteq\{1,\ldots,n\} \tag{11}\]  
From (11), using (9) it follows directly that the feasible \(n\)-dimensional SNR region of the considered multiple access scenario is characterized by the link SNRs at AP \(\{\gamma_{k}\}_{k=1}^{n}\) such that, for any chosen value of \(p\in\{1,\ldots,n\}\), the following inequalities are satisfied:  
\[\sum\limits_{m=1}^{p}\log_{2}\left(1+\gamma_{k_{m}}\right)\geq 2r\times[H\left(n \right)-H\left(n-p\right)]\;\;\;\;\;\;\forall\;\{k_{1},\ldots,k_{p}\}\subseteq \{1,\ldots,n\} \tag{12}\]  
In [25], it is shown that, if one solves (12) in a scenario with \(n=2\) sources and \(\rho=0.95\), the feasible SNR region shown in Figure 2 can be obtained. The target operational point of a feedback power control strategy can be represented as a point in this region. In Section 3., two feedback power control strategies for block faded scenarios will be proposed.