## 2 System Model in the Absence of Non-Idealities  
### Communication Scheme and Feedback Power Control  
Consider \(n\) spatially distributed nodes which detect (i.e., receive at their inputs) binary information sequences \(\boldsymbol{x}^{(k)}=[x_{0}^{(k)},\ldots,x_{L-1}^{(k)}]\), where \(k=1,\ldots,n\) and \(L\) is the signals' length (the same for all sources). The information signals are assumed to be temporally white with \(P(x_{i}^{(k)}=0)=P(x_{i}^{(k)}=1)=0.5\) and the following simple additive correlation model is considered:  
\[x_{i}^{(k)}=b_{i}\oplus z_{i}^{(k)} i=0,\ldots,L-1 k=1,\ldots,n\]where \(\{b_{i}\}\) are independent and identically distributed (i.i.d.) binary random variables and \(\{z_{i}^{(k)}\}\) are i.i.d. binary random variables with probability \(\rho\) to be 0 (and \(1-\rho\) to be 1). Obviously, if \(\rho=0.5\) there is no correlation between the binary information signals \(\{\mathbf{x}^{(k)}\}_{k=1}^{n}\), whereas if \(\rho=1\) the information signals are identical. Note that in this paper, similarly to previous studies [11, 12, 13, 14, 15], we refer to a transmission scenario where quantization and digitization are performed separately. In this case, the original source correlation may be converted into correlation among the bit sequences of the binary digitized signals, as shown, for example, in [22]. Eventually, the correlation model between bit sequences is univocally determined by the probability that two corresponding (on a time scale) bits are equal. Therefore, we simply adopt this correlation model. According to the chosen correlation model, the a-priori joint probability mass function (PMF) of the information signals at the inputs of the \(n\) sources at the \(i\)-th epoch (\(i\in\{0,\ldots,L-1\}\)), can be computed. After a few manipulations, one can show that [23]  
\[p(\mathbf{x}_{i}) = p(\mathbf{x}_{i}|b_{i}=0)p(b_{i}=0)+p(\mathbf{x}_{i}|b_{i}=1)p(b_{i}=1) \tag{1}\] \[= \frac{1}{2}\left[\rho^{n_{\rm b}}(1-\rho)^{n-n_{\rm b}}+(1-\rho) ^{n_{\rm b}}\rho^{n-n_{\rm b}}\right]\hskip 28.452756 pti=0,\ldots,L-1\]  
where \(\mathbf{x}_{i}=(x_{i}^{(1)},\ldots,x_{i}^{(n)})\) and \(n_{\rm b}=n_{\rm b}(\mathbf{x}_{i})\) is the number of zeros in \(\mathbf{x}_{i}\). Under the considered correlation model, it is straightforward to express the joint entropy \(H(n)\) of the \(n\)-dimensional vector \(\mathbf{x}_{i}\) emitted by the \(n\) sources at the \(i\)-th epoch as follows:  
\[H(n) = -\frac{1}{2}\sum_{n_{\rm b}=0}^{n}\binom{n}{n_{\rm b}}\left[\rho ^{n_{\rm b}}(1-\rho)^{n-n_{\rm b}}+(1-\rho)^{n_{\rm b}}\rho^{n-n_{\rm b}}\right] \tag{2}\] \[\hskip 28.452756 pt\cdot\log_{2}\left\{\frac{1}{2}\left[\rho^{n_{ \rm b}}(1-\rho)^{n-n_{\rm b}}+(1-\rho)^{n_{\rm b}}\rho^{n-n_{\rm b}}\right]\right\}\]  
In Figure 1, the overall model for the multiple access scheme with feedback is shown. The goal of the communication system is that of recovering, at the AP, the information signals \(\{\mathbf{x}^{(k)}\}_{k=1}^{n}\) with the lowest possible probability of error. Referring to the equivalent low-pass signal representation, we denote as \(\mathbf{s}^{(k)}\) the complex samples transmitted by the \(k\)-th source and as \(N\) the length of \(\mathbf{s}^{(k)}\). In the remainder of this work, we will assume that the same transmitting rate \(r=L/N\) is used at all sources: however, the proposed approach is general and can be applied also to scenarios where the transmitting rate varies from source to source. By \(\mathbf{\alpha}^{(k)}=[\alpha_{0}^{(k)},\ldots,\alpha_{N-1}^{(k)}]\) we denote the complex gain vector over the \(k\)-th link, which encompasses both path loss and fading, and by \(\mathbf{\eta}^{(k)}=[\eta_{0}^{(k)},\ldots,\eta_{N-1}^{(k)}]\) a complex AWGN vector. We assume a block fading model for the communication links between the sources and the AP: more precisely, the fading coefficient of each link is _constant_ for the entire duration of a single packet transmission, i.e., \(\alpha_{i}^{(k)}=\alpha^{(k)}\) for \(i=0,\ldots,N-1\).