## 2 System Model in the Absence of Non-Idealities
### Communication Scheme and Feedback Power Control
The fading coefficients are assumed to be independent from link to link and, on a single link, between consecutive packet transmissions (e.g., see note 1). Their amplitudes \(\{|\alpha^{(k)}|\}_{k=1}^{n}\) are assumed to be Rayleigh distributed with \(\mathbb{E}[|\alpha^{(k)}|^{2}]=1\). We denote as \(\mathbf{\
u}^{(k)}=[\
u_{0}^{(k)},\ldots,\
u_{N-1}^{(k)}]\) the binary (not modulated) codeword (\(\
u_{i}^{(k)}\in\{0,1\}\)) generated at the \(k\)-th node. For simplicity, we assume that binary phase shift keying (BPSK) is the modulation format, i.e., \(s_{i}^{(k)}=y_{i}^{(k)}\sqrt{2E_{\rm c}^{(k)}}\), where \(y_{i}^{(k)}=2\
u_{i}^{(k)}-1=\pm 1\) and \(E_{\rm c}^{(k)}\) is the energy per coded bit transmitted by the \(k\)-th node. Indicating by \(P_{\rm t}^{(k)}\) the transmit power at the \(k\)-th node, the transmitted bit energy in the \(k\)-th link can be written as \(E_{\rm c}^{(k)}=P_{\rm t}^{(k)}T_{\rm bit}\), where \(T_{\rm bit}\) is the bit duration. Since we are considering a block fading model, we assume that the link gains can be perfectly estimated at the AP (e.g., using a short preamble with pilot symbols). We preliminary consider a system with orthogonal links. This is meaningful for wireless sensor networking scenarios with reservation-based medium access control (MAC) protocols, such as time/frequency division multiple access (TDMA/FDMA). The use of these protocols allows to represent the multiple access channel as a set of parallel orthogonal channels [4]. Therefore, in these cases the assumption of orthogonality is realistic. However, since in some cases orthogonality may be partially lost (e.g., in the case of non-orthogonal code division multiple access (CDMA) or in presence of FDMA with overlapping bandwidths), in Section 5. we will investigate the case where distributed users transmit simultaneously to the AP and, consequently, there appears multiple access interference at the AP. Under the above assumptions, after matched filtering and carrier-phase estimation the real observable at the AP, relative to a transmitted sample, can be expressed as  
\[r_{i}^{(k)}=|\alpha^{(k)}|\sqrt{E_{\rm c}^{(k)}}y_{i}^{(k)}+\eta_{i}^{(k)} \qquad\quad i=0,\ldots,N-1\qquad k=1,\ldots,n \tag{3}\]  
where \(\eta_{i}^{(k)}\) is an AWGN variable with zero mean and variance \(N_{0}/2\). Upon reception of the signals transmitted from all sources, the goal of the AP is to reconstruct each information signal by exploiting the source correlation. In order to do this, JCD schemes for two-source scenarios with systematic channel coding at each source and iterative decoding at the AP have been proposed [12, 14, 15]. In all cases, the key operational principle of the iterative decoder is that of using a soft-output component decoder per source and, then, to pass the generated soft-output values on the systematic bits (properly weighed taking into account the source correlation) to the other decoders. In  
Figure 1: Multiple access scheme with feedback. Subsection 4.1., more details will be given and a general iterative decoder for an \(n\)-source scenario will be proposed. In Figure 1, the feedback channels are indicated as dashed lines. In this work, we preliminary assume that these channels are error-free (i.e., each source perfectly receives the power control command sent to it by the AP) and then analyze the impact of noisy feedback (i.e., each source may receive a power control command different from that sent by the AP). More details on power control strategies with ideal and noisy feedback will be given in Section 3. and Section 5., respectively.