## 4 Performance Analysis in the Absence of Non-Idealities
### Numerical Results  
In LDPC-coded scenarios, each of the source sequences is encoded using (i) a regular (3,6) LDPC code or (ii) an irregular LDPC code with double diagonal (DD) square submatrix in the parity check matrix [14]. The DD LDPC code provides a sort of implicit \"differential encoding\" effect which is in agreement with the design guidelines, presented in [32], for channel codes to be used in multiple access schemes. Both LDPC codes have a rate equal to 1/2 and \(L=1000\). Each component LDPC decoder at the AP performs a maximum number \(n_{\rm it}^{\rm int-max}=50\) of internal iterations, whereas the fixed number \(n_{\rm it}^{\rm ext}\) of external iterations between the decoders is set to 20. The LDPC codes are constructed in a _random_ fashion, according to the following algorithm, which exploits an idea similar to the progressive edge growth (PEG) algorithm presented in [33]. Some potential connections, denoted as _sockets_, are drawn for all the variable and check nodes. Then, for each variable node a socket is randomly chosen, among all the free sockets at the check nodes, and the connection is added only if a _cycle_ of a given (or lower) length is not created. In our case, the checked cycle length is equal to 6 for the regular LDPC code, whereas it is 4 for the DD LDPC code. In turbo-like coded scenarios, we consider a SCCC given by the concatenation, through a bit random interleaver, of an outer convolutional code with an inner convolutional code [34]. The decoder is based on a message passing (turbo decoding) algorithm, such that the extrinsic information is iteratively passed between the inner and the outer soft-input soft-output (SISO) decoders for a predefined number \(n_{\rm it}\) of iterations. The presence of a priori information coming from other decoders can be easily taken into account by feeding the a priori probabilities of information bits to the input of the outer SISO decoder in the form of LLRs. We consider the SCCC proposed in [25], which has been shown to perform very well in a scenario with JCD. More specifically, the SCCC is constituted by an outer 8-state rate-\(1/2\) non-systematic and recursive convolutional code characterized by the following generator matrix:  
\[G_{\rm outer}(D)=\left[\frac{1+D^{2}+D^{3}}{1+D+D^{3}},\;\frac{1+D+D^{2}+D^{ 3}}{1+D+D^{3}}\right]\]  
and by an inner 8-state rate-1/2 systematic and recursive code with the following generator matrix:  
\[G_{\rm inner}(D)=\left[1,\;\frac{1+D+D^{2}+D^{3}}{1+D^{2}+D^{3}}\right] \tag{25}\]  
The outer code is then punctured to obtain a rate equal to \(3/4\), while the inner code is punctured to obtain a rate equal to \(2/3\), so that the total rate of the SCCC is 1/2. The puncturing matrix \(P_{\rm o}\) for the rate \(3/4\) outer code is  
\[P_{\rm o}=\left[ \begin{array}{ccc}1&1&0\\ 1&1&0\end{array} \right]\]  
which punctures (or erases) one third of the parity bits. The puncturing matrix \(P_{\rm l}\) for the rate-\(2/3\) inner code is  
\[P_{\rm l}=\left[ \begin{array}{ccc}1&1\\ 1&0\end{array} \right] \tag{26}\]  
which erases one half of the parity bits. In both outer and inner convolutional codes, the systematic bits are not punctured. In this case, each component decoder performs a fixed number \(n_{\rm it}^{\rm int-max}=10\) of internal iterations, whereas the fixed number \(n_{\rm it}^{\rm ext}\) of external iterations between the decoders is set to 5 (e.g., see note 5). In all cases (LDPC-coded and SCCCed), we directly compare scenarios without feedback power control (W/o PC) and with feedback power control (W PC). Moreover, in the W PC case the performance is evaluated considering the two proposed power control strategies (balanced SNR and unbalanced SNR) and different values of the number \(n\) of sources (namely 2, 3, and 4). The maximum energy correction \(\Delta E_{\rm max}\) is set to 20 dB and \(\rho\) is set to 0.95. We have performed other simulations for different values of \(\rho\) and found similar results to those presented in this paper for values of \(\rho\) higher than 0.8.