## 4 Performance Analysis in the Absence of Non-Idealities
### Iterative Joint Channel Decoding at the AP
In a straightforward manner, one can rewrite \(P(y_{i}^{(k)})\) as  
\[P(y_{i}^{(k)})=\frac{1}{n-1}\underbrace{\left[P(y_{i}^{(k)})+\ldots+P(y_{i}^{(k )})\right]}_{\text{$n-1$ times}} \tag{22}\]  
Using Bayes' theorem [23], the probability \(P(y_{i}^{(k)})\) can be expressed as  
\[P(y_{i}^{(k)})=\sum_{y_{i}^{(\ell)}=\pm 1}P(y_{i}^{(k)},y_{i}^{(\ell)})=\sum_{y_{ i}^{(\ell)}=\pm 1}P(y_{i}^{(k)}|y_{i}^{(\ell)})P(y_{i}^{(\ell)})\hskip 28.452756 pt \ell=1,\ldots,N\quad\&\quad\ell\
eq k \tag{23}\]  
Approximating the a priori probability \(P(y_{i}^{(\ell)})\) in (23) with the _a posteriori_ reliability value, denoted as \(\hat{P}(y_{i}^{(\ell)})\), output by the \(\ell\)-th decoder (\(\ell\
eq k\)), from (23) one obtains:  
\[P(y_{i}^{(k)})\simeq\sum_{y_{i}^{(\ell)}=\pm 1}P(y_{i}^{(k)}|y_{i}^{(\ell)}) \hat{P}(y_{i}^{(\ell)})\hskip 28.452756 pt\ell=1,\ldots,N\quad\&\quad\ell\
eq k\]  
where  
\[\hat{P}(y_{i}^{(\ell)})=\left\{ \begin{array}{ll}\frac{e^{\mathcal{L}_{i, \mathrm{out}}^{(\ell)}}}{1+e^{\mathcal{L}_{i,\mathrm{out}}^{(\ell)}}}&\text{if }y_{i}^{(\ell)}=+1\\ \frac{1}{1+e^{\mathcal{L}_{i,\mathrm{out}}^{(\ell)}}}&\text{if }y_{i}^{(\ell)}=-1 \end{array} \right.\]  
where \(\mathcal{L}_{i,\mathrm{out}}^{(\ell)}\) is the soft-output a posteriori reliability on the \(i\)-th bit generated by the \(\ell\)-th decoder. At this point, we evaluate the conditional probability \(P(y_{i}^{(k)}|y_{i}^{(\ell)})\) in (23) d the _a priori_ distribution (rather than a posteriori reliability values). By applying Bayes' theorem, it follows that  
\[P(y_{i}^{(k)}|y_{i}^{(\ell)})=\frac{P(y_{i}^{(k)},y_{i}^{(\ell)})}{P(y_{i}^{( \ell)})}=2P(y_{i}^{(k)},y_{i}^{(\ell)})\]  
where we have used the fact that \(P(y_{i}^{(\ell)}=-1)=P(y_{i}^{(\ell)}=+1)=1/2\), since the BPSK symbols are supposed to be _a priori_ equiprobable. Finally, the probability in (22) can be approximated as follows:  
\[P(y_{i}^{(k)}) = \frac{1}{n-1}\left[\sum_{y_{i}^{(1)}=\pm 1}P(y_{i}^{(k)},y_{i}^{(1 )})+\cdots+\sum_{y_{i}^{(k-1)}=\pm 1}P(y_{i}^{(k)},y_{i}^{(k-1)})\right. \tag{24}\] \[\left.\qquad+\sum_{y_{i}^{(k+1)}=\pm 1}P(y_{i}^{(k)},y_{i}^{(k+1)})+ \cdots+\sum_{y_{i}^{(n)}=\pm 1}P(y_{i}^{(k)},y_{i}^{(n)})\right]\] \[\simeq \frac{2}{n-1}\sum_{ \begin{subarray}{c}\ell=1\\ \ell\
eq k\end{subarray} }^{n}\sum_{y_{i}^{(\ell)}=\pm 1}\underbrace{\hat{P}(y_{i}^{( \ell)})}_{\text{[from decoder $\ell$]}}\cdot\underbrace{P(y_{i}^{(\ell)},y_{i}^{(k)})}_{\text{[a priori source correl. ]}}\]  
where \(P(y_{i}^{(\ell)},y_{i}^{(k)})\) can be obtained by marginalization of the \(n\)-th dimensional a-priori joint PMF \(\{P(y_{i}^{(1)},y_{i}^{(2)},\ldots,y_{i}^{(n)})\}\) of the information sequences at the input of the sources (e.g., see note 4). The intuition behind (24) consists in modifying the input a priori probability of a single bit by taking into account, through a weighed average, the reliability values (on the same bit) generated by the other decoders. In particular, the weight of the reliability value generated by the \(\ell\)-th decoder is given by the joint a priori probability between the \(k\)-th and \(\ell\)-th decoders. It is immediate to observe that the proposed iterative JCD scheme has a complexity (measured in terms of basic operations, such as additions and multiplications) which, for a single external iteration, is a linear function of the number \(n\) of sources. In fact, there is a component decoder per source, so that there are \(n\) component decoders. The complexity required by each decoder, denoted as \(\mathcal{C}_{\rm dec}\), depends on the specific decoding algorithm under use, namely the SP algorithm in the presence of LDPC coding or the BCJR algorithm in the presence of the SCCC.