## 6 Extension to Scenarios with Fusion: CEO Problem
### Fusion Rule
It is of interest to evaluate the probability of decision error when the channel SNR becomes very high. In this case, in fact, the iterative JCD scheme allows to recover perfectly the effectively transmitted sequence, denoted as \(\boldsymbol{x}_{i}^{\mathrm{corr}}\). Therefore, it follows that:  
\[\lim_{\overline{E}_{c}/N_{0}\rightarrow\infty}\left[\prod_{j=1}^{n}P(x_{i}^{ (j)}|\mathcal{L}_{i}^{(j)})\right]=\left\{ \begin{array}{ll}1&\text{if } \boldsymbol{x}_{i}=\boldsymbol{x}_{i}^{\mathrm{corr}}\\ 0&\text{if }\boldsymbol{x}_{i}\
eq\boldsymbol{x}_{i}^{\mathrm{corr}} \end{array} \right.\]and the fusion rule (33) becomes  
\[\hat{b}_{i} = \underset{\hat{b}_{i}=0,1}{\mathrm{argmax}}\frac{P(\mathbf{x}_{i}^{ \mathrm{corr}}|b_{i})}{\sum_{b^{*}=0,1}P(\mathbf{x}_{i}^{\mathrm{corr}}|b^{*})} \tag{35}\] \[= \underset{\hat{b}_{i}=0,1}{\mathrm{argmax}}P(\mathbf{x}_{i}^{\mathrm{ corr}}|b_{i})\]  
Denoting \(n_{\mathrm{b}}=n_{\mathrm{b}}(\mathbf{x}_{i}^{\mathrm{corr}})\) as the number of zeros in \(\mathbf{x}_{i}^{\mathrm{corr}}\), the probability \(P(\mathbf{x}_{i}^{\mathrm{corr}}|b_{i})\) in (35) can be written, according to the correlation model presented in (2.1. ), as follows:  
\[P(\mathbf{x}_{i}^{\mathrm{corr}}|b_{i})=\left\{ \begin{array}{ll}(1-\rho)^{n_{ \mathrm{b}}}\rho^{n-n_{\mathrm{b}}}&\mbox{if}\,b_{i}=0\\ (1-\rho)^{n-n_{\mathrm{b}}}\rho^{n_{\mathrm{b}}}&\mbox{if}\,b_{i}=1\end{array} \right. \tag{36}\]  
By using (36), the decision strategy (35) becomes  
\[P(\mathbf{x}_{i}^{\mathrm{corr}}|b_{i}=0) \begin{array}{ll}\hat{b}_{i}=0\\ \begin{array}{ll}\times\\ \hat{b}_{i}=1\end{array}&P(\mathbf{x}_{i}^{\mathrm{corr}}|b_{i}=1)\\ (1-\rho)^{n_{\mathrm{b}}}\rho^{n-n_{\mathrm{b}}}&\begin{array}{ll}\times\\ \times\\ \hat{b}_{i}=1\end{array}&(1-\rho)^{n-n_{\mathrm{b}}}\rho^{n_{\mathrm{b}}}\\ \left(\frac{\rho}{1-\rho}\right)^{n-2n_{\mathrm{b}}}&\begin{array}{ll}\hat{b }_{i}=0\\ \times\\ \hat{b}_{i}=1\end{array}&1\end{array} \]  
from which one finally obtains:  
\[n_{\mathrm{b}} \begin{array}{ll}\hat{b}_{i}=1\\ \begin{array}{ll}\times\\ \hat{b}_{i}=0\end{array}&\left\lfloor\frac{n}{2}\right\rfloor\end{array} \]  
Note that if \(n_{\mathrm{b}}=\lfloor n/2\rfloor\) (this can happen only if \(n\) is even), the decision has to be randomly taken. In this case, we arbitrarily assume that \(\hat{b}_{i}=1\). Observing that  
\[P(n_{\mathrm{b}}=k|b_{i}=0) = \binom{n}{k}\rho^{n-k}(1-\rho)^{k}\] \[P(n_{\mathrm{b}}=k|b_{i}=1) = \binom{n}{k}(1-\rho)^{n-k}\rho^{k}\]  
the following limits hold:  
\[\lim_{\overline{E}_{\mathrm{c}}/N_{0}\to\infty}P(\hat{b}_{i}=0|b_ {i}=1) = P\left(n_{\mathrm{b}}<\left\lfloor\frac{n}{2}\right\rfloor|b_{i}=1\right)\] \[= \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor-1}\binom{n}{k}(1-\rho)^{n-k} \rho^{k}\] \[\lim_{\overline{E}_{\mathrm{c}}/N_{0}\to\infty}P(\hat{b}_{i}=1|b_ {i}=0) = P\left(n_{\mathrm{b}}\geq\left\lfloor\frac{n}{2}\right\rfloor|b_{i}=0\right)\] \[= \sum_{k=\lfloor\frac{n}{2}\rfloor}^{n}\binom{n}{k}(1-\rho)^{k} \rho^{n-k}\]Finally, the limiting value (for large channel SNR) of the probability of decision error (34) becomes  
\[P_{\rm e,lim}\triangleq\lim_{E_{c}/N_{0}\rightarrow\infty}P_{\rm e}=\frac{1}{2} \left[\sum_{k=0}^{[\frac{n}{2}]-1}\binom{n}{k}(1-\rho)^{n-k}\rho^{k}+\sum_{k=[ \frac{n}{2}]}^{n}\binom{n}{k}(1-\rho)^{k}\rho^{n-k}\right] \tag{37}\]  
The limiting probability of decision error in (37) corresponds to the probability of decision error in the presence of majority fusion, as typically observed in the realm of distributed detection [36], and does not depend on the channel SNR. Therefore, the probability (37) corresponds to a floor. Moreover, the final expression, at the right-hand side of (37), shows that the limiting probability of decision error _does not_ depend on the particular channel code under use. However, as it will be shown in the following subsection, the chosen channel code will affect the behavior of the probability of decision error above the limiting floor.