## 3 Optimal Decision-Level Fusion Criterion (O-DFC) and Its Simplified Form
### Optimal Decision-Level Fusion Criterion (O-DFC)  
In the distributed detection system as Figure 2 shows, signal processing is accomplished at the sensor and preliminary decisions are transmitted to the data fusion center. Then the global decision will be obtained as the output of this center, based on different DFCs. The optimal DFC (O-DFC) was derived by [PERSON] and [PERSON] [48]. Here, we review it at first and then extend it to our model in Figure 1.  
Assume a binary hypothesis testing problem, with two hypotheses:  
\[ \begin{cases}H_{0}:\text{the signal is absent}\\ H_{1}:\text{the signal is present}\end{cases}  \tag{11}\]  
Figure 1: The structure of decision-level fusion based MUD system model. Figure 2: Distributed detection system with decision-level fusion center. where the priori probabilities of these hypotheses are \(P(H_{0})=P_{0}\) and \(P(H_{1})=P_{1}\). Consider there are n detectors in Figure 2 and the observations of each detector are denoted by \(y_{i}\), \(i=1\), \(2\), \(\ldots\), \(n\) with the assumption of statistical independence. In addition, the conditional probability density function is \(p(y_{i}|H)\), \(i=1\), \(2\), \(\ldots\), \(n\), \(j=0\), \(1\). As Figure 2 depicts, each detector can make its own decision \(u_{i}\) by applying a decision rule g(\(y_{i}\)), \(i=1\), \(2\), \(\ldots\), \(n\), where  
\[u_{i}= \begin{cases}-1,&\text{if $H_{0}$ is decided}\\ +1,&\text{if $H_{1}$ is decided}\end{cases}  \tag{12}\]  
These local preliminary decisions \(u_{i}\) will be sent to the decision-level fusion center for the further processing. In general, the global decision is a function of these preliminary decisions as:  
\[u=f(u_{1},u_{2},\cdots,u_{n}) \tag{13}\]  
According to [48], the minimum probability of error criterion can be employed to derive the O-DFC, which can be summarized as:  
\[u=f(u_{1},u_{2},\cdots,u_{n})= \begin{cases}+1,&\text{if $a_{0}+\sum_{i=1}^{n}a_{i}u_{ i}>0$}\\ -1,&\text{otherwise}\end{cases}  \tag{14}\]  
where the weights are given by  
\[a_{0}=\log\frac{P_{1}}{P_{0}} \tag{15}\]  
\[a_{i}= \begin{cases}\log\frac{1-P_{u_{i}}}{P_{F_{i}}},&\text{if $u_{i}=+1$}\\ \log\frac{1-P_{F_{i}}}{P_{M_{i}}},&\text{if $u_{i}=-1$}\end{cases}  \tag{16}\]  
where \(P_{Fi}=\text{P}(u_{i}=+1|H_{0})\) is the false alarm probability and \(P_{Mi}=P(u_{i}=-1|H_{1})\) is the miss alarm probability of the \(i\)th detector, respectively. For more information of the detailed derivation, interested readers can refer to [48]. In our decision-level fusion based MUD model, however, these two hypotheses should be modified as:  
\[ \begin{cases}H_{0m}:-1\text{ is transmitted by the $m$th user}\\ H_{1m}:+1\text{ is transmitted by the $m$th user}\end{cases}  \tag{17}\]  
As a result, the false alarm probability \(P\_{Tim}=P(u\_{im}=+1|\,H\_{0}\,m)\) and the miss alarm probability \(P\_{Min}=P(u\_{im}=-1|\,H\_{1}\,m)\) of the \(i\)th sub-optimal detector for the \(m\)th user are both equal to its bit error rate (BER) \(P_{elim}\), we have that is \(P_{Fil}=P_{Min}=P_{elim}\), \(i=1\), \(2\), \(\ldots\), \(n\), \(m=1\), \(2\), \(\ldots\), \(K\) and \(K\) is the number of users in this multiuser system. For the equiprobable source assumption, we have \(a_{0}=0\). Therefore, the O-DFC in our model can be rewritten by \[u_{m}=f(u_{1m},u_{2m},\cdots,u_{m})= \begin{cases}+1,&\text{if }\sum_{i=1}^{n}a_{in}u_{ in}>0\\ -1,&\text{otherwise}\end{cases}  \tag{18}\]  
where \(u_{m}\) is the final decision for the \(m\)th user, \(u_{in}\) is the preliminary decision of the \(i\)th sub-optimal detector for the \(m\)th user, and  
\[a_{in}=\log\frac{1-P_{in}}{P_{in}} \tag{19}\]  
In particular, for the special case when there are it \(\{n\}=3\) detectors, a proposition has been proposed by [PERSON] _et al_. in [49].