# Combined Fusion Rules in Cognitive Radio Networks Using Different Threshold Strategies
## 2 Theoretical Background of Cognitive Radio Networks
### Detection Performance of Cooperative Sensing
#### 2.2.1 One-Bit Hard Decision Scheme  
A CR makes its hard decision as a one-bit, \(D\), by comparing the detected energy to the sensing threshold, \(\lambda\), as shown in Figure 1. The sensing threshold, \(\lambda\), is determined based on a predefined probability of detection. At the central entity, the gathered decisions are fused using the _K-out-of-N_ rule which makes a decision using only \(K\) observations out of \(N\) available observations in the system. A proper selection of fusion parameter, \(K\), plays an important role in optimizing the detection performance of the CSS. For instance, if \(K=1\), the _K-out-of-N_ operates as OR-Rule which significantly improves the global detection performance while reduces the spectrum efficiency. If \(K=N\), the _K-out-of-N_ operates as AND-Rule which significantly improves the spectrum efficiency while reduces the global detection performance. Therefore, to balance between improvement in detection performance and spectrum efficiency, \(K\) is selected as \(K=\left\lceil\frac{N}{2}\right\rceil\)) which is called half voting-rule, where \(\left\lceil.\right\rceil\) is a ceiling function. The global probability of false alarm, \(Q_{f}\), and the global probability of detection, \(Q_{d}\), at the central entity are computed using the generalized K-out-of N Rule.  
The generalized _K-out-of-N_ Rule provided in [26] is employed to compute the global probabilities of false alarm, \(Q_{f}\), and the global probability of detection, \(Q_{d}\), as below:  
\[Q_{f}=\sum_{i=K_{2}}^{C}\left\{\sum_{j=1}^{V_{K_{2}}}\left[\prod_{l=1}^{K_{2}} P_{f,j,l}\prod_{l=K_{2}+1}^{C}\left(1-P_{f,j,l}\right)\right]\right\}, \tag{5}\]  
\[Q_{d}=\sum_{i=K_{2}}^{C}\left\{\sum_{j=1}^{V_{K_{2}}}\left[\prod_{l=1}^{K_{2}} P_{d,j,l}\prod_{l=K_{2}+1}^{C}\left(1-P_{d,j,l}\right)\right]\right\}, \tag{6}\]  
where \(C\) is the number of participating members, i.e., CRs in a cluster or cluster heads in a CRN, \(P_{f,j,l}\) and \(P_{d,j,l}\) are the probability of false alarm and probability of detection of the \(i\)-th CR, \(V_{K_{2}}^{C}=C!/\left(K_{2}!\left(C-K_{2}\right)!\right)\), and \(!\) denotes the factorial, \(K_{2}=\left\lceil\frac{C}{2}\right\rceil\). Note that Equations (5) and (6) are used to find the global probability of detection and global probability of false alarm when the participating members have different, i.e., not equal, probabilities of detection and probabilities of false alarm.  
**Special case:** if a group of CRs, i.e., a cluster of \(N\) CRs, are close to each other, they experience the same radio conditions, then it can assume that \(P_{d1}=P_{d2}=\cdots=P_{d}\) and \(P_{f1}=P_{f2}=\cdots=P_{f}\), i.e.,  
Figure 1: The principle of One-bit hard scheme.  
the probability of detection and the probability of false alarm of all CRs in the cluster are the same [24], then Equations (5) and (6) are simplified as  
\[Q_{f}=\sum_{i=K}^{N}\binom{N}{i}P_{f}^{i}[1-P_{f}]^{N-i}, \tag{7}\]  
\[Q_{d}=\sum_{i=K}^{N}\binom{N}{i}P_{d}^{i}[1-P_{d}]^{N-i}, \tag{8}\]  
where \(K=\left\lceil\frac{N}{2}\right\rceil\) and \(N\) is the number of the CRs in the cluster.