# Combined Fusion Rules in Cognitive Radio Networks Using Different Threshold Strategies
## 2 Theoretical Background of Cognitive Radio Networks
### Detection Performance of Cooperative Sensing
#### 2.2.2 Two-Bit Softened Hard Scheme  
Unlike the one-bit scheme where the local decision is made using one threshold, the two-bit hard scheme employs three thresholds (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\)), to make the local decision as shown in Figure 2. The whole range of energy is divided into four regions where each region is allocated with a different weight value due to the distinction in the observed energies. Thus, each CR needs to exchange two bits to identify which region its observed energy belongs to. The detection criteria depends on the weighted summation given by \(S_{C}=\sum_{b=0}^{3}w_{b}S_{b}\) where \(S_{b}\) indicates the number of observed energies falling in region \(b\) and the weight values assigned to each region are given by \((w_{0}=0,w_{1}=1,w_{3}=L,w_{4}=L^{2})\); where \(L\) is a design parameter to be optimized. The weighted summation is compared to a predefined threshold, \(S_{C}=L^{2}\). The channel is detected as occupied if \(S_{C}\geq L^{2}\); otherwise, it is detected as idle.  
In the two-bit softened hard scheme, the three thresholds (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\)) which denote three local probabilities of false alarm (\(P_{f1},P_{f2},P_{f3}\)), respectively, are determined to satisfy a specific \(Q_{f}\). To meet the required \(Q_{f}\), \(S_{C}\) should be less than \(L^{2}\) which can be interpreted in Equation (9) where \(B=L^{2}-1\) and \(D_{b}=min\left\{\left\lfloor\frac{L^{2}-1-bw_{1}}{w_{2}-w_{1}}\right\rfloor,b\right\}\) with \(\lfloor.\rfloor\) denoting the floor function.  
Two design parameters \(\beta_{1}\) and \(\beta_{2}\) are defined as \(\beta_{1}=\frac{P_{f2}}{P_{f1}}\), \(\beta_{2}=\frac{P_{f3}}{P_{f2}}\). An exhaustive search is employed to determine \(L\), \(\beta_{1}\) and \(\beta_{2}\). Thereafter, the value of \(\rho\) which is given by \(\rho=\frac{P_{f1}}{1-P_{f1}}\) is determined using Equation (10).  
The total probability of detection is obtained likewise as in Equation (11); the values of \(L\), \(\beta_{1}\) and \(\beta_{2}\) are numerically calculated to maximize the average \(Q_{d}\) while meeting the required \(Q_{f}\).  
Figure 2: The principle of Two-bit softened hard scheme.  
When \(S\), \(Q_{f}\), \(\beta_{1}\), \(\beta_{2}\) and \(\rho\) are known, the values of \(P_{f1}\), \(P_{f2}\) and \(P_{f3}\) can be obtained while \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) can be determined in a subsequent manner from Equation (1). Then, \(P_{d_{1}}\), \(P_{d_{2}}\) and \(P_{d_{3}}\) of the Rayleigh fading channel can be obtained from Equation (4).  
\[1-Q_{f}=A_{1}\sum_{b=0}^{B}{S\choose b}(1-P_{f_{1}})^{S-b}, \tag{9}\]  
\[\left(1-Q_{f}\right)\left(1+\rho\right)^{S}=A_{2}\sum_{b=0}^{B}{S\choose b}, \tag{10}\]  
\[Q_{d}=1-A_{3}\sum_{b=0}^{B}{S\choose b}(1-P_{d_{1}})^{S-b}, \tag{11}\]  
where \(A_{1}=\sum_{d=0}^{D_{b}}{b\choose d}\left(P_{f_{1}}-P_{f_{2}}\right)^{b-d} \left(P_{f_{2}}-P_{f_{3}}\right)^{j}\), \(A_{2}=\sum_{d=0}^{D_{b}}{b\choose d}\times\left(1-\beta_{1}\right)^{b-d}\left( \beta_{1}-\beta_{1}\beta_{2}\right)^{d}\rho^{d}\), and \(A_{3}=\sum_{d=0}^{D_{b}}{b\choose d}\left(P_{d_{1}}-P_{d_{2}}\right)^{b-d} \left(P_{d_{2}}-P_{d_{3}}\right)^{d}\).