# On Optimal Cooperative Sensing with Energy Detection in Cognitive Radio
## 2 System Model
### Problem Formulation of Cooperative Sensing  
We investigate optimal sensing thresholds for sensing nodes using hard decision. Even though use of the soft decision achieves better performance than the hard decision, the burden of reporting overhead hinders the use of soft decision [28]. One may want to directly consider the \(k\)-out-of-\(N\) rule, but finding an globally optimal solution for the \(k\)-out-of-\(N\) rule is not very tractable unless the same average SNR value is assumed for cooperative sensing scenarios [16]. Moreover, [21; 22; 23] pointed out the very high computational complexity and impracticality of finding suboptimal \(k\). Thus, we first focus on AND and OR rules in this paper. The fusion center using the AND rule decides that PU is present when all cooperative sensing nodes report PU presence to the fusion center [9]. Then, the final detection and the final false alarm probabilities at the fusion center are simply given by  
\[P^{d}=\prod_{i=1}^{N}P_{i}^{d},\]  
\[P^{f}=\prod_{i=1}^{N}P_{i}^{f}.\]  
Based on the Neyman-Pearson criterion, our aim is to maximize the final detection probability \(P^{d}\) subject to the target final false alarm probability \(\overline{P^{f}}\), which is formulated as follows:  
\[\max_{\epsilon_{i},i\in\{1,\cdots,N\}} P^{d}=\prod_{i=1}^{N}P_{i}^{d},\] \[s.t. P^{f}=\prod_{i=1}^{N}P_{i}^{f}\leq\overline{P^{f}}. \tag{6}\]  
The fusion center using the OR rule decides that the PU is present when any cooperative sensing nodes report PU presence to the fusion center [9]. Then, the final detection and the final false alarm probabilities at the fusion center are simply  
\[P^{d}=1-\prod_{i=1}^{N}\left(1-P_{i}^{d}\right),\]  
\[P^{f}=1-\prod_{i=1}^{N}\left(1-P_{i}^{f}\right).\]Then, we can also use the OR rule to maximize the final detection probability \(P^{d}\) subject to the target final false alarm probability \(\overline{P^{f}}\) as follows:  
\[\max_{\epsilon_{i},i\in\{1,\cdots,N\}} P^{d}=1-\prod_{i=1}^{N}\left(1-P_{i}^{d}\right),\] \[s.t. P^{f}=1-\prod_{i=1}^{N}\left(1-P_{i}^{f}\right)\leq\overline{P^{f}}.\]  
Considering the symmetric mathematical structure of \(P_{i}^{m}\) and \(P_{i}^{c}\), an alternative optimization problem is formulated as follows:  
\[\min_{\epsilon_{i},i\in\{1,\cdots,N\}} P^{m}=\prod_{i=1}^{N}P_{i}^{m},\] \[s.t. P^{c}=\prod_{i=1}^{N}P_{i}^{c}\geq\overline{P^{c}},\]  
where \(\overline{P^{c}}\) is according given by \(1-\overline{P^{f}}\).  
If the globally optimal solution can be derived from the AND rule of (6), our optimization approach is also applicable to the OR rule because of its symmetric mathematical structure. AND and OR rules are two opposite extremes. For comparison, the existing literature [7; 11; 13; 28] have shown that dominant performance regions of both rules highly depend on system parameters such as \(\tau,\bar{\gamma},B\), and \(\overline{P^{f}}\) for different purposes. Herein we mainly focus on the existence of globally optimized sensing thresholds. Therefore, in the rest of paper, our problem formulation of cooperative sensing mainly exploits the mathematical structure of the AND rule so that we can provide a globally optimal solution to maximize the detection performance of cooperative sensing under different average SNRs.