# On Optimal Cooperative Sensing with Energy Detection in Cognitive Radio
## 3 Cooperative Sensing with Optimal Thresholds
### Cooperative Sensing with Optimal Thresholds  
In solving (6), the individual sensing threshold for sensing node \(i\) is \(\epsilon_{i}=\sqrt{4\tau B}\mathcal{Q}^{-1}\left(p_{i}^{f}\right)+2\tau B\) by (4). Then, we can use \(P_{i}^{f}\) as optimization variables instead of \(\epsilon_{i}\), and rewrite the optimization problem (6) by using (2) for analytical tractability:  
\[\underset{p_{i}^{f},i\in\{1,\cdots,N\}}{\max} \prod_{i=1}^{N}\mathcal{Q}\bigg{(}\frac{1}{\sqrt{1+\gamma_{i}}} \bigg{(}\mathcal{Q}^{-1}\big{(}P_{i}^{f}\big{)}-\ \frac{\gamma_{i}\sqrt{\tau B}}{2}\bigg{)}\bigg{)},\] \[s.t. \prod_{i=1}^{N}p_{i}^{f}\leq\overline{p^{f}}. \tag{7}\]  
With one further step, we define the _normalized_ threshold \(x_{i}\triangleq\frac{\epsilon_{i}-2\tau B}{\sqrt{4\tau B}}\) from \(\mathcal{H}_{0}\) of (1). Then, it can be easily drawn from (4) such that \(x_{i}=\mathcal{Q}^{-1}\left(P_{i}^{f}\right)\). After taking the logarithm of (7), the problem is equivalent to the following optimization problem:  
\[\underset{x_{i},\ y_{i},\ z_{i}\in\{1,\cdots,N\}}{\max} \sum_{i=1}^{N}\log\mathcal{Q}\left(y_{i}\right),\] \[s.t. y_{i}=\frac{1}{\sqrt{1+\gamma_{i}}}\bigg{(}x_{i}-\ \frac{\gamma_{i}\sqrt{\tau B}}{2}\bigg{)},\] \[\sum_{i=1}^{N}\log\mathcal{Q}\left(x_{i}\right)\leq\log\overline{ p^{f}}, \tag{8}\]  
where \(y_{i}\) is introduced as an auxiliary variable. The convex problem has three requirements: the objective function must be concave for the maximization problem; the equality constraint functions must be affine; and the inequality constraint convex functions must be non-positive [29]. Note that although the objective function of (8) is concave, (8) still remains a non-convex optimization problem. This is because when we make the inequality constraint functions convex, it becomes non-negative which cannot satisfy a standard form of convex optimization. Thus, the problem cannot be posed as convex optimization [29]. Hence, we now try to find the condition that can convexify the problem as presented in the following proposition. **Proposition 1**. : _Suppose that observations of cooperative sensing nodes follow the normal distribution as in (1) with different SNR \(\gamma_{i}\). Then, based on the Neyman-Pearson criterion, (8) can be convexified if_  
\[R\left(x_{i}\right)+x_{i}\geq\frac{1}{\sqrt{1+\gamma_{i}}}\left(R\left(y_{i} \right)+y_{i}\right),\quad\forall\ i\in\left\{1,\cdots,N\right\}, \tag{9}\]  
_where a function \(R(t)\) is defined as \(R(t)\triangleq\frac{\mathcal{Q}^{\prime}(t)}{\mathcal{Q}(t)}\)._  
Proof. : See Appendix A. **Remark 1**. : _Note that Proposition 1 serves as a sufficient optimality condition. As long as the sufficient optimality condition is satisfied, the problem can be convexified so that a globally optimal solution can be obtained._  
We notice that the sufficient optimality condition may not always hold for \(\forall x_{i}\in\mathbf{R}\). However, by considering all the practically meaningful scenarios, we rigorously verify that as long as \(P_{i}^{f}\) is on \([\epsilon,1-\epsilon]\) where \(\epsilon\) is infinitesimally small and sensing observations follow the normal distribution, the sufficient optimality condition holds so that a globally optimal solution is guaranteed in cooperative sensing, as described in Examples 3 and 4. **Example 3**. : _To investigate the practicality of Remark 1, in Figure 4 we plot both the left-hand side (LHS) and the right-hand side (RHS) of (9) with \(\tau=1\) ms and \(B=6\) MHz for various PU SNR \(\gamma_{i}\). As can be seen, LHS is greater than RHS, and thus (9) holds for \(\forall x_{i}\in[-10,10]\). Recall that \(x_{i}\) is a normalized version of \(\epsilon_{i}\) that determines the false alarm probability, i.e., \(P_{i}^{f}=\mathcal{Q}\left(x_{i}\right)\) and \(x_{i}\) has a region of interest depending on \(P_{i}^{f}\). Ideally, \(P_{i}^{f}\) should be any value on \([0,1]\). We notice that when (9) holds, the corresponding \(P_{i}^{f}\) covers almost all values on \((0,1)\).