# On Optimal Cooperative Sensing with Energy Detection in Cognitive Radio
###### Abstract
The authors of [18] proposed an on/off reporting mechanism to achieve robust cooperative sensing over an imperfect reporting channel as well as graceful degradation against sensor failures. However, the existing work [11; 12; 13; 14; 15; 16; 17; 18] assumed that the average SNR is _identical_ for all cooperative sensing nodes, and so is the sensing threshold. This assumption may not be realistic in practical systems. In addition to multipath fading and shadowing effects, there are several practical factors for heterogeneous cooperative sensing, such as geographically different positions (indoor or outdoor), inherent RF front-end sensitivity, obstacles, temperature variations, and so on. Thus, cooperative sensing performance highly depends on the heterogeneity of the received PU SNRs at individual SUs. Although previous work, such as [19], considered heterogeneous average SNRs for cooperative sensing nodes, all sensing nodes still used the same sensing threshold. In addition, the authors in [20; 21; 22; 23; 24] considered optimizing thresholds of cooperative sensing nodes based on different average SNRs. In [20], an iterative threshold selection algorithm was proposed to determine whether a specific sensing node can participate in cooperation or not, based on its contribution to detection performance. The work of [21] proposed a weighted decision rule by jointly optimizing the test threshold of likelihood-ratio and the local threshold of SUs. Although heuristic methods of optimizing two thresholds at a time did not guarantee the achievement of global optimal solutions, the proposed algorithm provided the highest throughput results over the non-jointly optimized schemes [21]. In [22; 23], a heuristic algorithm was proposed to achieve suboptimal thresholds at cooperative sensing nodes by minimizing false alarm probability subject to target missed detection probability. Different fusion rules with iterative algorithms were proposed to decrease the total error detection probability for both single channel and multichannel cooperative sensing [24]. However, even though the existing literature [20; 21; 22; 23; 24] has investigated distinct thresholds for cooperative sensing nodes according to their own average SNRs, they are suboptimal and _theoretically_ achieving the optimal detection performance still remains unsolved. Moreover, previous heuristic approaches would suffer from high computational complexity as the number of cooperative sensing nodes increases [20; 21; 22; 23; 24]. In this regard, we are motivated to theoretically develop a globally optimal solution for energy detection under a heterogeneous PU SNR environment. We summarize our key contributions as follows. Based on the Neyman-Pearson criterion we formulate an optimization problem to maximize the sensing performance when a hard decision is used under different average SNRs. It is found that this is a non-convex optimization problem, and thus we provide a condition that convexifies the original problem. We then verify that all _practically meaningful_ parameters satisfy the convexifying condition. Thus, our solution guarantees the maximized detection performance of cooperative sensing. Considering that using a single common threshold severely degrades the sensing performance when cooperative sensing nodes experience different average SNRs, our solution is promising. Our extensive simulations show that a globally optimized solution achieves an improvement of several orders of magnitude improvement under practical operating parameters such as the target probability of false alarm, different average SNRs, and the number of cooperative sensing nodes. The rest of this paper is organized as follows. In Section 2, the CR system model and problem formulation are introduced. Cooperative sensing with optimal thresholds is presented with a sufficient optimality condition in Section 3. We present numerical results under practical operating parameters in Section 4. Finally, we draw our conclusions in Section 5.