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Spectrum sensing is the key to coordinate the secondary users in a cognitive radio network by limiting the probability of interference with the primary users. Linear cooperative spectrum sensing consists of comparing the linear combination of the secondary users' recordings against a given threshold in order to assess the presence of the primary user signal. Simplicity is traded off for a slight suboptimality with respect to the likelihood-ratio test. Tuning the performance of linear cooperative radio sensing is complicated by the fact that optimization of the linear combining vector is required. This is accomplished by solving a nonconvex optimization problem, which is the main focus of this work. The global optimum is found by an explicit algorithm based on the solution of a polynomial equation in one scalar variable. Numerical results are reported for validation purposes and to analyze the effects of the system parameters on the complementary receiver operating characteristic. It is shown that the optimum probability of missed detection for a system with constant local signal-to-noise ratios (SNRs) and constant channel gain correlation coefficients can be expressed in closed form by a simple expression. Simulation results are also included to validate the accuracy of the Gaussian approximation. These results illustrate how large the number of sampling intervals must be in order that the Gaussian approximation holds.