If a specific PUs' spectrum utilization ratio is given, the error decision probability $P_e$ becomes a convex function changing with varying threshold. The proof of $P_e$ being convex is given as follow.follows:
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When the PUs are present in the channel, the CR system's corresponding error decision probability is the missed-detection probability $(1-P_d)$ that represents the PUs being detected as absent while they are actually present.
Hence, $\alpha(1-P_d)$ represents the error probability for PUs being present with spectrum utilization of $\alpha$. Similarly, $(1-\alpha)P_f$ represents the error decision probability for PUs being absent with probability $(1-\alpha)$.
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In this network, each CR user independently performs spectrum sensing via the common control channels; while the FC combines the sensing from different CR users to makesmake a decision on the presence or absence of the PU, via combining each CR user weights and the local statistics. Then, the global decision can be then transmitted.
The principle inspiration of the linear combining rule is that the probabilities of detection and false alarm rates are obtained in a closed-formclosed form. This gives insight into the system's parameters design,; whereas analyzing the detection performance is analytically not tractable for the likelihood ratio test basedtest-based detector.