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In this paper, we consider heterogeneous cognitive radio networks (CRNs) comprising primary channels (PCs) with heterogeneous characteristics and secondary users (SUs) with various sensing and reporting qualities for different PCs. We first define the opportunity as the achievable total data rate and its cost as the energy consumption caused from sensing, reporting, and channel switching operations and formulate a joint spectrum discovery and energy efficiency objective to minimize the energy spent per unit of data rate. Then, a mixed integer nonlinear programming problem is formulated to determine: 1) the optimal subset of PCs to be scheduled for sensing; 2) the SU assignment set for each scheduled PC; and 3) sensing durations and detection thresholds of each SU on PCs it is assigned to sense. Thereafter, an equivalent convex framework is developed for specific instances of the above combinatorial …