We analyze a wideband spectrum in a cognitive radio (CR) network by employing the optimal adaptive multiband sensing-time joint detection framework. This framework detects a wideband M-ary quadrature amplitude modulation (M-QAM) primary signal over multiple nonoverlapping narrowband Gaussian channels, using the energy detection technique so as to maximize the throughput in CR networks while limiting interference with the primary network. The signal detection problem is formulated as an optimization problem to maximize the aggregate achievable secondary throughput capacity by jointly optimizing the sensing duration and individual detection thresholds under the overall interference imposed on the primary network. It is shown that the detection problems can be solved as convex optimization problems if certain practical constraints are applied. Simulation results show that the framework under consideration achieves much better performance for M-QAM than for binary phase-shift keying or any real modulation scheme.