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Given the ever-growing demand for radio spectrum, cognitive radio has recently emerged as an attractive wireless technology. Since orthogonal frequency-division multiplexing (OFDM) is one of the major wideband transmission techniques, detection of OFDM signals in low-signal-to-noise-ratio scenario is an important research problem. In this paper, it is shown that cyclic prefix (CP) correlation coefficient (CPCC)-based spectrum sensing, which was previously introduced as a simple and computationally efficient spectrum-sensing method for OFDM signals, is a special case of the constrained generalized likelihood ratio test (GLRT) in the absence of multipath. As such, the performance of this algorithm degrades in a multipath scenario, where OFDM is usually implemented. Furthermore, by considering multipath correlation in the GLRT algorithm and employing the inherent structure of OFDM signals, a simple and low-complexity algorithm called the multipath-based constrained-GLRT (MP-based C-GLRT) algorithm is obtained. The MP-based C-GLRT algorithm is shown to outperform the CPCC-based algorithm in a rich multipath environment. Further performance improvement can be achieved by simply combining both the CPCC- and MP-based C-GLRT algorithms. A simple GLRT-based detection algorithm is also developed for unsynchronized OFDM signals, whose performance is only slightly degraded when compared with the synchronized detection in a rich multipath environment.