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Because of the powerful symbol error performance of multiuser maximum-likelihood (ML) detection, recently, there has been much interest in seeking effective ways of approximating multiuser ML detection (MLD) with affordable computational costs. It has been illustrated that for the synchronous code division multiple access (CDMA) scenario, the so-called semidefinite relaxation (SDR) algorithm can accurately and efficiently approximate multiuser MLD. This SDR-MLD algorithm, however, can only handle binary and quadratic phase shift keying (PSK) symbol constellations. In this sequel, we propose an extended SDR algorithm for MLD with -ary PSK (MPSK) constellations. For the synchronous CDMA scenario, the proposed SDR algorithm provides an attractive polynomial-time complexity order of 3 5 , where is the number of users. Simulation results indicate that the proposed detector provides improved symbol error performance compared with several commonly used multiuser detectors.Index Terms-M-ary phase shift keying, maximum likelihood detection, multiuser detection, relaxation methods, semidefinite programming.

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