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In this paper, we address the maximum-likelihood (ML) multiuser detection problem for asynchronous code-division multiple access (CDMA) systems with multiple receiver antennas in frequency-selective fading environments. Multiuser ML detection (MLD) in this case provides attractive symbol error performance, but it requires the solution of a large-scale combinatorial optimization problem. To deal with the computational complexity of this problem, we propose an efficient approximation method based on a block alternating likelihood maximization (BALM) principle. The idea behind BALM is to decompose the large-scale MLD problem into smaller subproblems. Assuming binary or quaternary phase shift keying (BPSK or QPSK) (which are often employed in CDMA), the combinatorial subproblems are then accurately and efficiently approximated by the semidefinite relaxation (SDR) algorithm-an algorithm that has been recently shown to lead to quasi-ML performance in synchronous CDMA scenarios. Simulation results indicate that this BALM detector provides close-to-optimal bit error rate (BER) performance. The BALM principle is quite flexible, and we demonstrate this flexibility by extending BALM to multicarrier (MC) multiuser systems. By exploiting the special signal correlation structure of MC systems, we develop a variation of BALM in which dynamic programming (DP) is used to solve the subproblems. It is shown using simulations that the BER performance of this DP-based BALM detector is as promising as that of the SDR-based BALM detector.