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In this paper, we present a unified mathematical framework to analyze the bit-error rate (BER) performance of general linear coherent multiuser receivers with diversity reception and imperfect channel estimation for doubly selective Rician-fading asynchronous code-division multiple-access channels. BERs of linear receivers with channel state information and data-aided channel estimation are analyzed, and both exact and low-complexity approximate BER evaluation formulas are presented. Furthermore, by using a Markov chain steady-state analysis, a tight BER approximation for receivers with decision-directed channel estimation is proposed. Numerical and simulation results verify the accuracy of the proposed BER evaluation methods.