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For a symbol-asynchronous (but chip-synchronous) single-cell code-division multiple-access (CDMA) system, we define a system-wide quantity called the total squared asynchronous correlation (TSAC) which, for arbitrary signature sets, depends on the users' delay profile. We develop a lower bound for TSAC that is independent of the users' delays. We show that if the signature set achieves this TSAC lower bound, then the user capacity of the asynchronous CDMA system using matched filters becomes the same as that of a single-cell synchronous CDMA system; in this case, there is no loss in user capacity due to asynchronism. We present iterative signature adaptation algorithms, which, when executed sequentially by the users, appear to converge to these optimum signature sequences; however, the existence, for all user delay profiles, of signature sequences achieving this lower bound remains a significant open problem.