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The paper addresses the problem of designing signature sequences for asynchronous CDMA channels with fixed signal delays. Assuming that users may be received at different power levels, the paper characterizes the so-called aperiodic WBE (A-WBE) sequences that are shown to maximize signal-to-interference ratios of the users. Using a compact matrix notation, we provide necessary and sufficient conditions for sequences to form a A-WBE sequence set and prove some important properties such sequence sets have. In some special cases, a method for constructing A-WBE sequences is presented.