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Optimum centralized demodulation of the independent data streams transmitted simultaneously by several users through a Code Division Multiple-Access channel is considered. Each user sends an arbitrary assigned signal waveform, which is linearly modulated by symbols drawn from a finite alphabet. If the users are asynchronous, the optimum multiuser detector can be implemented by a Viterbi algorithm whose time-complexity is linear in the number of symbols transmitted by each user and exponential in the number of users. It is shown that the combinatorial problem of selecting the most likely transmitted data stream given the sufficient statistics (sequence of matched filter outputs), and the signal energies and cross-correlations is nondeterministic polynomial-time hard (NP-hard) in the number of users. And it remains so even if the users are restricted to be symbol-synchronous.

The performance …