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Continuous-time equivalents of welch bound equality sequences

In this paper, the optimal signaling problem for multiple-access communications over Gaussian overloaded channels is considered with equal-power users having linear minimum mean-squared error (LMMSE) receivers. Unlike previous similar work on the sequence design for code-division multiple-access (CDMA) communications where discrete-time or vector channel models are predominantly employed, the overloaded channel is modeled as a strictly band-limited, continuous-time multiple-access channel corrupted by an additive white Gaussian noise. After being posed as a multi-input/single-output (MISO) joint transmitter and receiver optimization problem with the total mean-squared error as the objective functional, the variational problem is solved by using a frequency-domain approach. It is shown that the matched filters are the optimal receivers and that there exist continuous-time equivalents of Welch bound equality (CTE-WBE) sequences as the optimal transmit waveforms. The user capacity of the band-limited channel in terms of the admissibility of users into the system is also characterized by a necessary and sufficient condition on the channel load, the received signal-to-noise ratio, and the maximum signal-to-interference ratio.