Continuous-Time Equivalents of Welch Bound Equality Sequences

The theory developed on the sequence optimization for equal-power code-division multiple-access (CDMA) systems is generalized in this paper. Unlike previous works, where discrete-time or vector channel models are predominantly employed, the overloaded channels are modeled as strictly band-limited, continuous-time multiple-access channels corrupted by additive white Gaussian noise (AWGN). After being posed as a multiple-input/single-output (MISO) joint transmitter and receiver optimization problem with the total mean-squared error (MSE) as the objective functional to be minimized, the variational problem is solved by using a frequency-domain approach. It is shown that there exist continuous-time equivalents of Welch bound equality (CTE-WBE) sequences as the jointly optimum transmit waveforms and that the matched filters are the jointly optimum receivers. The user capacity of the band-limited channels is characterized by a necessary and sufficient condition for the admissibility of users into the system in terms of the channel load, the received signal-to-noise ratio, and the signal-to-interference-plus-noise ratio (SINR) requirement. It is also shown that CTE-WBE sequences achieve the lower bound on the continuous-time equivalent of total squared correlation (CTE-TSC).