Equivalence of CDMA, FDMA, and TDMA over Gaussian overloaded channels

It is often claimed that frequency-division multiple-access (FDMA) und time-division multiple-access (TDMA) are inferior to code-division multiple-access (CDMA) in the sense that they have hard limits on the user capacity while the performance of CDMA systems degrade gradually as the number of users increases. In this paper, it is shown that FDMA and TDMA systems with flat spectrum pulses are equivalent to optimum CDMA systems for multiple-access communications over Gaussian overloaded channels. Optimal transmit and receive waveforms are employed to minimize the output mean-squared errors of linear receivers, and equal-power users are assumed with strictly hand-limited continuous-time channels. Using a frequency domain approach, a necessary and sufficient condition is derived for the optimally of a multiple-access scheme. The continuous-time equivalents of Welch bound equality (CTE- WBE) sequences are defined and, as transmit waveforms, shown to comprise the optimal systems together with the matched filter receivers. The equivalence is verified by showing that FDMA and TDMA signals with flat spectrum pulses are CTE-WBE sequences.