Multiuser water-filling

We find the capacity region of a two-user Gaussian multiaccess channel with intersymbol interference (ISI) where the inputs pass through respective linear systems and are then superimposed before being corrupted by an additive Gaussian noise process. We give a novel geometrical method to obtain the optimal input power spectral densities and the capacity region. This method can be viewed as a nontrivial generalization of the single-user water-filling argument. We show that as in the traditional memoryless multiaccess channel, FDMA, with optimally selected frequency bands for each user, achieves the total capacity of the K-user Gaussian multiaccess channel with ISI. However, the capacity region of the two-user channel with memory is, in general, not a pentagon unless the channel transfer functions for both users are identical.