On MMSE properties of “good” and “bad” codes for the Gaussian broadcast channel

This work examines the properties of code sequences for the scalar Gaussian broadcast channel (BC). Specifically, the behavior in terms of the mutual information and minimum mean-square error (MMSE) functions for all signal-to-noise ratios (SNRs) is explored. It is shown that “good”, capacity achieving, code sequences must follow the behavior of a capacity achieving superposition code sequence, even if they use a different encoding-decoding scheme (such as “Dirty Paper Coding”). Necessary and sufficient conditions for reliable decoding in general and specifically for “good” code sequences for the scalar Gaussian BC, in terms of the MMSE and conditional MMSE functions, are derived. Finally, “bad” code sequences, that do not obtain the capacity of the scalar Gaussian BC, are examined. These codes are defined by an additional MMSE constraint at some other SNR. This constraint limits the amount of disturbance these codes may have on some unintended receiver at that SNR. The capacity region, given this constraint, is fully depicted.