The capacity of the multi-MMSE constrained Gaussian channel

We examine codes, over the additive Gaussian noise channel, designed for reliable communication at some specific signal-to-noise ratio (snr) and constrained by the permitted MMSE at K lower snrs. Extending the result of the single MMSE constrained code, we show that K-layers superposition codes attain the constrained capacity. Moreover, we prove that given a reliable code attaining the multi-MMSE constrained capacity, its MMSE and mutual information, as functions of snr, are completely defined. Thus, no other multi-MMSE constrained capacity achieving code attains better MMSE performance at unconstrained snrs.