On replica symmetry breaking in vector precoding for the Gaussian MIMO broadcast channel

The so-called ldquoreplica methodrdquo of statistical physics is employed for the large system analysis of vector precoding for the Gaussian multiple-input multiple-output (MIMO) broadcast channel. Focusing on discrete complex input alphabets, the transmitter is assumed to comprise a linear front-end combined with nonlinear precoding, that minimizes the front-end imposed transmit energy penalty. The energy penalty is minimized by relaxing the input alphabet to a larger alphabet set prior to precoding. The limiting empirical distribution of the precoder´s output, as well as the limiting energy penalty, are derived while harnessing what is referred to as the first order replica symmetry breaking (1RSB) ansatz. Particularizing to a ldquozero-forcingrdquo (ZF) linear front-end, and non-cooperative users, a decoupling result is derived according to which the channel observed by each of the individual receivers can be effectively characterized by the Markov chain u-x-y, where u is the channel input, x is the equivalent precoder output, and y is the channel output. An illustrative example is considered, based on discrete-lattice alphabet relaxation, for which the impact of replica symmetry breaking is demonstrated. A comparative spectral efficiency analysis reveals significant performance gains compared to linear ZF precoding in the medium to high E_b/N₀ region. The performance vs. complexity tradeoff of the nonlinear precoding scheme is also shortly discussed.