The “replica method” of statistical physics is employed for the large-system analysis of vector precoding for the Gaussian multiple-input multiple-output broadcast channel. The transmitter comprises a linear front-end combined with nonlinear precoding, minimizing transmit energy by means of input alphabet relaxation. For the common discrete lattice-based relaxation, the problem violates replica symmetry and a replica symmetry breaking (RSB) ansatz is taken. The limiting empirical distribution of the precoder's output and the limiting transmit energy are derived for one-step RSB. Particularizing to a “zero-forcing” (ZF) linear front-end, a decoupling result is derived. For discrete lattice-based relaxations, the impact of RSB is demonstrated for the transmit energy. The spectral efficiencies of the aforementioned precoding methods are compared to linear ZF and Tomlinson-Harashima precoding (THP). Focusing on quaternary phase shift-keying (QPSK), significant performance gains of both lattice and convex relaxations are revealed for medium to high signal-to-noise ratios (SNRs) when compared to linear ZF precoding. THP is shown to be outperformed as well. Comparing certain lattice-based relaxations for QPSK against a convex counterpart, the latter is found to be superior for low and high SNRs but slightly inferior for medium SNRs in terms of spectral efficiency.

There are a number of corrections for the above titled paper (ibid., vol. 58, no. 3, pp. 1413-1440, Mar. 2012). They are presented here.

We address the possibility of overloaded vector precoding in single user MIMO channels, i.e. the number of data streams is larger than the minimum of the number of antennas at transmitter and receiver side.We find that the convex vector precoding introduced in [1] allows for overloading while, with a certain probability, keeping the received signal free of interference. We find that the probability that overloading is not possible decays exponentially with the size of the system as long as the number of data streams is less than twice the minimum number of antennas. We give an explicit formula to calculate this probability for any antenna configuration in presence of correlated Rayleigh fading. Although overloading comes with the need for increased transmitted power, we show by means of the replica method that overloading up to 22% yields better spectral and power efficiency than without overload and spatial matched filter processing at the receiver.

Recently Muller et al. (IEEE J. Select. Areas Commun. 2008) used asymptotic methods from statistical physics to analyze non-linear vector precoding for MIMO broadcast channels. They proposed to extend BPSK input alphabets onto both real and complex supersets of the original constellation points. They showed that, as the optimization space is greater, lower energies are achieved when the extended alphabets are complex. In this work we use similar asymptotic methods and propose an alternative channel inversion technique which makes purely real alphabets perform as well as their complex extensions, which results in reduced complexity in the optimization process.