Multiuser Detection and Statistical Physics

We present a framework for analyzing multiuser detectors in the context of statistical physics. A multiuser detector is shown to be equivalent to a conditional mean estimator which finds the mean value of the stochastic output of a so-called Bayes retrochannel. The Bayes retrochannel is equivalent to a spin glass in the sense that the distribution of its stochastic output conditioned on the received signal is exactly the distribution of the spin glass at thermal equilibrium. In the large-system limit, the bit-error-rate of the multiuser detector is simply determined by the magnetization of the spin glass, which can be obtained using powerful tools developed in statistical mechanics. In particular, we derive the large-system uncoded bit-error-rate of the matched filter, the MMSE detector, the decorrelator and the optimal detectors, as well as the spectral efficiency of the Gaussian CDMA channel. It is found that all users with different received energies share the same multiuser efficiency, which uniquely determines the performance of a multiuser detector. A universal interpretation of multiuser detection relates the multiuser efficiency to the mean-square error of the conditional mean estimator output in the large-system limit.