Gaussian Channels: Information, Estimation and Multiuser Detection

This thesis represents an addition to the theory of information transmission, signal estimation, nonlinear filtering, and multiuser detection over channels with Gaussian noise. The work consists of two parts based on two problem settings—single-user and multiuser—which draw different techniques in their development. The first part considers canonical Gaussian channels with an input of arbitrary but fixed distribution. An “incremental channel” is devised to study the mutual information increase due to an infinitesimal increase in the signal-to-noise ratio (SNR) or observation time. It is shown that the derivative of the input-output mutual information (nats) with respect to the SNR is equal to half the minimum mean-square error (MMSE) achieved by optimal estimation of the input given the output. This relationship holds for both scalar and vector signals, as well as for discreteand continuous-time models. This information-theoretic result has an unexpected consequence in continuous-time estimation: The causal filtering MMSE achieved at SNR is equal to the average value of the noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is chosen uniformly distributed between 0 and SNR. The second part considers Gaussian multiple-access channels, in particular code-division multiple access (CDMA), where the input is the superposition of signals from many users, each modulating independent symbols of an arbitrary distribution onto a random signature waveform. The receiver conducts optimal joint decoding or suboptimal separate decoding that follows a posterior mean estimator front end, which can be particularized to the matched filter, decorrelator, linear MMSE detector, and the optimal detectors. Largesystem performance of multiuser detection is analyzed in a unified framework using the replica method developed in statistical physics. It is shown under replica symmetry assumption that the posterior mean estimate, which is generally non-Gaussian in distribution, converges to a deterministic function of a hidden Gaussian statistic. Consequently, the multiuser channel can be decoupled into equivalent single-user Gaussian channels, where the degradation in SNR due to multiple-access interference, called multiuser efficiency, is determined by a fixed-point equation. The multiuser efficiency uniquely characterizes the error performance and input-output mutual information of each user, as well as the overall system spectral efficiency.