Soft-Input Soft-Output Equalizers for Turbo Receivers: A Statistical Physics Perspective

Many algorithms in signal processing and digital communications must deal with the problem of computing the probabilities of the hidden state variables given the observations, i.e., the inference problem, as well as with the problem of estimating the model parameters. Such an inference and estimation problem is encountered, for example, in adaptive turbo equalization/demodulation where soft information about the transmitted data symbols has to be inferred in the presence of the channel uncertainty, given the received signal samples and a priori information provided by the decoder. An exact inference algorithm computes the a posteriori probability (APP) values for all transmitted symbols, but the computation of App-s is known to be an NP-hard problem, thus rendering this approach computationally prohibitive in most cases. We show in this paper how may of the well-known low complexity soft-input soft-output (SISO) equalizers, including the channel matched filter based linear SISO equalizers and minimum mean square error (MMSE) SISO equalizers as well as the expectation-maximization (EM) algorithm based SISO demodulators in the presence of the Rayleigh fading channel, can be formulated as solutons to a variational optimization problem. The variational optimization is a well-established methodology for low-complexity inference and estimation, originating from statistical physics. Importantly, the imposed variational optimization framework provides an interesting link between the APP demodulators and the linear SISO equalizers. Moreover, it provides a new set of insights into the structure and performance of these widely celebrated linear SISO equalizers while suggesting some fine tuning of them as well.