Cramér-Rao Lower Bound for Bayesian Estimation of Quantized MMV Sparse Signals

Abstract

We derive the Cramér-Rao lower bound (CRLB) on the mean squared error for Bayesian estimation of a row-sparse matrix based on quantized low-dimensional multiple measurement vectors (MMV). We impose a two-stage hierarchical circularly symmetric complex Gaussian prior parameterized by a diagonal precision hyperparameter matrix on the estimand. The precision hyperparameters themselves assume a non-informative conjugate hyperprior that induces a heavy-tailed Student’s t marginalized prior, which in turn promotes a sparse solution. We derive the joint probability distribution of the observables, estimand and the hyperparameters, and the associated Fisher information matrix (FIM). Due to the analytical intractability in computing the FIM, we resort to Monte Carlo numerical methods that closely approximate the FIM. We demonstrate that the CRLB is tight by considering a variational Bayes orthogonal frequency division multiplexing (OFDM) channel estimator in massive multiple-input multiple-output (MIMO) wireless communication systems with low-resolution analog-to-digital converters as an application and provide further insights on the numerical results.