On the Limitations of the Bayesian Cramér-Rao Bound for Mixed-Resolution Data

In this paper, we consider Bayesian parameter estimation in systems incorporating both analog and 1-bit quantized measurements. We develop a tractable form of the Bayesian Cram$\acute{\text{e}}$r-Rao Bound (BCRB) tailored for the linear-Gaussian mixed-resolution scheme. We discuss the properties of the BCRB and examine its limitations as a system design tool. In addition, we present the partially-numeric minimum-mean-squared-error (MMSE) and linear MMSE (LMMSE) estimators with a general quantization threshold. In our simulations, the BCRB is compared with the mean-squared-errors (MSEs) of the estimators for channel estimation with mixed analog-to-digital converters. The results demonstrate that the BCRB is not a tight lower bound, and it fails to accurately capture the non-monotonic behavior of the estimators' MSEs versus signal-to-noise-ratio (SNR) and their behavior regarding different resource allocations. Consequently, while the BCRB provides some valuable insights on the quantization threshold, our results demonstrate that it is not suitable as a practical tool for system design in mixed-resolution settings.