Lower and upper bounds on the minimum mean-square error in composite source signal estimation

The performance of a minimum mean-square error (MMSE) estimator for the output signal from a composite source model (CSM), which has been degraded by statistically independent additive noise, is analyzed for a wide class of discrete-time and continuous-time models. In both cases, the MMSE is decomposed into the MMSE of the estimator, which is informed of the exact states of the signal and noise, and an additional error term. This term is tightly upper and lower bounded. The bounds for the discrete-time signals are developed using distribution tilting and Shannon´s lower bound on the probability of a random variable exceeding a given threshold. The analysis for the continuous-time signal is performed using Duncan´s theorem. The bounds in this case are developed by applying the data processing theorem to sampled versions of the state process and its estimate, and by using Fano´s inequality. The bounds in both cases are explicitly calculated for CSMs with Gaussian subsources. For causal estimation, these bounds approach zero harmonically as the duration of the observed signals approaches infinity