Noise benefits parameter estimation in LMMSE sense

In this work, the effect of additive noise is studied in order to reduce the mean squared error (MSE) between the input parameter and its linear estimator constructed by the nonlinear system output. To improve the estimation performance, the optimal additive noise that minimizes the MSE of the noise enhanced linear minimum mean squared error (LMMSE) estimator is explored and determined. In addition, in the presence of prior information uncertainty, the estimation performances of the noise enhanced LMMSE are investigated under a constant constraint of the expected value of the output, and the corresponding algorithms are developed to find the optimal additive noise. Finally, two illustrative examples are provided to verify the theoretical results. The performance comparisons conducted between the LMMSE estimator without noise excitation and the optimal noise modified LMMSE estimator demonstrate that noise indeed improves the estimation accuracy under certain conditions.