Performance analysis for sparse based biased estimator: Application to line spectra analysis

Dictionary based sparse estimators are based on the matching of continuous parameters of interest to a discretized sampling grid. Generally, the parameters of interest do not lie on this grid and there exists an estimator bias even at high Signal to Noise Ratio (SNR). This is the off-grid problem. In this work, we propose and study analytical expressions of the Bayesian Mean Square Error (BMSE) of dictionary based biased estimators at high SNR. We also show that this class of estimators is efficient and thus reaches the Bayesian Cramér-Rao Bound (BCRB) at high SNR. The proposed results are illustrated in the context of line spectra analysis and several popular sparse estimators are compared to our closed-form expressions of the BMSE.