Uniformly Improving Maximum-Likelihood SNR Estimation of Known Signals in Gaussian Channels

The signal-to-noise ratio (SNR) estimation problem is considered for an amplitude modulated known signal in Gaussian noise. The benchmark method is the maximum-likelihood estimator (MLE), whose merits are well-documented in the literature. In this work, an affinely modified version of the MLE (AMMLE) that uniformly outperforms, over all SNR values, the traditional MLE in terms of the mean-square error (MSE) is obtained in closed-form. However, construction of an AMMLE whose MSE is lower, at every SNR, than the unbiased Cramér-Rao bound (UCRB), is shown to be infeasible. In light of this result, the AMMLE construction rule is modified to provision for an a priori known set S, where the SNR lies, and the MSE enhancement target is pursued within S. The latter is realized through proper extension of an existing framework, due to Eldar, which settles the design problem by solving a semidefinite program. The analysis is further extended to the general case of vector signal models. Numerical results show that the proposed design demonstrates enhancement of the MSE for all the considered cases.