  The minimum mean-squared error (MMSE) is one of the most popular criteria for
Bayesian estimation. Conversely, the signal-to-noise ratio (SNR) is a typical
performance criterion in communications, radar, and generally detection theory.
In this paper we first formalize an SNR criterion to design an estimator, and
then we prove that there exists an equivalence between MMSE and maximum-SNR
estimators, for any statistics. We also extend this equivalence to specific
classes of suboptimal estimators, which are expressed by a basis expansion
model (BEM). Then, by exploiting an orthogonal BEM for the estimator, we derive
the MMSE estimator constrained to a given quantization resolution of the noisy
observations, and we prove that this suboptimal MMSE estimator tends to the
optimal MMSE estimator that uses an infinite resolution of the observation.
Besides, we derive closed-form expressions for the mean-squared error (MSE) and
for the SNR of the proposed suboptimal estimators, and we show that these
expressions constitute tight, asymptotically exact, bounds for the optimal MMSE
and maximum SNR.
