On the Equivalence of Maximum SNR and MMSE Estimation: Applications to Additive Non-Gaussian Channels and Quantized Observations

Abstract

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.

Robust adaptive algorithms for underwater acoustic channel estimation and their performance analysis

“…Assuming that q m is independent from the received and noise signals, at the steady state, we have the following general variance relation for an adaptive filter with the error nonlinearity g(e m ) [25,39], 2Re E e *…”

“…In our analysis, we use the following assumptions: whenever L c is sufficiently large and the learning rate μ is sufficiently small [44]. Based on (20), Assumptions 1-3, and the Price's Theorem [25] for E[e * a,m csgn(e m )], it can be straightforwardly shown that [25,39] E |e a,m | 2 | g(e m ) = csgn(e m ) = ζ SA E |e a,m | 2 | g(e m ) = e m = ζ LMS E |e a,m | 2 | g(e m ) = e 3 m = ζ LMF .…”

“…Since in our final analysis, this case appears when the noise is Gaussian (non-impulsive), we assume that the noise power is σ 2 g . Furthermore, we use the assumption |e a,m | 2 σ 2 g [39], hence by substituting g(e m ) in (36) and simplifying the result, we obtain the following expression for EMSE:…”

“…In these analyses, in order to be consistent with the real world UWA channels, we assume that there are frequency and phase offsets as well as impulsive noise in the channel. For the first order algorithms, we avoid mentioning all of the intermediate steps of the analyses, instead we discuss the important steps, use the results of [39], and establish the final results based on them. However, for the second order analyses, we completely explain the steps since it is a nontrivial extension of the existing analysis methods.…”

Real-Time Generation of Standard-Compliant DVB-T Signals

“…In addition, power allocation algorithms can be exploited without changing the DVB-T receiver. Another possibility is the investigation of the effect of impulsive noise [34], non-Gaussian channels, and quantization errors [35], on the DVB-T received signal. Furthermore, by means of our SDR-based design, the DVB-T signal can first be received by a custom DVB-T receiver and then retransmitted to a single destination by a regenerative relay, nearby to the receiver.…”