The Gaussian assumption in second-order estimation problems in digital communications

This paper deals with the goodness of the Gaussian assumption when designing second-order blind estimationmethods in the context of digital communications. The low- andhigh-signal-to-noise ratio (SNR) asymptotic performance of the maximum likelihood estimator—derived assuming Gaussiantransmitted symbols—is compared with the performance of the optimal second-order estimator, which exploits the actualdistribution of the discrete constellation. The asymptotic study concludes that the Gaussian assumption leads to the optimalsecond-order solution if the SNR is very low or if the symbols belong to a multilevel constellation such as quadrature-amplitudemodulation (QAM) or amplitude-phase-shift keying (APSK). On the other hand, the Gaussian assumption can yield importantlosses at high SNR if the transmitted symbols are drawn from a constant modulus constellation such as phase-shift keying (PSK)or continuous-phase modulations (CPM). These conclusions are illustrated for the problem of direction-of-arrival (DOA) estimation of multiple digitally-modulated signals.

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting anyapproximation about the observation statistics or the a prioridistribution of the parameters. The proposed solution is obtainedminimizing the estimator variance subject to some constraints onthe estimator bias. The resulting optimal estimator is found todepend on the observation fourth-order moments that can be calculatedanalytically from the known signal model. Unfortunately,in most cases, the performance of this estimator is severely limitedby the residual bias inherent to nonlinear estimation problems.To overcome this limitation, the second-order minimum varianceunbiased estimator is deduced from the general solution by assumingaccurate prior information on the vector of parameters.This small-error approximation is adopted to design iterativeestimators or trackers. It is shown that the associated varianceconstitutes the lower bound for the variance of any unbiasedestimator based on the sample covariance matrix.The paper formulation is then applied to track the angle-of-arrival(AoA) of multiple digitally-modulated sources by means ofa uniform linear array. The optimal second-order tracker is comparedwith the classical maximum likelihood (ML) blind methodsthat are shown to be quadratic in the observed data as well. Simulationshave confirmed that the discrete nature of the transmittedsymbols can be exploited to improve considerably the discriminationof near sources in medium-to-high SNR scenarios.

Delayed delivery of second twin: a multicentre study of 35 cases.

OBJECTIVE: The aim of this study was to conduct a statistical analysis to determine the outcome of conservative treatment after delivery of a first fetus in multiple pregnancy and thus define new prognostic factors. STUDY DESIGN: Multicentre retrospective study involving 12 centers over a 10-year period. RESULTS: Twenty-eight twin pregnancies and seven triplet pregnancies which were managed conservatively. In twin pregnancies, 79% of the delayed-delivery fetuses survived; only 7% of the first delivered fetuses survived. The mean interval between deliveries was 47 days. No statistical difference was found concerning cerclage, antibiotic therapy, tocolysis and hospitalization. Earlier delivery of the first twin and premature rupture of membranes for the second twin were significantly related to a longer interval between deliveries. CONCLUSION: Delayed delivery in multifetal pregnancies can be successful if there are no contraindications and these pregnancies are managed in a tertiary perinatal center. Publications limited to successful cases have undoubtedly introduced some bias in assessment.

Nouveau dictionnaire universel. Tome second / par Maurice La Châtre

Collection : Archives de la linguistique française ; 205

Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers. Tome second, Azyme-Cezimbra / par une société de gens de lettres ; mis en ordre et publié par M. [Denis] Diderot,... et quant à la partie mathématique, par M. [Jean Le Rond] d'Alembert,..

Collection : Archives de la linguistique française ; 118