Theses: Sample Covariance Based Parameter Estimation for Digital Communications
J. Villares Piera


This thesis deals with the problem of blind second-order estimation in digital communications. In this field, the transmitted symbols appear as non-Gaussian nuisance parameters degrading the estimator performance. In this context, the Maximum Likelihood (ML) estimator is generally unknown unless the signal-to-noise (SNR) is very low. In this particular case, if the SNR is asymptotically low, the ML solution is quadratic in the received data or, equivalently, linear in the sample covariance matrix. This significant feature is shared by other important ML-based estimators such as, for example, the Gaussian and Conditional ML estimators. Likewise, MUSIC and other related subspace methods are based on the eigendecomposition of the sample covariance matrix. From this background, the main contribution of this thesis is the deduction and evaluation of the optimal second-order parameter estimator for any SNR and any distribution of the nuisance parameters.

A unified framework is provided for the design of open- and closed-loop second-order estimators. In the first case, the minimum mean square error and minimum variance second-order estimators are deduced considering that the wanted parameters are random variables of known but arbitrary prior distribution. From this Bayesian approach, closed-loop estimators are derived by imposing an asymptotically informative prior. In this small-error scenario, the best quadratic unbiased estimator (BQUE) is obtained without adopting any assumption about the statistics of the nuisance parameters. In addition, the BQUE analysis yields the lower bound on the performance of any blind estimator based on the sample covariance matrix.

Probably, the main result in this thesis is the proof that quadratic estimators are able to exploit the fourth-order statistical information about the nuisance parameters. Specifically, the nuisance parameters fourth-order cumulants are shown to provide all the non-Gaussian information that is utilizable for second-order estimation. This fourth-order information becomes relevant in case of constant modulus nuisance parameters and medium-to-high SNRs. In this situation, the Gaussian assumption is proved to yield inefficient second-order estimates.

Another original result in this thesis is the deduction of the quadratic extended Kalman filter (QEKF). The QEKF study concludes that second-order trackers can improve simultaneously the acquisition and steady-state performance if the fourth-order statistical information about the nuisance parameters is taken into account. Once again, this improvement is significant in case of constant modulus nuisance parameters and medium-to-high SNRs.

Finally, the proposed second-order estimation theory is applied to some classical estimation problems in the field of digital communications such as non-data-aided digital synchronization, the related problem of time-of-arrival estimation in multipath channels, blind channel impulse response identification, and direction-of-arrival estimation in mobile multi-antenna communication systems. In these applications, an intensive asymptotic and numerical analysis is carried out in order to evaluate the ultimate limits of second-order estimation.

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