The approximate maximum likelihood (AML) estimator by Harroy and Lacoume is generalized to virtually any source probability distribution. The new extended expression - extended ML (EML) - is based on a scatter-plot complex centroid which depends on the fourth-order statistics of the data, and which allows a geometric interpretation inspired by Bogner and Clarke. The EML performance is studied and an analytic expression for the estimator's asymptotic probability density function (pdf) is determined. With the analysis tools we develop, another closed-form formula by Comon (CF) is understood and linked to the AML and the EML. The latter is the best option.
The EML estimator is the closed-form solution to certain optimization problems, which allows the extension of a contrast function derived by Moreau, and the simplification of the associated analytic solution derived by Comon-Moreau.
AML, EML and CF suffer a performance degradation when a source statistic approaches zero. To overcome this shortcoming, another fourth-order estimator - alternative EML (AEML) - is proposed, and a combined estimation strategy is adopted. A suboptimal rule to decide between the EML and the AEML is derived and validated.
Extensions to scenarios of more than two signals are implemented, based on a Jacobi-like iteration strategy originally proposed by Comon. The resulting methods are successfully applied to a real problem: the extraction of the fetal electrocardiogram from maternal cutaneous potential recordings.
The compact centroid-based expressions permit the formulation of straightforward adaptive versions, whose high convergence speed and global convergence under mild conditions are remarkable.
A closed-form estimation family based on the data nth-order statistics is unveiled, which yields the EML at n=4. At n=3, a novel third-order estimator - third-order blind signal estimator (TOBSE) - is obtained and analyzed.
Through the so-called bicomplex numbers, some of the previous results are extended to complex-valued mixtures, evidencing an elegant connection between the real and the complex case.
Numerous computer experiments verify the theoretical results, make comparisons among the techniques considered, and contrast them with other non-analytic procedures. The computational complexity of the methods is also discussed.
In conclusion, this investigation provides the first unified comprehensive vision of closed-form higher-order estimators for BSS.