Optimisation of real functions of complex matrices for the adaptive estimation of complex sources

The second order Taylor series expansion (TSE) of scalar functions of complex matrices is explored in order to provide a new tool for gradient-based optimisation in the complex domain. The expansion is provided both in the augmented real and complex matrix spaces, as well as the multidimensional complex domain. The duality (isomorphism) between the augmented spaces is established and consequently the relation of the first- and second-order terms (gradient and Hessian) of the TSE in these spaces are introduced. Finally, a study of the trade-off between performance and computational complexity of algorithms for the estimation of complex sources in the two augmented spaces is performed.