Sinais complexos impróprios: filtragem adaptativa e gradiente natural

Abstract

This dissertation studies the behavior of complex signals and its symmetry (or the lack of it) with performance in adaptive filters. The study uses the following linear filtering techniques: the WLLMS algorithm (widley linear least mean square or augmented LMS), the augmented Newton’s method and the augmented natural gradient method.

WLLMS adaptive method is based on its usual LMS complex version. While the this last processes the signal through a linear structure, the augmented version uses the signal and its conjugated copy in a so-called widely linear structure. The optimization criterion is the mean squared error, whose gradient is used to adjust the filters coefficients.

The WLLMS method proved, in many cases, adequate in providing smaller errors with convergence speeds similar to its linear version, LMS.

Another technique, based on the usual Newton method, is the augmented Newton method. Developed in order to use the inverse of the augmented hessian matrix as a step factor in the gradient algorithm. The estimated model was constructed using the matrix inversion lemma. The optimization criterion is the same used in the WLLMS case, the mean squared error.

The augmented Newton method proved to be quite satisfactory in providing smaller errors. In the worst case, this presented a performance equivalent to that of the usual LMS method.

Finally, the so-called Augmented Complex Gradient, constructed as a complex real-case nota- tion, is presented. Using real metrics at background, it is possible to use complex notation with the same results as in the real case.