On the (non)existence of undesirable equilibria of Godard blind equalizers

Existing results in the literature have proved that particular blind equalization algorithms, including Godard algorithms, are globally convergent in an ideal and nonimplementable setting where a doubly infinite dimensional equalizer is available for adaptation. Contrary to popular conjectures, it is shown that implementable finite dimensional equalizers which attempt to approximate the ideal setting generally fail to have global convergence to acceptable equalizer parameter settings without the use of special remedial measures. A theory based on the channel convolution matrix nullspace is proposed to explain the failure of Godard algorithms for such practical blind equalization situations. This nullspace theory is supported by a simple example showing ill convergence of the Godard algorithm