Testing for the convergence of a linear decision directed equaliser

Adaptation algorithms for blind equalisation of communication channels suffer from the presence of local equilibria that may cause convergence to incorrect equaliser parameter settings. In this paper we consider an infinite impulse response (IIR) channel which is to be equalised by a linear transversal filter and a slicer connected in tandem, and use the correlation statistics of the sequence at the slicer output to determine if convergence to a desirable equilibrium has occurred. In particular, we show that if the input sequence to an IIR channel is binary, assuming values +or-1, and its autocorrelation sequence is known a priori, the slicer output has the same autocorrelation sequence if, and only if, the slicer output is a possibly delayed and/or sign-inverted replica of the channel input. A Neyman-Pearson test is constructed, based on this result, to make a decision as to whether or not the equaliser has converged to an open-eye parameter setting. A method for estimating a lower bound on the equalisation delay that draws upon a classical correlation based impulse response estimation technique is discussed. Finally, simulation examples are presented to verify the theoretical results and to illustrate their application.