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“Adaptive Blind Equalization”
by Yuanjie Chen and Chrysostomos L. Nikias
May 1992
This tutorial paper is focused on two topics, namely: (i) to describe systematic methodologies for selecting nonlinear transformations for blind equalization algorithms (and thus new types of cumulants), and (ii) to give an overview of the existing blind equalization algorithms and point out their strengths as well as weaknesses. It is shown in this paper that all blind equalization algorithms belong in one of the following three categories, depending where the nonlinear transformation is being applied on the data: (i) the Bussgang algorithms, where the nonlinearity is in the output of the adaptive equalization filter; (ii) the polyspectra (or Higher-Order Spectra) algorithms, where the nonlinearity is in the input of the adaptive equalization filter; and (iii) the algorithms where the nonlinearity is inside the adaptive filter, i.e., the nonlinear filter or neural network. We describe methodologies for selecting nonlinear transformations based on various optimality criteria such as MSE or MAP. We illustrate that such existing algorithms as Sato, Benveniste-Goursat, Godard or CMA, Stop-and-Go and Donoho are indeed special cases of the Bussgang family of techniques when the nonlinearity is memoryless. We present results that demonstrate the polyspectra-based algorithms exhibit faster convergence rate than Bussgang algorithms. However, this improved performance is at the expense of more computations per iteration. We also show that blind equalizers based on nonlinear filters or neural networks are more suited for channels that have nonlinear distortions.
The Godard or CMA algorithm is probably the most widely used blind equalizer in digital communications today due to its simplicity, low complexity and constant modulus property. Its main drawbacks, however, are slow convergence and no guarantee for global convergence starting from arbitrary initial guess. We present a new method for blind equalization, the CRIMNO algorithm (i.e., criterion with memory nonlinearity), which is shown to have the same advantages as Godard (simplicity, low complexity, constant modulus property) and yet guaranteeing much faster convergence. The CRIMNO algorithm is flexible enough to address blind deconvolution problems when the input sequence is colored.