A b s t r a c t the desired sequence is real and is a many-to-one when itThe error surface for the Constant Modulus Adaptive (CMA) algorithm is shown to be quadratic in the Parameter Kronecker Product Space, the space spanned by a basis vector equal to the kronecker product of the filter weight vector and its complex conjugate. As a result, closed-form analytical expressions similar to the Wiener type for the optimum filter coefficients can be derived. The regions of convergence as well as the regions of stability in terms of the autocorrelation sequences of the second and of the fourth order of the received signal can be expressed in this form. These expressions are valid for both the real and the complex arithmetic implementations of the CMA. Computer simulations to support these theoretical findings are compared to results published by others for real implementation of the algorithm.
I. I n t r o d u c t i o nSince its introduction in the early eighties, the CMA algorithm [1,2] has presented itself as a prime (if not the only) means of solving adaptive filtering problems where a reference signal or a template of the original transmitted sequence is not available but the signal is known to have a constant modulus. The attractive feature of this algorithm is that it seeks to restore the original signal by only restoring some of its known deterministic properties, namely the modulus, which has been degraded by passing through a communication channel and/or adding interference signals.There has been a systematic effort by Treichler and his colleagues in simulatin and in discovering new applications for the algorithm 73,4,5]. There has also been a recent effort by Johnson and his coworkers in investigating the existence of local minimas for the real-valued CMA [6].Despite all these efforts, however, the behavioral properties of this algorithm are not yet fully understood even in idealized situations. Until now, no complete formal proof of the convergence or stability of the algorithm has been formulated. The main difficulty rises from the fact that the proposed performance measure criterion is highly nonlinear in terms of the filter weight vector.As a step toward a more theoretical understanding of the CMA algorithm, the Kronecker product notation is used to express the cost function in the Parameter Kronecker Product Space, the space spanned by a basis vector 6 equal to the Kronecker product of the filter weight vector W and itself in the case of a real arithmetic implementation, or its complex conjugate W' in the case of a complex signal processing application. This change of notation is equivalent to a nonlinear mapping T of the original parameter space R to the parameter kronecker product space 0 . The nonlinear transformation T is a two-to-one map when is complex.The main advantage of this nonlinear transformation is the fact that the error performance measure turns out to be quadratic in the new weight vector 6, which simplifies the analysis greatly. As a result, closed form analytical expressions of the Wien...
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