On the Convergence of Normalized Constant Modulus Algorithms for Blind Equalization

Modulus Algorithms for Blind Equalization Constantinos B. Papadias e-mail: [email protected] Dirk T. M. Slock e-mail: [email protected] Institut EURECOM, 2229 route des Crêtes, Sophia Antipolis, 06560 Valbonne, FRANCE Abstract One of the most known classes of algorithms for blind equalization is the so-called class of Constant Modulus Algorithms (CMA's) [1], [2]. These adaptive algorithms use an instantaneous gradient-search procedure similar to that of the LMS algorithm in order to minimize a stochastic criterion that penalizes the deviations of the received signal's modulus with respect to the known modulus of the emitted input signal. However, as has been recently reported [4], [5], [6], [7], these algorithms might ill-converge if they are not properly initialized, due to false minima of their corresponding cost function. This holds even in cases where the equalizer can match exactly the inverse of the transmitting channel [6]. Recently, a variant of CMA algorithms, the so-called Normalized CMA (NCMA) has been introduced in [8] and a more general class of normalized CMA algorithms containing NCMA as its rst member has been introduced in [11]. These algorithms have a stable operation for any value of their stepsize in the (0,2) range, in contrast to unnormalized algorithms for which the range of stable stepsize values depends on the input signal's statistics and is very hard to determine. The choice of a big stepsize thus leads to a much faster convergence of normalized algorithms as compared to their unnormalized counterparts. In this paper we show that choosing a big stepsize may also help them circumvent the undesirable local minima of the algorithm's cost function thus avoiding the problem of ill-convergence.