DIAMOND CONTOUR-BASED PHASE RECOVERY FOR (CROSS)-QAM CONSTELLATIONS Marcos ´ Alvarez-D´ ıaz and Roberto L ´ opez-Valcarce Departamento de Teor´ ıa de la Se ˜ nal y las Comunicaciones Universidade de Vigo Vigo (Pontevedra) 36310, Spain E-mail: {malvarez, valcarce}@gts.tsc.uvigo.es ABSTRACT Two new iterative methods for blind phase estimation of QAM signals are presented. The algorithms seek the maxima of cer- tain cost functions derived from the dispersion of the de-rotated data with respect to a diamond-shaped contour, and their distinc- tive feature is the inclusion of a sign nonlinearity at every itera- tion. A connection between one of these methods and the stan- dard fourth-power method is presented. Simulations with cross- QAM constellations show that, when properly initialized, the new schemes present lower variance than the fourth-power method, and may even outperform Cartwright’s eighth-order method. Keywords: Carrier phase estimation, QAM constellations, synchronization. 1. INTRODUCTION We consider the problem of blind carrier phase acquisition in QAM digital communication systems. The detected data are assumed to be of the form r k = a k e jθ + n k , k =0, 1,...,L - 1, (1) where a k is the kth transmitted symbol, drawn equiprobably from a QAM constellation, and n k is the complex-valued kth noise sam- ple. The noise is assumed white Gaussian and circular with vari- ance σ 2 , and independent of the symbols. The goal is to identify θ without knowledge of the symbols a k . A classical approach to this problem results in the fourth-power method, ˆ θ4P = 1 4 arg - L-1 k=0 r 4 k . (2) The performance of the fourth-power method applied to square QAM constellations is considered acceptable (it is known to be the Maximum-Likelihood estimator as the SNR goes to zero [1]), while it is poor for cross QAM constellations [2]. Thus, alter- natives have been devised to improve phase detection for such constellations. Among them, one approach is to use higher or- der statistics of the observed signal. In [3], Cartwright presents a method using eighth-order statistics with improved performance with respect to the fourth-power method. The price paid in This work was supported by the Spanish Ministry for Education and Science (MEC) under project DIPSTICK (reference TEC2004-02551) and the Ram´ on y Cajal Program. Square contour Circular contour Diamond contour ¯ y ˜ y Fig. 1. Possible contours. Cartwright’s method (further referred to as C-VIII) is the higher computational cost. Nevertheless, other approaches are possible. For example, a joint blind adaptive equalization and phase recovery was presented in [4] based on the minimization of the dispersion of the equal- izer output with respect to a square contour. This idea can be used to devise blind phase estimators. In principle, different con- tours could be considered; Figure 1 illustrates three possibilities for which the following conditions and related cost functions can be written: • Circular contour (CM - Constant Modulus): |y k | 2 = ¯ y 2 k +˜ y 2 k = constant (3) JCM . = E |y k | 2 - γCM 2 ; (4) • Square contour (SC): max(| ¯ y k |, | ˜ y k |) = | ¯ y k - ˜ y k | + | ¯ y k +˜ y k | 2 = constant (5) JSC . = E (| ¯ y k - ˜ y k | + | ¯ y k +˜ y k |- γSC) 2 ; (6) • Diamond contour (DC): | ¯ y k | + | ˜ y k | = constant (7) JDC . = E (| ¯ y k | + | ˜ y k |- γDC) 2 , (8) where y k =¯ y k + j ˜ y k are the equalizer outputs, and γCM, γSC and γDC are appropriate constants.