1348 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005 Blind Phase Recovery for QAM Communication Systems Patrizio Campisi, Member, IEEE, Gianpiero Panci, Stefania Colonnese, and Gaetano Scarano, Member, IEEE Abstract—In this paper, a novel phase estimator that can be em- ployed for both square and cross Quadrature Amplitude Modu- lation (QAM) based digital transmission is presented. It does not need gain control and requires only the knowledge of the type of the transmitted symbol constellation, i.e., square or cross. It is based on the evaluation of the fourth power of the received data and the measurement of the orientation of the concentration ellipses of the bivariate Gaussian distribution having the same second-order moments. The analytical evaluation of the estimation error as well as of the asymptotic variance is provided. Experimental results outline the good performance of the estimator described here, which is supe- rior to that of well-known phase estimation methods. Finally, it is outlined how the eccentricity of the concentration ellipses can be used to devise a test for detecting the constellation type. Index Terms—Asymptotic variance, carrier phase recovery, con- stellation self-noise, square and cross QAM constellations. I. INTRODUCTION I N synchronous systems using high-speed signalling such as quadrature amplitude modulation (QAM), the phase recovery is a problem of paramount importance. For efficiency reasons, the phase estimation must be performed in a blind manner, that is, without using known training sequences. In the recent literature, several approaches for blind phase es- timation have been proposed. The blind phase recovery problem has been dealt with in [1] using higher order statistics. How- ever, the method requires that gain control should have been already performed. In [2], a modification of the method pre- sented in [1] is presented, and an estimator based on a set of fourth-order statistics, which does not need any gain control, is derived. In [3], the estimator described in [2] has been proved to be equivalent to the fourth-power estimator presented in [4]. This latter was demonstrated to approximate the maximum-like- lihood estimator in the limit of small SNR. In [5], an estimator based on eighth-order statistics gives improved performance for cross QAM systems with respect to the fourth power phase es- timator [2]. Moreover, less observed samples are needed. The fourth-power phase estimator’s performance based on a mod- ification of the transmitted constellation was presented in [6], Manuscript received October 8, 2003; revised April 26, 2004. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Prof. Fredrik Gustafsson. P. Campisi is with the Dipartimento Elettronica Applicata, Università degli Studi “Roma Tre,” I-00146 Roma, Italy (e-mail: campisi@uniroma3.it). G. Panci, S. Colonnese, and G. Scarano are with the Dipartimento IN- FOCOM, Università “La Sapienza” di Roma, I-00184 Roma, Italy (e-mail: gpanci@infocom.uniroma1.it; colonnese@infocom.uniroma1.it; scarano@in- focom.uniroma1.it). Digital Object Identifier 10.1109/TSP.2005.843702 whereas in [7], a nonlinear filtering is performed in order to re- tain only the received constellation points, which are more “re- liable” for phase estimation. Performance improvement can be gained resorting to data aided estimation; examples can be found in [7] and [8]. In this paper, a new blind phase estimator that does not require any gain control is presented. It is based on the evaluation of the fourth power of the received data and subsequent measurement of the orientation of the concentration ellipses of the bivariate Gaussian distribution having the same second-order moments. This novel blind phase estimator only requires the knowledge of the constellation type, i.e., square or cross. However, it is worth noting here that the detection of the constellation type can be achieved through a statistic test based on the measure- ment of the eccentricity of the concentration ellipses. In fact, as shown in Appendix B, square constellations always result in lower values of the eccentricity of concentration ellipses with respect to cross constellations. A completely blind phase esti- mator can be obtained using this test as a preliminary constella- tion-type detection stage. The paper is organized as follows. After having introduced the model of the received signal in Section II, the estimation of the phase rotation is described in Section III. The effects of a finite sample size are discussed in Section IV, where the the- oretical analysis concerning the estimation error and with the asymptotic estimation variance is also conducted. Performance comparison with existing estimators is reported in Section V, where the effect of the so-called constellation self-noise is also quantitatively analyzed. Section VI concludes the paper, and an- alytical details are left in the Appendices. II. DISCRETE-TIME SIGNAL MODEL The complex lowpass version of the received signal sampled at symbol rate after the analog receiver front-end can be written as follows: (1) where is the overall gain seen by transmitted symbols drawn from a QAM constellation normalized to have variance equal to 1. The unknown carrier phase offset that has to be esti- mated is denoted by . It is further assumed that is a realization of circularly symmetric complex Gaussian stationary process that is statisti- cally independent of . The signal-to-noise ratio (SNR) is defined as SNR , where is the noise variance. 1053-587X/$20.00 © 2005 IEEE