IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 10, OCTOBER 2002 301 A Class of Blind Phase Recovery Techniques for Higher Order QAM Modulations: Estimators and Bounds Yan Wang and Erchin Serpedin, Member, IEEE Abstract—This letter proposes a class of blind feedforward carrier-phase estimators for higher order quadrature ampli- tude-modulated transmissions. As an extension of the Viterbi and Viterbi (V&V) estimator, a constellation-dependent optimal matched nonlinear estimator is derived such that its asymptotic variance is minimized. The asymptotic variances of the optimal matched and V&V estimators are established in closed-form expressions and compared. Computer simulations are presented to corroborate the theoretical performance analysis. Index Terms—Asymptotic variance, carrier-phase recovery, QAM constellations. I. INTRODUCTION Q UADRATURE amplitude modulations (QAM) are cur- rently used in throughput-efficient high-speed communi- cation applications such as digital television systems. One of the problems associated with the use of QAM modulations is that of carrier acquisition, which for efficiency reasons must be performed without using preambles [2]–[4]. In this letter, a family of non-data-aided (NDA) or blind feed- forward carrier-phase estimators for higher order QAM modula- tions is proposed and its asymptotic (large sample) performance analyzed in a rigorous way. The proposed estimators represent a generalized form of the maximum-likelihood feedforward al- gorithm, which was originally proposed in [10] as a blind car- rier-phase estimator for fully modulated -PSK transmissions [7]. An optimal “matched” phase estimator that achieves the smallest asymptotic variance within this family of blind estima- tors is proposed and shown to improve the performance of the standard fourth-power estimator, especially at medium and high signal-to-noise ratios (SNRs). Computer simulations illustrate that the proposed optimal matched phase estimator outperforms the estimator proposed recently in [2]. II. PROBLEM FORMULATION The received signal is given by (1) Manuscript received March 11, 2002; revised July 15, 2002. This work was supported by the National Science Foundation under Award CCR-0092901. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Nicholas D. Sidiropoulos. The authors are with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: ser- pedin@ee.tamu.edu). Digital Object Identifier 10.1109/LSP.2002.804135. where is the sequence of zero-mean unit variance independently and identically distributed (i.i.d.) -QAM symbols; stands for the un- known carrier phase; and is a zero-mean circular white Gaussian noise process independent of and with vari- ance . The SNR per symbol is defined as SNR . Due to space limitations and the fact that the proposed matched estimator is constellation-dependent, we will present our study only for 16-QAM (square) and 32-QAM (cross) constellations in this letter. Because the input QAM constellation has quadrant symmetry, it follows that and that the estimate of presents four-fold ambiguity. Without any loss of generality, we assume that the unknown phase lies in the interval , . III. ESTIMATORS FOR 16-QAM CONSTELLATIONS Assuming a 16-QAM constellation with normal- ized unit energy, takes a value from the set with and . Consider the polar repre- sentation (2) and define the process via the nonlinear transformation (3) where is a general (arbitrary) nonlinear function. Conditioned on the transmitted signal , is nor- mally distributed with the probability density function (pdf) . Throughout the letter, the notation will stand for the pdf of certain random variables. Due to (2), it follows that (4) where and denote the amplitude and phase angle of , respectively. Based on (4), the joint and marginal pdfs 1070-9908/02$17.00 © 2002 IEEE