724 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 2, FEBRUARY 2006 Blind Identification/Equalization Using Deterministic Maximum Likelihood and a Partial Prior on the Input Florence Alberge, Mila Nikolova, and Pierre Duhamel, Fellow, IEEE Abstract—A (semi)deterministic maximum likelihood (DML) approach is presented to solve the joint blind channel identi- fication and blind symbol estimation problem for single-input multiple-output systems. A partial prior on the symbols is incor- porated into the criterion which improves the estimation accuracy and brings robustness toward poor channel diversity conditions. At the same time, this method introduces fewer local minima than the use of a full prior (statistical) ML. In the absence of noise, the proposed batch algorithm estimates perfectly the channel and symbols with a finite number of samples. Based on these considerations, an adaptive implementation of this algorithm is proposed. It presents some desirable properties in- cluding low complexity, robustness to channel overestimation, and high convergence rate. Index Terms—Adaptive algorithm, blind equalization, deter- ministic maximum likelihood method, joint estimation, prior knowledge. I. INTRODUCTION B LIND identification is an important problem in many areas and especially in wireless communications. Blind techniques present some advantages compared to the traditional training methods [1], [2]. First, the reduced need for overhead information increases the bandwidth efficiency. Furthermore, in certain communication systems, the synchronization between the receiver and the transmitter is not possible; thus training sequences are not exploitable. Finally, even if some training sequence exists, the combination of trained and blind tech- niques can often lead to improved performances, allowing fast tracking of time-varying channels, for example. Early approaches to blind equalization were based on higher order statistics of the received signal [3] since the second-order statistics of a scalar system output do not contain enough in- formation to identify a nonminimum phase system. Although these algorithms are robust and reliable in many cases, esti- mating high-order statistics usually requires a large number of data samples. Hence, their application in fast varying environ- ment is intrinsically limited. Tong et al. suggested a different option [4]. They proposed to introduce time or spatial diversity at the output. Then, the system considered is a single-input mul- tiple-output (SIMO) system. The SIMO equalization problem can be solved using second-order statistics only, as long as the Manuscript received April 5, 2004; revised March 26, 2005. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Helmut Boelcskei. F. Alberge and P. Duhamel are with Supelec/LSS, 91192 Gif-Sur-Yvette Cedex, France (e-mail: alberge@lss.supelec.fr; pierre.duhamel@lss.supelec.fr). M. Nikolova is with CMLA-ENS de Cachan and CNRS UMR 8536, 94235 Cachan Cedex, France (e-mail: nikolova@cmla.ens-cachan.fr). Digital Object Identifier 10.1109/TSP.2005.861787 subchannels do not share common zeros. In a fast fading envi- ronment, the statistical model of the input may not be available, or there may not be enough samples to find a reliable estimate of the statistics. In this kind of scenario, the problem may be solved by treating the input as a deterministic variable. Gener- ally, the resulting methods have the finite sample convergence property (i.e., the channel can be perfectly estimated using a fi- nite number of samples in noiseless situations). This is a desir- able property especially in packet transmission systems. In this paper, we focus on deterministic maximum likelihood (DML) methods since they have the additional advantage of being high signal-to-noise ratio (SNR) efficient [5]. Among the major contributions to DML methods, we can cite the two-step maximum likelihood (TSML) [6] and the iterative quadratic maximum likelihood (IQML) [7], both concentrating on channel estimation. Feder et al. proposed in [8] a dual algorithm to IQML which aims at estimating the symbols at each step. Unfortunately, the adaptive implementation of these methods is often cumbersome. Another DML method, the max- imum likelihood block algorithm (MLBA), has been proposed in [9]. The MLBA performs least squares estimation both in the channels and in the symbols in an alternating manner. This formulation permits to derive easily an adaptive algorithm (MLAA) as shown by the authors in [10]. The MLAA presents some nice properties including low-complexity in computation. However, it is not robust to the overestimation of the channel order and it has a limited ability to track time-varying channels. In this paper, we present a new algorithm that meets the fol- lowing four characteristics: adaptivity, low complexity, good speed of convergence, and robustness to the overestimation of the channel order. The proposed method consists of incorpo- rating prior information (related to the input signal) into the DML criterion. The two first properties follow from the MLAA- like structure of the algorithm and the last characteristics are a consequence of the use of the prior. Seshadri [11] and Gosh and Weber [12] first proposed to incorporate the finite alphabet prop- erties into DML to improve the accuracy of the estimates. Later, Talwar proposed the iterative least square with projection (ISLP) [13], which estimates the symbols first without taking the finite alphabet property into account and then projects the estimates onto the alphabet. The problem with these methods is that their convergence is not guaranteed in general and that the incorpora- tion of the finite alphabet property often increases the number of local minima. This is partially solved here by considering only a partial prior on the symbols in order to limit the number of ad- ditional spurious local minima (different from the global one). In the proposed approach, a continuous probability distribution function is used which reflects our prior knowledge on the input. 1053-587X/$20.00 © 2006 IEEE