IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 2273 An Efficient Block Adaptive Decision Feedback Equalizer Implemented in the Frequency Domain Kostas Berberidis, Member, IEEE, and Panos Karaivazoglou Abstract—In this paper, a new block adaptive decision feedback equalizer (DFE) implemented in the frequency domain is derived. The new algorithm is suitable for applications requiring long adap- tive equalizers, as is the case in several high-speed wireless commu- nication systems. The inherent “causality” problem appearing in the block adaptive formulation of the DFE equations is overcome by using tentative decisions in place of the unknown ones within each block. These tentative decisions are subsequently improved by using an efficient iterative procedure, which finally converges to the optimum decisions in a few iterations. This procedure is prop- erly initialized by applying a minimization criterion that utilizes all the available information. The whole algorithm, including the iterative procedure, is implemented in the frequency domain and exhibits a considerable reduction in computational complexity, as compared with the conventional DFE, offering, at the same time, a noticeable increase in convergence speed. Additionally, the level of the steady-state MSE, which is achieved by the new algorithm, is practically insensitive to the block length. Index Terms—Block adaptive filters, decision feedback equal- izer, frequency domain equalization. I. INTRODUCTION A MAJOR cause of performance degradation in many com- munication systems is the introduced intersymbol interfer- ence (ISI), due to the time-dispersive characteristics of the in- volved channels. The problem is particularly important in wire- less transmission systems, due to the multipath phenomenon. In these systems, under the same delay spread conditions, the higher the symbol rate, the longer the impulse response (IR) of the involved channel. Channels with long IR are encountered in HDTV broadcasting systems, in high-speed wireless networks (either fixed or mobile), in underwater communications, etc. In some cases, the involved channel IR is as long as several tens of s, in particular, the causal part (i.e., the part preceding the main signal), which may span up to several hundreds of symbol periods. Moreover, very often, the frequency response of the channel exhibits very deep nulls. A well-established equalization technique, which is very ef- fective in reducing the introduced ISI, and at the same time is capable of tracking the channel variations, is the adaptive deci- sion feedback equalizer (DFE) [1]–[4]. The DFE turns out to be Manuscript received September 27, 2000; revised May 8, 2002. This work was supported in part by the General Secretariat of Research and Technology of Greece under Grant PENED 99E 83 and by the Computer Technology Institute (CTI). The associate editor coordinating the review of this paper and approving it for publication was Dr. Rick S. Blum. The authors are with the Department of Computer Engineering and Informatics/C.T.I, School of Engineering, University of Patras, Patras, Greece (e-mail: berberid@ceid.upatras.gr; karaivap@ceid.upatras.gr). Publisher Item Identifier 10.1109/TSP.2002.801884. particularly suitable for multipath channels, which are charac- terized by an IR having usually a long causal part and a much shorter anticausal part. The postcursor ISI is almost perfectly cancelled by the feedback (FB) filter of the DFE, and the whole structure exhibits less noise enhancement effects as compared with linear equalizers since noise is now involved only in the output of the feedforward (FF) filter. In high-speed applications, as mentioned previously, a DFE with a large number of taps in the FB and FF filters is required. However, the implementation and real-time operation of such an equalizer is a difficult task, due to the increased complexity and the very small intersymbol period. Another important issue in adaptive equalization is the one of convergence speed. Fast converging equalizers are highly desirable since they require a reduced training sequence, thus offering a valuable saving in bandwidth. The issue of convergence is usually traded off with the issue of complexity. DFEs based on the recursive least squares (RLS) algorithm [5] exhibit very fast convergence, but unfortunately, they require a large number of operations per time step. On the other hand, the conventional adaptive DFE, which is based on the least mean squares (LMS) algorithm, has a much lower complexity as compared with the RLS-based DFE, but its convergence is very slow, especially in channels that contain spectral nulls. Moreover, even the “low” compu- tational burden of the LMS-based DFE turns out to be pro- hibitive in demanding applications. To overcome this difficulty, several efficient implementations of the LMS-based DFE have been proposed in literature (e.g., [6]–[10]). The works [6]–[9] are mostly based on the assumption that the channel IR has a discrete sparse form. The algorithms in the referenced works are possibly applicable to other channels as well, but in such a case, their complexity would become quite high. Moreover, in some cases, the efficiency depends strongly on the constellation used (as in [6]). The work in [10] is an exact time domain block adap- tive DFE in the sense that all filtering part quantities (including decisions) are identical to those obtained by the conventional symbol-by-symbol DFE. The price paid for this exactness is the rather complicated Winogrand-type algorithm used to solve the involved convolutions. Moreover, the algorithm has the same, with conventional DFE, convergence properties. A possible way to cope with both the complexity and conver- gence issues without any restrictive tradeoff is to develop block adaptive equalizers implemented in the frequency domain. Fre- quency domain adaptive (FDA) filters have been extensively studied in literature (e.g., [11] and [12]). As compared with sample-by-sample adaptive filters, the FDA filters exhibit lower complexity, faster convergence, and identical steady-state per- formance. However, most of the existing FDA filters are suit- 1053-587X/02$17.00 © 2002 IEEE