IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 6, JUNE 2003 885 RLS-Based Initialization for Per-Tone Equalizers in DMT Receivers Katleen Van Acker, Geert Leus, Marc Moonen, and Thierry Pollet Abstract—Per-tone equalization has recently been proposed as an alternative receiver structure for discrete multitone-based systems improving upon the well-known structure based on time-domain equalization. Fast initialization of all the equalizer coefficients has been identified as an open problem. In this letter, a recursive initialization scheme based on recursive least squares with inverse updating is presented for the per-tone equalizers. Simulation results show convergence with an acceptably small number of training symbols. Complexity calculations are made for per-tone equalization and for the case where tones are grouped. It is demonstrated with an example that in the latter case, initial- ization complexity becomes sufficiently low and comparable to complexity during data transmission. Index Terms—Asymmetric digital subscriber line (ADSL), discrete multitone (DMT), equalization, recursive least squares (RLS). I. INTRODUCTION D ISCRETE MULTITONE (DMT) has become an impor- tant transmission method, for instance, for asymmetric digital subscriber lines (ADSLs) [1]. A conventional equaliza- tion scheme for such a DMT modem consists of a time-domain equalizer (TEQ) which shortens the channel impulse response such that the total impulse response is shorter than the cyclic prefix, followed by a one-taps frequency-domain equalizer for each tone [3], [5], [8]. As an alternative to TEQ, per-tone equalization has been proposed in [6]. There is no TEQ involved in the system, but a -taps per-tone equalizer (PTEQ) is inserted for each tone separately. This scheme enables performing true signal-to- noise ratio (SNR)-optimization per tone, in contrast with the TEQ-based scheme, while complexity during data transmission is kept at the same level. Moreover, PTEQ has been shown to have a significantly reduced sensitivity to the so-called synchronization delay. This delay determines which received samples belong to the same received symbol. Paper approved by M. Chiani, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received September 20, 2000; re- vised July 15, 2002. The work of M. Moonen and G. Leus was supported by the Fund for Scientific Research, Flanders (FWO). This work was supported in part by the Belgian Prime Minister’s Office, Federal Office for Scientific, Technical and Cultural Affairs, under IUAP P4-02 (1997–2001): Modeling, Identification, Simulation and Control of Complex Systems; in part by the Concerted Research Action GOA-MEFISTO-666 of the Flemish Government; in part by the Flemish Institute for Scientific and Technological Research in Industry (IWT) [IRMUT (980271) and Advanced Internet Access (980316)]; and in part by Alcatel. This paper was presented in part at the European Signal Processing Conference, Tam- pere, Finland, September 2000. K. Van Acker and T. Pollet are with Research and Innovation, Alcatel Bell, 2018 Antwerpen, Belgium (e-mail: katleen.van_acker@alcatel.be; thierry.pollet@alcatel.be). G. Leus and M. Moonen are with the Katholieke Universiteit Leuven, ESAT-SISTA, 3001 Leuven, Belgium (e-mail: geert.leus@esat.kuleuven.ac.be; marc.moonen@esat.kuleuven.ac.be). Digital Object Identifier 10.1109/TCOMM.2003.813176 An initialization formula has been derived in [6] which is ap- plicable when a channel model is available, as well as signal and noise covariance matrices. This direct initialization is com- putationally intensive. Hence, there is a strong need for a less complex initialization algorithm. In this letter, a recursive initialization scheme based on so-called recursive least squares (RLS) with inverse updating is presented. It is demonstrated that this scheme achieves initialization with an acceptably small number of training symbols. As a significant part of the RLS computations can be “shared” among the different tones, this is achieved at an acceptable computational cost. II. PTEQ IN A DMT MODEM The concepts of PTEQ are briefly reviewed. For more de- tails, we refer readers to [6], where it is also shown that such an approach clearly outperforms the well-known TEQ (channel shortening) approach. The per-tone approach is based on transferring the TEQ oper- ations to the frequency domain (i.e., after the fast Fourier trans- form (FFT) demodulation) which results in a -taps PTEQ for each tone separately, fed with the outputs of a sliding FFT oper- ation. At first sight, multiple FFTs are needed for each symbol. But it is demonstrated in [6] that, for every -taps PTEQ, there exists a modified -taps PTEQ which has as its inputs the corre- sponding output of only one FFT and real difference terms. The modified equalizers then have two functions: they equalize the channel impulse response, and at the same time, incorporate the sliding Fourier transform computations, whereas the orig- inal equalizers only equalize the channel impulse response. Let us describe this in more detail. Suppose is the FFT size and is the length of the PTEQ. Collecting the received samples related to symbol period in the vector , the PTEQ for the th tone, denoted by the vector , then has as its inputs the th output of the FFT, de- noted by , and difference terms, denoted by , where is the identity matrix, is the all-zero matrix, and is the th row of the FFT matrix . The optimal modified equalizers, which give rise to maximum SNR on each tone , are then found by (1) 0090-6778/03$17.00 © 2003 IEEE