OPTIMAL DMT TRANSCEIVERS OVER FADING CHANNELS Yuan-Pei Lin Dept. Electrical and Control Engr. National Chiao Tung Univ. Hsinchu, Taiwan, R.O.C. ABSTRACT Recently DFT based discrete multitone modulation (DMT) systems have been widely applied to various applications. In this paper we study a broader class of DMT systems using more general unitary matrices instead of DFT matrices. For this class we will show how to design the optimal DMT systems over fading channels with colored noise. Examples will be given to show the improvement over the traditional DFT based DMT system. In addition we introduce a modified DFT based DMT system. The new system has the same complexity but better noise rejection property. zyxwvuts 1. INTRODUCTION Recently there has been considerable interest in ap- plying the discrete multitone modulation (DMT) tech- nique to high speed data transmission over fading chan- nels such zyxwvutsrq as ADSL and HDSL [1][2]. In the widely used DFT based DMT system (Fig. l), the channel is di- vided into a number of subchannels by using DFT ma- trices. High speed data transmission can be obtained at a relatively low cost [l]. In the DFT based DMT sys- tem, a certain degree of redundancy known as cyclic prefix is added to achieve IS1 free transmission over fading channels [1][2]. In [6] Kasturia et. a1 advance the DMT system using more general unitary matrices instead of DFT matri- ces. When the channel noise is AWGN, the authors show that the optimal transmitter and receiver are composed of eigenvectors of some Toeplitz matrices as- sociated with the channel impulse response. However for applications such as ADSL, the dominating noise source is usually crosstalk and the noise is colored [l]. In this paper, we will use a polyphase approach [2J to study the DMT system. Using this approach, we will derive a modified DFT based DMT system which has a better noise rejection property than the tradi- tional DFT based system at the same cost. Moreover THE WORK WAS SUPPORTED BY NSC 87-2213-E-009 052 AND BY NSC 87-2218E002-053, TAIWAN, R.O.C. See-May Phoong Dept. of EE zyxwv & Inst. of Comm Engr. National Taiwan Univ. Taipei, Taiwan, R.O.C. optimal transceiver for colored noise will be studied in details. In particular, we will show how to assign bits among the channel so that the total tranmitting power can be minimized for a given bit rate. Based on the optimal bit allocation the design of the optimal transceiver is derived. Futhermore we will see that al- though the DFT based DMT system is not optimal, for AWGN fading channels, its asymptotical performance approaches that of the optimal system when the num- ber of channels is large. 2. POLYPHASE REPRESENTATION OF DMT SYSTEMS Consider Fig. 2, where an M-channel DMT system is shown. Usually the channel is modelled as an LTI fil- ter zyxwvuts C(z) with additive noise zyxwv e(n). Assume that C(z) is an FIR filter of order L (a reasonable assumption af- ter channel equalization) and zyxw e(.) is a zero-mean WSS random process. For a given channel number M, the interpolation ratio N is chosen as N = M + L. As re- dundancy is introduced in this case, we say the system is over interpolated. The filters Fk(z) and Hk(z) are called transmitting and receiving filters respectively. In the DMT system, Fk(z) and Hk(z) have length 5 the interpolation ratio N. Using polyphase decomposition the DMT system can be redrawn as in Fig. 3 [2]. The tranmitter G is an N x M contant matrix; the kth column of G contains the coefficients of the transmitting filter Fk(z). The receiver S is an M zyxwv x N contant matrix; the kth row of S contains the coefficients of the receiving filter zyx H~(z). The matrix C(z) is an N x N pseudo circulant matrix [4] with the first column given by (COCl . . . cL 0 . . . o)T where c, is the channel impulse response. Perfect reconstructwn condition. From Fig, 3, we see that the overall transfer function of the DMT system is T(z) = SC(z)G. (1) 0-7803-5041-3/99 $10.00 0 1999 IEEE 1397