IEEE COMMUNICATIONS LETTERS, ACCEPTED TO APPEAR (JULY 2005) 1 Low-complexity Low-PAR Transmission for MIMO-DSL Daniel J. Ryan, Student Member, IEEE, Iain B. Collings, Senior Member, IEEE, and I. Vaughan L. Clarkson, Senior Member, IEEE Abstract— This letter considers clip-limited transmission over multiple-input multiple-output digital subscriber lines (MIMO- DSL). We show that a recent low complexity, low peak-to- average-ratio (PAR) single-input modulation technique can be applied to the case of multiple cross-talking channels in a bonded- DSL system. Unfortunately however the direct initialization procedure is computationally infeasible. In this paper, we pro- vide a novel low-complexity initialization procedure. Simulations confirm that the proposed approach has superior performance in clip-limited conditions, compared with both discrete matrix multitone and vectored discrete multitone. Index Terms— Digital subscriber line (DSL), peak-to-average ratio (PAR), MIMO, lattice codes. I. I NTRODUCTION Multiple-input multiple-output digital subscriber line (MIMO-DSL) systems have the ability to cancel crosstalk on neighboring twisted-pairs, and hence enable increased throughput. Current modulation techniques for MIMO-DSL include discrete matrix multitone (DMMT) [1, 2] and vectored discrete multitone (V-DMT) [3]. Both of these techniques involve bit-loading on frequency domain subcarriers and there- fore exhibit a high peak-to-average ratio (PAR). In this letter we take a fundamentally different approach to achieving coordinated bonded DSL transmission with low- PAR. We extend a recent single-input lattice-based transmis- sion technique which fits point-lattices within amplitude limits and totally avoids clipping [4]. As in the single-input case, the modulation and demodulation of the MIMO-extended lattice approach can be implemented with complexity comparable to the FFT used in DMT based techniques [5]. Unfortunately however, the initialization complexity of the directly extended lattice technique is prohibitively large in practice. Specifically, it is of order (CN ) 3 , where C is the number of parallel inputs and N is the length of the transmit symbol. The high complexity is due to a CN × CN MIMO channel matrix QR decomposition required at start up. In this letter, we present a new low complexity initialization for the MIMO-extended lattice scheme which requires only computations and storage of order C 3 NL, where L is the Manuscript received April 21, 2005. The associate editor coordinating the review for this letter and approving it for publication was Dr. Murat Uysal. D. J. Ryan and I. B. Collings are with the Telecommunications Lab, School of Electrical & Information Engineering, University of Sydney, NSW 2006, Australia, and also the CSIRO ICT Centre, Epping, NSW 1710, Australia. I. V. L. Clarkson is with the School of Information Technology & Electrical Engineering, University of Queensland, Qld. 4072, Australia. Digital Object Identifier 00.0000/LCOMM2005.000000 length of the channel impulse response. More specifically, we first observe that the MIMO-extended lattice scheme does not actually require the Q matrix from the QR decomposition, and we then present a low-complexity algorithm for calculating the R matrix which makes use of its banded block-Toeplitz structure. Simulations confirm that the MIMO-extended lattice scheme with efficient initialization has performance superior to both V-DMT and DMMT in VDSL channels with tight amplitude constraints. We show that the BER is significantly lower for clip-to-average ratios (CAR) in the range 0-14 dB. We also show that for a fixed clipping level the lattice approach allows higher data rates for a given target BER. II. MIMO- EXTENDED LATTICE TRANSMISSION Given C channels, we denote the multiple-input transmit- symbol vector x =(x ′ 1 ,..., x ′ N ) ′ (1) where (·) ′ denotes transpose, x k =(x k1 ,...,x kC ) ′ contains the time-domain amplitudes at each of the C transmitters at time k. Grouping the transmit symbols by time is crucial in allowing us to directly apply the single-input lattice-MCM modulation technique of [5] while maintaining the low com- plexity implementation. Let L be the maximum length of any channel impulse response or cross-channel impulse response. Assume that a guard-period of length N ≥ L samples is used between successive transmit-symbols. We allow the guard-period either to contain a cyclic prefix or zero transmit energy. The channel response matrix H is a C(N + L) × CN block-Toeplitz matrix in the zero-prefix case, and a CN ×CN block-circulant matrix in the cyclic prefix case. The first column of blocks in H can be written (H 0 ,..., H L , 0,..., 0) ′ where each H k is a C × C matrix containing the kth elements of the direct and crosstalk impulse responses. In either case the receive-symbol is given by y = Hf (x)+ η , (2) where η is an independent vector of zero-mean white Gaussian noise with variance σ 2 , and f (·) is a vector function represent- ing the transmitter nonlinearity such that the i th element of f (x) is clipped to +a or -a if x i >a or x i < -a respectively, and otherwise equals x i . 0000–0000/00$00.00 c  2005 IEEE