On the effect of estimation and quantization errors in downstream VDSL systems Marco Baldi, Franco Chiaraluce Università Politecnica delle Marche DIBET Ancona, Italy {m.baldi, f.chiaraluce}@univpm.it Roberto Garello Politecnico di Torino Dipartimento di Elettronica Torino, Italy garello@polito.it Marco Polano, Marcello Valentini Telecom Italia Torino, Italy marco.polano, marcello.valentini@telecom.it Abstract—We investigate the effect of non idealities in the diagonalizing precoder vectoring technique used for cancellation of the far-end crosstalk in downstream VDSL networks. The contribution of the paper is twofold: on one hand we provide analytical formulas to estimate the average bit rates achievable as a function of both relative and absolute estimation errors; on the other hand we show that, by adopting a smart quantization law, the number of quantization bits required for not exceeding a given transmission rate loss can be smaller than that foreseen by previous analyses. Keywords – VDSL; vectoring; channel estimation errors; quantization errors I. INTRODUCTION A great amount of work has been done to improve the performances of digital subscriber line (DSL) systems. The common idea, for most of the proposed solutions, is to overcome the limitations imposed by far-end crosstalk (FEXT), which is a major problem in very high speed networks, through the adoption of user coordination techniques [1]. In the downstream direction, which will be considered in this paper, coordination is possible as the transmitting modems are co-located at the central office. So, FEXT can be completely canceled, at least in principle, through a pre-distortion to apply, tone by tone, at each modem’s signal before its transmission. Using these discrete multi-tone (DMT) vectored transmission techniques, the achievable bit rates can be significantly increased with respect to non vectored solutions. However, some problems exist, that can make difficult to achieve nearly optimal performance. The first problem is the effect of estimation errors. Pre- distortion requires the knowledge of the channel transfer function; this is rather easy to obtain for the direct channels, but becomes more difficult for the crosstalk channels. The latter are not yet completely described and, although suitable models have been proposed, they require in depth verification. In this paper, we consider the adoption of the diagonalizing precoder (DP) proposed in [2], that has the advantage of an easy implementation and does not require modifications of the customer premise equipment. We develop a theoretical approach that permits to analytically determine the average bit rate achievable, as a function of the estimation error. Similarly to [3], we support the theoretical analysis with simulations, that allow a more complete characterization of the random variables involved, for example by evaluation of the cumulative distribution function. The second problem concerns the need to take into account the effect of finite word length in the representation of the precoder variables, i.e., to measure the impact of quantization errors. This issue has been faced only recently [4], but it is extremely important due to its influence on the performance complexity trade-off: coarse quantization can imply intolerable rate loss but, on the other hand, a large number of quantization bits can yield high hardware complexity and a great amount of memory needed for the precoding process. In [4], it has been shown that to obtain a capacity loss, due to quantization errors, below 1%, a 14 bits representation of the precoder entries is necessary. We will verify that, by adopting a suitable quantization law, the same loss can be ensured by using only 10 bits. The organization of the paper is as follows. In Section II we introduce the system and its relevant performance parameters in ideal conditions. In Section III we describe the channel model adopted and the statistical issues it involves. In Section IV we discuss the effect of the estimation errors, through theoretical arguments and simulations. The same is done, for the quantization errors, in Section V, where we also present the simple quantization law that permits to reduce the number of quantization bits. Finally, Section VI concludes the paper. II. IDEAL BEHAVIOR The block scheme of the considered vectored system, referred to the k-th downstream tone f k , is shown in Fig. 1. X k = [X k 1 , X k 2 , … , X k L ] T is an L-components vector collecting the symbols transmitted by L users on as many lines. H k is the L×L channel matrix; its (i, j)-th element, ij k H , represents the channel from transmitter j to receiver i. The diagonalizing precoder (DP) matrix at tone k is given by: 1 1 diag( ) k k k k - - =β ⋅ P H H , (1) with 1 row max diag( ) - ⎡ ⎤ β ⋅ ⎣ ⎦ H H  k k k i i . In (1), diag(H k ) is the diagonal matrix having elements H k 11 , … , H k LL . Also shown in Fig. 1, N k is the L-components vector describing the additive thermal noise.