Finite Word Length Effects on Transmission Rate in Zero Forcing Linear Precoding for Multichannel DSL

Abstract

Crosstalk interference is the limiting factor in transmission over copper lines. Crosstalk cancelation techniques show great potential for enabling the next leap in DSL transmission rates. An important issue when implementing crosstalk cancelation techniques in hardware is the effect of finite world length on performance. In this paper we provide an analysis of the performance of linear zero-forcing precoders, used for crosstalk compensation, in the presence of finite word length errors. We quantify analytically the trade off between precoder word length and transmission rate degradation. More specifically, we prove a simple formula for the transmission rate loss as a function of the number of bits used for precoding, the signal to noise ratio, and the standard line parameters. We demonstrate, through simulations on real lines, the accuracy of our estimates. Moreover, our results are stable in the presence of channel estimation errors. Finally, we show how to use these estimates as a design tool for DSL linear crosstalk precoders. For example, we show that for standard VDSL2 precoded systems, 14 bits representation of the precoder entries results in capacity loss below 1% for lines over 300m.

Simple Statistical Analysis of the Impact of Some Nonidealities in Downstream VDSL with Linear Precoding

“…For achieving a small rate loss, however, a large number of quantization bits may still be required. A typical value v ≥ 14 bits, identified in [20], is confirmed by the numerical example reported in Section 6.2.…”

“…In ideal conditions, that is assuming arbitrary precision, we have Δ k = D k = 0. Through simple algebra, the signal-tonoise ratio for the nth receiver at the kth tone in the presence of the quantization error is given by the following expression, that was already derived in [20]…”

“…By investigating the statistical properties of c n k , in the presence of quantization errors, it is possible to find the number of quantization bits needed to have a penalty smaller than a prefixed percentage. In this view, an indepth analytical work was done in [20], where a number of bounds were determined, and their reliability tested through simulations. In that paper, however, the elements of D k were modeled as random variables uniformly distributed in the range [−2 −v , 2 −v ], where v is the number of quantization bits adopted.…”

“…Moreover, as the precoding system is also affected by quantization errors, we can evaluate in the same way the effect of finite word length in the representation of precoder variables. This issue has been faced only recently in the literature [20], but it is extremely important due to its influence on the performance/complexity tradeoff: coarse quantization can imply an intolerable 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 [20], it has been shown that to obtain a capacity loss, due to quantization errors, below a prefixed small percentage, a 14 bits representation of the precoder entries is necessary.…”

“…This issue has been faced only recently in the literature [20], but it is extremely important due to its influence on the performance/complexity tradeoff: coarse quantization can imply an intolerable 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 [20], it has been shown that to obtain a capacity loss, due to quantization errors, below a prefixed small percentage, a 14 bits representation of the precoder entries is necessary. We will verify that by adopting a quantization law that exploits the row-wise diagonal dominant (RWDD) character of the downstream VDSL channel, the same loss can be reached by adopting a smaller number of bits.…”