Signal Processing - September 2017 - 150

Table 1. A summary of crosstalk cancelation techniques.
Algorithm

Performance

Comments

ZF-GDFE [9], [39]

Used in upstream. Requires ordering to optimize performance. Problem of error propagation in
the feedback loop.

MMSE [56]
ZF [10], [11]

Complexity

Used in downstream. Negligible power penalty due to the modulo operation. Requires
ordering to optimize performance.
Data rate

ZF-THP [9]

Performs marginally better than the ZF. Requires knowledge of noise covariance.
Requires channel inversion. Near optimal for lower frequencies.

AZF [12]

Does not require channel inversion. Near optimal for longer loops and lower frequencies.
Not suitable for typical G.fast applications.

AZF: approximate ZF.

region, which can be described by a set of 2 N-1 equations.
For example, for the two-user case, the rate region is given by
R 1 1 R mfb
1
R 2 1 R mfb
2
R 1 + R 2 # log 2 ^det ^I 2 + HSH H v -w2hh,

(3)

where S is a diagonal power matrix and R mfb
and R mfb
are the
1
2
matched filter bound (MFB), given later. However, when the
number of users is large and because in practical DSL implementations the coding scheme does not achieve capacity, it is
more convenient to use simple performance bounds. Leshem
and Zehavi [38] presented an efficient rate control for an MAC
subject to a power spectral density (PSD) mask for the transmitters in a multicarrier system and showed that there is no
need for upstream power control as long as the receiver can be
kept sufficiently linear.
Although the performance bounds presented in the following text are not achievable, they are quite tight upper bounds
in practical DSL scenarios. Hence, they allow us to quantify
the suboptimality of each algorithm by evaluating how close it
is to the bound. We demonstrate that, in DSL, better schemes
are very close to the bounds and, hence, are close to optimal.

Single-wire performance
The most intuitive approach is to compare the achievable performance to the case of a single-user transmission over a
single-wire pair. This performance will be denoted hereafter
as single-wire performance (SWP). In this case, the received signal for the ith user (the tested user) is given by
Yi = H ii X i + W i, where the single user is limited only by the
additive noise and attenuation of the channel. The additive noise
is assumed to be Gaussian distributed. With the assumption of
Gaussian distribution on transmit symbols, the Shannon capacity can be derived using the SNR expression. While the Shannon
capacity is achievable, it requires ideal signal processing and
hence is not realizable in practical systems. For this reason, it is
customary in DSL systems to model all the system imperfections by a single SNR-gap parameter, which is commonly
150

referred to as the Shannon gap. The Shannon gap includes all
types of imperfections, starting from amplifier noise and ending
with the use of a square QAM constellation instead of the theoretical Gaussian shaping. Thus, for a target probability of error
10 - 7, the SNR gap C = 10.75 dB is taken for DSL systems.
For a single tone of width T f , the SWP of the ith user is given
swp
as R i = D f log 2 ^1 + C - 1 Px, i | H i | 2 v w-,2i h, where Px, i is the
transmit signal power, and v 2w, i is the noise variance.
However, it is worth noting that the SWP is not an upper
bound on achievable performance. Although FEXT typically
degrades performance, in some instances the transmission
from different modems can be combined coherently through
crosstalk between the wires. The following section presents a
useful upper bound on the user rate.

MFB
The capacity achieved when a single user utilizes both direct
and all FEXT coupling for reception is commonly termed
the MFB, also known as the single-user bound in DSL terminology. As such, all the modems receive from single
users, and the received signal for the ith signal becomes
y = h i X i + w, where h i is the ith column of the channel
matrix H. The optimal processing of the received signal in
this case is known as a matched filter or maximal ratio combining. This receiver simply requires linear combining of the
received signals using Xt i = h iH y; thus, the achievable capacity for the ith user is
R mfb
= D f log 2 ^1 + C - 1 v w-,2i Px, i h i 2h.
i

(4)

Note that the power in this bound is the power allowed for a
single user and not all the power allowed in the network. This
is because the scenario is not practical and is simply used to
bound the performance for the multiuser case.
For VDSL systems, there is a marginal difference between
MFB and SWP performance, because the crosstalk channels are much smaller than the direct channel gains (see the
"Diagonal Dominance of the Channel Matrix" section for
more details). Because crosstalk increases with frequency,
the G.fast system already presents a notable gap between
the SWP and MFB.

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

Table of Contents for the Digital Edition of Signal Processing - September 2017