Signal Processing - September 2017 - 153

to transmit data without degradation in the presence of interference that is known to the transmitter. Equivalently, this
capacity can also be achieved using lattice precoding, as discussed by Erez et al. [46].
The derivation of the Gaussian BC capacity is beyond the
scope of this review article. Hence, as in the upstream, we
will focus on SWP and the MFB. The SWP for the downstream can be derived in a manner similar to the upstream.
The MFB in downstream transmission is the capacity when
all the modems transmit to a single user. The MFB for the
ith user is
R down
= D f log 2 ^1 + C - 1 v w-,2i Px, i h i 2h,
i

(10)

where h i is the ith row of the downstream channel matrix H
and Px, i is the allowed transmission power for each line (at the
considered tone). In the multiuser case, the single-user bound
can be achieved through nonlinear dirty paper coding.

Nonlinear precoding
As stated previously, optimal nonlinear processing can
asymptotically achieve the sum rate capacity in the downstream, using dirty paper coding or multidimensional lattice
precoding. However, both approaches require high implementation complexity and hence are not considered for
DSL. A simpler nonlinear scheme considered for DSL is the
Tomlinson-Harashima precoder (THP) [9], which can be
viewed as one-dimensional lattice precoding.
Recalling that FEXT is modeled through the channel matrix, (2),
the THP cancels the interference using the QR decomposition of
the conjugate transpose of the channel matrix, given by H H = QR,
where Q is a unitary matrix and R is an upper triangular matrix.
The precoding operation is divided into two parts. The modulated signal x first undergoes a precancelation of the interference
using the elements of R and a modulo operation, and it is then
rotated to cancel the channel rotation using the matrix Q.
To remove the interference associated with previous users
from the symbol of the mth user, the precoding operation on
the mth symbol evaluates
Xu m = X m -

m- 1

i= 1

6R H@i, m u
X i, m = 1gN.
6R H@m, m

(11)

Then, to lower the increase in required power, the symbol
undergoes a modulo operation:
Xu m = X m mod 2A,

(12)

where the modulo operation is defined such that its result
will be within a square with an edge of 2A centered at the origin of the complex plane. After collecting all symbols,
xu = [Xu 1, f, Xu N ] T , the resulting vector is rotated by applying
Q. Given that we used the QR decomposition of H H and that
the matrix Q is hermitian, the resulting received signal is
y = R H xu + w. By comparing the effect of the channel with
the precancelation operation in (11), it can be seen that all
interference between users is eliminated. The only step left is

to normalize the signal and reciprocate the modulo operation
by an additional modulo operation at the receivers. Thus, the
estimated symbol of the mth user is
Xt m = Ym mod 2A = ;X m + W m E mod 2A.
R m, m
R m, m

(13)

Thanks to the modulo operations, the THP can cancel all the
interference at almost no cost. This contrasts with linear precoding schemes (see the section immediately following) that can
also remove the interference, but at a power cost that can be
significant. The price of the modulo operation comes from the
effective change of the channel, so that the Gaussian signaling is
no longer optimal. This loss is called the shaping loss and is at
most 1.5 dB. Moreover, although Gaussian signaling maximizes
the achievable rate, all DSL schemes are limited to square QAM
modulations and hence lose most of this shaping loss regardless
of the interference cancelation method. Thus, the actual loss of
the modulo operation is negligible. For more information, see
"Tomlinson-Harashima Precoder."

Linear precoders
The equivalent of the MMSE canceler for the downstream is
typically termed the diagonal loading precoder or signal-toleakage ratio precoder [47]. As in the upstream, this precoder will typically perform slightly better than the ZF precoder,
using almost the same implementation complexity. However,
this precoder has not been studied extensively in the context
of DSL and hence will not be discussed here in more detail.
The linear ZF precoder is a simple technique that precompensates for the true symbol vector x with the inverse of the
channel matrix Fzf = H -1 diag (H) G, such that the precoded
signal vector becomes xu = Fzf x [11]. The diagonal scaling
matrix G is chosen such that the total transmit power on
each line satisfies the power mask constraint tightly; therefore, each element G ii can be different [48], [49]. (For more
information, see "Transmit Power Scaling in the Linear ZeroForcing Precoder.")

Legacy techniques for vectored VDSL systems
The extraordinary success of vectoring techniques for VDSL
systems has proved pivotal in moving toward the nextgeneration G.fast standard. However, most of the techniques
developed for VDSL vectoring relied on the diagonal dominance of the channel to enable low enough implementation
complexity [50], [51]. As discussed in the "Diagonal Dominance
of the Channel Matrix" section, diagonal dominance does not
hold over the G.fast frequency range. Nevertheless, in this
section we briefly discuss these methods to better depict the
challenges of the transition from VDSL to G.fast.

Crosstalk cancelation for VDSL
The research in [10] and [11] showed that the ZF-based linear
canceler/precoder yields a performance close to the single-user
bound. In contrast to the wireless MIMO, it is possible
to bound the performance of the ZF canceler using the

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

153

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