Signal Processing - September 2017 - 159

that describes novel approaches to dealing with strong crosstalk in the G.fast system [64]-[69].

Power normalization
Use of linear cancelation in the downstream transmission
requires an energy normalization that can guarantee the power
constraint for all lines. For the VDSL (low) frequencies, it was
shown that simple power normalization results in near-optimal
performance [11]. But this is not the case for G.fast frequencies, and power normalization can significantly reduce the
SNR when the channel matrix is ill conditioned. A single normalization factor b max = max i R Nj = 1 Fij 2 (i.e., gain scaling
G/b max ) has been shown to be near optimal for low frequencies [11].
In [48], the researchers adopted a simple two-stage power
normalization approach. In the first stage, the power of each
user was normalized according to its overall line power, while
the second stage was the simple overall power normalization of
VDSL. In a more effective approach, [64] optimized the perline power normalization scheme via a linear program that further improved the performance and balanced the rates among
the users.

Ordered successive interference cancelation
The nonlinear interference cancelation schemes for the downstream do not require power normalization, as their power
increase is negligible (e.g., because of the modulo operation in
THP). However, the issue of user ordering for successive
interference precoding becomes very important. As the channel gains depend on the decoding order, the users precoded
earlier experience higher data rates, while those precoded later
achieve lower data rates, creating undesirable rate variance
among users. Several works (e.g., [65] and [66]) have suggested novel approaches for user ordering that both improve the
total throughput and reduce the rate variance. In the upstream,
the ordering affects the performance of the ZF-GDFE, which
strongly depends on the detection ordering due to the error
propagation effect [39].
Compared to wireless communications, the G.fast requires
more complicated processing, as it simultaneously serves
more users and with higher user rates. On the other hand, the
low rate of change in the channel and in user activity allows
a longer processing time for this optimization problem. The
problem becomes even more complicated as we take the whole
bandwidth into account and try to jointly optimize the complete network. This is discussed in the section that follows.

Joint multiuser crosstalk cancelation and dynamic
spectrum management
Dynamic spectrum management (DSM) techniques were
shown to improve the performance of DSL systems (e.g.,
[70]-[73]) by improving the resource allocation between all
users over the system bandwidth. DSM has evolved mostly
independently of multiuser crosstalk cancelation, and both
were separately considered as different levels of coordination
in DSL systems [9], [10].

Novel research approaches for G.fast consider a joint signal
and spectrum coordination. The joint implementation of crosstalk cancelation and spectrum management can be formulated
as an optimization problem that needs to derive the optimal
precoder and decoder and optimal power allocation subject to
a given set of constraints. Practical G.fast systems typically face
three power constraints.
1) The PSD mask, Pk,mask, is a constraint that represents the
maximum PSD allowed at tone k and is defined by regulation and prevents G.fast from allocating too much power in
certain frequency bands.
2) The transmit power per line, Pline, is a constraint that
accounts for the limited range of transmit amplifiers.
3) The bit cap, bcap, is a constraint for bit loading, reflecting
the limited capabilities of the modulator, which does not
allow constellation sizes greater than a threshold.
It is noted that similar optimization problems have been
studied extensively for wireless networks [74]-[77]. However,
the proposed algorithms from wireless systems are not directly applicable to the G.fast context because of the large numbers of users in DSL systems, the constraint of per-line power
Pline, and the PSD mask Pk,mask (i.e., the existing per-antenna
power constraint). The joint sum-rate optimization of the precoder and the powers can be obtained, for example [68], as
max
Fk, p k

/ / } k,n (Fk, p k), 6k
n

diag (Fk F kH) # p k,mask, 6k

s.t.

/ diag (Fk F kH) # p line, 6k
k

} k, n (Fk, p k) # b cap, 6k = 1, gK, 6n = 1, gN, (17)

where } k, n (Fk, p k) is the achievable rate for the line n and
tone k for a given set of power allocation vectors p k ! R 1 # N
and precoding matrix Fk ! C N # N in the desired interference cancelation scheme. The vectors p k,mask ! R 1 # N and
p line R 1 # N denote the PSD masks and per-line powers for N
users at tone k, respectively.
The case of MMSE precoding was solved using a closedform solution for the precoders, given the power allocation
per tone per line and using the uplink-downlink duality [68].
Recently, the MMSE precoding was proposed in the nonlinear
THP framework [69]. The algorithms to solve these optimization problems are quite complex for G.fast systems. However, these techniques are useful, as they provide a computable
bound to compare computationally efficient algorithms for
other optimization problems.
To reduce the algorithm complexity, the DSM algorithms
can be applied with a fixed canceler structure, such that the
transmit power optimization minimizes the residual crosstalk
and improves the performance. It was shown [9], [10] that the
application of ZF-GDFE or linear ZF canceler decouples the
power allocation problem and essentially becomes independent
for each user. In this scenario, an iterative water filling (IWF)
algorithm can be applied for each user to selfishly maximize

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

159

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