This document summarizes research on achievable data rates for hybrid copper/fiber networks using G.fast technology. It finds that linear precoding methods like zero-forcing perform well for shorter copper line lengths, while nonlinear methods have advantages for longer lines. The work analyzes performance losses from implementation limitations and proposes optimizing the transmit spectrum to improve achievable rates by incorporating these limitations into the optimization process. Rate-reach curves are generated based on a statistical channel model and constraints from the G.fast standard.

1. Technische Universität München
Achievable Rates with Implementation Limitations for G.fast-
based Hybrid Copper/Fiber Networks
ICC 2015 - Symposium on Selected Areas in Communications (ICC15 SAC)
Rainer Strobel, Michael Joham and Wolfgang Utschick
c 2015 IEEE. Personal use of this material is permitted. However, permission to reprint/re-
publish this material for advertising or promotional purposes or for creating new collective
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of this work in other works must be obtained from the IEEE.
Department of Electrical Engineering and Information Technology
Associate Institute for Signal Processing
Univ.-Prof. Dr.-Ing. Wolfgang Utschick

2. Achievable Rates with Implementation Limitations
for G.fast-based Hybrid Copper/Fiber Networks
Rainer Strobel∗‡, Michael Joham∗, Wolfgang Utschick∗
∗Fachgebiet Methoden der Signalverarbeitung, Technische Universität München, 80290 München, Germany
{rainer.strobel,joham,utschick}@tum.de
‡Lantiq Beteiligungs-GmbH & Co. KG, 85579 Neubiberg, Germany, rainer.strobel@lantiq.com
Abstract—Hybrid copper/fiber networks bridge the gap be-
tween the fiber link and the customer by using copper wires
over the last meters. This solution combines energy efficiency
and low cost of the copper access network with high data rates
of a fiber connection. However, the fiber to the distribution point
(FTTdp) network must prove its ability to convey data at fiber
speed over copper wire bundles under the spectral constraints
of the copper access network.
This work investigates achievable data rates of the FTTdp
network. It provides an analysis of the sources of performance
loss in a system implementation due to complexity limitations.
Methods to improve achievable rates are shown, that are based
on incorporating the limitations in the optimization process.
Achievable data rates are analyzed in terms of rate vs.
reach curves, based on a statistical channel model and the
ITU standard G.fast. The results indicate that optimized linear
methods perform well on shorter lines, while nonlinear methods
have advantages for long lines.
I. INTRODUCTION
The main requirements to the next generation broadband
access network are increasing data rates, energy efficiency,
and higher flexibility for the deployment of access nodes.
A hybrid network [1] of fiber links to a remotely powered
distribution point box (DP box) which is connected to the
customer premises equipment (CPE) via traditional copper
wires, as shown in Fig. 1, shall satisfy these requirements.
CPE
CPE
CPE
Fiber
DP
Copper
CPE
CPE
Fig. 1. Hybrid copper fiber network
To address increasing consumer demand on higher data
rates, ITU is working on the next generation copper access
technology called G.fast [2]. Based on the spectral require-
ments defined in [3] and the knowledge of the channel
This work has been founded by the research project “FlexDP - Flexible
Breitband Distribution Points”, funded by the Bayerische Forschungsstiftung.
characteristics of short copper wires at high frequencies [4],
outer bounds on achievable rates of copper links in hybrid
access networks are shown. Fiber links do not limit capacity
in the hybrid network because their capacity is far beyond that
[5]. Implementable FTTdp systems do not achieve channel
capacity due to various limitations in copper and fiber trans-
mission. The fiber transceivers, using GPON [6] or 10G EPON
[7] technology shall not limit overall performance, because
they are dimensioned with respect to the copper rates.
This paper compares linear zero-forcing, Tomlinson Ha-
rashima precoding (THP), and theoretical bounds for wireline
MIMO communication under spectral mask and per-line sum-
power constraints. A low complexity gradient-based solution
for spectrum optimization with linear precoding is presented.
After a brief discussion of THP, the methods are compared
in multi-line rate vs. reach simulations under realistic condi-
tions for an implementation. The simulations show that linear
precoding is well suited for short lines.
II. SYSTEM MODEL
The key methods used for fast wireline access in the FTTdp
network are summarized in [8] using a frequency spectrum
from 2 MHz to 106 or 212 MHz with DMT [9] multi-carrier
modulation and synchronized time division duplexing.
Energy efficiency is very important for the G.fast DP
because they do not have a local power supply, but are fed
with power from the subscribers via the copper wires.
A. Architecture
This work investigates the downlink transmission from a
multi-port DP. The FTTdp network according to [10] consists
of small access nodes, which are placed close to the sub-
scribers and supply a limited number of customers. The focus
is on optimization of linear precoding which is compared to
the achievable data rates of nonlinear techniques.
Every subscriber uses a set of subcarriers for data trans-
mission over the crosstalk channel described by the matrix
H(k)
∈ CL×L
for subcarrier k and a system with L lines.
The linear precoder matrix P (k)
∈ CL×L
at the DP box is
used to pre-compensate crosstalk between the lines.
The gain-scaling is collected in diagonal matrices S(k)
=
diag(s
(k)
1 , . . . ,s
(k)
L ) ∈ RL×L
which are inverted at the receiver
side. The operation diag(.) transforms a vector into a diagonal

3. matrix and a diagonal matrix into a vector. The receivers also
apply the diagonal equalizer G(k)
= diag(g
(k)
1 , . . . ,g
(k)
L ) ∈
CL×L
to compensate channel distortion.
The system model is given by
ˆu(k)
= S(k)−1
G(k)
H(k)
P (k)
S(k)
u(k)
+ n(k)
(1)
for each subcarrier k and linear precoding. The transmit and
receive signal vectors are u(k)
∈ CL
and ˆu(k)
∈ CL
,
respectively. The transmit signals are assumed to be statis-
tically independent, zero-mean, unit power QAM signals. The
receivers experience additive white Gaussian noise (AWGN)
n(k)
∼ NC(0,σ2
I). The corresponding block diagram is
shown in Fig. 2.
H ˆu
n1 s−1
1
PS
˜p1
u
g1
˜p2
˜pL
.
.
.
n2 s−1
2
g2
nL s−1
L
gL
.
.
.
Fig. 2. Downstream system model representing one subcarrier
The channel matrix H(k)
is square because there is one
transmitter and one receiver for each subscriber, connected
by one twisted pair. Due to complexity limitations, the linear
zero-forcing precoder [11] satisfying
G(k)
H(k)
P (k)
= I (2)
is preferred, where I is the L×L identity matrix. The precoder
is assumed to be a pseudo-inverse of the channel matrix scaled
to unit diagonal as indicated in Fig. 2
P (k)
= [H(k)
]+
· diag(diag([H(k)
]+
))−1
(3)
where []+
denotes the Moore-Penrose pseudo-inverse. The
normalization to unit diagonal is useful in vectored wireline
systems to separate the system into a crosstalk canceler
˜P (k)
= P (k)
− IL which handles the crosstalk couplings
and a per-line signal processing part. Direct channel scaling is
implemented with the scale matrix S which is known to the
receivers as proposed by the G.fast standard.
B. Spectral Constraints
Two power constraints must be satisfied for data transmis-
sion over copper wires [3]. One is the power spectral density
(PSD) of the signal on the line which is limited by a certain
PSD mask, defined by regulation. This translates into a power
limit p
(k)
mask ∈ RL
per line l and per subcarrier k at the precoder
output, i.e.,
diag P (k)
S(k)
P (k)
S(k)
H
≤ p
(k)
mask. (4)
It can be seen that the spectrum is controlled via the matrix
S(k)
= diag(s
(k)
1 , . . . ,s
(k)
L ) as shown in Fig. 2.
The second constraint comes from limited capabilities of
the transmit amplifier. It gives a maximum per-line sum-power
vector psum ∈ RL
for L lines according to
K
k=1
diag P (k)
S(k)
P (k)
S(k)
H
≤ psum (5)
for a system with K subcarriers.
C. Channels
The channel model from [4] with a 0.5mm PE loop accord-
ing to [12] is used to create channel matrices which corre-
spond to cable binders with different line length and different
numbers of twisted pairs for each binder. The individual pairs
within the binder also have different length from the DP to
the customer premises, as the system setup in Fig. 1 shows.
D. Complexity Constraints
There are various limitations which make a data transmis-
sion system implementable, but cause a gap between channel
capacity and achievable rates.
1) Precoding: Dirty paper coding (DPC) [13] is the coding
scheme that achieves channel capacity [14]. While DPC is
too complex for realistic systems, there exist implementable
nonlinear techniques like THP which approximates DPC.
Linear zero-forcing precoding [11] are preferable in terms
of complexity. Achievable rates of nonlinear techniques will
be compared to linear precoding with an additional spectrum
optimization step as described in Subsec. III-B.
2) Constellation Mapping: Analog-to-digital conversion
and constellation mapping and de-mapping is done with a
limited digital resolution which results in a maximum number
bmax of bits per channel use. G.fast supports up to 12 bit per
line and per carrier, using a 4096-point QAM constellation.
It must be noted that a finite set of constellations is used,
supporting only an integer number of bits to be transmitted.
3) Channel Coding and Bit Error Rate Constraints: Limi-
tations in channel coding and bit error rate (BER) constraints
further reduce the achieved rates compared to channel capacity.
These are collected in a SNR gap Γ [15] which is incorporated
into the bit rate computation. For the Reed-Solomon and
Trellis codes used in G.fast, the SNR gap can be assumed to be
Γ = 9.8 dB [16]. The target BER for wireline communication
is 10−7
plus 6 dB SNR margin to protect against impulsive
noise. Redundancy added by the code Roh trellis is 0.5 bits per
carrier for Trellis and ηRS = 237/255 for Reed-Solomon code.
4) Modulation, Duplexing and Protocol Overhead: In
G.fast, the time division duplexing (TDD) scheme, the cyclic
prefix of DMT modulation and channel estimation symbols
and control channels cause some overhead. This is included
into the rate calculation with an efficiency η and an overhead
rate Roh to compute he actual rates. With a standard configu-
ration, this reduces the data rate by η = 0.965 (2 SYNC and
8 guard out of 288 symbols) and up to Roh = 13 Mbit/s for
overhead channels. The cyclic prefix overhead is considered
in the rate calculation by an increased symbol duration tsym.

4. E. Sum-Rate Objective Function
The channel matrix can be split into one vector h
(k)
l ∈
CL
per subscriber H(k)
= h
(k)
1 , . . . ,h
(k)
L
T
. Similarly, the
precoder matrix for zero-forcing precoding is split into one
column vector p
(k)
l ∈ CL
per subscriber, i.e. P (k)
=
[p
(k)
1 , . . . , p
(k)
L ].
With the precoder and equalizer definition from (2) and (3),
the achievable bits per channel use for line l and subcarrier k
reads as
b
(k)
l = log2 1 + h
(k),T
l p
(k)
l s
(k)
l
2
(Γσ2
)−1
(6)
and the corresponding line rate is
Rl = η
1
tsym
K
k=1
b
(k)
l − Roh (7)
with tsym including the cyclic prefix and windowing. Note that
(7) includes the overhead as described in Subsec. II-D4.
With a limited modulation alphabet b
(k)
l ≤ bmax (see
Subsec. II-D2), the actual number of bits ˆb
(k)
l per line l and
subcarrier k is given by
ˆb
(k)
l = min b
(k)
l , bmax (8)
which is incorporated into the rate optimization for the achiev-
able data rates of the G.fast system. (8) indicates that the
actual rate is a discrete function. For a discussion of the
corresponding combinatorial optimization problem, see [17].
In [18] it is shown that the rounding operation in (8) can
be ignored for spectrum optimization when the SNR gap Γ
is selected accordingly. Therefore, (6) is used as an objective
function for the following steps, while reported data rates are
calculated using (8).
III. TRANSMIT SPECTRUM SHAPING
Besides other coding and framing parameters, the transmit
spectrum in combination with precoding is an important
parameter to maintain system performance. In the system
according to Fig. 2 and (1), spectrum shaping is implemented
using the gain matrices S(k)
. The optimization problem is
formulated in terms of power vectors x(k)
∈ RL
with
x
(k)
l = s
(k)
l
2
(9)
to end up with linear constraints.
A. Power Scaling
In VDSL Vectoring systems [19], a suboptimal precoding
strategy applies linear zero-forcing precoding and power scal-
ing to satisfy the spectral mask [20].
For this simplified approach, the power values are found by
x(k)
= 1L · min
l
p
(k)
mask l P (k)
P (k)
H −1
ll
(10)
where []ik denotes the element from row i and column k
of a matrix and 1L is the L-dimensional all-ones vector. An
improved method is solving a linear program and maximizing
the received power, as presented in [21].
B. Sum-Rate Optimal Spectrum
Work on the sum-rate maximization with the given con-
straints is available in [22], where it is proposed to solve
the optimization problem in an iterative power allocation
algorithm. However, the method does not consider precoded
communication systems. For linear precoding in the wireless
context under per-antenna power constraints, the work of [23]
presents a water-filling-like algorithm for the two-users case
and [24] proposes a method based geometric programming.
These approaches on the per-antenna power constraint do not
consider an additional group-wise sum-power. The described
algorithms are not directly applicable to high-dimensional
problems as they occur in the G.fast context.
We focus on the optimization of the complexity constrained
sum-rate of a DP. For linear zero-forcing precoding, the limited
modulation alphabet can be translated into a power constraint
per subcarrier p
(k)
bmax ∈ RL
, which is given by
p
(k)
bmax l = 2bmax
− 1
Γσ2
h
(k),T
l p
(k)
l
2 (11)
such that x
(k)
l ≤ p
(k)
bmax l. The constrained sum-rate maximiza-
tion problem can be written as
max
x(1),...,x(K)
L
l=1
Rl ∀k : A(k)
x(k)
≤ d(k)
∀ k = 1, . . . ,K
K
k=1
P (k)
⊙ P (k)
∗
x(k)
≤ psum (12)
where ⊙ is the Hadamard product, i.e., the element-wise
product of matrices. The per-subcarrier constraints are linear
inequality constraints which are collected in constraint matri-
ces A(k)
and the constraint vector d(k)
A(k)
=


P (k)
⊙ P (k) ∗
IL
−IL

 d(k)
=



p
(k)
mask
p
(k)
bmax
0L


 (13)
where 0L is the L-dimensional all-zeros vector. This compact
formulation includes the spectral mask, the limited modulation
alphabet and the positiveness of the transmit power.
C. Gradient-Based Optimization (Optimized LP)
For wireline communication, the transmission channel
varies slowly over time. Therefore, the precoder matrices P (k)
also change over time. A projected gradient-based adaptive
algorithm is used to follow small changes of the channel.
Furthermore, the described algorithm scales well for systems
with thousands of subcarriers and many lines.
Lagrange duality allows to separate the problem into per-
subcarrier sub-problems which can be solved efficiently. Start-
ing with the Lagrangian function
Φ(x,λ) =
L
l=1
−Rl+λT
K
k=1
P (k)
⊙ P (k)∗
x(k)
− psum
(14)

5. where the Lagrangian multipliers are collected in λ ∈ RL
and
the rate definition from (6) and (7). The corresponding dual
function Θ(λ) is given by
Θ(λ) = min
x(1),...,x(K)
L
l=1
−Rl + λT
K
k=1
P (k)
⊙ P (k)∗
x(k)
−p(k)
sum s.t. A(k)
x(k)
≤ d(k)
∀k = 1, . . . ,K (15)
where the per-subcarrier constraints are comprised in the
inequalities A(k)
x(k)
≤ d(k)
.
To calculate the minimum in (15) with a gradient method,
the gradient step
x
(k)[t]
grad l = x
(k)[t−1]
l + (16)
αx x
(k)[t−1]
l + Γ h
(k),T
l p
(k)
l
−2
σ2
−1
−
L
v=1
λv|p
(k)
vl |2
with a step size αx is performed for each time instance t.
The projection moves the power allocation outside the
constraint set. For the projection back onto the constraint set,
a set Iviolated of row indices of violated constraints is defined.
For a certain constraint j ∈ Iviolated, the projection
x[t+1]
= x
[t]
grad + aj · (aT
j aj)−1
(dj − [Ax
[t]
grad]j) (17)
with the corresponding row aj ∈ RL
of the constraint set
matrix A = [a1, . . . ,a3L]T
can be done. (17) is performed
for the index that minimizes the distance to the constraint set
e =
3L
j=1 | max(aT
j x[t+1]
−dj,0)|2
. This is repeated until the
constraints are satisfied within a certain precision.
The dual optimization problem of finding the Lagrangian
multipliers λ is given by
max
λ
Θ(λ) s.t. λ ≥ 0 (18)
which can also be solved with a gradient method.
As shown in [25], the gradient step for Θ(λ) is
λ
[t]
grad = λ[t]
+ αλ
K
k=1
P (k)
⊙ P (k)
∗
x(k)
− psum
(19)
followed by the projection step
λ
[t+1]
l = max λ
[t]
grad l, 0 . (20)
The gradient-based algorithm can be used for initialization as
well as for permanent updates. Fixed step sizes of αx = 1e−5
and αλ = 2 were used.
D. Precoder Column Reduction
Whenever one or more of the power values x
(k)
l of the
optimal solution become zero, the corresponding column of
the precoder matrix P (k)
can be set to zero which results in
additional degrees of freedom for the matrix inversion.
The work of [26] shows methods to use these degrees of
freedom and find the optimal pseudo-inverse. This paper’s
simplified approach is based on the pseudo-inverse as defined
in (3). The algorithm searches the appropriate set of active
subcarriers by successively removing the weakest subcarrier
until all subcarriers have sufficient SNR to achieve the mini-
mum bit loading, e.g. bmin = 1.
E. Theoretical Limits
The presented optimization approach is compared to outer
bounds channel capacity of the FTTdp network. Methods to
compute sum-capacity of the MIMO broadcast channel under
the given constraints are known, see e. g. [27]. However,
standard convex solvers cannot be applied to the given problem
due to the problem size of jointly optimizing several thousand
covariance matrices with some tens of thousands of linear and
semidefinite constraints.
To provide an outer bound, the per-line sum-power con-
straint is removed. Without this constraint, the approach of
[27] is used to compute the MIMO broadcast sum capacity
C
(k)
sum for every subcarrier k independently by
C(k)
sum = min
Σ(k)
max
Φ(k)
log2 det I + H(k)
Φ(k)
H(k),H
(Σ(k)
)−1
s.t. [Φ(k)
]ll ≤ p
(k)
mask i∀i = 1, . . . ,L
diag(Σ(k)
) = σ2
1L
Φ(k)
0 Σ(k)
0 (21)
with the transmit covariance matrix Φ(k)
and a worst case
noise covariance matrix Σ(k)
. The individual single user upper
bounds are given by
C
(k)
l = max
Φ(k)
log2 det 1 + h
(k),T
l Φ(k)
h
(k),*
l σ2
s. t. [Φ]
(k)
ii ≤ p
(k)
mask i∀i = 1, . . . ,L, Φ(k)
0. (22)
The outer bound for the rate region is given by the rate
vectors r = [R1, . . . ,RL]T
which satisfy
Rsato = r : Rl ≤
K
k=1
C
(k)
l ;
L
l=1
Rl ≤
K
k=1
C(k)
sum . (23)
A similar outer bound can be defined for the case of a
limited modulation alphabet to be
Rmod = r : Rl ≤
K
k=1
min(C
(k)
l ,bmax);
L
l=1
Rl ≤
K
k=1
C(k)
sum .
(24)
This represents the upper bound for achievable data rates of
nonlinear coding.
F. Nonlinear Precoding
Especially for G.fast implementations using up to 212 MHz,
nonlinear precoding is discussed as an alternative to linear
methods. The reference implementation used for nonlinear
precoding is a zero-forcing Tomlinson Harashima precoder
according to [28]. Spectrum optimization is mentioned in [28],
but the proposed method is based on assumptions for the
channel matrices which do not hold for G.fast channels.
The nonlinear precoder consists of a feedback matrix P
(k)
b
and a feed forward matrix P
(k)
f The feedback signal uback is
given by
u
(k)
back = P
(k)
b u
(k)
mod. (25)

6. The nonlinear operation is performed according to
u
(k)
mod = mod u
(k)
back + S(k)
u(k)
(26)
where mod (.) is the modulo operation. The precoding
output signal y(k)
is given by
y(k)
= P
(k)
f u
(k)
mod (27)
To show the capabilities of nonlinear precoding, an upper
and a lower bound are simulated. The upper bound is calcu-
lated by ignoring the per-line sum-power constraint of (5) and
taking only the spectral mask constraint (4) into account. Data
rates are approximated according to [28] using
b
(k)
TH l = log2 1 + x
(k)
l (Γσ2
|g
(k)
l |2
)−1
(28)
for bit loading calculation and limiting the bit rates to the
upper bound as shown in (8). This approach ignores that TH
precoding causes a power increase for small constellations.
Computation of the lower bound takes into account the
power increase by a power scaling. The per-line sum-power
constraint is satisfied by limiting the spectral mask to the
appropriate maximum value. The precoder input power x for
THP is given by
S(k)
= IL min
l
p
(k)
mask lE y
(k)
i y
(k)∗
i
−1
. (29)
Data rates are evaluated in a data transmission simulation with
the BER and coding constraints from II-D3.
IV. SIMULATIONS
Performance evaluation is done on random cable bundles
matching the topologies with up to 16 pairs and up to 250 m
line length. Background noise is AWGN with −140 dBm/Hz
power spectral density. The limit transmit PSD from the G.fast
standard for the 212 MHz profile with 4 dBm per-line sum-
power is used.
A. Performance Evaluation
Performance is reported in rate vs. reach curves. The high
frequency channels of the FTTdp network cause a significant
spread of the achieved data rates for a specific line length.
The rate-reach curves show average data rates at a specific
line length range for various channel realizations. The method
allows a performance comparison that depends on the line
length.
B. Results
Fig. 3 compares channel capacity with the actual data rates
when different implementation limitations are considered.
Rsato serves as an upper bound. A single rate point from the
region must be selected to create the rate vs. reach curve. For
Fig. 3, the sum-rate optimal point with the same ratio of data
rates as the single user bounds, or the closest sum-rate optimal
point to that, is used. It indicates that 2 Gbit/s are the limit for
lines at 200 m length and 3 Gbit/s are possible at 75 m line
500
1000
1500
2000
2500
3000
50 100 150 200 250
AverageRate/Mbit/s
Line length/m
Sato Bound Rsato
Nonlinear Precoding Rmod
Linear Precoding
Channel Code Γ
TDD/Framing
Fig. 3. Capacity and achievable rates including different implementation
limitations
length with the given spectral mask on a twisted pair line
with highly optimized transceivers.
The performance gap caused by the limited modulation
alphabet (II-D2) is shown by the limited bound Rmod which is
the theoretical upper bound achievable by nonlinear precoding
methods. Optimized linear methods (II-D1) neglecting imple-
mentation limitations achieve the performance as shown by the
line marked with “Linear Precoding”. Including the impact of
suboptimal coding (II-D3) with the SNR gap Γ, the overhead
Roh trellis and the code rate ηRS (II-D4) reduces performance to
the blue curve. Finally, taking into account framing overhead,
TDD overhead and DMT modulation gives the lowest curve.
The results indicate that for longer lines, improved channel
coding methods other than Reed-Solomon and Trellis coding
may give a significant performance gain. Nonlinear precoding
seems to be beneficial for longer lines with more than 100 m
length.
400
600
800
1000
1200
1400
1600
1800
2000
50 100 150 200 250
AverageRate/Mbit/s
Line length/m
Optimized LP
THP (ignore limitations)
THP (implementation)
Power Scaling
Fig. 4. Effect of spectrum optimization on achievable rates
Fig. 4 shows the performance of the proposed spectrum
optimization method. Without per-line spectrum optimization,
when the power scaling approach of (10) is used, the perfor-
mance of zero-forcing precoding is very poor. The optimized
linear precoding improves performance for short and long
lines. Comparing with the upper bound of THP without sum-
power limit and ignoring the power increase due to the modulo
operation shows the potential of nonlinear methods.

7. But a simple power scaling approach for THP, where the
transmit power for each carrier is reduced such that the output
power does not violate the spectral mask and the sum-power
limit results in poor performance on longer lines. This must
be resolved to be able use THP efficiently.
0
0.2
0.4
0.6
0.8
1
400 600 800 1000 1200 1400 1600 1800 2000
FR(R)
Rate R/Mbit/s
Power Scaling 100m
Tomlinson Harashima 100m
Optimized Linear 100m
Power Scaling 200m
Tomlinson Harashima 200m
Optimized Linear 200m
Fig. 5. Rate distribution for 100 m and 200 m line length
For two selected line lengths of 100 m and 200 m, the
cumulative distribution functions FR(R) of the achieved data
rates R are shown in Fig. 5. This allows to evaluate not only
the average rates, but also the worst case and best case rates
for the given network topology. For each length, the achieved
data rates of power-scaling are compared to the results of
the presented optimized approach. The THP implementation
is shown for comparison. The results show that not only the
average rates improve with the proposed method, but also the
worst case and best case rates.
V. CONCLUSION
Simulation results indicate good performance for optimized
linear methods on shorter lines. But for longer lines at high
frequencies, nonlinear methods potentially give a significant
performance gain.
The implementation of Tomlinson Harashima precoding
which was used as reference, performs well compared to linear
zero-forcing with power scaling. But it does not achieve the
same performance as the optimized linear methods. Therefore,
further research on spectrum optimization methods for non-
linear precoding are of high interest and the algorithm from
Subsec. III-B gives a good starting point.
Non-zero-forcing methods may also achieve higher per-
formance in practical scenarios. While this paper focuses
on the sum-rate maximization problem, power minimization
is another optimization problem of high interest for FTTdp
network. Minimizing power consumption with respect to rate
constraints is another optimization problem of practical rele-
vance for this system.
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