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REVISED UPSTREAM POWER BACK-OFF FOR VDSL
Driton Statovci, Tomas Nordstr¨
om, Rickard Nilsson ∗
Telecommunications Research Center Vienna (ftw),
Donau-City-Straße 1/3, A-1220 Vienna, Austria
Emails:{statovci, nordstrom, nilsson}@ftw.at
Vladimir Oksman
Infineon Technologies North America,
485 Route 1 South, Iselin NJ 08830, USA
Email: Vladimir.Oksman@infineon.com
ABSTRACT
Accurate upstream power back-off (PBO) parameters are
needed by operators deploying very high-speed digital sub-
scriber line (VDSL) modems. Although a standardized PBO
method for VDSL exist, the standard gives little or no guid-
ance to an operator how to establish these optimized PBO
parameters for its particular network and customers. In this
paper, we present an efficient algorithm based on the Nelder-
Mead simplex search which calculates optimized upstream
PBO parameters. To make the PBO parameter calculation in-
dependent of the network scenario we present a new method
for establishing worst-case far-end crosstalk (FEXT) noise,
which is based on virtual modems.
1. INTRODUCTION
Very high-speed digital subscriber line (VDSL) is one of the
latest introduced DSL technology, which currently utilizes
frequencies up to 12 MHz. It uses frequency division du-
plex (FDD) transmission scheme in order to avoid near-end
crosstalk (NEXT) noise between VDSL systems. Further-
more, for robustness reasons, current standardized VDSL sys-
tems use two frequency bands for each transmission direction,
i.e., four band plans are employed.
CO/Cabinet
l
l
1
U
Rx−1
Tx−U
Tx−1
Rx−U
Fig. 1. Illustration of near-far problem in VDSL.
Power back-off (PBO) is used in VDSL to solve the near-
far problem in the upstream transmission direction, as illus-
trated in Fig. 1. With upstream PBO, modems located close to
central office (CO) or cabinet should reduce their transmitted
power spectral densities (PSDs) in the upstream direction in
order to improve the performance of modems located further
away.
∗This work was partially financed by the Austrian Kplus programm
Many PBO methods have been proposed for VDSL, as
described by Schelstraete in [1] and the references therein.
However, standardization bodies have agreed to use the ref-
erence PBO method [2] where different reference PSDs have
been defined for each upstream band. The actual parameters
used for the reference PBO in the current VDSL standards
were established by Schelstraete [1] and Oksman [3]. They
both used a kind of exhaustive search to find optimized PBO
parameters, which is time consuming. To circumvent this
problem, we show how to calculate the PBO parameters by
using the Nelder-Mead simplex algorithm [4].
To make the calculation of PBO parameters independent
of the network scenario a worst-case far-end crosstalk (FEXT)
noise concept has been introduced [1]. However, we have dis-
covered that this concept does not always represent the worst-
case, especially for discrete multi-tone (DMT) based VDSL
systems. Therefore, we present a new improved way to estab-
lish the worst-case FEXT noise, which is based on a concept
of virtual modems.
The paper is organized as follows: Section 2 briefly de-
scribes some preliminaries concerning VDSL systems; Sec-
tion 3 shows our improved method to calculate the worst-case
FEXT noise; Section 4 presents the proposed algorithm to
find the optimized PBO parameters; and Section 5 summa-
rizes the major findings in this paper.
2. PRELIMINARIES
The upstream bitrate of a VDSL system is calculated, based
on Shannon’s formula, as
R=fU
log 1+SNR(f)
Γdf , (1)
where fis the frequency, fUis the set of frequencies used in
the upstream direction, Γis the signal-to-noise ratio (SNR)
gap, and SNR(f)is the received signal-to-noise ratio. The
SNR can be expressed as
SNR(f)= PRx(f)
PTotN(f)=|H(f)|2PTx(f)
PF(f)+PBGN(f),(2)
IV633142440469X/06/$20.00©2006IEEE ICASSP2006
where PRx(f)is the received signal PSD, PTotN(f)is the to-
tal noise PSD at the receiver. PTx(f)represents the transmit
PSD, |H(f)|2is the cable insertion loss, PF(f)is the FEXT
noise from other VDSL systems, PBGN consists of alien noise
and any other type of background noise. The NEXT noise is
avoided due to the FDD transmission. The FEXT noise de-
pends on the modems transmit PSDs and the FEXT crosstalk
couplings, which are typically quite random in nature.
However, within VDSL standardization a conservative 99%
worst-case crosstalk coupling model is used
PF(f)=KFN0.6f2lx|H(f,l)|2PTx(f),
where lxis the coupling length, Nis the number of disturbing
VDSL modems, and |H(f,l)|2is the insertion loss of disturb-
ing modems. The constant KFis empirically determined by
ETSI to be −45 dB at 1 MHz, when the frequency fis ex-
pressed in MHz and the length lin km [2].
2.1. The Reference PSD Method for PBO
The reference PSD method was developed after observing
that many PBO methods could be described by a certain de-
sired received PSD. This reference PSD, PR(f), which deter-
mines the maximum received PSD, is a parameterized func-
tion of frequency that can be designed to meet certain ob-
jectives. One such common objective is the maximum reach
for a predefined set of bitrates. Even if almost any shape of
PR(f)is conceivable, for practical reasons it was decided in
the standardization process [1, 3] to select a reference PSD
model expressed as
PRdBm(f)=α+βf, [dBm/Hz],(3)
where fis given in MHz, and αand βare the parameters that
are free to be determined in order to maximize the reach. It
was also decided that independent reference PSDs should be
assigned for each upstream band.
In addition, modems need also adhere to a maximum al-
lowed transmit PSD, Pmax(f). Hence, the transmit signal
PSD of a particular user nis given by
PTx,n(f)=minPR(f)|H(f, ln)|−2,Pmax(f).(4)
The optimized reference PSDs depends on the alien noise,
the maximum transmit PSD mask, cable types, the network
topology, and the services (bitrates) the operator wants to of-
fer. Their influence on the reference PSDs is analyzed in [1].
To make the reference PSDs independent of any partic-
ular network scenario Schelstraete [1] proposed to use the
worst-case FEXT noise model. That is, the reach for a par-
ticular set of PBO parameters will be based on the scenario
that gives the worst-case FEXT. We can then use the follow-
ing cost function, which minimizes the maximum between the
reach without and with PBO of all protected bitrates, to find
the optimized reference PSDs:
y=minmax
i{lNoPBO(Ri)−lPBO (Ri)},(5)
where Ridenote the bitrates for which the reference PSDs are
optimized; lNoPBO(Ri)denotes the reach without PBO and
collocated disturbers; and lPBO (Ri)denotes the reach with
PBO and worst-case FEXT. A similar approach was used in
[1, 3] to find the optimized reference PSDs.
There are of course another ways to define the optimiza-
tion criteria for the cost function yin (5). For instance, we can
minimize the differences between lNoPBO(Ri)and lPBO(Ri)
for different Risuch that all bitrates are protected equally or
differently based on some constraints. However, in this paper
we restrict ourself to (5), since we think it is a good optimiza-
tion strategy.
3. WORST-CASE FEXT NOISE
From (1) and (2) one can easily see that for the case when all
frequencies can be utilized for transmission, the worst-case
performance appears when the integral of the FEXT noise is
the greatest. This worst-case FEXT noise, PF-WC, was in [1]
calculated by assuming that all disturber modems are collo-
cated. Depending on the the loop length of victim modem
lVand the disturbing (collocated) modems l0the following
expression for the worst-case FEXT noise has been used:
PF-WC(f)=KFN0.6f2lVPR(f)if lV≤l0
KFN0.6f2l0PR(f)if lV>l
0
,(6)
where l0is determined by the maximum of
Φ(li)=fU
φ(li,f)df (7)
=fU
KFN0.6f2limin PR(f),|H(f,li)|2Pmax(f)df .
(8)
Thus, Φ(li)<Φ(l0)for all li=l0.
The value l0can be found by integrating (8) with PR=
PR-1U and PR=PR-2U for the first and second upstream
bands, since the reference PSDs are independently defined for
both upstream bands. Superscripts 1U and 2U denote the first
and second upstream bands, respectively. For instance, for
PR-1U dBm =−60−17√fand PR-2U dBm =−60−12√fwe
have found that the length l0= 612 m causes the worst-case
FEXT, for which length the PSD of FEXT noise is shown in
Fig. 2 with dashed line. Assume now that we are searching
the loop reach for a low bitrate. Due to high loop attenuation,
modems will utilize, for example, only the frequencies of the
first upstream band. Thus, the FEXT noise that is determin-
ing the reach depends only on the PSD of FEXT noise on the
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0 2 4 6 8 10 12
−140
−135
−130
−125
−120
−115
−110
Frequency (MHz)
FEXT noise from 20 distrubers (dBm/Hz)
l0 (612m)
l01 (893m)
l02(611m)
Fig. 2. FEXT noise from 20 disturbers for reference PSDs:
PR-1U dBm =−60 −17√fand PR-2U dBm =−60 −12√f.
first upstream band. When calculating the length that causes
the worst-case FEXT on only the first upstream band we have
found that l01 = 893 m. For this length the PSD of FEXT
noise is shown in Fig. 2 with solid line and is approximately
2 dBm/Hz above the case when l0= 612 m. For illustra-
tion purposes in Fig. 2 is shown also the length l02 = 611 m
that causes the worst-case FEXT only on the second upstream
band.
CO/Cabinet
l
l
l02
01
V
Rx−2U
Rx−1U
Rx−V
Tx−2U
Tx−1U
Tx−V
Fig. 3. Our proposed scheme to calculate the worst-case
FEXT noise, by introducing virtual modems for each up-
stream band
To deal with the problems described in previous paragraph
we propose to use ‘virtual modems’ which only transmit in a
single upstream band and each of them is placed as a worst-
case disturber in a particular band. In Fig. 3, Tx-1U and Tx-
2U denote disturbing ‘virtual modems’, which transmit only
in the first and second upstream bands, respectively. Now, for
the two-band case the worst-case FEXT noise is calculated as
PF-WC(f)=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
af2lVPR-1U(f)+PR-2U(f)if lV≤l01 ,l
02
af2l01PR-1U (f)+lVPR-2U (f)if l01 <l
V<l
02
af2lVPR-1U(f)+l02 PR-2U (f)if l02 <l
V<l
01
af2l01PR-1U (f)+l02 PR-2U(f)if lV≥l01 ,l
02
(9)
where a=KFN0.6,l01 and l02 are the lengths for which (8)
achieves its maximum in the first and second upstream bands,
respectively. It is worth mentioning that even this proposed
scheme does not represent the true worst-case environment,
since due to constraint (4), Φ(li)≤Φ(l0)in (7) always holds
but not φ(li,f)≤φ(l0,f)for any f. Theoretically, the net-
work scenario that causes the worst-case noise can be built
as in Fig. 3 with the number of virtual modems equal to the
number of subcarriers used in the upstream and each of them
transmitting only in one subcarrier. However, this will make
the computation very challenging and furthermore with our
proposed scheme we are very close to the network scenario
that causes the worst-case FEXT noise.
4. THE OPTIMIZATION ALGORITHM
Previous attempts [1, 3] to find the α’s and β’s in (3) for
both upstream bands, which minimize the cost function in (5)
use some form of exhaustive search. This strategy makes the
search for the optimized reference PSDs very challenging and
time consuming due to a large search space for α’s and β’s.
Instead, we propose to use the Nelder-Mead simplex algo-
rithm [4] to search for α’s and β’s. The pseudo-code of the
proposed algorithm is listed as Algorithm 1.
Algorithm 1 PBO optimization based on Nelder-Mead for
two upstream bands
Initial Values
Ri{Bitrates to protect}
x=[α1U,β
1U,α
2U,β
2U]
Main Function
repeat
y, x=NelderMead(@ReachDiff ,x)
until the specified accuracy have been reached
Function y=ReachDiff (Ri)
Find lNoPBO (Ri)for all Ri
Find l01 and l02
for each Rido
Test Ri0{Test if Rican be achieved for zero loop
length}
if Ri0true then
Find lPBO (Ri,α
1U,β
1U,α
2U,β
2U)
{For each tested length, PF-WC is calculated as in (9)}
else
lPBO (Ri,α
1U,β
1U,α
2U,β
2U)=0
end if
end for
y=minmax
i{lNoPBO(Ri)−lPBO (Ri)}
The Nelder-Mead algorithm starts with the single initial-
ization point X0=xwhich has Ddimensions (for two up-
stream bands D=4) and then the Nelder-Mead simplex al-
gorithm constructs an initial simplex with D+1points. The
additional Dpoints are calculated by
Xd=X0+λed,for d=1...D (10)
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where the ed’s are Dunit vectors and λis a constant. For our
case the search works well with any λvalue between 0.05 and
0.1. Then, depending on the outcomes yof ReachDiff func-
tion, the simplex figure is changed according to the Nelder-
Mead algorithm as explained in [4] until the diameter of the
simplex and the difference between the two minimum values
of yhave reached the specified accuracies.
Since the power-sum FEXT is a concave function of loop
length, the lengths l01 and l02 can be found by using the
golden section search algorithm. To find the reach lNoPBO the
bisection line search algorithm or any another line search al-
gorithm can be used, because for that case the SNR and there-
fore also the bitrate are decreasing functions of loop length.
However, the simple bisection search can fail to find the reach
lPBO, since for specific α’s and β’s there are cases where the
bitrate is a constant function of loop length. This arises for the
reaches that lie above the lengths where virtual modems are
placed and below the length when the received PSD begins
to be lower than the reference PSD due to Pmax constraint.
For all these reaches the noise is not increased and the re-
ceived signal is the same, which results in equal bitrates. If
the bitrate that we search for lies exactly in that flat area the
bisection will fail to find the maximum reach. However, the
bisection algorithm can be extended in a straightforward way
to deal with such cases.
It should also be noted that our algorithm can still find
optimized PBO parameters when the worst-case FEXT is cal-
culated without virtual modems as in (6). This is due to the
fact that also for this case the bitrate is non-increasing func-
tion of loop length.
4.1. Simulation Results and Discussions
The proposed algorithm can be used with any number of up-
stream bands. However, for easy comparison with the pre-
viously published results, we have used simulation parame-
ters according to ETSI’s VDSL standard. Thus, we use Γ=
12.3dB as the SNR gap, cable TP100, and the FEXT noise
from 20 disturbers. We also selected for our simulations the
band plan 997, which uses two upstream bands.
Nelder-Mead algorithm is an ad-hoc optimization method,
which finds the global maximum of a function if it is concave
and a local maximum near the initialization point if it is non-
concave. Therefore, by selecting a ‘good’ initialization point
X0we reduce the number of iterations and find a good local
maximum for non-concave functions. We have noticed that
the proposed algorithm performs well if α’s are initialized to
the maximum PSD values and β1U and β2U are initialized to
the insertion losses of reaches without PBO and collocated
modems for the lowest and highest bitrates, respectively.
For the case when α’s are fixed and we search only for
optimized β’s the cost function in (5) is piecewise concave
(there are flat areas). This can be shown by plotting (5) for
all combination of β’s. Furthermore, as can be seen from the
bitrate/reach plots in Fig. 4 the performance for fixed and
varying α’s are nearly similar. Therefore, we propose to fix
α’s and to search only for the optimized β’s, since this strat-
egy substantially reduces the number of iterations.
0 250 500 750 1000 1250 1500
0
5
10
15
20
25
30
Loop length (m)
Bitrate (Mbit/s)
Equal-length distrubers
For PS D s : PR−1U
=−60 −20.99√f,PR−2U
=−60 −16.18√f
For PS D s : PR−1U
=−43.58 −29.56√f,PR−2U
=−96.56 −4.31√f
Fig. 4. Upstream bitrates for noise model E for equal-length
disturbers and for reference PSDs optimized to protect bi-
trates: 3, 6, and 12 Mbit/s.
5. CONCLUSIONS
In this paper we presented an efficient algorithm to calculate
the optimized parameters for PBO in VDSL, which uses the
Nelder-Mead simplex search [4]. We also developed a new
method to calculate the worst-case FEXT noise, which is es-
pecially important for DMT-based VDSL systems. The high
efficiency of our algorithm allows deployment in DSL trans-
mission systems with more than two upstream bands, which
will be the case for VDSL2. This efficiency also allows oper-
ators to optimize the PBO parameters for their networks, i.e.,
cables, noises, and selected VDSL type.
6. REFERENCES
[1] S. Schelstraete, “Defining upstream power backoff for
VDSL,” IEEE Journal on Selected Areas in Communica-
tions, vol. 20, no. 5, pp. 1064–1074, May 2002.
[2] ETSI, “Transmission and Multiplexing (TM); Access
transmission systems on metallic access cables; Very
high speed Digital Subscriber Line (VDSL); Part 1:
Functional requirements,” Standard TS 101 270-1, Ver-
sion 1.3.1, ETSI, Jul. 2003.
[3] V. Oksman, “Optimization of the PSD REF for upstream
power back-off in VDSL,” ANSI T1E1.4 contribution
2001-102R1, Feb. 2001.
[4] J. A. Nelder and R. Mead, “A simplex method for func-
tion minimization,” Computer Journal, vol. 7, pp. 308–
313, Jul. 1965.
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