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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)

IV635

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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