D. Zanatta Filho, R. R. Lopes, R. Ferrari, M. B. Loiola, R ...

THE CAPACITY OF BINDERS FOR MIMO DIGITAL SUBSCRIBER LINES

D. Zanatta Filho, R. R. Lopes, R. Ferrari, M. B. Loiola, R. Suyama, G. C. C. P. Simoes,C. Wada, J. M. T. Romano

School of Electrical and Computer Engineering

University of Campinas - UNICAMP

{daniloz,rlopes,rferrari,mloiola,rsuyama,gsimoes,crisw,romano}@decom.fee.unicamp.br

B. Dortschy and J. Rius i RiuEricsson – ASP Lab

{boris.dortschy,jaume.rius.i.riu}@ericsson.com

ABSTRACTCrosstalk is one of the main limiting factors in the data

rates achievable by digital subscriber line (DSL) systems,and several algorithms have been proposed to mitigatethis impairment. In this paper, we compare the capacityof binders under different crosstalk-mitigating techniques.When computing capacity, we also compare two differentpower constraints: either on the total power in the binderor on the power in each pair. We will see that, for thescenarios considered in this paper, the fact that the signalsare jointly processed in one or both ends of the DSLlink leads to roughly the same performance, which is farsuperior to that of systems with no cooperation betweenthe users. Both power constraints also lead to similarachievable rates.

I. INTRODUCTION

Digital subscriber line (DSL) systems exploit the twistedpairs traditionally used for phone services to transmithigh-rate data services such as internet access. Twistedpairs from different users are normally grouped in binders,which may contain as many as 50 pairs. Due to theproximity of these pairs, their signals are magneticallycoupled, generating crosstalk between the users. Crosstalkis one of the main impairments of DSL systems, and oneof the main factors limiting the achievable data rates ofthese systems.

Currently deployed digital subscribe line (DSL) systemsuse a single wire pair to transmit data between the centraloffice (CO) and the end user. Furthermore, the users are notprocessed jointly at the CO. In other words, current DSLsystems essentially transmit through single-input single-output (SISO) channels. In these cases, little can be doneto mitigate crosstalk; perhaps the only choice is to allocatespectrum to different users so as to minimize the impactof crosstalk.

It has long been recognized that jointly processing thesignals from different users may improve performance ofDSL systems [1], [2]. These works exploit the fact that theCO has access to the signals of all the pairs in a binderto mitigate crosstalk using signal processing techniques,achieving rates close to the optimum. Note that [1], [2]assume that the signals are jointly processed only at theCO side. In fact, these works assume that, at the costumerpremises (CP), users have access to a single pair, and thusmay process only its own signal.

However, users often have access to more than onepair. This if fairly common in Europe, and may alsooccur in connections between central offices and remoteterminals. In these cases, we have a multiple-input andmultiple-output (MIMO) DSL system. In a MIMO system,the signals in the wire-pairs may be jointly processedat both ends of the links, enabling the use of crosstalk-mitigating techniques that are even more powerful thanthose proposed in [1], [2].

In this paper, we compute the capacity of several bindersbased on different signal processing algorithms. Whencomputing the capacity, we also compare two differentpower constraints. First we use the traditional waterfill-ing constraint, which limits the total power transmittedin the binder. We also use an alternative constraint ofmore practical interest: a per-pair power constraint. Thislimitation reflects the fact that the signal in each pairmust go through a power amplifier, whose output power islimited by practical considerations.

II. CHANNEL MODEL AND PROBLEMSTATEMENT

In this paper, we consider an ideal MIMO-DSL sys-tem, in which all the lines in a binder can be jointlyprocessed both at the receiver and at the transmitter. Weassume that frequency-division duplexing is used, so that

SBrT © 717

VI International Telecommunications Symposium (ITS2006), September 3-6, 2006, Fortaleza-CE, Brazil

there is no near-end crosstalk. Furthermore, as with allDSL systems, the use of orthogonal-frequency divisionmultiplexing transforms the time-dispersive channel intomultiple parallel channels, called subchannels or tones,with no intersymbol-interference. In this case, the channeloutput at a given tone, y is given by

yk = Hkxk + nk, (1)

where xk is a vector containing the signals transmittedfrom each pair, Hk is a matrix that models the crosstalkbetween the lines and nk represents the additive whiteGaussian noise (AWGN) with covariance matrix σ2

kIN .Several attempts were made to mathematically model the

matrix Hk on each tone [3]. In this paper, we employ whatis perhaps the most accurate model in the literature, the onedeveloped by Bin Lee in his doctoral thesis [4]. Based onBin Lee’s model, the aim of this paper is to compare thecapacity achieved by systems employing different typesof crosstalk mitigating techniques and different spectrumallocation algorithms. We will also use these algorithms todetermine the capacity of an actual binder whose channelmatrix was measured.

III. CROSSTALK-MITIGATING ALGORITHMS

In this section, we describe the three signal processingalgorithms used in this paper to mitigate crosstalk. Thefirst one attempts no cancelation whatsoever, and is termedno signal processing (NSP). The second is based on theinversion of the channel matrix, and is called zero-forcing(ZF), since it forces the crosstalk to zero. The third is basedon the singular-value decomposition (SVD) of the channelmatrix.

Note that the techniques used in this paper assume per-fect channel knowledge. Furthermore, all these techniquescan be seen as the multiplication of the transmitted signalby a preprocessing matrix before transmission and/or themultiplication of the received signal by a receive matrix.Thus, all schemes can be represented as in Fig. 1. In thisfigure, N represents the number of pairs in the binder, xn

k

is the information symbol transmitted in the k-th tone ofthe n-th user, xn

k (ynk ) is the signal actually transmitted

(received) in the k-th tone of the n-th pair, and ynk is

the received signal in the k-th tone of the n-th user. Notethat this linear transceiver scheme creates a virtual channelbetween the transmitted and received signals for the users:

yk = RkHkPkxk + nk, (2)

where nk represents zero-mean additive Gaussian noisewith covariance matrix Cnk

= RkRHk .

III-A. No Signal Processing

In this case, we assume that there is no cooperationbetween the users at either end of the DSL link. In other

��

��

� � � � � � � �

� �

� � � � � � � �

� � � � � � �

��

� �

� ��

� ��

��

��

��

�

Fig. 1. Tone k of a DSL system with a general lineartransceiver.

words, ynk depends only on yn

k , and xnk depends only on xn

k .As a consequence, the receiver and preprocessing matricesfor all tones must be diagonal, and no attempt can be madeto mitigate crosstalk. On the other hand, the receiver maycompensate for the direct channel attenuation. To that end,we make the n, n entry of Rk equal to the inverse of thechannel attenuation for the n-th pair at the tone, i.e.,

[Rk]n,n = ([Hk]n,n)−1. (3)

Also, we assume that the preprocessing matrix in Fig. 1 isthe identity matrix for all tones.

III-B. Zero Forcing

The ZF algorithm assumes that the signals are jointlyprocessed only at the CO, and that a single pair isavailable to each user. Now, in the downstream the COis the transmitter, while it is the receiver in the upstream.Thus, for downstream transmission we can only performZF precoding, while ZF processing at the receiver is allthat is available for upstream transmission. In both cases,following [2], we use the inverse of the channel matrixfor processing, so that the equivalent channel in (2) is theidentity matrix.

A well-known drawback of ZF techniques is powerenhancement. In the downstream, a ZF precoder is used,which causes an SNR penalty by artificially increasing thepower transmitted in each line. This power penalty is takeninto account in the power allocation algorithms presentedin the next section. On the other hand, in the upstream,the use of a ZF receiver causes noise enhancement, sincethe equivalent noise in (2) now has a covariance matrixgiven by H

−1k H

−Hk . This SNR penalty is easily taken into

account in the power allocation algorithms. Both upstreamand downstream penalties may lead to a capacity loss.However, as we will see, the capacity of ZF schemes issimilar to that of other methods, implying that both typesof power enhancement have negligible effect.

III-C. Singular-Value Decomposition

In the fully-coordinated case, both the CO and CP canjointly process all the DSL pairs. This case corresponds

SBrT © 718


Table I. MIMO Transceiver Structures - DownstreamSVD ZF NSP

Pk Vk H−1k

IN

Rk UHk

IN diag�H−1

k

�

Cn σ2kIN σ2

kIN diag

�H−1

k

�2

gnk

��Σk

�n,n

��2

1 1

to a single-user MIMO case, where the CO communicateswith a single-user CP, both of them equipped with as manymodems as used pairs. In this fully-coordinated case, anSVD-based solution was implemented. We now describethis solution.

Let Hk = UkΣkVHk be the SVD of the channel matrix

Hk. Then, the precoding and receiver matrices are given byVk and U

Hk , respectively, yielding an equivalent channel

given by Σk. Since this is a diagonal matrix, we see thatcrosstalk has been eliminated.

It is interesting to notice that the receiver matrix isunitary, so that it does not change the noise statistics,i.e., Cnk

= σ2kIN . The precoding matrix is also unitary,

which implies that the total power (sum of the power inall pairs) before and after precoding are the same. In otherwords, the SVD effectively decouples all the users, with nonoise enhancement and no increase in the total power. Thisobservation leads to a simple power allocation algorithm,the well-known waterfilling solution described in the nextsection. On the other hand, the power in a specific pairbefore and after precoding are different. Thus, if there is aper-line power contraint, the precoding operation must beaccounted for. This can be done with the optimal powerallocation algorithm described in the next section.

III-D. Summary

The signal processing algorithms described in this sec-tion are summarized in table I for downstream transmissionand in table II for the upstream. In these tables, the entrygn

k corresponds to the magnitude-squared of the gain ofthe equivalent direct channel in (2) for the n-th pair andthe k-th tone, i.e.,

gnk = |[RkHkPk]n,n|

2. (4)

This value will be used by the power allocation algorithmsin the next section.

IV. POWER ALLOCATION ALGORITHMS

Traditionally, optimal power allocation is determinedunder the assumption that the total power to be transmittedis limited. In the MIMO-DSL case, this is equivalent toconstraining the sum of the powers in all the pairs in a

Table II. MIMO Transceiver Structures - UpstreamSVD ZF NSP

Pk Vk IN IN

Rk UHk

H−1k

diag�H−1

k

�

Cn σ2kIN σ2

k

�H−1

kH−H

k

�diag

�H−1

k

�2

gnk

��Σk

�n,n

��2 1 1

binder. However, it may be of practical interest to actuallyconsider a power constraint per line. This is due to thepower limitation at the amplifier of each modem, as wellas restrictions to avoid excessive interference with othersystems. We begin this section with a brief presentation ofthe waterfilling solution [3], which achieves the channelcapacity when the total transmit power is limited. Then, wepresent the optimal power allocation algorithm [2], whichdetermines the optimal power allocation for a precodedsystem when there is a per-line power constraint.

IV-A. Waterfilling

As mentioned in section III-C, the use of an SVD-basedtransceiver yields several parallel and independent AWGNchannels. In this case, the noise variance and the totalpower transmitted in both the equivalent and the actualchannels are the same. The problem of power allocationin this case is standard, and its solution is the well-known waterfilling solution [3]: the power allocated to theinformation symbol xn

k is given by

snk =

[λ − Γ

σ2k

gnk

]+

, (5)

where

[x]+ �{

x, if x ≥ 0

0, if x < 0, (6)

and where Γ stems from the use of the gap approximation[3]. In (5), λ represents the water level, which must bechosen so that the total transmit power is equal to themaximum value allowed, P . In other words, λ is chosenso that ∑

n

∑k

snk = P. (7)

Finally, it should be pointed out that snk is the power

allocated to the symbols at the k-th tone of the n-th userbefore the precoding matrix (see Fig. 1).

IV-B. Optimal Power Allocation with Preprocessing

In this section, we describe a power allocation algorithmfor a system based on a linear preprocessing algorithm.This algorithm, known as optimal power allocation (OPA)

SBrT © 719


ChannelTone k

Modulator1

Tone k

Modulatorn

x��

˜

x ��

˜

Source

x��

x ��

Modulator1

Precoder 1

Tone 1

ModulatorN

x ��

˜

x ��

˜

x ��

x ��

P1

Power constraint (P� � � )

Power constraint (P� � � )

ChannelTone 1

Fig. 2. Detail of the precoding matrix and the powerconstraint per line.

and presented in [2], limits the power transmitted on eachline.

Consider the precoding operation

xk = Pkxk, (8)

shown in detail in Figure 2. The power of each informationsymbol xn

k is denoted by snk and the power of the precoded

symbols xnk is denoted by sn

k . The optimal power allocationproblem can be then stated as follows{

s∗nk

}n=1...N

k=1...K= arg max

snk

, ∀k,n

∑n

∑k

bnk

(sn

k

)(9)

subject to ∑k

snk ≤ P, ∀n

snk ≥ 0, ∀n, k ,

(10)

where bnk

(sn

k

)is the maximum bit rate that can be trans-

mitted using power snk .

Using the gap approximation [3] and a diagonalyzingprecoder such as the ZF or the SVD, the capacity in (9) isgiven by

bnk

(sn

k

)= log2

(1 +

gnk sn

k

σ2kΓ

)(11)

where gnk is the square of overall gain between transmitted

symbol xnk and the received symbol yn

k , Γ is the SNRgap, and σ2

k is the noise variance at tone k after receiverprocessing (refer to Tables I and II).

Also, considering the elements of the preprocessingmatrix p

n,mk � [Pk]n,m, the output power for each modem

is given bysn

k =∑m

|pn,mk |

2sm

k . (12)

Combining both equations, the original optimization prob-lem becomes{

s∗nk

}n=1...N

k=1...K= arg max

snk

, ∀k,n

∑n

∑k

log2

(1 +

gnk sn

k

σ2kΓ

)(13)

Table III. Optimal Power Allocation with Precodingrepeat

∀n, k: snk

=

�1

�

mλm|p

m,nk |

2 − Γσ2

k

gnk

�+

∀n, k: λn =

�λn + µ

��k

�m

��pn,m

k

��2 smk

− P

��+

until convergence

subject to ∑k

∑m

|pn,mk |

2sm

k ≤ P, ∀n

snk ≥ 0, ∀n, k .

(14)

The solution of the above optimization problem can befound by using Lagrange multipliers. In [5], Cendrillonproposes the iterative algorithm shown in Table III to findthe optimum solution.

V. SIMULATIONS

In this section we present simulation results that com-pare the capacity achieved by the different transceivers andpower allocation algorithms described in the previous sec-tion. In particular, we computed the capacity of SVD, ZFand no processing transceivers with OPA, and of an SVDtransceiver with total power constraint, both for upstreamand downstream channels. In the sequel, we describe thetwo scenarios simulated and present the capacity results.

V-A. First Simulation: results for the Bin Lee’s channelmodel on MIMO-VDSL

In this simulation, the rate as a function of the reach wasevaluated for a MIMO-VDSL channel generated using BinLee’s model. We used a binder with 10 VDSL pairs, with4096 tones. The wire diameter is 0.5 mm (24-AWG). Thetarget bit error rate (BER) was set to 10−7 or less, thecoding gain was set to 3.8 dB and the noise margin to 6dB, which leads to an effective gap of Γ = 9.8−3.8+6 =12 dB. In accordance with the VDSL standards [6], [7],the tone spacing ∆f is set to 4.3125 kHz and the DMTsymbol rate fs to 4 kHz. Background noise is composedof white thermal noise with a PSD of -140 dBm/Hz andthe available transmitted power for each line is +11 dBm.

Figure 3 depicts the rates achieved by the three process-ing algorithms implemented (no processing, ZF and SVD)using OPA, as well as an SVD algorithm using total powerconstraint. The results are plotted as a function of the reach.One can see that the SVD performance is a little betterthan the ZF. Also, there is no difference between the SVDresults obtained using a per-line power constraint (OPA)and a total binder power constraint. This is because thedirect channel and the crosstalk for all the ten pairs isalmost the same, which leads to a similar SNR value in all

SBrT © 720


the cables. In other words, the channels for all pairs aresimilar, so there is no reason to allocate more power to oneline than to another. Finally, we observe that all techniquesoutperform by far the results obtained without any signalprocessing, especially for short distances.

100 200 300 400 500 600 700 800 900 10000

200

400

600

800

1000

1200

1400

1600

1800

2000Capacity − Downstream

Reach ( m )

Rat

e (

Mbp

s )

NO PROCESSING − OPASVD − OPAZF − OPASVD − TOTAL POWER

(a) Downstream Rate.

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800Capacity − Upstream

Reach ( m )

Rat

e (

Mbp

s )

NO PROCESSING − OPASVD − OPAZF − OPASVD − TOTAL POWER

(b) Upstream Rate.

Fig. 3. Rate as a function of reach for no processing, SVDand ZF algorithms using OPA power allocation and SVDusing total power.

V-B. Second Simulation: MIMO-ADSL2+ results forthe measured channel

The measured channel data was generated for a MIMO-ADSL2+ binder with 10 pairs for two distances: 500meters and 1500 meters. The capacity of this binder wasevaluated for the following signal processing algorithms:no signal processing, SVD and ZF. To highlight the effect

of noise in the algorithms, we measured the capacity as afunction of the noise power. As per the ADSL2+ standard[8], 512 tones were used, the tone spacing ∆f is setto 4.3125 kHz and the DMT symbol rate fs to 4 kHz.Background noise is composed of white thermal noise andthe available transmitted power for each line is +11 dBm.Our computations are based on the gap approximation: thetarget bit error rate (BER) is 10−7 or less, the coding gainwas set to 3.8 dB and the noise margin to 6 dB, whichleads to an effective gap of Γ = 9.8 − 3.8 + 6 = 12 dB.

In Figure 4 we can notice that, for a large rangeof noise power, using signal processing algorithms likeSVD and ZF produces the same data rate as using nocrosstalk-mitigating algorithm. The similar performancefor all algorithms can be explained by the fact that thechannel matrices for all tones are almost diagonal, since thecrosstalk is very small for the ADSL2+ frequency range.In fact, the only difference in performance happens forthe shorter binder at low noise levels, where the crosstalkbecome the dominant impairment. We suppose that adifference would also be observed for the longer cable,but for a noise level below the range we considered. Noticethat the difference in the upstream transmission is even lesspronounced than in downstream.

VI. CONCLUSIONS

In this paper, we have computed the capacity as afunction of the distance for binders of twisted pairs, as usedin DSL systems. We have used several signal processingand power allocation algorithms. We have shown thatcrosstalk mitigation techniques may yield significant gainsover systems that do not attempt to cancel crosstalk.However, our simulation results suggest that the capacity ofa system that jointly processes signals at both ends of thecommunications link is similar to the capacity of a systemthat performs joint processing only at the CO side. Finally,the results in this paper also indicate that imposing a per-line power restriction incurs almost no capacity penaltyunder the simulated scenarios. This is because all the pairssee similar channels, so that there is no reason to allocatemore power to one pair than to another. In other words,waterfilling allocates similar powers to all the pairs, so thatit approximately observes a per-line power restriction.

ACKNOWLEDGMENTS

We acknowledge the financial support receivedfrom the Research and Development Centre, EricssonTelecomunicacoes S.A., Brazil; from the Fundacao deAmparo a Pesquisa do Estado de Sao Paulo, FAPESP;from the Coordenacao de Aperfeicoamento de Pessoalde Nıvel Superior, CAPES; and from the EuropeanCommission IST 6th Framework and the Swedish Agencyfor Innovation Systems, VINNOVA, through the IST

SBrT © 721


−180 −170 −160 −150 −140 −130 −120 −1100

100

200

300

400

500

600

Noise Power (dBm/Hz)

Capacity (Mb/s)

No ProcessingZFSVD

500m

1500m

(a) Upstream Rate.

−180 −170 −160 −150 −140 −130 −120 −1100

100

200

300

400

500

600

700

Noise Power (dBm/Hz)

Capacity (Mb/s)

No ProcessingZFSVD

500m

1500m

(b) Downstream Rate.

Fig. 4. Rate as a function of noise power for no process-ing, SVD and ZF algorithms using OPA for the MIMO-ADSL2+ measured channels.

- MUSE and the Eureka - Celtic BANITS projectsrespectively.

VII. REFERENCES

[1] G. Ginis and J. M. Cioffi, “Vectored transmission fordigital subscriber line systems,” Journal on SelectedAreas in Communications, vol. 20, no. 5, pp. 1085–1104, June 2002.

[2] R. Cendrillon, G. Ginis, E. Van den Bogaert, andM. Moonen, “A Near-Optimal Linear CrosstalkCanceler for VDSL,” IEEE Transactions on SignalProcessing, Submitted 2004.

[3] T. Starr, J. M. Cioffi, and P. J. Silverman, Understand-ing Digital Subscriber Line Technology, Prentice HallCommunications Engineering and Emerging Technolo-gies Series. Prentice Hall, 1999.

[4] B. Lee, Binder MIMO Channels, Ph.D. thesis, StanfordUniversity, Stanford, USA, November 2004.

[5] R. Cendrillon, Multi-User Signal and Spectra Co-ordination for Digital Subscriber Lines, Ph.D. thesis,Katholieke Universiteit Leuven, Leuven, Belgium, De-cember 2004.

[6] “Very High Speed Digital Subscriber Line (VDSL);Part 1: Functional Requirements,” ETSI Std. TS 101270-1, 2003.

[7] “Very-high bit-rate Digital Subscriber Lines (VDSL)Mettalic Interface,” ANSI Std. T1.424, 2004.

[8] “ITU-T Recommendation G.992.5, Asymmetric digitalsubscriber line (ADSL) transceivers extended band-width ADSL2 (ADSL2plus),” 2005.

SBrT © 722


D. Zanatta Filho, R. R. Lopes, R. Ferrari, M. B. Loiola, R ...

Documents

Transcript of D. Zanatta Filho, R. R. Lopes, R. Ferrari, M. B. Loiola, R ...