General Framework and Algorithm for Data Rate Maximization in DSL Networks

General Framework and Algorithm for Data Rate Maximization in DSL Networks

Rodrigo B. Moraes, Member, IEEE, Paschalis Tsiaflakis, Member, IEEE, Jochen Maes, Senior Member, IEEE, and Marc Moonen, Fellow, IEEE

Abstract— In this paper, we treat the combined signal and spectrum coordination problem in digital subscriber line (DSL) networks with linear design for transmitters and receivers. The transmission is modeled as a multitone MIMO system where each user has a number of transceivers and there is coordination between sets of users on the transmitter and on the receiver sides.

We consider the possibility of an asynchronous transmission, i.e.

when the transmission of DMT blocks for different users is not aligned in time. This gives rise to inter-carrier interference. Our objective is the maximization of the weighted sum of users’ data rates subject to power constraints. Although this problem is well known in the literature, previous works have always based their designs on strong assumptions about the network infrastructure.

In this paper, we propose a general framework and algorithm that apply for any infrastructure, including any number of users, any number of transceivers, any number of tones, any kind of coordination on both the transmitter and on the receiver sides, and synchronous or asynchronous transmission. We also do not assume any special structure of the channel matrix. Our algorithm is seen to perform very well and is polynomial time solvable.

Index Terms—DSL, crosstalk, optimization, MIMO.

I. INTRODUCTION

M

ULTI-INPUT, multiple-output (MIMO) processing has constituted a paradigm shift in the way communication systems are designed. It has captured the attention of re- searchers and the telecommunication industry since the 1990’s,

Manuscript received July 4, 2013; revised November 27, 2013 and January 29, 2014. The editor coordinating the review of this paper and approving it for publication was S. Galli.

This research work was carried out at the ESAT Laboratory of the KU Leuven, in the framework of the KU Leuven Research Council PFV/10/002 (OPTEC); the Bilateral Scientific Cooperation between Tsinghua Univer- sity & KU Leuven 2012-2014; the FWO project G091213N ‘Cross-layer optimization with real-time adaptive dynamic spectrum management for fourth generation broadband access networks’; the Belgian Programme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office, ‘Belgian network on Stochastic modeling, analysis, design and optimization of communication systems’ (BESTCOM) 2012-2017; and the Concerted Research Action GOA-MaNet. The scientific responsibility is assumed by the authors.

R. B. Moraes and P. Tsiaflakis were with the STADIUS Center for Dynam- ical Systems, Signal Processing and Data Analytics, Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium. They are now with Access Research Domain, Alcatel-Lucent Bell Labs, Antwerp, Belgium (e-mail: rodrigo.moraes@alcatel-lucent.com, paschalis.tsiaflakis@alcatel-lucent.com).

J. Maes is with the Access Research Domain, Alcatel-Lucent Bell Labs, Antwerp, Belgium (e-mail: jochen.maes@alcatel-lucent.com).

M. Moonen is with the STADIUS Center for Dynamical Systems, Signal Processing and Data Analytics, Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium (e-mail:

marc.moonen@esat.kuleuven.be).

Digital Object Identifier 10.1109/TCOMM.2014.030214.130507

when it was first studied. The technology is an undeniable success, and within a decade it has evolved from a theoretical concept to practical implementation [1].

The initial research efforts in MIMO communications were focused mainly on single-user systems, but the focus quickly evolved to multi-user scenarios [2]. In these scenarios, the extra spatial dimensions serve the purpose of spatial separation of the different users, providing the means for interference mit- igation and thus an improved channel utilization. Multi-user MIMO is classically divided into three situations, depending on how much coordination there is on the receiver and on the transmitter sides. A system with full transmitter coordination is called a broadcast channel (BC), whereas a system with full receiver coordination is called a multiple access channel (MAC). When there is no inter-user coordination neither on the transmitter nor on the receiver side, the scenario is referred to as an interference channel (IC).

Most of the research and standardization activities when it comes to MIMO technology have focused on wireless trans- mission. However, the same paradigm can be applied to any multi-transceiver scenario where there are cross channel gains between all transmitters and all receivers.¹A digital subscriber line (DSL) binder fits such a description. In DSL, multiple users transmit over closely packed copper pairs. Because of electromagnetic radiation, a signal transmitted in a given pair leaks to the neighboring pairs. This phenomenon is known as crosstalk. Crosstalk has been traditionally identified as the main source of performance degradation for such systems, but, with MIMO processing, crosstalk can be used as a means to improve performance. After all, crosstalk contains signal energy that can be detected on the other side of the network [3]. It is the processing of the signals that defines whether crosstalk is beneficial or detrimental to performance.

By far the world’s favorite means of broadband access, DSL counts more than 400 million subscribers and a market share of more than 70% [4]. Although it is predicted that DSL will eventually be replaced with optical fiber even for home use, it is expected that DSL will be around as an important market technology for decades.

To expand its lifetime as much as possible and to keep competitive, DSL technology has been evolving in two main directions. First, as a result of the expansion of the optical

1We define a transceiver as a generalization of the concept of DSL line. A transceiver is connected to a physical channel that can be a direct mode or common mode of a copper pair or the phantom mode of two copper pairs or one wire in split wire signaling. In wireless parlance, ‘antenna’ is a similar concept.

0090-6778/14$31.00 c 2014 IEEE

fiber network, copper lines are getting shorter. In the future, it is foreseen that DSL will be responsible for bridging the last couple of hundred meters from the fiber-fed last distribution point to the customer premisses equipment (CPE).

Accordingly, standardization bodies have been working on new types of DSL better suited for shorter lines. E.g. future generations standards include the G.fast, which will go into the market around 2016 and is designed to work on copper lines that are at most a couple of hundred meters long. The second direction of evolution results from the decade-long research activities that aim at amplifying DSL’s advantages from a signal processing perspective. This body of work is called dynamic spectrum management (DSM) and its main goal is to deal with crosstalk interference—in the DSL context, DSM that involves MIMO processing is often called vectoring or DSM level 3 [5]. We remark that MIMO processing in DSL also encompasses the possibility of the utilization of common mode (CM), phantom mode (PM) or split wire (SW) transmission [3], [6]. While the first direction of evolution is about the physical infrastructure of the access network (i.e. bringing the network hardware closer to the CPE), DSM techniques rely on intelligent coordination and processing of signals (i.e. expanding the functionality of the network software). It has been repeatedly shown in the literature that applying DSM leads to formidable gains.

Given a DSL network, depending on the kind of coor- dination on the transmitter and on the receiver sides, e.g.

receiver-only, transmitter-only, etc., a BC, a MAC, an IC or a combination of them can be used as a model. Several papers have focused on the design of the transmission and reception strategies for such scenarios, e.g. [5], [7]–[15]. However, it is observed that these references base their designs on strong assumptions about the network infrastructure. We give four examples of these assumptions.

• First, most previous work considers a synchronous trans- mission case, i.e. the situation when all users’ discrete multitone (DMT) blocks are aligned (i.e. time synchro- nized) at the receivers. This is not necessarily the case.

Users who are in different physical locations or who belong to different service providers are difficult to synchronize. The result of asynchronous DMT transmis- sion is inter-carrier interference (ICI) [16]–[19], which complicates the problem significantly.

• Some previous references [5], [7], [8], [12] assume that in DSL coordination is only possible on the central office (CO) side of the network.²In other words, there is either a BC scenario for downstream transmission or a MAC scenario for upstream transmission. We believe this is too restrictive. E.g. it can be that a number of DSL lines arrive at the same box on the CPE if the connection serves a large residential building. This allows for some limited coordination on the CPE side as well. Plus, if two copper pairs that arrive on the CPE use PM transmission, then there is a three transceiver system that can be coordinated on the CPE side.

2By ‘coordination on the CO side of the network’ we mean that coordi- nation can be done on the CO itself, in a fiber-fed cabinet in the street, a distribution point or in the basement of a large building.

• Another assumption is that every user uses only one line (or one transceiver). This is not always the case. There are places where it is not uncommon that there are two DSL lines connected to a user. In some places, quads are popular. A quad is a group of four copper wires twisted together that serve a single customer. In these cases, pair bonding, CM, PM or SW transmission can then be used.

• A final common assumption is that DSL channels have the so-called property of column-wise or row-wise di- agonal dominance [7], [8], [12]. However, this only holds true for the case when sources of noise other than crosstalk are spatially white.

We believe there is a much richer range of interesting scenarios than what hitherto has been considered. Telephone networks have evolved differently through the decades in different parts of the world, giving rise to complex networks that have each their characteristics. Many of these networks would probably not fit on the scenarios considered previously in the literature. Modern DSL networks are likely better represented by an abundant set of different hybrid scenarios, where users can have multiple transceivers and elements of IC, MAC and BC are present. In this work we consider a general scenario that encompasses all these hybrid situations as special cases.

To the best of our knowledge, no work up to now has developed a general framework and a corresponding algorithm that apply to this general scenario. This is the goal of this paper. We develop a general system framework that includes MAC, BC and IC and any combination of them as a special case (we consider only linear transmission schemes). We propose an algorithm similar to the one in [20] that works for all cases, including any number of users, any number of transceivers, any number of tones, any kind of coordination on both the transmitter and on the receiver sides, and synchronous or asynchronous transmission. Through numerical simulations, the algorithm is seen to perform very well and is shown to be polynomial time solvable.

We organize this paper as follows. Section II presents the system model, the notation, the problem of interest and previous solutions. In Section III we derive and present our proposed approach. Section IV contains some simulation results and Section V presents final remarks.

We use lower-case boldface letters to denote vectors, upper- case boldface letters for matrices and calligraphic letters for sets (e.g. a, A and A). We use I_A as the identity matrix of size A, 0_A×B as the A× B matrix of zeros, R+ as the set of non-negative real numbers, (·)^T as the transpose, (·)^H as the Hermitian transpose, (·)^∗ as the complex conjugate, E [·] as expectation, tr

·

as trace, | · | as determinant and diag {a}

as the matrix with a vector a on the main diagonal.

II. SYSTEMMODEL ANDPROBLEMSTATEMENT

A. System model and notation—Synchronous case

This is the only section in this paper where we specifically treat the synchronous transmission situation. That is so because the aim of this section is to present the notation we use and to see the effects of adding coordination on the transmitter and receiver sides of the network. To consider the asynchronous

situation here would be too cumbersome. All conclusions from this section are readily extendable to the more general asynchronous case.

We consider an N user DSL system with DMT modula- tion with K Δ_f-spaced tones. We consider a system where upstream and downstream transmission are separated (with, e.g. time or frequency division duplexing), hence the crosstalk we consider is far end crosstalk (FEXT).³ We denote the set of users by N = 

1, . . . , N

and the set of tones by K =

1, . . . , K

. We let p^k_nbe the transmit power of user n on tone kand we organize these values in the matrix P∈ R^K×N+ . The nth column of P, denoted by p_n =

p¹_n . . . p^K_nT

∈ R^K+, contains the power allocation of user n in all tones. The kth row of P, p^k =

p^k₁ . . . p^k_N

∈ R^N+, represents the power allocation of all users in tone k. User n has A_n transceivers.

Every user belongs to a group both on the transmitter side and on the receiver side. Inside a group, users can apply coordinated MIMO processing. We define each group as a set. For the grouping on the transmitter side, we define G_i^tx, i = 1, . . . , I ≤ N. For the grouping on the receiver side, we similarly defineG_q^rx, q = 1, . . . , Q≤ N. Here I and Q denote the number of groups on the transmitter and on the receiver sides, respectively. A user can only be in a single group both on the transmitter and on the receiver sides, i.e. if n ∈ G^tx_i then n /∈ G_q^tx, q = i and if n ∈ G_q^rx then n /∈ G_i^rx, q = i.

We also define the number of transceivers per group on the transmitter and on the receiver sides respectively as

A_Gtx

i = 

n∈G_i^tx

A_n (1)

A_Grx

q = 

n∈G_q^rx

A_n (2)

As already mentioned, throughout this paper we focus on a linear design for both transmitters and receivers and treat in- terference as noise. All channel gains are considered perfectly known. Taking this into account, we obtain the received signal vector for groupG_q^rxon tone k and the estimated signal vector for user n on tone k respectively as

y^k_Grx q = vec

n∈G_q^rx[y^k_n] (3)

ˆ

x^k_n = R^k_ny_G^krx

q , n∈ G_q^rx (4)

Here y^k_Grx

q ∈ C^AGrxq is a concatenation of the received signal vectors y^k_n∈ C^An of the users that belong toG_q^rx. For exam- ple, if G_q^rx={1, 2, 3}, then y^k_Grx

q = [(y₁^k)^T (y^k₂)^T (y₃^k)^T]^T. In (4), we have ˆx^k_n ∈ C^An. Eqs. (3) and (4) are specified in detail later with some examples. For now, we just mention that they depend on five variables:

• x^k_n ∈ C^An, the transmitted signal vector for user n on tone k.

• T^k_n ∈ C^{AGtxi ×An}, the transmit matrix for user n on tone k, where n∈ G^tx_i .

• R^k_n ∈ C^An×AGrxq , the receive matrix for user n on tone k, where n∈ G^rx_q .

3The system model can be straightforwardly generalized to a situation where upstream and downstream transmission are jointly optimized, even- tually taking near end crosstalk (NEXT) into account.

• H^k_n,j, n, j ∈ N , the channel matrix on tone k between the transmitter of user j and the receiver of user n. The size of H^k_n,j depends on the groups on both transmitter and receiver sides, i.e. H^k_n,j∈ C^AGqrx×AGitx, where q and i are such that n∈ Grx^q and j∈ Gtxⁱ .

• z^k_Grx

q ∈ C^AGrxq , a vector of circularly symmetric zero mean complex Gaussian noise. It is a concatenation of vectors similar to (3), i.e. z^k_Grx

q = vec_n∈Gqrx[z^k_n], where z^k_n∈ C^An. We assume that x^k_n has A_n parallel data streams, but some of these streams can have rate of zero. We also assume E

x^k_n(x^k_n)^H

= I_An. The noise vector z^k_Grx

q is assumed to be spatially white with covariance matrix E

z^k_Grx q (z^k_Grx

q )^H

= I_AGrx

q , q = 1 . . . , Q.

We now specify (3) and (4) in three examples. The main point of these examples is to show that H^k_n,j depends on the type of coordination on both sides of the channel and that it can be viewed as a concatenation of matrices relating to the MIMO IC, that we define as H^k_n,j∈ C^An×Aj ∀n, j .

The first example is depicted in Fig. 1. This is a system with three users and three tones. This is a pure MIMO IC, i.e. every group contains a single user. This is the situation where there is a minimum amount of coordination. We have Q = I = N = 3andG_n^tx=G_n^rx={n}, n ∈ N . In this case, we write

y^k_Grx

n = y^k_n= 

j∈N

H_n,jT^k_jx^k_j + z^k_n (5)

ˆx^k_n= R^k_nH^k_n,n

 

 H^k_n,n

T^k_nx^k_n+

j=n

R^k_nH^k_n,j

 

 H^k_n,j

T^k_jx^k_j + R^k_nz^k_n (6)

We define H^k_n,j, n, j ∈ N as the channel matrix that comes between R^k_n and T^k_j. In this first example, for all tones we have H^k_n,j = H^k_n,j, n, j ∈ N ; A_G^tx_n = A_Grx

n = A_n and T^k_n, R^k_n∈ C^An×An, n∈ N .

The second example is depicted in Fig. 2. This is a system with three users, with, say A1 = 2, A2 = 3 and A3 = 1, and three tones. There are two groups on both transmitter and receiver sides, i.e. Q = I = 2. We have G1^tx = {1} and G2^tx = {2, 3}; and on the receiver side G1^rx = {1, 2} and G2^rx={3}. Using (1) and (2), we obtain A_G^tx₁ = 2, A_Gtx

2 = 4, A_Grx

1 = 5and A_G2^rx = 1. As a consequence, for all tones we have T^k₁ ∈ C^2×2, T^k₂ ∈ C^4×3, etc. Notice that this system does not fit exactly neither the MAC, nor the BC, nor the IC case. This is a scenario with elements of BC, MAC and IC.

To calculate the equivalent channel matrices, we take into account the matrices H^k_n,j ∈ C^An×Aj, n, j ∈ N and k ∈ K.

As mentioned in a previous paragraph, these are the channel matrices for the case of a pure MIMO IC. The received signal for, say, group 1 is given by

y^k_Grx

1 =

y^k₁ y^k₂

= H^k_1,1

H^k_2,1

T^k₁x^k₁+

H^k_1,2 H^k_1,3 H^k_2,2 H^k_2,3

T^k₂x^k₂ +

H^k_1,2 H^k_1,3 H^k_2,2 H^k_2,3

T^k₃x^k₃+ z^k₁

z^k₂

. (7)

The estimated signal for, say, user 2 is given by applying (4),

tone 3

1

T1

1 2 1 2T p

1

T3

A3

A2

tone 3

1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

tone 2 tone 1

1

R1

1

R2

1

R3

3 tones 3 tones

A₁

tone 2

1

T1

1 2 1 2T p

1

T3

tone 1

1

T1

1

T2

1

T3 1

ˆx1

+

1

ˆx2

1

ˆx3

1

x1

1

x2

1

x3 1

x1

1

x2

1

x3 1

x1

1

x2

1

x3

User 1

User 2

User 3

1

ˆx1

1

ˆx2

1

ˆx3 1

ˆx1

1

ˆx2

1

ˆx3

Fig. 1. Illustration of a MIMO IC. For this case, there is only one user on every group in both the transmitter and on the receiver sides. Here, we have Q = I = N = 3 and G^tx_n = G^rx_n = {n}, n = 1, . . . , 3.

tone 3

1

R1

1

R2

1

R3

tone 3

1

R1

1

R2

1

R3 1

ˆx1

+

1

ˆx2

1

ˆx3

1

x1

1

x2

1

x3 1

x1

1

x2

1

x3 1

x1

1

x2

1

x3 1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

1

H11 H¹₁₂ H¹₁₃

1

H21 H¹₂₂ H¹₂₃

1

H31 H¹₃₂ H¹₃₃

User 1

User 2

User 3 3 tones

3 tones

A1

1

ˆx1

1

ˆx2

1

ˆx3 1

ˆx1

1

ˆx2

1

ˆx3

tone 2

1

R1

1

R2

1

R3 1

R1

1

R2

1

R3

tone 1

1

R1

1

R2

1

R3

1

T1

1

T2 T₃¹

1

T1

1

T2 T₃¹

tone 1

A3

A2

tone 1

1

T1

1

T2 T₃¹

Fig. 2. Illustration of a hybrid MAC, BC and IC. For this case, we have on the transmitter sideG₁^tx= {1} and G₂^tx= {2, 3}; and on the receiver side G₁^rx= {1, 2} and G^rx₂ = {3}.

i.e.

ˆ x^k₂ = R^k₂

H^k_1,1 H^k_2,1

 

 H^k_2,1

T^k₁x^k₁+ R^k₂

H^k_1,2 H^k_1,3 H^k_2,2 H^k_2,3

 

 H^k_2,2

T^k₂x^k₂

+ R^k₂

H^k_1,2 H^k_1,3 H^k_2,2 H^k_2,3

 

 H^k_2,3

T^k₃x^k₃+ R^k₂z^k_Grx 1 . (8)

Again we use the definition of H^k_n,j, n, j ∈ N , i.e it is the channel matrix that comes between R^k_n and T^k_j. Here we see that with added coordination, H^k_n,j becomes a concatenation of the matrices H^k_n,j. The concatenation depends on the grouping on both the receiver and transmitter sides.

The third example is that of a three user MIMO BC. This system is represented by G^tx = {1, 2, 3}, i.e. I = 1, and G_n^rx={n}, n ∈ N , i.e. Q = 3. If A_n= 2∀n, then A_G^tx = 6 and A_G^rx_q = 2, s = 1, 2, 3. For this case, we for the equivalent channel matrices as

H^k_1,n=

H^k_1,1 H^k_1,2 H^k_1,3

, n∈ N (9)

H^k_2,n=

H^k_2,1 H^k_2,2 H^k_2,3

, n∈ N (10) H^k_3,n=

H^k_3,1 H^k_3,2 H^k_3,3

, n∈ N (11) After these three examples, we can now focus on the general case. We write (4) as

xˆ^k_n= R^k_nH^k_n,nT^k_nx^k_n+ R^k_n

j=n

H^k_n,jT^k_j + R^k_nz^k_Grx

q , n∈ G_q^rx. (12) Because of the structure with groups and because the H^k_n,j are defined as functions of the H^k_n,j, (12) includes any kind

of transmitter and receiver coordination. On one extreme, we have a MIMO IC, as explained in the first example. On the other extreme, there is only one group both on the transmitter and on the receiver side, i.e.G^tx =G^rx ={1, . . . , N}. This is the case with full two-sided coordination. It is often called a MIMO point-to-point system. With our formulation, every case between (and including) these two extremes is possible.

As a rule of thumb, we remark that coordination on the transmitter side makes H^k_n,j ‘wider’ (more columns) than H^k_n,j, and that coordination on the receiver side makes H^k_n,j

‘taller’ (more rows) than H^k_n,j.

B. System model and notation—Asynchronous case

An asynchronous transmission scenario occurs when the DMT blocks of the different users are not aligned in time.

We demonstrate this with the example of Fig. 3, where two users (denoted n and j), each with two transceivers, interfere with each other. Their respective DMT blocks are offset by β_n,j, 0≤ βn,j ≤ 1, as shown in the figure. Such a situation gives rise to ICI, which complicates the problem significantly.

With ICI, transmission on a given tone k of an interferer influences not only the corresponding tone k of a victim, but all neighboring tones as well.

The bulk of the system model described in Section II continues to be valid for the asynchronous case, including the definition of groups on the transmitter and receiver sides and the fact that more coordination makes the channel matrices increase in size. We assume that all users inside a group either on the transmitter or on the receiver sides are synchronized.