CA2488871A1 - System, method and computer program for optimization in digital subscriber lines for multi-user spectrum balancing - Google Patents

Abstract

The present invention provides a method and an apparatus for the dynamic control of the transmit power spectral densities in a bundled digital subscriber line (DSL) system. The disclosed method enables dynamic control of the transmit power spectra so as to provide efficient minimization of mutual interference and efficient maximization of achievable data transmission rates in a DSL bundle. The disclosed method for dynamic control of the transmit power spectra provides an improvement in computational complexity and transmission rates.

Description

SYSTEM, METHOD AND COMPUTER PROGRAM FOR OPTIMIZATION
IN DIGITAL SUBSCRIBER LINES FOR MULTI-USER SPECTRUM
BALANCING
Background Numerous digital communication networks and related systems (or "digital communication systems") are known that include copper wiring. The copper wiring typically consists of twisted pairs (also known as "loops" or "lines"). Such digital communication systems include, for example, DSL, but also ISDN, HDSL, ADSL, VDSL, and Local Area Networks such as Ethernet. These systems commonly include a transceiver of some type (such as a modem) that connects a computer (such as a personal computer) to the digital communication system. The transceiver generally also incorporates copper wiring.
The copper wiring (such as in telephone lines) is often bundled to provide service to one user, or to permit service to multiple users over the copper wiring.
Communication over the digital communication system in such situations, however, is often negatively affected by well known factors such as "crosstalk", which refers to interference caused by the electromagnetic coupling between neighbouring pairs of copper wire. Crosstalk affects the signal carried across the copper wire pair.
Another problem with the current multiple user digital communication systems is power control. In a typical interference limited digital communication system, each user's performance depends not only on its power allocation, but also on the power location of all other users.
Also, demand for higher data rates is increasing, and digital communication systems are moving toward higher frequency bands, where the crosstalk problem is more pronounced. Accordingly, spectral compatibility and power control are important issues in the design and deployment of such digital communication systems. In particular, power allocation in this context requires not only for the total amount of power allocated for each user to be optimized, but also the power allocation for each user in each frequency to be optimized. This is particularly the case, in VDSL

systems where interference from a transmitter that is closer to the central office can overwhelm the signal at a transmitter that is farther from the central office, when communication is attempted with the central office from the farther transmitter.
Numerous prior art solutions have been devised for addressing the aforesaid problems. One category of such solutions involves the control of physical layer signals by a single service provider (whether central or by means of distributed control linked to the single service provider) for the purpose of coordination of transmitted signals in manner that favours optimal service over the digital communication system.
United States Patent Application No. 2003/0086514 A1, assigned to The Board of Trustees of the Leland Stanford Junior University ('514) discloses a dynamic spectral control method commonly known as the iterative water-filling process.
Specifically I 5 'S 14 describes a method whereby a central office collects information regarding the line, signal, and interference characteristics for each of a plurality of communication lines; a model is created for such line, signal and interference characteristics; signals between transmitters and receivers are synchronized; and signals are processed using the model to remove interference from such signals. A particular method and related algorithms for enabling the foregoing are also described in '514.
One of the disadvantages of the '514 solution is that the particular methods described for controlling the interference affecting signals in a digital communication system is that the data transmission rates that are provided by the solution described in '514 are less than optimal.
Another prior art solution generally known as "optimal spectrum balancing" is described in a prior publication "Optimal Multiuser Spectrum Management for Digital Subscriber Lines", authored by R. Cendrillon, M. Moonen, J. Verliden, T.
Bostoen, and W. Yu, published in the Proceedings of the 2004 IEEE International Conference on Communications, 20-24 June 2004. The particular methods described in this publication for achieving spectrum balancing are relatively difficult to implement in a digital communication system environment because such methods are difficult to deploy in a computationally efficient way.

What is required therefore is a system, method and computer program for spectrum balancing in an interference-affected digital communication system that provides improved data transmission rates and is computationally efficient to deploy, especially in multi-user digital communication system environments.
Summary of the Invention The present invention relates to a method, and a system and computer program for deploying the method, for dynamically optimizing digital communication systems such as a DSL system. In an embodiment of the present invention, the optimization method enables the efficient maximization of the joint transmission rates of multiple DSL lines in a DSL bundle subject to a total power constraint for each line.
A method for optimal spectrum balancing is provided that consists of iterative per-tone optimization, whereby each user iteratively optimizes a joint objective function by operation of Lagrange multipliers. Specifically, a g function (particularized below) is iteratively evaluated by approximation.
Also, a method for dynamically optimizing a digital communication system (such as a DSL system) is provided, the method including the steps of (1) applying a dual optimization method (i.e, that updates the dual variables) for mufti-user orthogonal frequency-division multiplex systems, such as the particular subgradient method described below, and (2) applying the iterative method of the present invention that involves the evaluation of a function g (particularized below) by approximation.
It should be understood that other dual optimization methods can be used in (1) such as an ellipsoid method or an analytic centering cutting plane method.
The maximization problem (in the course of optimization) can take various forms, including but not limited to the formulation in equations (12) and (21) described below. The optimization method described in this invention can be realized in a computationally efficient way.

The update in accordance with (1) can be done on all dual variables at the same time, or the update can be done on a subset of variables at a time. In the subgradient method, described in one particular embodiment of the present invention, the subgradient update equation (10) described below may be used with a wide range of step sizes. In one implementation of the update equation, the step size s' in step l is chosen to be equal to an arbitrary constant divided by a number equal to l raised to kth power, where k can take values such as'/z, 1, 2 or other values.
Exact evaluation of g in (2) above is generally computationally prohibitive.
One particular method for approximate evaluation is described below under the heading "Iterative and Near-Optimal Spectrum Management". This method is iterative and it enables g to be evaluated approximately with a lower complexity as compared to an exact evaluation.
In another embodiment of the present invention, the optimization.methods of the present invention enable the optimal usage of the frequency spectrum among the upstream and downstream transmissions of multiple DSL lines in the same bundle. A
detailed description of this aspect of the invention can be found below under the heading "Optimal Frequency Planning".
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In yet another embodiment of the present invention, the optimization method of the present invention enables the optimal usage of mufti-user crosstalk cancellation units.
A detailed description of the related methods is provided below under the heading "Partial Crosstalk Cancellation in Vector DSL".
Brief Description of the Drawings The present invention can be readily understood by the following detailed drawings:
Fig. 1 is a schematic diagram of a set of twisted pair telephone lines controlled by a power spectral density optimization unit.
Fig. 2 is a graphical depiction of the power spectral density control method.
This provisional patent covers methods to evaluate dual function g (~,,, ~,2, ..., ~,K), such as the iterative method described, and methods to update dual variables (~,,, ~.2, ..., ~.K) such as the subgradient method described herein.
Fig. 3 illustrates time-sharing property implies zero duality gap.
Fig. 4 illustrates rate region for the two-user ADSL lines.
Fig. 5 illustrates topology of the two-user ADSL lines.
Fig. 6 illustrates loop topology for two downstream ADSL users.
Fig. 7 illustrates rate region for two downstream ADSL users.
Fig. 8 illustrates OBS (left) and IOSB (right) power spectral densities for two distributive ADSL users at equal rate. Power spectral densities for both CO-based (top) and RT-based (down) lines are plotted.
Fig. 9 illustrates loop topology for 10-User VDSL.
Fig. 10 illustrates rate region for upstream for 10-User VDSL.
Fig. 11 illustrates loop topology for 5-User VDSL.
Fig. 12 illustrates rate region for 5-User full duplex VDSL.
Fig. 13 illustrates downstream (top) and upstream (bottom) power spectral densities for 5-user full duplex VDSL at equal rate. The power spectra depend on the ordering in iteration in IOSB. Downstream-upstream order is on the left, and the upstream-downstream order is on the right.
Fig. 14 illustrates power spectral densities for 10-user full duplex VDSL at maximum minimal rate.

In an orthogonal frequency-division multiplex (OFDA~I) system, the frequency domain is partitioned into a large number of tones. Data transmission takes place in each tone independently. The overall system throughput is the sum of individual rates in each frequency tone. The design constraints are typically linear but coupled across all the tones. The design problem involves the optimization of the overall performance subject to design constraints. For example, the optimal bit and power allocation problem is often formulated as follows. Let H(n), P(n) and N(n) denote the channel frequency response, the transmit power spectral density and the noise power spectral density at tone n, respectively. The optimization problem is:
N P{r)Hz(n.) maximize ~ log Cl + N(n) ~ (1) n=1 N
subject to ~ P(n) < P
n=1 P(n) > 0.
The above problem has a well-known solution called "water-filling" . Efficient solution exists in this case because the objective function is concave in the optimizing variable P(n).
However, not all optimization objectives are concave. The multiuser bit and power allocation is such an example. In this case, several OFDM transmitters interfere with each other, and the sum rate rnaxirnization problem becomes:
x ~' Pk(n)H~~(n) rnax ~ ah ~ log 1 +
k=i n-r N(n) + ~~~k H~~(n)P~(n) N
s.t. ~ P,~(n) < Pk ~ = 1, . . . , h' (2) P,~(~z) > 0, k; = 1. . . . K
where H~,~(n) is the channel transfer function from system j to system ~: in tone n, Pk(r) is the power allocation for user k; in tone n, each user has a separate power constraint.
Because the objective function is not concave in Pk(~a), the optimization problem is difficult to solve. Previous approaches (e.b. itc;rative water-filling) us<-a Io;uristics to c'lerivc: suboptimal solutions.
In a prior work. an exact. "Opt.imal Spectrum Balancing" algorithm to solve this problem was proposed by Cendrillon. et, al. Tl-ie basically idea is as follows. Forrn the Lagnangian of the optimization problem (2):
x N Pk(n)Hkk(n) max aK ~ ~ log 1 + N(rt) + ~~~k H ~(n)P~ (n) K N
+ ~ ~,~ (Px - ~ Pk(n)1 k=I \ n=I /
s.t. I'k(-n,) > 0, k: = 1. . . . , K.
Solve the Lagrangian for each set of positive and fixed (.~I; ~ ~ ~ , ~K).
Then, the solution to the original pr oblenl may be found by an exhaustive search over the a-space. ,>,k is increased or decreased depending on whether ~n I Pk(rr,) is greater or less than Pk. When the process converges, either ak = 0 or ~n I Pk(n) = Pk for each k. In this case, the Lagrangian objective is identical to the original objective, thus solving tile original problem.
This Lagrangian approach works because of the following. First, for a fixed ak, the objective decouples into N independent problems corresponding to the N frequency tones.
Thus, solving the dual problem requires a much lower computational complexity as compared to the original problem.
Second, .~k represents the price of power for user k. A higher price leads to a lower power usage.
Thus, as a function of .~k, the optirTlal ~~ r Pk(n) is monotonic in ~k. An exhaustive search over the ~-space can then be perforn ued using bisection on each .~k. This is essentially an exhaustive search over all power usages. It leads to tine global optimum regardless of whether the original problem is convex. However, with K users, K loops of bisections are involved, one for each ak. Therefore, the cornputat,ional complexity of optimal spectrum balancing, a,ltlnough linear in N, is exponential in I~C.
~'l%llerl the rmmber of users is large, the complexity becomes prohibitive.
Tllis invention eliminates the exponential complexity in Lagrangian search, and generalizes the algoritl-un for other optimization problems in rnultiuser OFDA1 system design.
Toward this end, we show that true optimal spectrum balancing algorithm belongs to a class of dual optimization methods.
Contrary to genes al non-convex problems, the duality gap for rnultiuser OFDM
optimization always tends to zero as the number of frequency tones goes to infinit~~ regardless of whether the optimization problem is convex. This approach leads to more efficient ~-search methods.
Further, we show that the general tl-leory is applicable to many other areas of OFDM systerrl design. Optirrlal frequency planning a.nd optimal cornplexit.v a,lloca,t,ion in vectored digital subscriber line systems are some of these examples. In true second part of this invention, we further reduce the complexity of the q optimization method by near-optimal methods for spectrum balancing.
I. Dual Optirrrization Methods A. Duality Gap Consider an optimization problem in which both the constraints and the objective function consist of a large number of individual functions. corresponding to the N frequency tones:
N
maximize ~ f~(xn) (4) ~.=r N
subject to ~ hn(xn) < P, n=r where fn(~) is a scalar function which is not necessarily concave, and h~(~) is a vector-valued function that is not necessarily~ convex. P is a vector of constraints. Also, there may be other (possibly integer) constraints irxrplicit in the problem. The idea of the dual method is to solve (4) via its Lagrangian:
N N
L(:L~, ~) _ ~.ln(~r~) -E- ~T ' P - ~ I~,n(a;n) ~.=1 n=r where .~ is a vector, and "~'' denotes vector dot product. Note that the Lagrangian decouples into a set of N smaller problems, so optimizing the Lagrangian is much easier than solving' (4). Define the dual objective g(~) as the solution to the following:
g(a) = max L(x.~, .~) (6) ~n The dual optimization problem is:
minimize g(~) ( r ) subject to .~ > 0.
When fn(x~) is concave and h.,~(x~) is convex, standard convex optimization results guarantee that tire primal problerrr (4) acrd the dual problem (7) have the same solution.
When convexity does not bold, the dual problerrr provides a solution wlniclr is an upper bound to tire solution of (4). Tlue upper bound is not always tight, and the di$~erence is called the "duality gap''.
In multiuser OFDI\~I design, convexity often does not. hold. However, it is usually the case that.
the following ''tune-sharing" proper ty is satisfied:

Definition 1: An optimization problem of the form (4) satisfies the time-sharing property if the following holds: Let xn and y~ be optimal solutions to the problem with P = P~
and P = Py, respectively. Then, for any 0 < v < 1, there exists a set of zn such that ~n h~,(z~) < vP~+(1-v)Py, and ~ f~.(zn) ~ v ~ fn(xn) + (1 - v) ~ fn(yn)~
This property is clearly satisfied if time-division multiplexing may be implemented. (Throughout the document, the channels are assumed to be time invariant.) The frequency tones can then be assigned to x~ for v percentage of the time and y~ for (1 - v) percentage of the time.
In practical OFDM
systems in which channel conditions in adjacent tomes are similar and there are a large number of frequency tones. the time-sharing property can be satisfied with frequency-sharing. This is true because time-sharing can be approximately implemented by interleaving x~ and y~ in the frequency dorrrain. As N --> oo, frequency-sharing is equivalent to time-sharing.
Note that the concavity of fn(x~) and the convexity of h~(x,~) and all other constraints imply time-sharing but not vice versa. Time-sharing is alwa.~~s satisfied regardless of convexity as long as N is sufficiently large and f,~ ~ ~ ~ fn.,.~ are sufficiently similar for small values of k (and likewise for Icn - ~ ~ )a.~+~.) This is the case in almost all OFDM systems as the subchannel width in OFDM systems is chosen so that. tine channel response in adjacent subchannels are approximately the same.
The main result of this section is that the tune-sharing property implies that the duality gap is zero. Specifically, if an optimization problem satisfies the time-sharing property, then it ha,s zero duality gap, i.e. the primal problem (4) and the dual problem (7) have the same solution.
The proof of tloe above result is previously known if (4) is convex. Fig. 3 illustrates the proof when convexity does not hold but time-sharing does. The first diagram illustrates a function that satisfies the tune-sharing property. The solid line plots the optimal (~ Iz~ (xn), ~
f.~(xn)) as the constraint P
varies. The intersection of the curve with the vertical axis where ~ hn(x~) =
P is the optimal value of the primal objective. Clearly larger P leads to higher objective value, so the curve is increasing.
More importantly, the curve is concave because of the time-sharing property.
Now, consider a fixed tangent, line with slope ~. By the definition of L(~, x~,), the intersection of the tangent line with the vertical axis is precisely g(.~). This allows the minimization of tlne dual problerrr to be visualized easily. As a varies, g(.1) achieves a minimum at. exactly the rnaxirrmrn value of the primal objective.
Thus, the duality gap is zero. (The second diagram illustrates a case where time-sharing property does not 1-~old. In this case, the minimum g(~) is strictly larger tluan the maximum ~ fn(x~).) ~1 ~ f,c~.~) 5lape=a . a~
gca) ,, .:, I. = s, P ~ h~aT;,) P
~ h" ~x;~) Fig. 3. Time-sharing property implies zero duality gap.
The main consequence of the above result is that as long as the time-sharing property is satisfied, even a non-convex optimization problem can be solved by solving its dual. The dual problem is typically much easier to solve because it usually lies in a lower dimension.
~rther, g(a) is convex regardless of the concavity of f~(x~). (This is because L(x~, ~) is linear in ~ for each fixed xn, and g(~) is the maximum of linear functions and is therefore convex.) Tlnus, any gradient-based algorithm is guaranteed to converge. Note tluat the optimization of g(~) requires an efficient evaluation of g(.~). This usually involves an exhaustive search over the primal variables.
However, as g(a) is unconstrained and it decouples into N independent sub-problems, such an exhaustive search is much more manageable.
B. Dual ll.~Ietluods The optimal spectrum balancing algorithm solves L(xn, a) exhaustively for all possible values of a.
The multiuser spectrum optimization problem (2) consists of K constraints, and successive bisection on each component of a would yield the primal optimum. The main innovation of this invention is that we can take advantage of the duality relation and solve the dual objective g(.~) instead. By using am cHicicnt search of ~, the computational efficiency of the optimal spectrum balancing can be improved drastically.
The main difficulty in deriving an efficient direction for ~ is that g(~) is not necessarily differentiable. Thus, it does not always have a gradient. Nevertheless, it is possible to find a search direction based on what is called a subgradient. A vector d is a subgradient of g(~) at ~ if for alt a' Subgradient is a generalization of gradient for (possibly) non-differentiable functions. Intuitively, d is a subgradient if the linear function with slope d passing through (.~, g(.~)) lies entirely below g(~).
For g(.1) defined in (6), the following choice of d N
d=p-~hn(xn) ".=r satisfies the subgradient condition (8). The subgradient search suggests that ~ should be increased if ~,~ 1 h.~(xn) > P and decreased otherwise. Here, ~ represents a price for power. Price should increase if the constraint is violated. In fact, ~ updates can be done systematically. The following update rule ~t+r - ~t _ St p _ ~ ~"~(x~,) (10) is guaranteed to converge to the optimal .~ as long as st is chosen to be sufficiently small. Here, st is a scalar. In one implementation of the update equation, the step size st may be chosen to be equal to an arbitrary constant divided by a number equal to l raised to kth power, where k can take values such as 0.5, 1, 2 or other values. Or, st may be chosen to be a.
sufficiently small constant.
The above update is called the subgradient method.
The search of optimal ~ can also be dOlle uslllg ellipsoid method, where the search for the a is done using an ellipsoidal search. The search of optimal a can also be done using analytic center cutting plane method, or a combination of above methods.
By the result described previously, the minimum g(~) is also equal to the maximum ~ ,/',~(xn).
Thus, the solution to the dual problem immediately yields the optimal solution to the original problem.
The crucial difference between the update equation (10) a,nd that suggested by Cendrillon is that ( 10) updates all components of ~ at the same time. Instead of doing bisection on each component individually, the subgradient method collectively finds a suitable direction for all components of ~
at once. This eliminates the exponential complexity in ~-search.

t3 However, note that the evaluation of g(~) is still exponential in K. This is probably inevitable if an exact solution to the non-convex optimization problerrr is desired. For practical problems, however, sub-optimal rrrethods in evaluating g(a) often exist.
II. Applications A. Multiuser Spectrum Management VVe now return to the rrrultiuser optimal spectrum management problem. In digital subscriber line applications, electromagnetic coupling induces crosstalk between adjacent lines. The goal of optimal spectrum management. is to find a set of power allocations (Pr(n), ~ ~ ~ , PK(n)) so that a target rate-tuple is satisfied. There is generally a trade-off between the achievable data rates of different users. Such a trade-off can be represented in a rate region defined as the set of all achievable rates (Rr, ~ ~ ~ , Rx). When the channel transfer function is a slow varying function of n, the spectrum optimization problem satisfies the time-sharing property.
In this section, we formulate a novel optimization problem that characterizes the boundary of the rate region. The objective is the nraxirnization of a base rate R subject to a fixed ratio between Rk and R. for each k, = 1, ~ ~ ~ , K. More, specifically, we rnay insist that Rr : RZ
: - ~ - : RK = ,fir : ,~2 : - : ~jx, where Pk (n) Hkk(n) Rk _ ~ log 1 + N(n) + ~~~k H ~(n)P~(n) ' (11) Then, the maximization problem becomes ma,x R (12) s.t. Rk > RkR
N
~'k(~-) < Pk, k = l, . . . , K
n=r pk(n) >_ 0, ~; = 1. . . . h-Here, the variables ~jk directly represent the ratios of service rates among the diff~ererrt users.
The dual function for (12) can be written as follows:
9(wr, ~ . . , ~K~ ~1. . . . , ,~K) - max (13) P," R
h x N
R+~~k(Rk-~kR)+~~k Pk-~pk(~Z) k=1 k=r n=r s v- a - ~ -6 OSM - FWI Lagranpien Search 8 v d OSM - Reducs0 Compexiry Search q B- Itaralive Waler-Fix \ \G
G
N 0.
D
o b .
y b User t ADSI Downsueam Rats (MDps) Fig. 9. Rate region for the two-user ADSL lines I OK feet CO RT
7K feet I OK feet Fig. 5. Topology of the two-user ADSL lines Collecting terms, we see that the maxirr>ization involves a term (1 -~Wk,3k)R. Since R is a free variable to be optimized, the maximization leads to R = oo if (1 - ~ wk~ik) >
0 and R = 0 if (1 - ~ wk~3k) < 0. Thus, non-trivial solution exists only if (I - ~ wk,Qk) =
0.
It. is now straightforward to apply the technique developed in the previous section to derive a subgradient search for the minimization of g(wl, ~ - ~ , wx, ~1. - ~ ~ , ax).
The idea is the following. First, solve the maximization problem ( 13) for a fixed set of (wl, - - - , wK. y . -- - , .fix) with ( 1-~ wk,Qk) = 0.
This is done using exhaustive search in each tone separately and it yields a set of power allocation Pk(~n.) and achievable rates R.k. The maximum R can be found as R = mink Rk/,Cjk. The subgradient method can now be used to update wk and .~k:
wok ] - [wk - Sk (Rk - ~kR~,+ (14) N +
t~ ~' - ~ pk(n) n=7 'dote that the new wk may no longer satisfy ~ wk~k = 1. Renormalization is needed to project wk.

back to the proper subspace ~~+i ~c+i - W k . (16) ~k ~~~ la~
As long as s~ and t~ is suf~cicntly small. the subgradient algorithm is guaranteed to converge. This sub-gradient algorithm improves the computational complexity of the optimal spectrum balancing algorithm described by Cendrillon. No bisection is needed. The number of times that g(wk, J~~) is evaluated is polynomial in K.
Note that the evaluation of g(wk, ~,~), if done exhaustively, still has a complexity exponential in K.
However, for the spectrum optimization problem, experimental results suggest that lower complexity search algorithms often work well. Fig. 4 shows the rate region for a two-user ADSL system with a configuration shown in Fig. 5. Both the full implementation of optimal spectrum balancing and a reduced complexity gradient search are shown. Their performances are very similar, and both outperform iterative water-filling significantly.
B. Optimal Frequency Planning The optimal spectrum balancing algorithm is 'applicable to many other areas of OFDM system design. For example, in a wireless multiuser OFDI\~Z system, different users are often allocated to different sets of tones. The optimal power and bit allocation problem is essentially the spectrum management problem (12) with an additional constraint that only one user occupies each tone:
Pk(n)P~(n) = 0 tlk ~ 7 Previous solutions to tluis problem rely on a relaxation of the non-convex constraint. As Theorem 1 in Section II slows. this problem can instead be efficiently solved in the dual domain. The same subgradient, updates as in the previous section apply here. The constraint, P~(n)P~(n) = 0 for all k~
and .j is incorporated in tlue evaluation of the dual function. Theorem 1 guarantees that. the dual solution is identical to the primal solution.
In fact. the complexity of this problem is strictly sub-exponential. The evaluation of the dual g(~,~~,, ~k) involves an exhaustive search in It possible power allocations.
Its complexity is therefore linear in li .

C. Partial Crosstalk Cancellation in Vector DSL
Future digital subscriber line applications are expected to implement crosstalk cancellation and precoding to further improve the data rates in twisted-pair transmission.
Multiple transmitters and multiple receivers at the central office can be regarded as a single entity.
Crosstalk cancellation can be done in a similar way as echo cancellation.
A typical DSL bundle consists of 50 to 100 twisted pairs. Cancelling all crosstalk involves 50 x 50 to 100 x 100 matrix processing, which is beyond the computational complexity constraints of current digital signal processors. On the other hand, in a 50-pair DSL bundle each twisted-pair has only a limited number of nearest neighbours. Thus, we expect that the cancellation of only a few pairs would achieve most of the benefits. Furthermore, crosstalk is frequency dependent. The crosstalk level is low in low frequency bands, so cancellation in these frequency bands has limited utility. On the other hand, in very high frequency bands, the data rates are already small. Thus, data rate improvement due to crosstalk cancellation is most noticeable in the mid-frequency range.
Given a complexity constraint, how to choose the best combination of lines and tones in which to implement crosstalk cancellation is an interesting problem. This problem was previously solved by greedy algorithms. However, the greedy solution assumes a fixed transmit spectrum level. In this section, w~e formulate a more realistic problem that jointly performs line/tone selection and spectrum optimization.
The basic setup is the same as the optimization problem (12) except the evaluation of Rk now takes the following form:
R,~ _ ~ log 1 + N(7i) + ~i~k G.ik(n)1'~(n) . (17) where G~~(n) = Hkj(n) except where crosstalk cancellation takes place, in which case G~~(n) = 0.
The total number of places where G,~~ (n) = 0 represents the number of crosstalk cancellation units that can be implemented. Tluis number is typically constrained by an implementation limit. More formally, ~r 1{Hkj('~»G'kj(~.)1 C C (18) n=1 k~7 where 1{1 is an indicator function and C is a constant representing the complexity constraint over all tones and all users.

Clearly (17) rrray be solved using the dual formulation. The complexity constraint is the same as the power constraints. As long as exhaustive search within each tone can be done with manageable complexity, the optimization over tire N tones only adds a polynomial factor.
III. Brief Summary of the First Part of the Invention The main result in the first part of the invention is that many optimization problems in OFDM
design can be decoupled in a tone-by-tone basis via the dual method. It is shown that the time-sharing property is always satisfied when the number of tones is large, and when the time-sharing property is satisfied, the duality gap becomes zero r egar Bless of whether the original problem is convex. This observation leads to efficient dual optimization techniques such as the subgradient method. As long as the evaluation of the dual objective for each tone may be done with manageable complexity, the entire problem may be solved efficiently. This principle is applicable to a wide range of OFDM design problems. Multiuser spectrurrr optimization, frequency planning and line/tone selection in reduced complexity crosstalk cancellation are some of these examples. In the second part of this invention, methods for evaluating the dual objective is described in more detail.
IV. Low Complexity Spectrum Balancing AMethods In a digital subscr fiber line system, rnult.iple copper pairs are bundled together. The electromagnetic coupling b~aval<-x:n t1 : coppc;r pairs eaizsc.s crosstallc interference, whicIu leas long been idr~utified as tlm primary source of line impairment in DSL deployments. Current DSL systems use a static spectrum management (SSM) approach where a fixed transmit power spectral density is applied for each line regardless of the loop topology or user service requirements. The performance projection under SSM
is based on tire levels of worst-case crosstalk interference.
Future generation of DSL services are envisioned to utilize d5-namic spectrum management (DSM).
DSM gives each line an ability to adapt to its loop environment and service requirements, and it leas the potential to drastically improve the achievable r a,tes and ser vice r anges of current DSL systems.
The crosstalk problem is the most severe when the channel transfer functions are heavily unbalanced. This is the situation in downstream ADSL systems where a remote optical network unit.
(ONL:) is deployed (ONLT is usually located much closer to the customer premise modems served from the central office) and in upstream VDSL systems where some of the upstream transmitters may be much closer to the central office than others. Power back-off methods are traditionally applied in these cases.
Digital Multi-tone (DMT) is the modulation format used in almost all digital subscriber line standards. In a DMT system, the frequency spectrum is divided into many parallel subchannels.
Different amount of power and different number of bits can be assigned in each subchannel. This gives DSL applications great flexibility in performing spectrum optimization.
In particular, spectral shaping may be done in a tone-by-tone basis for each line individually.
However, precisely also because of this flexibility, the number of variables in a spectrum optimization problem is the product of the number of users If and the cumber of frequency tones N. Further, the objective function in the spectrum optimization problem is non-convex. Thus, a brute-force search-based optimization has a computational complexity that is exponential in KN, which is intractable.
Iterative water-filling is one of the first low-complexity spectrum optimization techniques that. takes advantage of the ability for DSL modems to perform spectral shaping. In the iterative water-filling algorithm, each user iterativeiy maximizes its own achievable rate by performing a single-user water-filling with the crosstalk interference from all other users treated as noise.
The single-user water-filling process is a convex optimization process and has a complexity of O(N log(N)).
Thus, each iteration of the iterative water-filling process has an O(KN log(N)) complexity.
However, the iterative water-filling process does not seek to find the global optimum for the entire binder. Instead, each user participates in a non-cooperative game, and the convergence point of the iterative water-filling process corresponds to a competitive equilibrium. Although not optimum; the iterative water-filling algorithm has been shown to significantly outperform static spectrum management schemes. Further, iterative water-filling can be implemented distributively.
Recently. Cendrillon proposed an optimal spectrum balancing (OSB) algorithm which finds the true global optimal solution to the spectrum optin uization problem. The OSB
algorithm transforms the spectrum optimization problem into the dual domain by forming the Lagrangian dual of the primal optimization. As pointed out earlier, the class of spectrum optimization problems for digital subscriber lines has the special property that the primal and the dual optimization problems yield the carne solution even when the primal problem is non-convex. As the dual problem has a much lower dimension; the computational complexity of solving the dual problem is much lower. It. can be shown that the OSB algorithm has a computational complexity that is linear in the number of frequency tones N. The optimal spectrum balancing algorithm can provide a significant performance improvement as compared to iterative water-filling.
However, the computational complexity of the optimal spectrum balancing (OSB) algorithm, although linear in N, is still exponential in the number of users h'.
Implementation experience shows that the complexity of the OSB algorithm becomes unmanageable when the number of users in the binder is larger than two.
The main objective of this part of the document is to describe an appropriate middle ground between iterative water-filling and optimal spectrum balancing. The goal is to take advantage of both the dual formulation of the optimal spectrum balancing algorithm and the competitive (thus low-complexity) nature of iterative water-filling. Toward this end, this document proposes an iterative spectrum balancing technique that achieves almost all the gain of optimal spectrum balancing while having a computational complexity that is comparable to iterative water-filling.
The computational methods proposed here has a wider irnplicatio~ beyond that of DSL
applications. The optimal power and bit loading problem for wireless orthogonal frequency-division multiplex (OFDM) systems leas been extensively studied in the past. A low-complexity near-optimal solution to this class of long-standing problems can lead to many other applications.
V. System Model The achievable data rates in a K-user DMT-based DSL system are computed as follows:
N
Rk = T ~ b~ (19) n=1 where k is the user's index, n is the tone's index, N is the total number of frequency tones, T is the symbol period. bk denotes the achievable bit rate for user k in tone n, and it is computed as n b~ = Iog2 1 -I- . ~ Sk n n (20) ~k + ~i~k ai,k'~i where Sk is the transmit power for user k in tone n, Qk is the normalized channel noise for user k in tone n, and ai k is the normalized crosstalk transfer function from the ith user to the kth user in tone n. Channel noise and crosstalk transfer function are normalized by T/
~Hk ~z, where r is the SNR gap for the system and ~ Hk ~2 is the kth user's direct channel transfer function in tone n..
The spectrum optimization problem in a multiuser DSL system is formulated as the maximization of a weighted sum rate of all participating users subject to power constraints K
max ~ wkRk s.t. Pk < Pk dk (2I) Si ,...,SK -k=1 where Pk is the kth user's power constraint. The weights wk > 0 are chosen so that ~~ 1 wk = 1.
The total power used by user k is computed as N
Pk = Of ~5,~. (22) ~,=i Here ~ f is the frequency width of the DMT tones. The weights {w1, w2, . . . , wK~ are the priority put on the users.
In a two-user system, (21) reduces to rnax mRl + (1 - m)R2 s.t. Pk < Pk b'k: (23) si~S2 By varying w between 0 and 1, an achievable rate region can be generated.
VI. Optimal Spectrum Balancing The main idea of the optimal spectrum balancing (OSB) is to solve the constrained optimization problem (21) in the dual domain. Instead of an exhaustive search over all possible bit allocations and over all frequency tones, the optimal spectrum balancing algorithm fixes dual variables (.~I, ~ ~ ~ , ax), corresponding to each of the K power constraints, and forms the dual objective function:
9(~i, ~ . . ~ ~K) K K
- max ~ mkRk - ~ ak(Pk - Pk) {s; ,...,sxln ~ km k=~
m N K nr - max ~ wk~b~ -~~k ~Sk -Pk {Si ,...,Sj~}~ i k-1 ~=1 k=1 n-1 N K K
- ~ max ~ (u~kb;~ - ~kS~ ) '+ ~ akPk~ (24) n n n-1 S~ ,...,Sly k_1 k=1 Note that the evaluation of g(~1, ~ ~ ~ ; ~K) is now decoupled in a tone-by-tone basis. Therefore, if B is the maxirrmrn number of bits that can be loaded in each tone, the evaluation of g(~1, ~ ~ ~ . aK) requires O(.NBK) operations. Although still exponential in K, this is nevertheless a significant computational saving as compared to the O(BI''f~) operations needed for an exhaustive search over all frequency a~
tones.
In the first part of this invention, we showed that even when the original problem (21) is a non-convex optirrrization problem, the dual objective is always convex. Further, the minimal value of 9(~r, ~ ~ ~ , ax) over all positive a's is equal to the optimal solution of (21):
x min g(~r, . . . ~,K) = rrrax ~ wkRk. (25) ~1,... ,aK s; ,...,sK
k=1 This crucial observation enables a subgradient search method to be implemented for the spectrum optimization problem. In particular, the number of subgradient steps needed to reach a global optimal solution is a polynomial function of the number of dimensions, which is K.
The above statement is rrreaningful, however, only if the function g(al, ~ ~ ~
,.1K) can be evaluated efficiently. Unfortunately, as seen in (24), the evaluation of g(~r, ~ ~ ~ , fix) is exponential in K.
Therefore, the evaluation of g(~r, ~ ~ ~ , aK) is the computational bottleneck of the optimal spectrum balancing algorithm. Computational experience shows that optirrral spectrum balancing algorithm is impractical when the nurrrber of users is larger than two.
VII. Iterative Near-Optimal Spectrum Management The main innova.t.ion of this part of the invention is an efficient algorithm that enables g(ar, ~ ~ ~ , fix) to be approximately evaluated with a complexity that is linear in li . Recall that the evaluation of 9(~r, ~ ~ ~ , fix) involves the tone-by-tone optimization of the following function:
K
maxn ~ (wkb~ - ~kS~ ) ~ max h(S~ , . . . SK) (26) s; ,...,sK k_r s; ,...,sK
Our main idea is that the optimization of h(Si ; ~ ~ ~ , SK) may be done in an iterative water-filling fashion via coordinate descent. For each fixed set of (~1, ~ ~ ~ , .fix), our proposed approach first finds the optirrral S~ while keeping S2 . ~ ~ ~ ; SK fixed, then optimizes S2 keeping all other S~ fixed, then S3 . ~ - ~ . S'~ , tluen S~ , S2 . ~ ~ ~ , and so on. Such an iterative process is guaranteed to converge because each iteration strictly increases the objective function. The convergence point is guaranteed to be at Ieast a local maximum for h(S~ , ~ ~ ~ , Sk ).
This new aL~proach differs from iterative water-filling im the following two kev aspects. First..
unlike the iterative water-filling algorithm where each user maximizes its own rate in each step of the iteration, the above algorithm optimizes an objective function drat includes the joint rates a~
of all users. Thus, the new algorithm has the potential to reach the social optimum. Second, the power constraint in the iterative water-filling process is handled in an ad-hoc basis, while the new algorithm proposed here dualizes the power constraint in an optimal fashion.
The correct values of the dual variables are then used in a sub-gradient search. We called this approach the Iterative OSB (IOSB) algorithm. The performance of the IOSB algorithm is near-optimal as compared to the optimal spectrum balancing method.
The computational complexity of this new iterative approach is significantly Lower than that of optimal spectrum balancing (OSB) algorithm proposed in. In the evaluation of h(Si , ~ ~ ~ , S~ ), each iteration has a computational complexity that is linear in K. Let Ti be the number of iterations needed in the evaluation of each h(Si , ~ ~ ~ , Sk ). Let T2 be the number of subgradient updates needed in the optimal spectrum balancing algorithm. The total computational complexity of IOSB
is O(TiT2BNK). Computational experience suggests that Tl and T2 are not strong functions of N
and K. The overall complexity is therefore approximately linear in K. This is significant as K may be large in realistic DSL deployment scenarios. Table I summarizes the computational complexity comparison. (Here, T3 is the number of iterations needed in iterative water-filling.) Algorithm Computational Complexity Exhaustive Search O(B ) Optimal Spectrum Balancing_ O(T1NB ) Iterative OSB O(T1T2BNK) Iterative ~'~~ater-FillingO(T3KN log(N)) TABLE I
Computational Complexity Analysis VIII. Performance Evaluation This section shows drat the proposed iterative algorithm has a near-optimal performance as compared to optimal spectrum balancing. This is verified with extensive simulation. In the following simulation, all DSL lines are 26-AWG twisted pairs with a background noise level of -140dBm/Hz.
Users are assumed to be symbol synchronized so that the sidelobe interference is not in effect. Also.
no spectral masks are enforced.

12k feet 3k feel Optical FILer RT
CO
13k feet Fig. 6. Loop Topology for Two Downstream ADSL Users A. 2-User ADSL Downstream The first set of simulations examines a 2-user ADSL downstream distributive environment with both users having a loop length of 12k feet and with a crosstalk distance of 3k feet. No other disturbers are assumed to exist in the binder. The loop topology is shown in Fig. 6. Such a distributive environment is expected to benefit significantly from dynamic spectrum management because of its highly unbalanced crosstalk channels. The power constraint for each user is set to 20.4dBm as defined in the ADSL standard. For fair comparisons, the number of iterations and initial ~ settings are identical for OSB and IOSB in simulations.
Fig. 7 shows the achievable rate regions of OSB, IOSB, iterative water-filling and static spectrum management algorithms. As can be seen in the figure. the rate regions for OSB
and IOSB are almost identical to each other. Both outperform iterative water-filling significantly. Interestingly, although the a.clrievable rates of OSB and IOSB are identical, the optimal spectra obtained from the two algorithms can be different. Fig. 8 shows the downstream spectra obtained from the two algorithms.
The frequency band belowr 38UkHz is shared by both direct channels for downstream transrrrission, and all tones that are beyond this range are occupied by a single user only.
The frequency division multiplex (FDM) is optimal because of the strong far-end crosstalk (FEXT) generated by the RT
modem. The main difference between the spectra of OSB and IOSB is in the FDM
region. Both power spectral densities (PSDs) essentially achieve the same rates because many equivalent permutations of frequency tones in the FDM regions are possible.
B. 10-user VDSL Upstream The next set of simulations examines a 10-user VDSL upstream environment with two loop lengths:
five; linc;s are at 2kft acrd the other five are at 4lcft. All twisted pairs are connected to tire same Central Office (CO) as shown in Fig. 9. The 998 frequency ba.ndplan is used in the simulation with the upstream bands at 3.75MHz to 5.2>\MHz and 8.2MHz to l2MHz. The power constraint on each a~
osa P

I ~r IOSB

-!
.~,~,r IWF

T .
.
SSM

a 1 .

Z 1 ..
.

O i ~ z a a s s GO User Downstream Rele (MOps) Fig. 7. Rate Region for Two Downstream ADSL Users .,~ _"
I_ c ~..,I t_ Cou.~
P
.m ~ -p° I
a az o.v o.e , a ~~ o o.l a3 oe , is t..
.,aw~,nwer .wwtwu~
.b - onu u..:
i- ow, i .b _b a ..1 a 03 0A Pe , t3 a o 0.1 O~ i ,3 a w~weYIMW I t.ywq.nwlYl Fig. 8. OS$ (left) and IOSB (right) power spectral densities for two distributive ADSL users at equal rate. Power spectral densities for both CO-based (top) and RT-based (down) lines are plotted.
co 141an Fig. 9. Loop Topology for IO-User VDSL
line is Il.5dBm. Again, as shown in Fig. 10. the rate region achieved by IOSB
algorithm is almost identical to that achieved by OSB.

per v ,s . ...
a a i2 x y 0 0.5 7 1.5 2 2.5 3 J.5 4 4.5 4k /eel Users UpSUeam Rare (MCpa) Fig. 10. Rate Region for Upstream for 10-User VDSL
C. 5-user VDSL F1r11 Duplex The current VDSL standard uses a fixed frequency bandplan (i.e. 998) to separate upstream and downstream. This is not optimal because no overlapping of upstream and downstream transmissions is allowed. In this set of simulations, we explore the achievable rate-region and the optimal power allocations with overlap spectra for full duplex transmission in a VDSL
environment. The simulation setup consists of 5 users with the same Loop length (3k feet long) in the same binder. As the loop characteristics for the five users are identical, this is essentially a two-user scenario between upstream and downstream. Perfect echo cancellation is assumed. The near-end crosstalk (NEXT) is modelled in addition to the far-end crosstadk (FEXT). The downstream transmission has a power constraint of 11.5dBm, and the upstream transmission has a power constraint of 14.5dBm, in accordance to VDSL standard.
Fig. 12 shows the achievable rate regions obtained from the OSB and IOSB
algorithms, As can be seen, the performance of IOSB is very close to that of OSB, although IOSB
is clearly a sub-optimal algorithm. Furthermore, it observed that the solution provided by IOSB
is not unique.
The non-uniqueness of this algorithm is exposed by choosing a different order of users during the iteration procedure in IOSB. IOSB gives slightly different rate regions for different iteration orders.
Interestingly, no particular order has a rate region that is completely superior to the rate regions of all other orders. In addition, as seen in PSD plots ordering affects the power spectral densities as well. Fig. 13 shows the PSD pairs corresponding to the downstream-upstream ordering and the 3k feet CU Users' Terminals Fig. 11. Loop Topology for 5-User VDSL
~ iOSB
~ iOSB
v~
E 25 ~7~
~' ~ 20 ~.
N
15 , ~ "
\W
\.\ a \.\

0 5 10 15 20 25 30 35 40 4;
VDSL DownsW am Rate (Mbps) Fig. 12. Rate region for 5-User Full Duplex VDSL
Algorithm Downstream Upstream Rate Rate IOSB (Down-Up)23.87Mbps 23.67Mbps IOSB (Up-Down)23.93Mbps 23.87Mbps OSB 29.71\Mbps 29.7A~Ibps TABLE II
Mlaximum Equal Rate for 5-User Full Duplex VDSL
upstrearrr-downstream ordering. As can be seen, a narrow low frequency spectrum is always shared by both directions. In tire high frequency range, frequency-division duplex (FDD) separates the upstream and the downstream. FDD is optimal in the high frequency range because of the strong NEXT interference. Interestingly, if the downstream-upstream ordering is used in IOSB, the resulting frequency division follows an Up-Down-Up pattern. The situation is completely reversed when the upst.rearn-downstream ordering is used. Tyre upstream-downstream ordering produces a FDD solution that. follows Dowrr-Up-Down pattern.

a~-_ oe , a o ~ .,m ,bu i m . i i "~~ n . i i i - o x . . rs ,. a n a x . a rs a n a w.w.~.r,~i ~.a.~wn~r ~.m.a ,-w. of i-w. vm, b p .m ~ .,m a x ~~ ~ a ,x a ,a ,a o x . a rs ,e ,e ,e ..a.~w,rem...~"vra, Fig. 13. Downstream (top) and upstream (bottom) power spectral densities for 5-user full duplex VDSL at equal rate. The power spectra depend on the ordering in iteration in IOSB. Downstream-upstream order is on the left, and the upstream-downstream order is on the right.
D. 10-user VDSL Full-Duplex In this final set of simulations, we explore the full duplex transmission of a 10-user VDSL scenario with the same topology as in Fig. 9, but allowing overlapping spectra. Perfect echo cancellation is again assumed. The OSB algorithm is not computationally practical in this case.
Table III compares the performance of the proposed IOSB algorithm with that of iterative water-filling (IWF). Iterative water-filling is able to support a minimal data rate of 12.2A~Ibps, while IOSB
is able to achieve at least 15.2Mbps. A minimum gain of at least 2.8Mbps is possible.
The power spectral densities obtained from the IOSB algorithm are shown in Fig. 14. Interestingly, a shall low frequency band is shared by all four transmitters with full duplex operation. In the middle frequency band, frequency-division mutiplex (FDD) separates upstream and downstream transmissions of the 2kft and 4kft users. The high frequency band is used exclusively by the 2kft lines. Again, frequency division duplex (FDD) is used there. This type of optimal spectrum usage is non-obvious and is channel and user data rates dependent.
'I~ansmitter IOSB IWF

4k ft Downstream15.2Mbps12.2Mbps 4k ft Upstream15.2Mbps12.4Mbps 2k ft Downstream21.8Mbps12.7Mbps 2k ft Upstream25.2Mbps12.5Mbps TABLE III
INaximum Aiinimum Rate for 10-User Full Duplex VDSL

a$
-2p 4k tt Users Downslnam .
_-g0 ~o-g0 ~(700 p720 2 g 12 -20 Fr ra -4p- - -2k tt Users DownshHm B
_gp -BO

rt00 ft720 f~tao 0 s t0 2 a t2 a to a ~

-60 4k users U
slrsam ~

laO

D g 10 f, Fr -20 wnc --40 M 2k - tt Users U
Uwm D g 10 2 B t2 4 to Frpwncy (MHy Fig. 14. Power spectral densities for 10-user full duplex VDSL at maximum minimal rate IX. Summary of the Second Part of the Invention The second part of the invention comprises of low-complexity and near-optimal spectrum balancing algorithm for digital subscriber line applications. As compared to previous optimal spectrum balancing methods, the new algorithm offers a significant complexity reduction. The complexity is reduced from an exponential complexity in the number of users to a linear complexity. The rrrain idea of tl-le algorithm is an iterative evaluation of the Lagrangian function in the optimization step.
Simulation results show that the performance of the new algorithm is very close to that of the optimal algorithm. The proposed iterative algorithm is a. significant step forward in making optimal spear unr balancing practical.
The proposed iterative algorithm has a wider implication beyond that of digital subscriber lines.
The proposed algorithm can be easily applied to the adaptive bit, power and subcarrier allocation problems for wireless applications whenever multiuser orthogonal frequency division multiplex (OFDM) is used.

Claims (3)

1. A method of controlling a digital communication system having a plurality of communication lines on which signals are transmitted and received, the signals being affected by interference during transmission, each of the communication lines being used by a user of the digital communication system and having linked thereto at least one transmitter and at least one receiver, the method comprising the steps of:
(a) calculating a set of optimal transmit power spectral densities and frequency bands used by each transmitter using a Lagrangian dual method, comprising the steps of:
(i) initializing the values of a plurality of applicable Lagrangian dual variables; and (ii) evaluating a Lagrangian function by enabling each user to optimize its power spectral density and bandwidth allocation by maximizing a joint transmission data rate objective based on the user's own transmission rate and on the transmission of some or all other users of the digital communication system;
(b) updating the dual variables by applying a subgradient update method, an ellipsoid update method, an analytic center cutting plane update method, or other dual update methods, or a combination of the foregoing dual variable methods; and (c) controlling the transmit power spectral density and frequency band of each transmitter using such optimal transmit power spectral densities and frequency bands.

2. The method of claim 1, whereby the digital communication system is a DSL
system.

3. The method of claim 1, whereby the set of optimal transmit power spectral densities and frequency bands is calculated on a approximate basis.