Maximizing the Sum Rate in Symmetric Networks of Interfering Links

Abstract

We consider the power optimization problem of maximizing the sum rate of a symmetric network of interfering links in Gaussian noise. All transmitters have an average transmit power constraint, the same for all transmitters. This problem has application to DSL, as well as wireless networks. We solve this nonconvex problem by indentifying some underlying convex structure. In particular, we characterize the maximum sum rate of the network, and show that there are essentially two possible states at the optimal solution depending on the cross-gain (√ε) between the links, and/or the average power constraint: the first is a wideband (WB) state, in which all links interfere with each other, and the second is a frequency division multiplexing (FDM) state, in which all links operate in orthogonal frequency bands. The FDM state is optimal if the cross-gain between the links is above 1/√2. If √ε <; 1/√2, then FDM is still optimal provided the SNR of the links is sufficiently high. With √ε <; 1/√2, the WB state occurs when the SNR is low, but as we increase the SNR from low to high, there is a smooth transition from the WB state to the FDM state: For intermediate SNR values, the optimal configuration is a mixture, with some fraction of the bandwidth in the WB state, and the other fraction in the FDM state. We also consider an alternative formulation in which the power is mandated to be frequency flat. In this formulation, the optimal configuration is either all links at full power, or just one link at full power. In this setting, there is an abrupt phase transition between these two states.

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 9, SEPTEMBER 2010 4471
Maximizing the Sum Rate in Symmetric
Networks of Interfering Links
Sibi Raj Bhaskaran, Stephen V. Hanly, Member, IEEE, Nasreen Badruddin, Member, IEEE, and
Jamie S. Evans, Member, IEEE
Abstract—We consider the power optimization problem of max-
imizing the sum rate of a symmetric network of interfering links in
Gaussian noise. All transmitters have an average transmit power
constraint,thesameforalltransmitters.Thisproblemhasapplica-
tion to DSL, as well as wireless networks. We solve this nonconvex
problembyindentifyingsomeunderlyingconvexstructure.Inpar-
ticular,wecharacterizethemaximumsumrateofthenetwork,and
showthatthere areessentiallytwopossiblestatesattheoptimalso-
lution depending on the cross-gain ???? between the links, and/or
the average power constraint: the first is a wideband (WB) state,
in which all links interfere with each other, and the second is a fre-
quency division multiplexing (FDM) state, in which all links op-
erate in orthogonal frequency bands. The FDM state is optimal if
the cross-gain between the links is above ????. If?? ?
FDM is still optimal provided the SNR of the links is sufficiently
high. With?? ?
but as we increase the SNR from low to high, there is a smooth
transition from the WB state to the FDM state: For intermediate
SNRvalues,theoptimalconfigurationisamixture,withsomefrac-
tion of the bandwidth in the WB state, and the other fraction in the
FDM state. We also consider an alternative formulation in which
thepowerismandatedtobefrequencyflat.Inthisformulation,the
optimal configuration is either all links at full power, or just one
link at full power. In this setting, there is an abrupt phase transi-
tion between these two states.
?
??, then
?
??, the WB state occurs when the SNR is low,
Index Terms—Power control, resource allocation, spectrum al-
location, sum rate maximization, interference mitigatio.
I. INTRODUCTION
W
andinterferencebetweenlinks.Inthispaper,wefocusprimarily
on the management of the second problem using optimized
power allocation. The problem of interference also arises in a
DSL wireline access network, and our results are applicable to
this system as well, perhaps even more so given that we only
treat the time-invariant setting in the present paper. We pose a
IRELESS networks are plagued by two key problems
not encountered in wireline networks: multipath fading
Manuscript received July 17, 2009; revised April 20, 2010. Date of current
version August 18, 2010. This work was supported in part by ARC Grant
DP0984862 and in part by NUS Grant R-263-000-572-133.
S. R. Bhaskaran is with the Department of Electrical Engineering, IIT
Bombay, Powai Mumbai 400 076, India (e-mail: bsraj@ee.iitb.ac.in).
S. V. Hanly is with the Department of Electrical and Computer Engineering,
NUS, Singapore, 117576 (e-mail: elehsv@nus.edu.sg).
N. Badruddin and J. S. Evans are with the Department of Electrical and Elec-
tronic Eng., University of Melbourne, Parkville, Vic. 3010 Australia (e-mail:
nasreenb@ee.unimelb.edu.au; jse@ee.unimelb.edu.au).
Communicated by R. A. Berry, Associate Editor for Communication Net-
works.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIT.2010.2054652
power allocation problem in which the objective is to maximize
the total rate achieved in the network. Each link has to choose
a transmit power spectrum, but the choice impacts not only the
rate achievable on the desired link, but also the rates achievable
on the remaining links of the network.
Unlike traditional power control formulations, in which rate
targets are constraints of the problem [1], the rate maximization
formulation that we consider in this paper provides a more
challenging nonlinear, nonconvex optimization problem. In
this paper, we focus on networks in which the channel transfer
functions are time-invariant and frequency flat; otherwise,
the problem is infinite dimensional and computationally in-
tractable [2].
Recently, progress has been made on time-invariant networks
characterized by frequency flat channel responses (i.e., those
that can be represented by one-parameter channel gains), but
under the assumption that the power allocation itself is time
invariant and frequency flat, with maximum power constraints
on the links [3]–[5]. We will also address this problem in
Section IV, but first we consider the less constrained version
of the problem, in which there are average power constraints
on each link, but no peak power constraints, and the frequency
response is not mandated to be frequency flat. This problem is
known as the “spectrum management problem” or “spectrum
balancing problem” in the literature [6], [7].1
Although the spectrum balancing problem is not itself
convex, we exhibit an underlying convex structure that arises
in symmetric networks, and this structure helps us identify the
optimal solution. We show that the optimal power spectrum
always consists of a relatively small number of modes, where
a mode is a chunk of spectrum in which the power spectral
density of all links is constant. Thus, the optimal total power
spectrum is piecewise constant [8]. In this paper, we charac-
terize the optimal solution precisely for the case of symmetric
interfering links: We provide the bandwidths of the modes, and
the power allocation for each link in each mode.
The general characteristic of the optimal solution is that it
involves at most two states: a frequency division multiplexing
(FDM) state, or a wideband (WB) state in which per-link power
allocationsareflatacrossthefrequencyband.Insomescenarios,
depending on the cross-gain factor and the signal to noise ratio,
the optimal configuration is a mixture of these two states.
In Section IV, we impose the constraint that the transmit
power spectrum of each link be flat across the frequency band.
1The spectrum balancing problem is usually posed for frequency selective
channels, notfor the frequency flatchannels consideredin the presentpaper, but
ourapproachcanstillbeinterpretedas“spectrumbalancing”forthisspecialized
problem.
0018-9448/$26.00 © 2010 IEEE

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4472IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 9, SEPTEMBER 2010
This is in line with assumptions made in several other works in
the literature [3]–[5]. Again, exploiting the underlying convex
structureofthisproblem,weprovidethecompletesolutionfora
symmetric network of
interfering links. Moreover, we iden-
tify an interesting phase transition phenomenon for large net-
works that occurs as the interference cross-gain parameter
in the formulation below) crosses a certain threshold, and we
explicitly characterize this threshold.
II. RELATED WORK
The problems investigated in this paper are special cases of
“spectrum management” [9], otherwise known as “spectrum
balancing” [10]. There has been intensive interest in these
problems for about a decade, mostly in the context of digital
subscriber lines (DSL). More recently, there has been interest in
applications to wireless communications, including cognitive
radio [8], wireless mesh networks [11], and similar problems
were investigated much earlier for cellular systems [12].
In applications to DSL, the focus is a multi-user communi-
cation system, in which each link provides interference to the
others. Typically, each link is time-invariant (appropriate for
DSL applications), and there is a set of time-invariant transfer
functions from each transmitter to each receiver in the network.
There are
desired links (transmitter-receiver pairs) but for
each receiver, there are
other transmitters, each providing interference to the desired
receiver. Each transmitter has a power constraint, and all links
share the same frequency band. The spectrum management
problem is to shape the spectrum of each link to meet the power
constraints, but also to maximize some objective function, such
as the sum of the rates on each link [2]. Various other objective
functions have also been considered in the literature [2].
Inthetime-invariantsetting,onecanformulateageneralsum-
rate maximization problem subject to power constraints
undesired cross-links from the
(1)
where
is the average power constraint for the th link, and
is the noise power spectral density at frequency
fortunately, this problem is infinite dimensional, and NP-hard
as we increase the number of links [2].
A key reason for the difficulty in solving (1) is the noncon-
vexity of the objective function. A common approach is to re-
strict attention to a finite number of subcarriers, as in Orthog-
onal Frequency Division Multiplexing (OFDM). This make the
problem finite dimensional, but does not overcome the compu-
tational difficulties: The general problem is NP hard, as we in-
creasethenumberofsubcarriers,forafixednetworkoflinks[2].
It is also NP hard as we increase the number of links, for a fixed
number of subcarriers [2]. One is, therefore, motivated to seek
suboptimalapproachestothegeneralproblem,althoughonecan
seek exact solutions in small problem instances [13], [14].
Suboptimal approaches for the general problem include
game-theoretic methods, including iterative water filling [15];
. Un-
high SNR approximations, and the use of Geometric pro-
gramming methods [16], [17]; methods of successive convex
approximation [18]; and dual decomposition/column genera-
tion methods [11].
Another approach is to impose further structure on the
problem to reduce the inherent complexity. It is shown in [8]
that even in the continuous spectrum balancing formulation,
the problem is inherently finite dimensional provided that the
transfer functions are finite dimensional. For example, in the
case of flat fading channels, the dimensionality is no more than
, whereis the number of links. In the present paper,
we restrict the class of problems infinitely more, to the very
special case in which all the desired channel gains are unity,
and all the cross gains are the same fixed value,
this symmetric model is far too specialized to be directly
relevant to applications, we can completely characterize the
optimal solution, for any
, and the solution is elegant and
insightful. We hope that techniques developed for the solution
will prove more generally useful, perhaps in the development
of suboptimal methods for more general classes of problems,
or exact methods for more specialized scenarios.
A paper closely related to our own is [13]. Our paper con-
siders an arbitrary number of links, but the channel gains are
completely symmetric. In [13], there are only two links, but the
channel gains are arbitrary. The symmetric version of the two
link problem is also solved in [14].
The discrete version of the sum-rate maximization problem,
with a fixed number of subcarriers, has been studied by many
authors. The special case of one subcarrier leads to considera-
tion of the following optimization problem:
. Although
(2)
where
is the power allocated to link , and
link
to link . This is problem
is different. We solve the symmetric version of this problem in
Section IV, but the general problem, as stated here, is NP-hard
[2]. The two link version of this general problem was solved
in [3].
The optimization problem (2) is motivatedby the fact that the
single link capacity of a discrete time Gaussian noise channel is
nats/symbol, when the SNR is
in (2) can be shown to be achievable using the Gaussian input
distribution in a classical random coding argument. Each trans-
mitter selects i.i.d. symbols, with transmitter
tribution
for its codeword symbols. This approach is
further supported by the mismatched decoding results of [19],
which show that these rates are achievable using minimum dis-
tance decoding, even if the codebooks of the interferers are not
selected randomly as above, which shows that the Gaussian in-
terference model is robust.
The sum-rate in (2) is by no means optimal. One simple ex-
tension that can increase capacity is to allow each transmitter to
modulatethevarianceoftheGaussianinputoverdifferenttrans-
mitted symbols, whilst satisfying the long-term average power
is the number of links,the power constraint,
is the channel gain from
in [2], although our notation
. The sum-rate
using the dis-

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BHASKARAN et al.: MAXIMIZING THE SUM RATE IN SYMMETRIC NETWORKS OF INTERFERING LINKS4473
constraint. This approach is taken in [11], [12]. For linear, time-
invariant, continuous-time channels, one can equivalently vary
the power spectral density across the frequency band. By al-
lowing the input spectrum to vary across frequency (“spectrum
balancing”), rather than restrict it to be frequency flat (white),
we can get an improvement in network capacity, even in the
caseoftime-invariant,frequencyflatchannels[8].Inthepresent
paper, we treat the spectrum balancing problem in Section III,
and relegate the flat input spectrum problem to Section IV.
The sum-rate can be increased further, in some cases, by se-
lecting a different, non-Gaussian, random coding distribution.
In general, the optimal distribution for a given problem instance
is unknown, although it is known to be Gaussian when the in-
terference level is sufficiently low [20]. In the present paper, we
will always use the Gaussian distribution to obtain our achiev-
able rates.
The literature we haveconsidered thus far makes theassump-
tion that each decoder treats the signals from the other links as
Gaussian noise. As remarked above,this is a robust assumption,
and it is reasonable to make this assumption when the decoders
do not have access to the codebooks of the interfering links.
However, if the codebooks are known, then the decoder may be
abletodecodeandcancelsomeoftheinterference,whichcanbe
much better than treating it as Gaussian noise, especially when
the interference to signal ratio is not so low [21]. This motivates
the more fundamental question: what are the absolute limits of
communication in networks of interfering links, in the sense of
Shannon? It turns out that the capacity region of even the most
simple symmetric two link network (of the class investigated in
this paper) is unknown, although there has been recent progress
in obtaining inner and outer bounds to the capacity region. In
particular, the inner and outer bounds are now quite close, with
a gap of no more than 1 bit/sec/Hz, irrespective of SNR, in the
two link, Gaussian interference channel [21]. One interesting
recent result is that it can be optimal to treat the interference
as Gaussian noise, when the interference to noise ratio is suffi-
ciently low [20].
Inspiteoftherecentprogressonthefundamentalinformation
theory, the present paper is focused on the interesting and im-
portant problem of spectrum balancing in Gaussian interference
networks, in which interference is treated as Gaussian noise.
III. SPECTRUM BALANCING PROBLEM
In this paper, we will adopt the convention that “power spec-
tral density” refers to the one-sided version, and thus we will
focus attention on the positive frequencies. In the following, we
denote the Shannon capacity of a discrete time, additive white
Gaussian noise (AWGN) channel, with SNR
where
, by,
We begin with a spectrum balancing problem for a symmetric
network of interfering links, in which the transmitters are band-
limited, with symmetric average power constraints: Consider a
real, base-bandmodelof
communicationlinks,eachof band-
width
Hz, and each link is individually an additive white
Gaussian noise channel (AWGN) with common noise power
Fig. 1. Symmetric network of interfering links, where ? ??? ? ? ??? ?
??? ??? ? ? ???.
spectral density of
canassumewithoutlossofgeneralitythat
The
links interfere with each other, as depicted in Fig. 1, and
weassumeanadditivemodelfortheinterferencebetweenlinks.
We first write down some well known achievable rates for
thenetwork.Onecancomputemutualinformationsbetweenthe
transmitter and receiver of each link, assuming that each trans-
mitter uses a stationary Gaussian process to generate the signal.
Classical random coding arguments then show that these mu-
tualinformationratesare achievable.Denote thereal,stationary
Gaussian process transmitted on link
signal on link
is, where
at the receivers. By re-scaling powers, we
Watts/Hz.
by. The received
(3)
is white Gaussian noise of power spectral density
is the cross-gain between the links of the network. If
process
haspowerspectraldensity
able rate on link
is given by [22]
,
and
thenanachiev-
We impose the power constraint that for all
The problem we address is that of computing the max-
imum achievable sum capacity of this network, under the
above assumptions, which reduces to finding the optimal input
spectra for the links, as expressed in the following optimization
problem:
Problem 3.1: Find the input spectra that achieve the max-
imum in the following program:
(4)
(5)
Clearly this problem is a highly specialized one, with fre-
quencyflatchannelgains(i.e.,nofading)andsymmetricallinks.
Therefore, one should expect a relatively simple solution, per-
haps a wide-band (WB) solution with frequency flat power al-
locations across the links. However, if the cross-gain parameter,
,islarge,itmightbebettertotakeafrequencydivisionmulti-
plexing (FDM) approach, to avoid the interference between the

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4474 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 9, SEPTEMBER 2010
Fig. 2. Two phases and their mixture for the symmetric network of ? links.
links. Due to the symmetry of the model, both approaches allo-
cateequalpowertoalllinks,andintheFDMapproach,eachlink
isallocatedasub-bandofwidth
cationwithinthesub-band.Thesetwoapproachesareillustrated
in Fig. 2, where one is called the “WB phase”, and the other is
called the “FDM phase”. A natural question to ask is: which
phase is the best at a given SNR and level of the cross-gain pa-
rameter,
?. A deeper question is to examine the optimality or
otherwise of these two approaches. We will examine both ques-
tions inthissection, butwe startwith thefirst question about the
relative performance of WB versus FDM.
Hz,withaflatpowerallo-
A. WB Versus FDM
Comparing the performance of WB and FDM is very nat-
ural as these are the two standard approaches used in modern
wirelesssystemstohandlemultipleaccessinterference.FDMis
the classical approach used in traditional radio applications, as
wellasinnarrowbandcell-phoneradionetworkswithfrequency
re-use partitioning between cells. WB is the approach taken in
Qualcomm’s code division multiple access (CDMA) networks,
and is presently used in wideband CDMA 3G networks.
Which approach is better in the symmetric network of inter-
fering links? To answer this question, define the functions
andby
(6)
(7)
Note that
channel at SNR =
of a discrete time Gaussian link that receives interference from
other links, as in the symmetric Gaussian network
is the capacity of a discrete-time AWGN
, and is times the capacity
model. Both are in units of nats per channel use. It follows that
is the achievable sum rate (treating interference
as noise) of all links in the WB model, and
achievable sum-rate of all links in the FDM model.
Let
denote the curve defined by the function
. The following two lemmas characterize the relative perfor-
manceof WB versusFDM,across differentvalues ofcross-gain
and.
is the
,
Lemma 3.1: If
i.e., the curve
, then there exists a unique
then for all,
lies entirely above curve. However, if
such that
(8)
(9)
(10)
i.e.,
the point where they cross.
Proof: See Appendix A.
Lemma 3.1 answers the first question concerning the relative
meritsofWBversusFDM,andisillustratedinFig.3forthecase
. However, Fig. 3 also illustrates the fact that neither
scheme is necessarily optimal: for
both schemes are beaten by a mixture of the two. This observa-
tion holds in general, as shown by the following lemma:
is abovefor, belowfor, and is
in the interval,
Lemma 3.2: If
that touches both
and
and
then there is a unique tangent curve
at two points, namely
(see Fig. 3). Since both
are strictly concave, it follows that for all
and
, with
(11)

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BHASKARAN et al.: MAXIMIZING THE SUM RATE IN SYMMETRIC NETWORKS OF INTERFERING LINKS4475
Fig. 3. FDM versus WB when two curves cross, two user case.
with strict inequality for
touches
at
, and, since the tangent
, we have that
(12)
Thus, both curves lie below this unique tangent line. For
, there is a unique supporting tangent to the curve
point
, and both
for all
at the
andlie below this line, i.e.,
(13)
Similarly, for
to the curve
below this line, i.e., for all
, there is a unique supporting tangent
at the point
,
, both andlie
(14)
Proof: See Appendix A.
We conclude that the situation depicted in Fig. 3 holds for
all
. In other words, when
phase (FDM or WB) can be optimal when
terval
, as one can do better by time sharing between
the phases, or by doing the sharing in the frequency domain as
illustrated in the mixture plot in Figure 2.
This analysis leads to the more fundamental question about
optimality for the symmetric network of interfering links. We
have so far highlighted three distinct schemes to handle inter-
ference: pure FDM, pure WB, and a hybrid scheme that is a
mixture of the two. The following subsection will demonstrate
that the optimal scheme in the symmetric model is always one
of these three candidates, but the particular one depends on the
cross-gain parameter,
, and/or the SNR.
, neither pure
lies in the in-
B. Optimal Scheme
The main result of this section is Theorem 3.1, which sup-
plies the solution to Problem 3.1. In the theorem, the term
is defined by
(15)
where, in the last case, i.e.,
we define
to be the unique number in
.
and,
such that
Theorem 3.1:
The optimal value in Problem 3.1 is
bits/sec. If
optimal value is achievable by dividing the spectrum into
equal sized bands, and allocating each link one of the bands
(FDM). Each link uses power spectral density
in its allocated band. If
optimal value is achievable via the WB approach: all the links
share the entire band, and each link uses power spectral den-
sity
across the wide band. Otherwise,
for some unique
case, rate
is achievable with
cated as follows. One band, of width
the
links, and each link uses power spectral density
in this band. The sumrate achieved in this band is
bits/sec. The remaining bandwidth,
into
sub-bands, and each sub-band is allocated to one of the
links. In its own sub-band, a link uses power spectral density
, and obtains a rate of
in this sub-band.
In summary, there are essentially two distinct states for the
system, FDM or WB. Which one is optimal depends on
,and,inanintermediatescenario,theoptimalconfiguration
is a mixture of the two.
or, the
and then the
and
. In this
bands allo-
Hz is shared amongst
Hz is split evenly
bits/sec
and
C. Proof of Theorem 3.1
The achievability of
fied.Theissueweaddressinthissectionistheconverse,namely,
that there is no other strategy that can beat
Itcanbeshownthattheoptimumcanbeachievedwithspectra
that are piece-wise constant: There are at most
intervals in
with each link having constant power spec-
tral density within each interval [8]. This can be proven via
Caratheodory’s convexity theorem [23], and is a consequence
of the dimensionality of the problem. Here, the dimension of
the problem is
, since there are
has to choose a power level, and the sum-rate provides an addi-
tional dimension. Thus, the following problem is equivalent to
Problem 3.1
Problem 3.2: Let
izedbandwidths
can be immediately veri-
.
disjoint
links, each of which
. Find the normal-
andpower
to solve
levels
(16)
(17)
(18)
Before attempting to solve this problem, we remark that at
first sight it does not appear to be a very simple problem to
solve,as it is nota convexproblem. Forthis reason,we beginby