1102 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

Simultaneous Water Filling inMutually Interfering Systems

Otilia Popescu, Member, IEEE, Dimitrie C. Popescu, Senior Member, IEEE,and Christopher Rose, Fellow, IEEE

Abstract— In this paper we investigate properties of simul-taneous water filling for a wireless system with two mutuallyinterfering transmitters and receivers with non-cooperative cod-ing strategies. This is slightly different from the traditionalinterference channel problem which assumes that transmitterscooperate in their respective coding strategies, and that interfer-ence cancellation can be performed at the receivers. In this non-cooperative setup, greedy capacity optimization by individualtransmitters through various algorithms leads to simultaneouswater filling fixed points where the spectrum of the transmitcovariance matrix of one user water fills over the spectrum ofits corresponding interference-plus-noise covariance matrix, andin our paper we study the properties of these fixed points. Weshow that at a simultaneous water filling point the eigenvectorsof transmit covariance matrices at each receiver are aligned, andidentify three regimes which correspond to simultaneous waterfilling that depend on the interference gains: a) complete spectraloverlap, b) partial spectral overlap, and c) spectral segregation.These imply that the transmit covariance matrices will be whitein regions of both overlap and segregation, but not necessarilywhite overall. We also consider performance as a function ofinterference gain and show that complete spectral overlap is astrongly suboptimal solution over a wide range of gains.

Overall, our results suggest that for strong mutual interfer-ence, an effort should be made to do joint decoding over receiverssince such collaboration can provide large capacity increases.For moderate interference, distributed and/or centralized conflictresolution algorithms would be most effective since more complexcollaborative methods do not afford much improvement andstrictly greedy methods such as water filling perform poorly,while for weak interference a laissez faire approach seemsreasonable.

Index Terms— Capacity, interference channel, MIMO, multi-base systems, water filling.

I. INTRODUCTION

IN unlicensed bands, spectrum is shared by all transmittersand receivers, and decoding information from a given trans-

mitter by the intended receiver must cope with interference

Manuscript received July 4, 2005; revised March 3, 2006 and June 3,2006; accepted June 5, 2006. The associate editor coordinating the reviewof this letter and approving it for publication was K. K. Wong. This workwas supported in part by the National Science Foundation thorugh grant ITR–0312323, and by the Texas Higher Education Coordinating Board AdvancedResearch Program grant 010115-0013-2006. This paper was presented in partat the GLOBECOM 2003 and 2004 conferences.

O. Popescu was with the Wireless Information Network Laboratory (WIN-LAB), Rutgers University, 73 Brett Road, Piscataway, NJ 08854-8060 USA(e-mail: otilia@winlab.rutgers.edu).

D. C. Popescu is with the Department of Electrical and Computer Engi-neering, Old Dominion University, 231 Kaufman Hall, Norfolk, VA 23529USA (e-mail: dpopescu@odu.edu).

C. Rose is with the Wireless Information Network Laboratory (WINLAB),Rutgers University, Technology Center of New Jersey, 671 Route 1 South,North Brunswick, NJ 08902-3390 USA (email: crose@winlab.rutgers.edu).

Digital Object Identifier 10.1109/TWC.2007.05464.

from “alien” transmitters. Under certain conditions (such asco-located receivers, or receivers connected to a common highspeed backbone network) a wireless communication systemcan be modeled as a multiple access channel [1, p. 388], forwhich a vast body of literature offers numerous methods ofdecoding information in the presence of interference [2]–[7].That is, if receivers are allowed to collaborate, then they canbe regarded as one large receiver with multiple inputs andmultiple outputs (MIMO), and results established for MIMOsystems pertain [7], [8].

However, when no cooperation among receivers is assumed,information from a given transmitter must be decoded at itsintended receiver in the presence of interference from othertransmitters. This is an instance of the interference channel[1, p. 382] whose complete characterization is still an openproblem. We note that an early formulation of the interferencechannel problem is due to Shannon [9], and was followedby results obtained decades later by Ahlswede [10], Carleial[11]–[13], Sato [14]–[16], Han and Kobayashi [17], and Costa[18]. More recent results have been obtained in the context ofMIMO systems with multiple transmitters that interfere witheach other by Bengtsson [19] and Blum [20].

Nonetheless, mutual interference is a fact of life in wirelesssystems and especially in unlicensed bands. Thus, researchinto practical systems has cast the mutual interference problemin a competitive context where “opposing” systems seek tomaximize their performance. For instance, Yu et al.[21], [22]define a non-cooperative game in which transmitters competefor data rates, and each transmitter’s objective is greedyperformance maximization regardless of other transmitters inthe system. From this perspective it is shown that for a systemwith two transmitters and receivers that do not collaborate,the individual performance of both transmitters is optimizedby a simultaneous water filling distribution of transmittedpowers [21], [22], which may be achieved by application of theiterative water filling procedure established originally in [23].According to Yu et al. the simultaneous water filling pointis a Nash equilibrium solution for the system [21], [22], andin general multiple Nash equilibria are possible for a givensystem. We note that sometimes a socially optimal solutionmay not be a Nash equilibrium [24, p. 18].

In our paper we consider a similar setup as Yu et al. [21],[22] consisting of a system with two transmitters with non-collaborating receivers. Similar to [21], [22] we assume thatthe two mutually interfering transmitters have non-cooperativecoding strategies such that interference cancellation can not beperformed at the receivers. This is slightly different than the

1536-1276/07$25.00 c© 2007 IEEE

POPESCU et al.: SIMULTANEOUS WATER FILLING IN MUTUALLY INTERFERING SYSTEMS 1103

traditional information-theoretic approach of the interferencechannel problem, and in this approach, greedy capacity opti-mization by individual transmitters through various algorithms(like iterative water filling [21]–[23] or greedy interferenceavoidance [25]–[27]) converges to simultaneous water fillingfixed points. We investigate the structure of possible simul-taneous water filling fixed points, and derive a relationshipbetween the spatial distribution of the transmitters and re-ceivers (roughly characterized by their corresponding gains)and the set of potential simultaneous water filling fixed points.We also define socially optimum points for such systems,and determine when these correspond to simultaneous waterfilling.

The paper is organized as follows. We define the problemformally in Section II. In Section III we analyze simultaneouswater filling solutions and derive the spectral structures of mu-tual water filling points in Section IV. In Section V we assumeGaussian signaling by both transmitters and define/illustratethe simultaneous water filling region for a symmetric system.We also introduce the notion of collective capacity (in lieuof the usual information theoretic “sum capacity”) as a globalperformance measure. We discuss weak/moderate and stronginterference in Sections V-A and V-B respectively, focusing onoptimum system performance and when simultaneous waterfilling is optimum. In Section VI we explore different systemcontrol strategies for weak, moderate and strong mutual in-terference and also provide bounds on relative improvementin comparison to completely collaborative systems wherereceivers can pool information to decode transmitted infor-mation. We close with a summary of results in Section VII.

II. SYSTEM DESCRIPTION

Consider a wireless system with two transmitters and tworeceivers as depicted in Fig. 1, which is similar to thosetreated in related work by Carleial [12], [13], Costa [18],and Sato [15], as well as in the recent work by Yu [21],[22]. The two transmitters and receivers coexist in an arbitraryreal-valued signal space of dimension N implied by finitecommon bandwidth and signaling interval constraints [28], butwe do not assume the same representation for the signal spacein terms of a single set of orthonormal basis functions forboth transmitter-receiver pairs. Rather, we assume that eachtransmitter uses a different set of orthonormal basis functionsfor the signal space, and denote the two sets of orthonormalbasis functions by

F (t) =

⎡⎢⎢⎢⎢⎢⎢⎣

f1(t)...

fn(t)...

fN (t)

⎤⎥⎥⎥⎥⎥⎥⎦ and H(t) =

⎡⎢⎢⎢⎢⎢⎢⎣

h1(t)...

hn(t)...

hN(t)

⎤⎥⎥⎥⎥⎥⎥⎦ (1)

The use of two sets of basis functions by the two transmittersmay be due to the fact that each of the transmitters may use adifferent wireless standard which is a reasonable assumptionin the case of mutually interfering transmitters operating inunlicensed bands. Nevertheless, since both F (t) and H(t)span the same signal space, they can be related by a lineartransformation represented by an N × N unitary matrix U,

1 2R

2T

R

T21gg1

11

Fig. 1. A system with two transmitters and two receivers. g1 is theinterference gain corresponding to transmitter 1’s signal at “alien” receiver 2and g2 is comparably defined. Gains to “home” receivers are normalized to1.

such that F (t) = UH(t) and H(t) = U�F (t). We note thatknowledge of this matrix is not required, since – as it willbe seen in Section III – it is not relevant for our analysis ofsimultaneous water filling points.

We assume synchronization between transmitters at bothreceivers, and frequency flat communication channels char-acterized only by scalar gains. The gain between a giventransmitter and its intended receiver is normalized to 1,and we denote by g1 the interference gain corresponding totransmitter 1’s signal at receiver 2 and g2 the interference gaincorresponding to transmitter 2’s signal at receiver 1.

Letting x�(t) be the signal sent by transmitter � = 1, 2, thesignals at the two receivers are

r1(t) = x1(t) +√g2x2(t) + n1(t)

r2(t) =√g1x1(t) + x2(t) + n2(t)

(2)

where n�(t) is the additive white Gaussian noise corruptingthe signals at receiver � = 1, 2. For simplicity we assume thatnoise characteristics are the same at both receivers, with zeromean and power spectral density η0. We note that the signalsfrom the two transmitters can be written in terms of the basisfunctions in equation (1) as

x1(t) = F�(t)x1 = H�(t)U�x1

x2(t) = H�(t)x2 = F�(t)Ux2

(3)

with x1 and x2 being the corresponding signal vectors. Thetransmitted power spectral density for transmitter � is Sx�

(f)and the corresponding transmitted power is

P� = E[x2� (t)] =

∫ ∞

−∞Sx�

(f)df � = 1, 2 (4)

By projecting the received signals in equation (2) onto thebasis functions F (t) and G(t) at receiver 1 and receiver 2respectively, we obtain the corresponding signal vectors

r1 = x1 +√g2Ux2 + n1

r2 =√g1U�x1 + x2 + n2

(5)

with x1 and x2 the signal vectors sent by the two transmitters,and n1 and n2 white Gaussian noise vectors with covariancematrices E[n1n�

1 ] = E[n2n�2 ] = η0IN . Let X� = E[x�x�

� ],� = 1, 2, be the transmit covariance matrices of the twotransmitters, with the transmitted powers

P� = E[‖x�‖2] = Trace [X�] � = 1, 2 (6)

1104 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

We assume each receiver treats interference from the otheras (colored) Gaussian noise. Thus, capacities expressed instandard units of [bits/transmission] [1] can be written as [23]

C1 =12

log |R1| − 12

log |g2UX2U� + η0IN |

C2 =12

log |R2| − 12

log |g1U�X1U + η0IN |(7)

where log(·) denotes the base 2 logarithm, and R1 and R2

are the correlation matrices at each receiver expressed as

R1 = E[r1r�1 ] = X1 + g2UX2U� + η0IN

R2 = E[r2r�2 ] = g1U�X1U + X2 + η0IN

(8)

In this framework, capacity can be optimized by individualtransmitters through various algorithms (like iterative waterfilling [21]–[23] or greedy interference avoidance [25]–[27])which converge to simultaneous water filling fixed pointswhere the spectrum of the transmit covariance matrix ofone user water fills over the spectrum of its correspondinginterference-plus-noise covariance matrix. In our paper wewill not focus on a specific algorithm by which such fixedpoints are reached, and note that these have been discussed in[21], [22], [26], [27]. Rather, we concentrate on the spectralstructure of user transmit covariance matrices at simultaneouswater filling fixed points and their performance.

In the following section we will show that the greedycapacity optimization implied by simultaneous water fillingleads to transmit covariance matrices that imply a diagonalstructure on the correlation matrices of the received signalsat the two receivers, which is similar to classical waterfilling solutions. Then, we will show that unlike classicalwater filling spectra for single or multiple transmitters witha single receiver, mutually water-filled spectra are in this caseconstrained to three different classes – completely overlapped,segregated, and partially overlapped. Furthermore, the spec-trum in each region must be white although the spectra Xi

are not necessarily white themselves.

III. SIMULTANEOUS WATER FILLING

Simultaneous water filling is formally defined by Yu [21]as a generalization of the single-user water filling to multipleuser scenarios. For the considered system with two mutuallyinterfering transmitters described in the previous section, asimultaneous water filling point is reached when one trans-mitter maximizes its corresponding capacity by regarding theother transmitter’s interfering signal as Gaussian noise anddistributes its transmit energy over the signal space accordingto a single-user water filling scheme [1, p. 253], and viceversa. As noted in the previous section, simultaneous waterfilling for the considered system with two transmitters andreceivers can be achieved by using algorithms like iterativewater filling [21], [22] or interference avoidance [26], [27],which are not discussed here since they lie outside the scopeof the paper. We note that these algorithms yield the transmitcovariance matrices X1 and X2 whose spectral structure willbe investigated in the subsequent sections of the paper.

The main result of this section is to show that at asimultaneous water filling point the two transmit covariance

matrices have the same eigenvectors from the perspective ofeach receiver, that is, at each receiver the eigenspaces of thesignals from the two transmitters are aligned. We state thisresult formally as a theorem:

Theorem 1 (Alignment): At a simultaneous water fillingpoint, at a given receiver, the signals from the intendeduser and the interfering user are aligned along the sameeigenvectors.

Proof: Let us express the transmit covariance matri-ces of the two transmitters in terms of their eigenvectorsand eigenvalues using the spectral factorization theorem [29,p. 104],[30, p. 296] as

X1 =N∑

i=1

αiψiψ�i = ΨAΨ�

X2 =N∑

i=1

βiφiφ�i = ΦBΦ�

(9)

where Ψ and Φ are the matrices of eigenvectors {ψi}Ni=1

and {φi}Ni=1 for X1 and X2, respectively. A and B are the

corresponding diagonal eigenvalue matrices, with eigenvaluesα1 ≥ . . . ≥ αN and β1 ≤ . . . ≤ βN , in decreasing andincreasing order of their magnitudes, respectively. We notethat eigenvalues are real and non-negative since the X� arecovariance matrices, and are therefore positive semidefinte.

From the perspective of a given transmitter, the othertransmitter is Gaussian interference, and, in order to achieveminimum mutual interference, at a simultaneous water fill-ing point the eigenvectors of its transmit covariance matrixmust align with the eigenvectors of the interference-plus-noise covariance matrix [21], [23], [31], [32]. Therefore, ata simultaneous water filling point, at receiver 1 the transmitcovariance matrix X1 of transmitter 1 and interference-plus-noise covariance matrix g2UX2U�+η0IN will have the sameeigenvectors specified by Ψ. This implies that Ψ diagonalizesthe interference-plus-noise covariance matrix and we can write

Ψ�(g2UX2U� + η0IN )Ψ︸ ︷︷ ︸diagonal

= g2Ψ�UX2U�Ψ + η0IN︸ ︷︷ ︸diagonal

(10)We note that since the left hand side is a diagonal matrix, andthe second term in the right hand side is also a diagonal matrix,then the first term in the right hand side must be a diagonalmatrix as well, which implies that UX2U� is diagonalized bythe same set of eigenvectors Ψ as X1. Thus, at a simultaneouswater filling point, at a receiver 1 the signals from the intendedtransmitter 1 and the interfering transmitter 2 are aligned alongthe same eigenvectors specified by Ψ. We can show similarlythat at recevier 2 the eigenvectors of U�X1U and X2 are thesame and equal to Φ.

Corollary 1 (Diagonal Structure): At a simultaneous waterfilling point, at a given receiver, the received signal correlationmatrix is diagonal.

Proof: Multiplying the received signal correlation ma-trices R1 and R2 in equation (8) with their correspondingeigenvector matrices Ψ and Φ, and using the alignment result

POPESCU et al.: SIMULTANEOUS WATER FILLING IN MUTUALLY INTERFERING SYSTEMS 1105

in Theorem 1 implies

Λ1 = Ψ�R1Ψ = A + g2B + η0IN

Λ2 = Φ�R2Φ = B + g1A + η0IN

(11)

which shows that the Λi are diagonal matrices.

We conclude this section by noting that the eigenvalues ofthe transmit covariance matrix of a given transmitter mustalso satisfy a water filling distribution [1, p. 253] over thoseeigenvectors of the corresponding interference-plus-noise co-variance matrix with minimum eigenvalues.

IV. SIMULTANEOUS WATER FILLING STRUCTURES

As mentioned in the previous section, a simultaneous waterfilling point can be obtained by application of various algo-rithms, which will yield a set of transmit covariance matricesX1 and X2 corresponding to the signals sent by the twotransmitters. In this section, we discuss the possible spectralstructures that correspond to these covariance matrices. Wenote that in light of the diagonal structure result proved inthe previous section, we focus the discussion on the diago-nal eigenvalue matrices A and B corresponding to the twotransmit covariance matrices X1 and X2, which are the onesthat determine the spectrum of the simultaneous water fillingstructures.

The three possible spectral structures for simultaneouslywater-filled transmit covariance matrices are:

1) Complete overlap of the two spectra, when all dimen-sions of the signal space are utilized by both transmit-ters.

2) Incomplete overlap of the two spectra, when only asubset of signal dimensions is shared, leaving somedimensions for exclusive use by a single transmitter.

3) No overlap of the two spectra, when transmitters placetheir signal energy in orthogonal signal subspaces.

We derive the properties of such spectra in the following threesections.

A. Complete Overlap Between Transmitted Spectra

Suppose the transmit covariance matrices for both trans-mitters span the entire signal space, which corresponds tocomplete overlap in signal space between the transmittedspectra. Water filling requires that eigenvalue matrices inequation (11) be scaled identity matrices, that is

A + g2B + η0IN = c1IN

B + g1A + η0IN = c2IN

(12)

where c1 and c2 are the “watermarks” corresponding to thetwo transmitters, and are obtained using the trace constraintsin equation (6) as

c1 =P1 + g2P2

N+η0 and c2 =

P2 + g1P1

N+η0 (13)

With the exception of the case g1 = g2 = 1 when the twomatrix equations implied by (12) and (13) are identical, thesolution of the system of matrix equations (12) is

A =P1

NIN and B =

P2

NIN (14)

Since all eigenvalues of transmit covariance matrices for bothtransmitters are equal in this case, the water filling structure isalso unique and implies that the transmit covariance matricescorresponding to the two transmitters must be scaled identitymatrices.

When g1 = g2 = 1 multiple solutions are possible such thatthese satisfy the undetermined matrix equation

A + B + η0IN = cIN with c =P1 + P2

N+ η0 (15)

We conclude this section by noting that complete spectraloverlap is possible for any values of the interference gains g1and g2, and is unique except for the case of g1 = g2 = 1when many solutions are possible.

B. Incomplete Overlap Between Transmitted Spectra

Now suppose that transmitter i ∈ {1, 2} covariance matrixat a simultaneous water filling point spans ki dimensions ofthe signal space, such that k1 + k2 > N and the two spectraoverlap in k1 +k2−N dimensions. We note that ki coincideswith the rank of transmitter i covariance matrix, and is fixedonce the simultaneous water filling point is reached.

From transmitter 1’s perspective, water filling implies thatthe first k1 values of Λ1 in equation (11) must be identicaland equal to the “water level” L1 in transmitter 1’s subspace.This water level is determined from the trace constraints as

L1 =

Trace [X1] + g2

k1∑i=1

βi

k1+ η0

=

P1 + g2

k1∑i=1

βi

k1+ η0

(16)

Equation (16) along with the water filling condition [1, p. 253]implies that the eigenvalues of the transmit covariance matrixof transmitter 1 can be expressed as

αi = (L1 − g2βi − η0)+ i = 1, . . . , N (17)

where the notation (x)+ denotes the positive part of x [1,p.253] that is (x)+ = x if x ≥ 0 and (x)+ = 0 if x < 0. Thisimplies that

αi ={ L1 − g2βi − η0 for i = 1, . . . , k1

0 for i = k1 + 1, . . . , N (18)

and X1 is expressed only in terms of its first k1 eigenvectorsas

X1 =k1∑

i=1

(L1 − g2βi − η0)φiφ�i (19)

Similarly, from the perspective of the transmit covariancematrix of transmitter 2, water filling implies that the last k2

values of Λ2 must be identical. Proceeding as for transmitter 1we have

L2 =

P2 + g1

N∑i=N−k2+1

αi

k2+ η0 (20)

1106 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

with eigenvalues of the transmit covariance matrix of trans-mitter 2 expressed as

βj = (L2 − g1αj − η0)+ j = 1, . . . , N (21)

or

βj ={

0 for j = 1, . . . , N − k2

L2 − g1αj − η0 for j = N − k2 + 1, . . . , N(22)

The transmit covariance matrix of transmitter 2 is written thenonly in terms of its last k2 eigenvectors as

X2 =N∑

j=N−k2+1

(L2 − g1αj − η0)φjφ�j (23)

We note that incomplete overlap between the two spectrain signal space does not imply a unique solution since thedimensions (k1, k2) of the two subspaces spanned by thetransmit covariance matrices of the two transmitters are notunique. These various solutions may be obtained by usingdifferent initializations in the algorithms that generate thetransmit covarince matrices X1 and X2.

At this point we distinguish two cases of incomplete overlapbetween the two transmitter spectra:

1) Partial overlap with k1, k2 < N , where no transmitcovariance matrix spans the entire signal space.

2) Nesting partial overlap with k1 = N and k2 < N , wherethe transmit covariance matrix of transmitter 1 spansthe entire signal space, and the spectrum of the transmitcovariance matrix of transmitter 2 appears “nested” withit).

1) Partial Overlap: k1, k2 < N : In this case trans-mitter 1’s signal occupies alone N − k2 dimensions of thesignal space, transmitter 2’s signal occupies alone N − k1

dimensions of the signal space, and the two signals overlap ink1 +k2−N dimensions. The watermark from the perspectiveof transmitter 1 (transmitter 2) is determined by its largesteigenvalue, α1 (βN ), and the water filling structure imposesthe following constraints:

α1 = α2 = . . . = αN−k2 == αN−k2+1 + g2βN−k2+1 = . . . = αk1 + g2βk1

(24)

βN = βN−1 = . . . = βk1+1 == g1αk1 + βk1 = . . . = g1αN−k2+1 + βN−k2+1

(25)

In addition, the water filling condition requires also that thewatermark L1 (L2) be always less than or equal to the inter-ference plus noise levels on the dimensions that are not waterfilled [1, p. 253]. This implies the following relationships forthe largest eigenvalues of the two transmit covariance matrices

α1 ≤ g2βN and βN ≤ g1α1 (26)

which further implies that the incomplete overlap case of thewater filling structure is possible only if the interference gainssatisfy

g1g2 ≥ 1 (27)

By solving the system of equations (24) – (25) one cansee that when g1g2 > 1, all the eigenvalues can be written in

terms of the largest eigenvalues α1 and βN as⎧⎪⎨⎪⎩

α1 = α2 = . . . = αN−k2

αN−k2+1 = . . . = αk1 =g2βN − α1

g1g2 − 1αk1+1 = . . . = αN = 0

(28)

and ⎧⎪⎨⎪⎩

βN = βN−1 = . . . = βk1+1

βN−k2+1 = . . . = βk1 =g1α1 − βN

g1g2 − 1β1 = . . . = βN−k2 = 0

(29)

Thus, when g1g2 > 1 the eigenvalues of the transmit covari-ance matrix of each transmitter can have only two nonzerovalues: one for the unshared dimensions of the signal spaceand another for the shared dimensions of the signal space.

When g1g2 = 1, we can denote

g1 = g and g2 =1g

(30)

such that the two systems of equations (24) – (25) becomeidentical

gαn + βn = gα1 = βN n = N − k2 + 1, . . . , k1 (31)

In this case it is no longer possible to find expressions for eacheigenvalue in terms of the largest eigenvalues α1 and βN asbefore, and theoretically any values that satisfy equation (31)will also satisfy the required water filling conditions.

2) Nested Partial Overlap: k1 = N and k2 < N : Inthis case we assume with no loss of generality that trans-mitter 1’s signal occupies all N dimensions of the signalspace while transmitter 2’s signal occupies only k2 dimensionsof the signal space, such that the two signals overlap inthe k2 dimensions of the signal space that are occupiedby transmitter 2. Thus, in this case the transmit covariancematrix of transmitter 1 spans the entire signal space and thetransmit covariance matrix of transmitter 2 spans only a propersubspace. That is, transmitter 2’s signal space is completelycontained (or nested) in transmitter 1’s signal space. Thederivation in previous section applies with minor changes, andthe water filling conditions in equation (26) imply in this casethat

g2βN + αN = α1 and βN + g1αN ≤ g1α1 (32)

which corresponds also to g1g2 ≥ 1.When g1g2 > 1 the eigenvalues of the transmit covariance

matrix of the first transmitter are given in this case by⎧⎨⎩

α1 = α2 = . . . = αN−k2

αN−k2+1 = . . . = αN =g2βN − α1

g1g2 − 1(33)

while those of the transmit covariance matrix of the seconduser are given by{

βN = βN−1 = . . . = βN−k2+1

β1 = . . . = βN−k2 = 0 (34)

When g1g2 = 1 the eigenvalue expressions for the over-lapped dimensions N − k2 + 1, . . . , N cannot be determineduniquely, and similar to the previous case, any values thatsatisfy the water filling conditions are possible.

POPESCU et al.: SIMULTANEOUS WATER FILLING IN MUTUALLY INTERFERING SYSTEMS 1107

C. No Overlap: k1 + k2 = N

Finally, suppose that the transmit covariance matrices ofthe two transmitters span orthogonal subspaces of the signalspace at a simultaneously water filling point, that is, assumethat transmitter 1 signals in a subspace of dimension k andtransmitter 2 signals in the remaining N − k dimensions ofthe signal space. We note that k coincides with the rankof transmitter 1 covariance matrix, and is fixed once thesimultaneous water filling point is reached.

Using the trace constraints in equation (6) we obtain{α1 = α2 = . . . = αk =

P1

kα1 ≤ g2βN

(35)

and {βk+1 = . . . = βN−1 = βN =

P2

N − kg1α1 ≥ βN

(36)

which implies thatg1g2 ≥ 1 (37)

andP1

P1 + P2g2≤ k

N≤ g1P1

g1P1 + P2(38)

Thus, in general the solution for k is not unique, and can beany integer in the interval

k ∈[

P1

P1 + P2g2N,

g1P1

g1P1 + P2N

](39)

We note that as the interference gains increase the range of kexpands since the lower limit decreases when g2 increases andthe upper limit increases when g1 increases. We also note thatfor the particular case corresponding to g1g2 = 1 there maybe no solution for k in the discrete representation considered.This is because when g1g2 = 1 we can replace for exampleg2 = 1/g1 in the lower bound so that this becomes equal tothe upper bound or vice-versa, and if the resulting numberis not an integer there will be no solution for k in this case.Nevertheless, as N → ∞ there will always exist an integerupper/lower bound in equation (39) when g1g2 = 1 whichimplies a unique solution for k.

We conclude this section by formally summarizing therelationship between number of possible simultaneous waterfilling solutions and the interference gain values g1 and g2 inthe following theorem:

Theorem 2 (Overlap): The interference gain product g1g2arbitrates the types of spectral overlap possible with mutuallywater-filled spectra.

1) g1g2 ≥ 1: the transmitted spectra may overlap, and mustbe “white” in both the overlapped and non-overlappedregions. There can be many such sets of spectra.

2) g1g2 < 1: each transmitted spectrum must be white andspan the entire signal space – a unique solution.

Proof: The first statement of the theorem is a rehash ofthe observations presented in Sections IV-B and IV-C wherewe have shown that incomplete overlap, as well as separationof transmitted spectra correspond to g1g2 ≥ 1. For the secondstatement, we note that when g1g2 < 1 neither incompletespectral overlap, nor spectral separation are possible since the

interference gain product is less than 1 in this case. Thus,when g1g2 < 1 only complete spectral overlap is possible.This solution is unique since in this case all dimensions of thesignal space are spanned by both transmit covariance matrices.

Using distance as a proxy for interference gain values{g1, g2} the fixed point classifications of Theorem 2 arerepresented pictorially in Table I.

V. THE SIMULTANEOUS WATER FILLING REGION

We define the simultaneous water filling region as the setof all possible capacity pairs (C1, C2) under the assumptionof Gaussian signaling by each transmitter. We note that thesimultaneous water filling region depends on the interferencegain values g1 and g2 since these affect the capacity values(see equation (7)) as well as the possible simultaneous waterfilling structures (see Table I).

Using physical distance as a proxy for the interference gainvalue (i.e., farther away implies lower value) and equal trans-mitted powers, the scenario g1g2 < 1 corresponds roughly to“weak/moderate interference”, in which the distance from atleast one transmitter to its associated receiver is smaller thanits distance to the “alien” receiver – i.e., the interfering signalat the alien receiver is weaker than the desired signal. Thescenario g1g2 > 1 roughly corresponds to “strong interfer-ence” where a given transmitter is close to the “alien” receiverassociated to the other transmitter, and interferes strongly withits transmission.

To illustrate the simultaneous water filling region and itsdependence on interference gain values, we consider a par-ticular example of a symmetric system with N = 100 signalspace dimensions, gains g1 = g2 = g, transmitted powersP1 = P2 = 10, and background noise level η0 = 0.01. Forthese numerical values the signal-to-noise ratio (SNR) at thereceiver

ρ =P

Nη0(40)

is ρ = 10 or 10 dB. Fig. 2 shows various simultaneous waterfilling regions corresponding to different values of g for theconsidered system.

Referring to Fig. 2, our previous analysis indicates thatfor g < 1 the simultaneous water filling region consistsof a single point, and this is shown for several values ofg < 1 in the upper left plot of Fig. 2. We note that as gincreases getting closer to 1 the two capacities decrease, andthe simultaneous water filling point gets closer to the origin.We also note that in the case of g < 1 we distinguish atruly weak interference scenario (e.g. points with g < 0.18 inthe figure2) for which simultaneous water filling implies goodsystem performance with capacities relatively close to thosein the no interference case, as well as a moderate interferencescenario for which performance of simultaneous water fillingdegrades with capacities in this case far away from those inthe no interference case.

1Except for the case g1 = g2 = 1 when many equilibrium points are alsopossible as discussed in Section IV-A.

2The value of g = 0.18 was obtained using equation (51) in Section V-A.

1108 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

TABLE I

Equilibrium PointsOverlap g1g2 > 1 g1g2 = 1 g1g2 < 1

T12T1 RR 2 1RT2

1

R2

T

2T 2R1T1R

Complete unique unique1 uniqueIncomplete many many -

None many unique -

0 0.5 1 1.5 20

0.5

1

1.5

2

C

C2

g < 1

g = 0.9 g = 0.5

g = 0.1

[bits/transmission/dimension]

[bits

/tran

smis

sion

/dim

ensi

on]

g = 0.18

weak interference moderate interference

no interference

(a)

0 0.5 1 1.5 20

0.5

1

1.5

2g=1

C1

C2

Social Optimum

Complete Overlap

[bits/transmission/dimension]

[bits

/tran

smis

sion

/dim

ensi

on]

no interference

(b)

0 0.5 1 1.5 20

0.5

1

1.5

2g=2

C1

C2

[bits/transmission/dimension]

[bits

/tran

smis

sion

/dim

ensi

on]

no interference

SocialOptimum

(c)

0 0.5 1 1.5 20

0.5

1

1.5

2g=100

C1

C2

[bits/transmission/dimension]

[bits

/tran

smis

sion

/dim

ensi

on]

no interference

SocialOptimum

(d)

Fig. 2. Simultaneous water filling region as a function of interference gain. N = 100 signal space dimensions, interference gains g1 = g2 = g, transmittedpowers P1 = P2 = 10, and noise floor η0 = 0.01.

The case of g ≥ 1 corresponds to strong interference, andis illustrated in the remaining three plots of Fig. 2. In thiscase the simultaneous water filling region consists of multiplepoints which can be classified in three categories according tothe three types of spectral overlap discussed in the previoussection:

• A most interior point corresponding to complete overlapof the spectra of the two transmit covariance matricesat which both capacities have minimum values sincetransmitters interfere in all dimensions of the signal spaceat this point. The most interior point moves closer to theorigin as g increases showing that capacities decrease dueto the increased interference from the “alien” transmitter.

• An outer border containing all points that correspondto spectral separation of the two transmit covariancematrices for which the two transmitters do not interferewith each other. As g increases, the outer border expandsfrom a single point in the case g = 1 to more points thatcorrespond to more potential partitions of the signal spacebetween the two transmitters.

• An interior region containing all points between the

most interior point and the outer border that correspondto incomplete spectral overlap for which transmittersinterfere only in some dimensions of the signal space.

We note that the points that make up the simultaneous waterfilling region have capacity values which are always less thanthe capacities without interference, which can be obtained bysetting the interference gains equal to zero. When g = 0,signals sent by the two transmitters do not interfere with eachother and the corresponding capacities are

C� =N

2log

(1 +

P�

Nη0

)� = 1, 2 (41)

and the point implied by (C1, C2) – (1.73, 1.73) on the plots inFig. 2 – is always above the simultaneous water filling region.According to Carleial [13] this point is achievable in the caseof strong interference by subtracting interference from thedesired signal and decoding it as in the absence of interference,but it does not correspond to a simultaneous water filling pointsince it lies outside of the simultaneous water filling region.

To evaluate the performance of the system overall, we

POPESCU et al.: SIMULTANEOUS WATER FILLING IN MUTUALLY INTERFERING SYSTEMS 1109

introduce the sum of capacities

C = C1 + C2 (42)

which we call the collective capacity in order to distinguishit from the information-theoretic sum capacity used whenreceivers can collaborate [6], [7]. This measure is usefulin the case g1g2 ≥ 1 (where multiple simultaneous waterfilling points are possible) to evaluate which point is a mostdesirable from a global perspective. It is also useful in thecase g1g2 < 1 (when the simultaneous water filling point isunique) to compare it with other spectral partitions which maybe more desirable for the system.

We define the social optimum as the point for which thecollective capacity C is maximized. We note that, when g1g2 =1 the outer border of the simultaneous water filling regionconsists of a single point where both transmitters maximizetheir capacities, and which corresponds to complete spectralseparation in signal space. At this point both individual andcollective interests are satisfied since the collective capacityC is also maximized at this point, and we note that completespectral separation corresponds to a social optimum in thiscase.

What is perhaps most interesting about the montage ofFig. 2 is that from a system design perspective, the weakand strong interference cases are least problematic. For weakinterference (shown in upper left plot of Fig. 2), mutualinterference levels are comparable to or below the noise floorso mutually water-filled solutions will provide good systemperformance with capacities relatively close to those in theno interference case. For strong interference (shown in theremaining three plots of Fig. 2 for g = 1, 2, 100) there existsa mutual water filling solution which is also optimum froma collective capacity standpoint. This solution corresponds tosignal space partitioning between transmitters, and is situatedon the outer border of the simultaneous water filling regionas shown in the corresponding plots of Fig. 2. In addition,in the case of strong interference all the other points onthe outer border of the simultaneous water filling region,although suboptimal, correspond to collective capacity valuesclose to the optimal value [26], [33]. In contrast, for moderateinterference (shown in upper left plot of Fig. 2 along withweak interference) simultaneous water filling leads to capacityvalues that are much below those in the no interference case,and transmitters would be better served if they agreed (or wereforced) to signal in different regions of the signal space. Weexplore these various levels of interference in the next sections.

A. Weak and Moderate Interference

In his doctoral dissertation [21] Yu approaches the sys-tem with two mutually interfering transmitters and non-cooperating receivers from a game-theoretic perspective, andmodels this instant of the interference channel problem as anon-cooperative game in which the two transmitters competefor maximizing their capacities. Using this approach it isshown that when interference gains g1g2 < 1, simultaneouswater filling represents a Nash equilibrium point of the non-cooperative interference channel game [21]. In game theory, aNash equilibrium is defined by a set of strategies such that

each player’s strategy is an optimal response to the otherplayers’ strategies [24, p. 11]. From this perspective, a Nashequilibrium is reached for the Gaussian interference channelgame if and only if a simultaneous water filling solution isproduced by both transmitters, and the optimal signaling strat-egy of each transmitter is to water fill the signal space whileregarding the interfering signal of the other transmitter as noise[21]. We note that a Nash equilibrium is said to be Paretodeficient (or non-Pareto-optimal) if at least one player woulddo better and the other one would do no worse by switching toa different strategy [34, p. 52]. Such Nash equilibria are notnecessarily efficient in that there exist cooperative strategieswhere both players achieve better returns. Classical examplesin this sense are the Prisoner’s Dilemma [34, p. 51] or titfor tat strategies [35]. For g1g2 < 1 we see from Fig. 2that capacities achieved by transmitters under simultaneouswater filling decrease as interference gains increase, and wewill show that the Nash equilibrium implied by simultaneouswater filling in this case can be either efficient or inefficientdepending upon the level of interference. Specifically, we willexplore the relative efficiencies of simultaneous water fillingand signal space partitioning (or segregation) under moderateand weak interference.

With simultaneous water filling each transmitter achievescapacity

Cwf1 =

N

2log

(1 +

P1

g2P2 +Nη0

)

Cwf2 =

N

2log

(1 +

P2

g1P1 +Nη0

) (43)

and the collective capacity is

Cwf = Cwf1 + Cwf

2

=N

2log

(1 +

P1

g2P2 +Nη0

)

+N

2log

(1 +

P2

g1P1 +Nη0

) (44)

For simplicity let us assume a symmetric system with P1 =P2 = P , g1 = g2 = g, and the raw SNR ρ as defined inequation (40), for which we obtain

Cwf = N log(

1 +ρ

ρg + 1

)(45)

With signal space partitioning, capacities are given by

Csp1 =

k

2log

(1 +

P1

kη0

)

Csp2 =

N − k

2log

[1 +

P2

(N − k)η0

] (46)

so the collective capacity is

Csp = Csp1 + Csp

2 =k

2log

(1 +

P1

kη0

)

+N − k

2log

[1 +

P2

(N − k)η0

] (47)

1110 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

k

bits

/tran

smis

sion

/dim

ensi

on

Cwf

C1sp C

2sp

Csp

= C1sp+C

2sp

Fig. 3. Capacity variations as a function of signaling subspace width formoderate interference. N = 100 signal space dimensions, transmitted powersP1 = P2 = 10, interference gains g1 = g2 = 0.5 and noise floor η0 = 0.01.

with k the number of dimensions occupied by transmitter 1.Once again for simplicity we assume a symmetric system toobtain

Csp =k

2log

(1 +

ρ

k/N

)(

1 +ρ

1 − k/N

) +N

2log

[1 +

ρ

(1 − k/N)

]

(48)which is maximized by k = N/2 which implies

C∗sp =

N

2log [1 + 2ρ] (49)

We plot values Csp1 , Csp

2 , and Cwf1 = Cwf

2 = Cwf fora symmetric system with N = 100 signal space dimensions,transmitted power P = 10, moderate interference gain g =0.5 and noise power η0 = 0.01 for k ranging from 1 to99 dimensions3 in Fig. 3 and see that as the number ofsignal space dimensions k occupied by transmitter 1 increases,Csp

1 increases and Csp2 decreases – an obvious result. Less

obvious, we see that for a wide range of values for k, bothtransmitters achieve higher capacities Csp

1 and Csp2 if they

partition the signal space than the capacity Cwf correspondingto the simultaneous water filling solution. We also see thatthe collective capacity when transmitters partition the signalspace Csp does not vary widely with k. We thus form theimpression that for moderate interference, both transmitterscan often do much better under segregation than they wouldunder simultaneous water filling, thus suggesting that the Nashequilibrium point implied by simultaneous water filling is aninefficient one.

Of course, the illustration begs the question of what consti-tutes moderate vs. weak interference. To this end we note thatsimultaneous water filling collective capacity C∗

wf depends onboth the interference gain g and the (raw) transmitter SNRρ, whereas signal space partitioning collective capacity Csp

3The values k = 1, respectively k = 99, correspond to the extreme casesin which transmitter 1, respectively transmitter 2, signal in only one signaldimension.

depends only on ρ. We can plot these two capacity surfacesas a function of g and ρ as in Fig. 4 (shown on the next page):the left hand side plot has a linear scale for g, while the righthand side plot has a logarithmic scale for g and “magnifies”the interval g < 0.5. Qualitatively, we note that only forsmall values of the interference gain g does simultaneouswater filling outperform segregation. Otherwise, the collectivecapacity under signal space partitioning is larger than forsimultaneous water filling. This observation can be quantifiedby comparing the two capacity expression in equation (45)and equation (49)

N

2log [1 + 2ρ]

segregate><

share

N log(

1 +ρ

ρg + 1

)(50)

which simplifies to

ρ

segregate><

share

1 − 2g2g2

(51)

Since the raw SNR ρ is positive, equation (51) is always satis-fied for 0.5 ≤ g < 1, and one can define moderate interferenceas corresponding to interference gain values in the interval[0.5, 1), for which segregation outperforms simultaneous waterfilling, and is always preferable when g > 0.5. In general thecritical value of g for a given SNR ρ is

g∗ =√

1 + 2ρ− 12ρ

(52)

and for any g > g∗, segregation is preferable. For instance,for the numerical example considered in Fig. 2, with ρ = 10,we have g∗ = 0.18.

We close this section by noting that, assuming moderateinterference and segregation, the necessary condition for max-imizing the collective capacity in equation (47) is [26], [36]

P1

k=

P2

N − k(53)

and is similar to the result obtained in [37] using a differentperformance criterion than the collective capacity used in thiswork.

B. Strong Interference

As can be seen from Fig. 2, “strong interference” (g1g2 ≥1) implies multiple solutions for which capacities of bothtransmitters can vary widely. The most interior point of thesimultaneous water filling region corresponds to completespectral overlap in signal space, the points inside the regioncorrespond to incomplete spectral overlap, and points on theouter border correspond to spectral separation in signal space.For g = 1 the outer border of the water filling region becomesa single point, and as g increases, the outer border expandsand more points corresponding to spectral separation in signalspace are possible. Furthermore, the point corresponding tocomplete spectral overlap moves closer to the origin as gincreases. The points inside the water filling region are sub-optimal with respect to achievable capacity since the largestcapacity is achieved by points on the outer border. The worstcase scenario is complete spectral overlap.

POPESCU et al.: SIMULTANEOUS WATER FILLING IN MUTUALLY INTERFERING SYSTEMS 1111

00.10.20.30.40.50.60.70.80.91

−200

2040

60

0

2

4

6

8

10

ρ [dB]

Gain g

bits

/tran

smis

sion

/dim

ensi

on

Csp Cwf

(a)

−30−25−20−15−10−50

−200

2040

60

0

2

4

6

8

10

ρ [dB]

Gain g [dB]

bits

/tran

smis

sion

/dim

ensi

on

Csp Cwf

(b)

Fig. 4. Water filling and separation collective capacity as a function of interference gain and noise level for a symmetric system with M = 2 transmitters.N = 100 dimensions, user power P = 10, interference gain ranges from g = 0.001 to g = 1, and background noise level η0 ranges from 10−6 to 10 (sothat ρ ranges from −20dB to 50dB).

However, the fact remains that under strong interference,points on the boundary of the simultaneous water fillingregion are stable under greedy assumptions. Furthermore,from a system perspective, the collective capacity varies onlymoderately as a function of the amount of signal space allotedto one transmitter over another as seen in Fig. 3. So the mainobjective from a system control standpoint is to find algorithmsor methods which nudge transmitters toward the simultaneouswater filling boundary [26], [33].

VI. DISCUSSION

Our results have interesting system control implications. Forinstance, under weak interference (g1g2 1 and moderatesignal to noise ratio in a symmetric system), we have foundthat essentially ignoring interference and water filling overthe entire signal space provides good performance for eachtransmitter and maximum collective capacity as well. Understrong interference (g1g2 ≥ 1), socially optimal segregatoryNash equilibria exist alongside strongly suboptimal but equallystable water-filling Nash equilibria. While the suboptimalequilibria are undesirable, the stability of the optimal equi-libria hold out hope that algorithms could be devised whichreward transmitters who limit their spectral use and punishthose who spread too widely. Under moderate interference,(g1g2 < 1), the water filling Nash equilibria can be stronglysuboptimal from a collective capacity standpoint and moresocially optimal segregatory solutions are inherently unstable.

Since all these interference regimes seem probable inunlicensed wireless environments populated by current andemerging technologies, it seems prudent to ask what types ofcontrol mechanisms should be developed for efficient systemoperation.

In environments where interferers must repeatedly interact(as opposed to one-time interactions) there are a variety ofdistributed control ideas which can be brought to bear –strategies such as “tit for tat” or “generous tit for tat” [38],[39]. These ideas are currently being explored in the contextof mutually interfering systems for both moderate and stronginterference [33], [36].

Alternatively, one could envision a “spectrumserver/advisor” [40], [41] to which systems might turn

to help mediate spectrum use. The effective spectrum serverwould monitor local environments, determine the type andstrength of interference, and advise systems on the mostefficient use of spectral resources or at least on iterativestrategies which led to improved performance.

This more global view of a spectrum advisor begs a com-parison of collaborative versus non-collaborative approachessince one could imagine that in certain circumstances (suchas strong interference) collaborative decoding could produceimpressive capacity gains. That is, in a world increasinglypopulated by smart wideband radios, the best advice mightbe to cooperate [3]–[7] as opposed to finding ways to in-dependently share the signal space. Likewise, there are alsoscenarios where the increased complexity of collaborationor explicitly coordinating spectrum use might not be worthmeager best-case capacity gains.

To this end, consider the example in Fig. 5 which cor-responds to a symmetric system with two transmitters forweak, moderate, and strong interference. In both plots ofFig. 5 we have shown the maximum sum capacity line of theGaussian MIMO multiple access channel that corresponds to acollaborative scenario computed as in [8] for transmit covari-ance matrices that maximize sum capacity in the collaborativescenario [7].

For moderate and strong interference (g > 0.5), we notethat there is a significant difference between collaborativecapacities corresponding to maximum sum capacity in thecollaborative scenario and the non-collaborative capacities inachieved on the outer border of the simultaneous water fillingregion. The reason for this disparity is obvious since increasedg implies more power capture by the “alien” receiver. Incontrast, for weak interference the difference in performanceis much less pronounced so that from a systems perspective,there is little to be gained by employing more sophisticatedmulti-system collaboration or interference channel codingmethods – we would be better served by finding ways to helptransmitter to avoid one another.

VII. SUMMARY AND CONCLUSIONS

We have considered a simple non-collaborative wirelesssystem with two transmitters and two receivers with flat

1112 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 3, MARCH 2007

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

C2 [b

its/tr

ansm

issi

on/d

imen

sion

]g < 1

C1 [bits/transmission/dimension]

SWF moderate interference(g = 0.5)

SWF weak interference(g = 0.1)

collaborative maximum sum capacity g=0.5

collaborative maximum sum capacity g=0.1

(a)

0 1 2 3 4 50

1

2

3

4

5

C1 [bits/transmission/dimension]

C2 [b

its/tr

ansm

issi

on/d

imen

sion

]

g = 20

SWF region boundary

no interference

collaborative maximum sum capacity

with separated users

(b)

Fig. 5. A comparison of capacity performance between collaborative scenario and non-collaborative Simultaneous Water Filling (SWF) scenario for mutuallyinterfering systems. Transmitted powers P1 = P2 = P = 10, noise level η0 = 0.01, and interference gains g1 = g2 = 0.1 (weak interference),g1 = g2 = 0.5 (moderate interference), and g1 = g2 = 20 (strong interference).

communication channels, in which greedy performance op-timization by individual transmitters leads to simultaneouswater filling solutions. We have investigated properties ofdifferent solutions in relation to the interference gains betweentransmitters and receivers, and have identified their spectralstructure: completely overlapped where each transmitter’sspectrum is white, partially overlapped where each transmit-ter’s spectrum is white in the overlap region and also whitein regions where there is no overlap but not white overall,and completely separated where each transmitter spectrumis white in its respective subspace. In addition, we havedefined the simultaneous water filling region as the set ofpairs of achievable capacities for all possible simultaneouswater filling points, for given transmitted powers, interferencegains, and signal space dimensions, and the collective capacityas a global performance measure which allowed a globalperspective on system performance. We have also provideda more detailed investigation of “weak”, “moderate”, and“strong” interference, focusing on when simultaneous waterfilling achieves optimal system performance and when it doesnot.

For weak interference, simultaneous water filling is indeedoptimal since mutual interference levels are comparable to orbelow the noise floor. For moderate interference, water fillingcan be strongly suboptimal, and signal space partitioningoffers much better performance. For strong interference, si-multaneous water filling can result in socially optimal resourcesharing. However, without external guidance toward the outerborder of the simultaneous water filling region, there are manystable suboptimal “traps” in which the competing systemscould be caught. These considerations led to the notion of aspectrum server/advisor which could help mutually interferingsystems to better utilize shared spectrum.

To better understand design tradeoffs, non-collaboration wascompared by means of numerical examples to complete col-laboration where receivers pool information for joint decoding.For strong interference, both systems would be best served byfinding ways to pool information and collaborate as opposed topursuing various signal space separation strategies [26], [33],[36], or even sophisticated joint coding/decoding methods

[3], [4] which would only increase capacity incrementally.In contrast, for moderate to weak interference, signal spacepartitioning affords good performance and the incrementalgains to be had by collaboration are relatively small. So somemeans of keeping transmissions out of each others’ way seemsto be indicated. Such methods might be centralized as in aspectrum “advisor/server” [40], [41] or could be distributedas in tit for tat or other repeated game strategies [33], [36],[38], [39], [42]–[44].

ACKNOWLEDGMENTS

The authors wish to thank the Associate Editor, Dr. Kit K.Wong, and the three anonymous reviewers for their construc-tive comments on the paper.

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Otilia Popescu (S’03-M’05) received the Engineer-ing Diploma and M.S. degree in 1991 from thePolytechnic Institute of Bucharest, Romania, and thePhD degree from Rutgers University in 2004, all inElectrical Engineering. Her research interests are inthe general areas of communication systems, controltheory, and signal processing. She has worked forUniversity of Texas at Dallas, University of Texasat San Antonio, Rutgers University, and PolitehnicaUniversity of Bucharest, and has been a member ofthe technical program committee for IEEE GLOBE-

COM 2006 and CAMAD 2006.

Dimitrie C. Popescu (S’98-M’02-SM’05) receivedthe Engineering Diploma and M.S. degrees in 1991from the Polytechnic Institute of Bucharest, Roma-nia, and the Ph.D. degree from Rutgers Universityin 2002, all in Electrical Engineering. His researchinterests are in the areas of wireless communica-tions, digital signal processing, and control theory.He is currently an Assistant Professor in the Depart-ment of Electrical and Computer Engineering, OldDominion University. Between 2002 and 2006 hewas with the Department of Electrical and Computer

Engineering, the University of Texas at San Antonio. He has also worked forAT&T Labs in Florham Park, New Jersey, on signal processing algorithmsfor speech enhancement, and for Telcordia Technologies in Red Bank, NewJersey, on wideband CDMA systems. He has served as technical programcommittee member for IEEE GLOBECOM 2006, WCNC 2006, and VTC2005 Fall conferences. His work on interference avoidance and dispersivechannels was awarded second prize in the AT&T Student Research Sympo-sium in 1999. He has co-authored, with Christopher Rose, a monograph bookon interference avoidance methods for wireless systems published in 2004 byKluwer Academic Publishers.

Christopher Rose (M’77-SM’05-F’07) received theB.S. (1979), M.S. (1981) and Ph.D. (1985) degreesall from the Massachusetts Institute of Technologyin Cambridge, Massachusetts. Following graduateschool, Dr. Rose joined AT&T Bell Laboratories inHolmdel, N.J. as a member of the Network SystemsResearch Department. He is currently a professorof Electrical and Computer Engineering at RutgersUniversity in New Jersey an Associate Directorof the Wireless Information Networks Laboratory(WINLAB), as well as a Henry Rutgers Research

Fellow. He is an editor for the ACM Wireless Networks (WINET) journal,the Elsevier Computer Networks Journal, and has served on many conferencetechnical program committees and was technical program co-chair for Mobi-Com’97, Co-chair of the WINLAB Focus’98 on the U-NII, the WINLABBerkeley Focus’99 on Radio Networks for Everything and the BerkeleyWINLAB Focus 2000 on Picoradio Networks. His current technical inter-ests include mobility management, novel mobile communications networks,applications of genetic algorithms to control problems in communicationsnetworks and most recently, interference avoidance methods using universalradios to foster peaceful coexistence in what will be the wireless ecology ofthe recently allocated 5GHz U-NII bands.