Stron Interference and Spectrum Warfare Otilia opescu and Christopher Rose WILAB Ruters University 73 Brett Rd., iscataway, J 8854-86 Email: {otilia,crose}@winlab.ruters.edu Dimitrie C. opescu Department of Electrical Enineerin University of Texas at San Antonio 69 Loop 64W, San Antonio, TX 7849-669 Email: dpopescu@utsa.edu Abstract We consider a wireless system with multiple userbase pairs randomly distributed in some reion, in which users try to reedily optimize their performance without any exchane of information between bases, and for which a fixed point is reached. This is a ash equilibrium point for the system and corresponds to a simultaneous water fillin solution. In this paper we focus on systems with stron interference for which the simultaneous water fillin solution implies not an unique fixed point but a set of fixed points, amon which the information theoretic capacities of users vary widely. We propose a dynamic ame to move the system from a possible suboptimal point to a better simultaneous water fillin fixed point, eventually to the optimal point. I. ITRODUCTIO In a wireless information network multiple users (transmitters) and base stations (receivers) are distributed over some eoraphical reion, and users are assined to particular bases as in a typical cellular setup. However, in spite of user assinment to particular bases, the shared nature of the spectrum that characterizes wireless systems implies that all users will interfere with each other leadin to an interference channel scenario which is mostly an open research topic (see [3, p. 38] and references therein). Recent research [7] approaches the Gaussian interference channel from a non-cooperative ame theoretic perspective in which users compete for data rates, and each user s objective is reedy performance maximization reardless of other users in the system. From this perspective it is shown [7] that for a system with two users and bases as depicted in Fiure U U B Fi.. A system with two users and two base stations. is the ain correspondin to user s sinal at base, is the ain correspondin to user s sinal at base. Gains to the own base are normalized to. performance of both users is optimized by a simultaneous water fillin distribution of user powers, which represents a ash equilibrium solution for the system. A detailed analysis of the water fillin solution set relatin eoraphic distribution of users and bases (as characterized by user-base ains) with the set of potential ash equilibria for the system is presented in [6]. Usin a sinal space approach B reference [6] shows that three sinal space confiurations are possible at a fixed point: complete overlap between users in all the dimensions, partial overlap between users (when only some dimensions of the sinal space are shared), and no overlap between users (when users reside in orthoonal subspaces). Table summarizes the results in [6] and relates the number of potential ash equilibria and overlap scenarios to the relative ains of users to bases. TABLE I Equilibrium oints Overlap > = < B B U U B U B U B U U B Complete unique unique unique Incomplete many many - one many unique - Reference [6] defines also the simultaneous water fillin reion as the set of all possible fixed points achievable with simultaneous water fillin, for iven user power, ains and sinal space dimension. In this paper we focus on the case correspondin to which is the stron interference case and for which the simultaneous water fillin solution corresponds to multiple ash equilibria. We note that the case < which corresponds to weak interference [] has an unique ash equilibrium point [6], [7] and is considered in more detail in [5]. We analyze the simultaneous water fillin reion in the case of stron interference and present a procedure that moves the system from any possible low performance point to a better point, eventually to the optimum point. We use as lobal performance measure the collective capacity [6] to show that the most important improvement in performance is achieved by movin the system on the border of the simultaneous water fillin reion, correspondin to orthoonal users in sinal space. In terms of the collective capacity all points on the border offer comparable performance. II. STROG ITERFERECE AD SIMULTAEOUS WATER FILLIG The case corresponds to a physical situation in which the distance from a iven user to the base to which it is associated is larer than the distance to the other base for which the iven user s transmission is not intended and creates interference. This situation corresponds to a stron 83
interference case since usually, with equal user transmitted power, the interferin sinal at any base station is stroner than the actual desired sinal, and is illustrated in Fiure. B U U B C.5.5 =.9 < =.5 =..5.5 C = C.5.5 Complete = Social Optimum Overlap.5.5 C = Fi.. The stron interference case. A user is closer to the base to which it produces interference, than to the base to which its transmission is intended. C.5 areto Boundary C.5 Let i be the transmitted power for user i and i be the spectral heiht of the additive white Gaussian noise that corrupts the sinal received at base station i. If the two users were not interferin then they can transmit at rates R i C i = ( lo + ) i i =, () i The capacity reion will be the rectanle defined by the oriin (, ) and points (,C ), (C, ), and (C,C ), and any rate pair (R,R ) in this reion is achievable. Furthermore, Carleial has shown [] that if interference is stron enouh, then it can be subtracted and the capacity reion of the considered two-user system with stron interference is identical to the capacity reion of the system without interference. This is true when ains satisfy [] i j + j i, j =, i j () i A simultaneous water fillin solution for the considered system is satisfied when each user distributes its power accordin to a traditional water fillin allocation [3, p. 53] reardin interference from other users as Gaussian noise. From a ame theoretic perspective a simultaneous water fillin solution corresponds to a ash equilibrium for the system and is unique only in the case of weak interference which corresponds to < [5] [7]. In the case of stron interference, when, the simultaneous water fillin solution corresponds to multiple ash equilibria for the system, and the simultaneous water fillin reion was defined [6] as the set of all possible user capacities achievable with simultaneous water fillin distribution, for iven user power, ains and sinal space dimension (as implied by fixed communication bandwidth and sinalin interval constraints). Fiure 3 presents a typical simultaneous water fillin reion for a symmetric system with equal user ains = =, sinal space dimension =, user power = and backround noise with = at each base. From Fiure 3 one can see that in the case of weak interference the water fillin reion consists of a sinle point (upper left plot), while in the case of stron interference the reion consists of multiple points amon which user capacities vary widely. More insiht into the water fillin structure that corresponds to these points can be found in [6]..5.5.5 C Fi. 3..5.5.5 C Simultaneous water fillin reion for a symmetric system. We note that the water fillin reion is inside the rectanular capacity reion with no interference, and that in the case of stron interference, when, the simultaneous water fillin reion consists of multiple points amon which information theoretic capacities of the two users vary widely. The most interior point of the simultaneous water fillin reion in this case corresponds to a complete user overlap in sinal space, the points inside the reion correspond to incomplete user overlap in sinal space, and points on the outer border correspond to user separation in sinal space. For =the outer border of the water fillin reion reduces to a sinle point, and as increases, the outer border expands and more points correspondin to user separation in sinal space are possible. Furthermore, the point correspondin to complete user overlap moves closer to the oriin as increase. The points inside the water fillin reion are suboptimal with respect to achievable capacity since the larest user capacity is achieved by points on the outer border, and the worst situation corresponds to the complete user overlap in sinal space case for which users achieve the lowest capacities. The simultaneous water fillin reion has an ellipsoidal border, and the coordinates of a fixed point on the border are iven by the user capacities. Since points on the border correspond to user separation in sinal space user capacities can be expressed as C i = k i lo ( + i k i ) i =, k + k = (3) where k i is the dimension of the sinal subspace occupied by user i. The distance between the oriin and a iven point on the border is r = C + C (4) and varies as the point moves alon the border. For a symmetric system (equal user powers, ains, and backround noise) this distance is maximum when users reside in orthoonal subspaces of equal dimension k = k = /. The point 84
r WF maximum radius 9 8 7 6 5 4 3 =.5.. =. =5 = 5 5 5 3 35 4 n number of users Fi. 4. The variation of r max in equation (7) as a function of number of users and noise level for = =. eak value for =.5,, 5, corresponds to n =, 4,, 39. with maximum distance from the oriin corresponds also to the socially optimal solution for the system defined in terms of the collective capacity, which is a lobal measure used to characterize the overall system performance and was defined in [5], [6] as the sum of user capacities C = C + C (5) For the symmetric system considered this corresponds to that fixed point on the border with equal user capacities C = C. This analysis can be extended to an arbitrary number of user-base pairs n for which users are separated in sinal space. User capacities will be expressed as in equation (3) with i =,...,n and k +...+ k n =, and the distance from the oriin to a point on the ellipsoidal border of the simultaneous water fillin reion is iven in this case by r = n Ci (6) Maximum r is achieved also for equipartition of sinal space amon users, that is for k i = /n, and is written as r max = ( n lo + n ) (7) Fiure 4 shows the variation of r max for different backround noise levels and number of user-base pairs n. In all cases, the plot illustrates a peak value after which r max decreases slowly as the number of user-base pairs increases. We note that as the number of users increases the border of the simultaneous water fillin reion moves closer to the oriin, and the simultaneous water fillin reion dwindles. We conclude our analysis of the simultaneous water fillin reion for the stron interference case by notin that a socially optimal solution implies separation of users in sinal space and corresponds to a simultaneous water fillin i= distribution with no user overlap. Such a solution can be obtained usin various alorithms which yield simultaneous water fillin distributions [5], [7], [8]. However, reardless of what alorithm is used, it is not known a priori to what type of simultaneous water fillin solution these alorithms may convere, and suboptimal simultaneous water fillin distributions in which users overlap (partially or totally) in sinal space are also possible in this case. Thus, in order to reach a socially optimal solution in the stron interference case we may need to aument the alorithm which drives the system to a simultaneous water fillin solution with an extra procedure desined to move the system toward the areto boundary of the simultaneous water fillin reion. III. SECTRUM WARFARE: ADYAMIC GAME FOR ERFORMACE IMROVEMET From the perspective of maximizin individual user capacity, it is desirable for a iven user to reside alone in as lare a sinal subspace as possible. Thus, in the case of a simultaneous water fillin distribution in which users overlap in sinal space a iven user miht employ aressive strateies which seek to drive other users out of one or more sinal space dimensions in order to ain sole residence in those dimensions. This can be achieved throuh a dynamic ame (or competition) that moves the system from interior points of the simultaneous water fillin reion toward the areto border which contains the socially optimal solution, and we propose such a ame in the sequel. For this ame the time is quantized by equal-duration epochs in which one user acts as leader and the other as follower. Durin one epoch the leader and follower make only one move each, which can be either aressive () or passive (retreat). Each user s payoff is the capacity achieved at the end of the epoch. One epoch starts with an aressive move () of the leader for one or more sinal dimensions, and requires that the leader deploy some enery in the tareted dimensions. The amount of enery used and the size of the tareted subspace are both part of the leader s (er s) stratey. In response to the leader s move, the follower reacts in a rational way by applyin a water fillin procedure: estimates the interference covariance and adjusts its transmit covariance accordinly. Thus, unlike the leader s move, the follower s response is completely predictable iven the leader s move. We note that in eneral the leader s move results in a spectral distribution of enery which is not water fillin. Thus, the leader will suffer an immediate decrease in information theoretic capacity after its move. However, the assumption is that the follower will redistribute its enery in a rational way and retreat from the sinal dimension(s) tareted by the leader, which will result in hiher capacity for the leader. In the followin epoch users take turns and the leader of the previous epoch becomes follower and viceversa. Fiure 5 depicts an epoch of the ame in which user is the leader and s sinal dimension k which is shared with user. For clarity, the fiure illustrates the enery 85
User erspective User erspective k k k User Attack User erspective k k k User User erspective k k k k Fi. 5. User s for dimension k shared with user. Fi. 6. User s for dimension k occupied by user only. distribution from both user and user perspectives. The performed by user is better seen from the perspective of user : dependin on the amount of enery user deploys in dimension k user is forced to retreat from that dimension either partially or totally, and in Fiure 5 we assumed that enouh enery is used in the to force user to vacate dimension k. Fiure 6 depicts an epoch of the ame in which user is the leader and s sinal dimension k which is occupied only by user. We assumed aain that enouh enery is used in the to force user to vacate dimension k. In a similar way, the leader could multiple sinal space dimensions usin the same basic method taxin enery from certain dimensions and placin that enery in the tareted sinal dimensions. The maximum number of sinal space dimensions that the leader can attempt to occupy alone can be computed, based on its power and on the interference that the follower creates. For example, for equal user power and ains ( = = and = =) a iven user actin as leader cannot hope to command more than half the sinal space. Tryin to command a larer portion will enable the other user actin as follower, to water fill over some part of the leader s sinal space durin its retreat, thus resultin in lower capacity for the leader at the end of the epoch. Assumin equal power budets, for iven ains and (with ) and sinal space fraction x = k/ occupied solely by user, a separated but mutually water filled confiuration is stable if + x (8) + For =the space partitionin for complete separation is iven by x = = (9) + + (which corroborates the specific = =example). This implies a sinle possible stable point. Thus, user can taret, attain and stably hold at most fraction + of the sinal space. It is important to note that this unique partition of the sinal space represents also the fixed point that maximizes both user capacities, and consequently the collective capacity, thus corresponds to the social optimum for the system. For symmetric systems this point corresponds to equal partition of the sinal space. Thus, for systems with =, the dynamic ame moves the system from any interior suboptimal fixed point of the simultaneous water fillin reion (with users overlappin over some reion of the sinal space) to the socially optimal point. For > the fraction of the sinal space in which user can reside with complete separation, x in equation (8), does not have a unique solution but a set of solutions. For example, for = = = each user can command between one third and two thirds of the sinal space. The larer the ain value the larer the solution interval. As opposed to the case in which =, in this case there is no fixed point on the simultaneous water fillin reion border where both user capacities are maximized. For a symmetric system the social optimum is still the point on the border with C = C, but none of the user capacities are maximized at this point. Thus, for > if a water fillin procedure ends up in an interior point of the simultaneous water fillin reion (with users overlappin in some sinal subspace) the dynamic ame described will move the system on the border of the simultaneous water fillin reion, separatin the users in sinal space, in only one epoch. However, the point on the border where the system is moved is not necessary the socially optimal point. As a consequence, different dynamic ames can be desined based on the number of sinal space dimensions that the leadin user tarets to ain durin one epoch of the ame. We describe here a reedy procedure with users tryin to maximize their performances for which the dynamic ame is desined such that at each epoch, the leadin user tarets the maximum number of sinal space dimensions from which a full retreat by the follower can be enforced without leavin some other portion of their sinal space open to water fillin procedure performed by the follower. We illustrate such a competition in Fiure 7 for a symmetric system with equal ains = = and user powers = =.For =one user can occupy between one third and two thirds of the sinal space at a simultaneous water fillin point on the areto border of the simultaneous water fillin reion, and when the user acts as leader it will taret maximum occupancy of two thirds of the sinal space and will drive the follower to reside in only one third of the sinal space. However, when users chane turns, the follower will act as leader and taret maximum occupancy of two thirds of the sinal space. Thus, owin to the perfect symmetry, the competition enters a limit cycle with users alternatin between occupancy of 86
User erspective / /3 /3 8 7 6 5 4 C +C C / /3 User Attack User erspective / /3 /3 3 C user user user user user 4 6 8 steps Fi. 8. Capacity variations durin one epoch of the dynamic ame for =, = =, =. USER USER = Fi. 7. /3 /3 Competin for maximum sinal space occupancy User Attack k Fi. 9. k Sinal space partition for two orthoonal users. one third and two thirds of the sinal space. We note that the averae performance is improved by movin the system from any interior point to a point on the areto boundary, and this is accomplished in only one epoch for a two-user system, and that the cyclic behavior does not brin any more improvements. Fiure 8 shows capacity variations for the ame played in the particular case presented in Fiure 7. Circles indicate the capacity values correspondin to the simultaneous water fillin fixed points, and the squares indicate capacity values durin one epoch of the ame, after the leader s move and before the follower s response. The initial point on the raph corresponds to the socially optimum point, with each user residin in half of the sinal space, and we included it as a reference point. Each user s capacity variation shows that the leader s capacity decreases after its aressive move, but the decrease in capacity for the leader is not sinificant. However, the decrease in capacity suffered by the follower is sinificant. After the follower s water fillin move the capacity of the leader will be sinificantly increased. Fiure 8 illustrates also the variation of the collective capacity durin the ame. We note that variations of the collective capacity amon different points on the areto boundary of the simultaneous water fillin reion are insinificant, and the collective capacity at the socially optimal point where users occupy one half of the sinal space each is almost identical to the collective capacity at the extreme points where one user occupies one third of the sinal space and the other user occupies two thirds of the sinal space. We also note that the aressive move of the follower durin one epoch has a sinificant neative impact on the collective capacity, but this is compensated by the follower s water fillin move which restores the collective capacity value. From a collective point of view we can conclude that, once a point on the areto boundary of the simultaneous water fillin reion has been reached, continuin the ame will not brin improvement anymore, but has a neative impact due to the lare variations in capacity experienced by users. Thus, the dynamic ame may be desined such that the competition is stopped as soon as capacity oscillations occur. In order to ensure fairness in resource allocation some refined procedures may be necessary to lead the system to the socially optimum point. These procedures may be based on chanin the rational behavior of users such that, once limit cycles occur users bein taretin less sinal dimensions for occupancy. Since such behavior could be exploited by reedy users in the lon run, the ame can be aumented with tit for tat strateies [4] which reward socially acceptable behavior, and punish reedy users. IV. THE BOUCIG REGIO We saw in the previous section that competin only for maximum rate users lead the system into limit cycles, in which each user is bouncin between maximum and minimum sinal space occupancy. We will call this reion the bouncin reion, and for the two user symmetric case considered in the previous section this represents between one third and two thirds of the sinal space. In this section we look more closely to this reion and to the parameters that may affect it. We note that a point on the border of the simultaneous water fillin reion is characterized by orthoonal users in sinal space, as shown in Fiure 9. Let i be the power of user i. The simultaneous water fillin conditions k k and k k () 87
imply that the bouncin reion for user is iven by k + () + and the bouncin reion for user is iven by k + () + Because of the symmetry of the two expressions we can work with no loss of enerality with the expressions for user. The bouncin reion defined by the inequalities in equation () can be rewritten in terms of user power ratio ρ = / as ρ k ρ + ρ (3) + ρ This implies that the reion varies as a function of user ains and power ratio. As ain i and/or power i increases the area that user i can control extends. The width of the bouncin reion is iven by k = ρ + ρ ρ (4) ρ + and is maximized when ρ = /. Thus, for a symmetric system with equal user powers and equal ains, the bouncin reion is maximum. We note that the larer the bouncin reion, the larer the user capacity variation alon the border of the simultaneous water fillin reion. From this perspective, the symmetric case illustrated before is the least favorable one for the dynamic ame proposed. We also note that a narrower bouncin reion is preferable since it makes capacity variations less sinificant which in turn implies that the exact location of the point on border becomes less important. For a two user system with equal user powers the bouncin reion shrinks to a sinle value for systems with =, and implies k = = and k + + = = + + (5) This point corresponds to the social optimum in this case. In the symmetric case corresponds to = = =as illustrated in Fiure 3. The results can be extended for systems with more than two users-base pairs. Simultaneous water fillin conditions for a system with M user-base pairs, with i the power of user i and ij the ain from user i to base j imply the bouncin reion for user i to be i M j= k i ji j i M (6) j= ij j After some alebra it can be shown that for systems with equal user powers the bouncin reion of each user reduces aain to a sinle point if all cross-products ij ji =, such that k i = M j= (7) ji This corresponds to the socially optimum point for the multiuser system. In the case of a symmetric system with multiple user-base pairs, for which all user powers are equal, ains to the own base ii =, and ains to other bases ij =, i j, the width of the bouncin reion is iven by k = + M = +(M ) (M )( ) [+(M )][+(M ) ] (8) and decreases as the number of user-base pairs M increases. The conclusion of this bouncin reion analysis is that the symmetric case with two user-base pairs has the larest bouncin reion. This implies a worst-case scenario in which the competition for maximum sinal space occupancy between users leads to limit cycles. However, we have seen that the performance on any point on the border of the simultaneous water fillin reion is very close to optimum. Thus, it is not crucial for the system to reach the socially optimum point once the border has been reached, and most improvement in performance is achieved by movin the system from interior points of the simultaneous water fillin reion to a point on the border. Dependin on the particular water fillin structure correspondin to the interior point, this can usually be accomplished in a few steps (only one step for a two-user system). After the system has reached the border, the effort (in terms of enery and time consumed) in movin it alon the border may not be worth the extra ain in collective capacity. V. COCLUSIO In this paper we analyzed the simultaneous water fillin solution for a system with two user-base pairs with stron interference correspondin to relative ains. Multiple simultaneous water fillin fixed points are possible for this system for which individual and collective performances vary widely. Thus, simultaneous water fillin alone does not necessary imply optimum resource sharin in this case. In order to improve system performance we propose a dynamic ame that moves the system to better simultaneous water fillin fixed points, eventually to the socially optimum point. REFERECES [] A. B. Carleial. A Case Where Interference Does ot Reduce Capacity. IEEE Tran. on Information Theory, (6):569 57, Sept. 975. [] M. Costa. On the Gaussian interference channel. IEEE Transactions on Information Theory, 3(5):67 65, September 985. [3] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley- Interscience, ew York, Y, 99. [4] R. Gibbons. Game Theory for Applied Economists. rinceton University ress, rinceton, ew Jersey, 99. [5] O. opescu. Interference Avoidance for Wireless Systems with Multiple Receivers. hd thesis, Ruters University, Department of Electrical and Computer Enineerin, 4. Thesis Director: rof. C. Rose. In proress. [6] O. opescu and C. Rose. Water Fillin May ot Good eihbors Make. In roceedins 3 IEEE Global Telecommunications Conference - GLOBECOM 3, volume 3, paes 766 77, San Francisco, CA, December 3. [7] W. Yu. Competition and Cooperation in Multiuser Communication Environments. hd thesis, Stanford University, Department of Electrical Enineerin,. rincipal Advisor: rof. J. M. Cioffi. [8] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi. Iterative Water-Fillin for Gaussian Vector Multiple-Access Channels. IEEE Tran. on Information Theory, 5():45 5, Jan. 4. 88