is always convex on the line created by the power constraint equality. In the case ... rate of a single user over many frequency carriers satisfies the time sharing ...
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Convex Structure of the Sum Rate on the Boundary of the Feasible Set for Coexisting Radios Kandasamy Illanko, Alagan Anpalagan and Dimitri Androutsos Department of Electrical and Computer Engineering Ryerson University, Toronto, Canada.
Abstract—The power allocation that maximizes the sum rate of transceivers operating in the same frequency band is a difficult non-convex problem. In our earlier work [1], we proved that for transceivers operating under a total power constraint, the maximum sum rate occurs at the boundary of the feasible set formed by the hyper plane representing the power constraint. This finding is nontrivial considering that we are dealing with an interference limited system. In this paper, we study the convex structure of the sum rate on the boundary of the power constraint hyper plane. For two transceivers, we prove that the sum rate is always convex on the line created by the power constraint equality. In the case of three transceivers, we identify a region in the middle of the plane created by the power constraint equality, where the sum rate is concave. This is significant because it is in the middle of the boundary plane that the power allocation can be expected to be fair to all three users. We also provide a power allocation protocol and an algorithm that distribute the power among the transceivers with fairness. Simulation results are provided to support the theorems proven in the paper as well as to demonstrate the convergence of the algorithm to the global maximum sum rate. Results of the algorithm are compared with solutions based on Game theory.
I. I NTRODUCTION Wireless systems sharing the same unlicensed band in 802.11 networks have become widespread inside buildings, and outside in urban centers. The ways in which spectrum is to be managed in the future [2], [3], and the new standards, such as 802.15, are only going to make these scenarios of multiple transceivers operating in the same spectra even more ubiquitous. A key parameter in such systems is the amount of power employed by a particular transceiver. While increasing the power used by a transceiver will increase its data rate, it will also increase the amount of interference experienced by any other transceiver operating in the same frequency band. This trade off leads to the following question: Given the channel characteristics for a number of transceivers operating under a total power constraint in the same band, what distribution of power will maximize the total capacity? The simplest way to model multiple transceivers operating in the same frequency band is to use the Gaussian interference channel - first investigated by Shannon [4]. However, even after almost 50 years, the capacity of the Gaussian interference channel remains an open problem [5], [6]. Communication engineers concerned with optimizing the throughput of the Gaussian interference channel have circumvented this difficulty by analyzing what is called the sum rate, instead of the capacity. This sum rate is obtained by applying Shannon’s original formula for capacity to each user separately, while
considering the interference from the other users as noise (Fig. 1). Even then, finding the power allocation that maximizes the sum rate remains a difficult problem. The difficulty is due to the fact that the sum rate is not concave in the powers. This lack of a convex structure excludes the direct use of the Lagrangian dual technique because the duality gap [7] may not be zero. A breakthrough was achieved by Yu and Lui [8] while investigating resource allocation for DSL systems modeled by a multicarrier Gaussian interference channel. They showed that if an optimization problem satisfies a condition called “time sharing” then the duality gap is zero regardless of the convex structure of the objective. They further showed that the sum rate of a single user over many frequency carriers satisfies the time sharing condition as the number of carriers approaches infinity. However, maximizing the sum rate of many users over a single carrier remains an open problem. Geometric programming [9] and monotonic optimization [10] have been proposed to tackle the numerical difficulty involved in finding a solution. While the sum rate or system throughput of many users transmitting over a single carrier is not concave in the power levels of the users, a single user’s rate or throughput is indeed concave in that user’s power. This fact together with the promise of a distributed and fair solution has led many researchers to turn to Game theory for a solution1 . It should be noted that work that uses Game theory to find an “optimal power allocation” does not maximize the system throughput, rather it maximizes individual user’s throughput or utilities. Yu and Cioffi [11] looked at the two user Gaussian interference channel from a Game theory point of view and established that if each user’s sole objective was to maximize its own data rate, a unique Nash equilibrium [12] could be achieved. Etkin et al. [13] considered secondary user pairs operating on a frequencyflat unlicensed band and used Game theory to investigate whether efficiency and fairness could be obtained with selfenforcing rules. Xu et al. [14] extended this work to frequency selective bands. A system of two users operating on two unlicensed channels was modeled as a Gaussian interference channel by Clemens and Rose [15]. They used Game theory ideas and Genetic algorithm to investigate self imposed power control strategies that lead to socially acceptable global solutions. Leshem and Zehavi [16] considered an N -user Gaussian 1 The concavity or quasi-concavity of the individual user’s throughput or utility guarantees a Nash equilibrium.
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interference channel to show that a Nash equilibrium as well as a Nash bargaining solution could be computed using convex optimization techniques. Work that deals with the problem of power allocation in a much broader context (for example, that limits interference to the licensed users) can be found in [17][20]. Our program of research focuses on maximizing the sum rate or system throughput of a number of transceivers operating on the same frequency band under a total power constraint. In our previous work [1], we proved that the power distribution that maximizes the sum rate lies on the boundary of the feasible set formed by the hyper plane created by the power constraint. In this paper, we analyze the convex structure of the sum rate restricted to the plane formed by the power constraint equality. We first show that in the case of two transceivers, the sum rate is convex on the line representing the power constraint. For three transceivers, we prove that there is a region in the middle of the power constraint plane where the sum rate is concave. This result is significant because it is in the middle of the boundary plane that the power allocation can be expected to be fair to all three users. We then propose a protocol for the two user case, and an algorithm for the three user case, that find the power allocation that maximizes the sum rate, while being fair to the users. We present simulation results that verify our theorems, compare the performance of our algorithm to solutions based on Game theory, and demonstrate the convergence of the algorithm to the global maximum sum rate. Section II presents the system model and the optimization problem. Theorems that articulate the convex structure of the sum rate on the boundary are developed and proven in Section III. A power allocation protocol and an algorithm that exploit the convex structure of the sum rate are presented in Section IV followed by Section V, which contains results of simulations that mimic our algorithm in action and verify our theorems. Section VI concludes the paper. N1
T1
a11
R1
a12
a21
a13 a22
T2
T3
Fig. 1.
N2
a32
a23 a33
R2 a31
N3
R3
Multiple users in an unlicensed band.
II. S YSTEM M ODEL Consider N transceivers operating in the same unlicensed frequency band under a total power constraint P modeled using the Gaussian interference channel shown in Fig. 1. User
i employs transmitter i to communicate with receiver i but receiver i experiences interference from all other transmitters. aij denotes the channel coefficient between transmitter j and receiver i, and pi denotes the power used by transmitter i. The transmission rate ci of user i is aii pi ci = log2 1 + Ni + j=i aij pj where Ni stands for the additive white Gaussian noise (AWGN). Our goal is to determine the power distribution p = (p1 , p2 , ...pN ) among the users that maximizes the sum rate of the N users. We wish to solve the following optimization problem. arg max R = p
N
ci
i=1
subject to N
(1)
pi ≤ P, and pi ≥ 0 for all i.
i=1
The N difficulty in solving (1) is due to the fact that the objective i=1 ci is not concave in pi . In our previous work we made some progress by proving a theorem thatasserted that the N solution to (1) occurs at the hyper plane i=1 pi = P. We now present this theorem (The reader is referred to [1] for the proof). Theorem 1: For transceivers operating on a Gaussian interference channel with a total power constraint, given the channel coefficients, the maximum sum rate occurs at a point on the boundary of the feasible set formed by the hyper plane created by the power constraint. In the next section we analyze N the convex structure of the sum rate on the hyper plane i=1 pi = P . III. C ONVEX S TRUCTURE AT THE B OUNDARY In this section we develop and prove the Theorems that articulate the convex structure of the sum rate on the boundary of the feasible set. We begin with the two user case. A. Two Users For two users the sum rate can be written as a11 p1 a22 p2 + log2 1 + . R = log2 1 + N + a12 p2 N + a21 p1 Letting N1 = N/a11 , N2 = N/a22 , a = a12 /a11 , and b = a21 /a22 , p1 p2 + log2 1 + (2) R = log2 1 + N1 + ap2 N2 + bp1 According to Theorem 1, the maximum sum rate occurs on the line p1 + p2 = P . We will now prove that the sum rate R is convex in (p1 , p2 ) when the pair is restricted to the line p1 + p 2 = P . Theorem 2: Given the channel coefficients, the sum rate of two transceivers operating on a Gaussian interference channel, restricted to the power constraint equality, is convex in the powers.
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Proof: On the line p1 + p2 = P , p1 and p2 can be parameterized by p1 = P/2 + r and p2 = P/2 − r with −P/2 ≤ r ≤ P/2. Substituting in (2), P/2 + r R = log2 1 + N1 + a(P/2 − r) P/2 − r + log2 1 + . N2 + b(P/2 + r) This can be rewritten as R = log2 (C1 + Ar) − log2 (C2 − ar) + log2 (C3 − Br) − log2 (C4 + br) where C1 = N1 + (a + 1)P/2, A = 1 − a, C2 = N1 + aP/2, C3 = N2 + (b + 1)P/2, B = 1 − b, and C4 = N2 + bP/2. Twice differentiating R with respect to r, and omitting the multiplicative constant 1/ ln 2, Rrr = −
A2 a2 + 2 (C1 + Ar) (C2 − ar)2 B2 b2 − + . (C3 − Br)2 (C4 + br)2
We will now show that the magnitude of the first term on the right is smaller than that of the fourth term, and the magnitude of the third term is smaller than that of the second term. Please note that b2 C4 b A2 < ⇔ < 2 2 (C1 + Ar) (C4 + br) C1 A b N2 + bP/2 < That is, N1 + (a + 1)P/2 1−a After ignoring N1 and N2 , which are orders of magnitude smaller than the terms they add to, the above inequality reduces to the statement a > 0, which is true. The inequality that states that the magnitude of the third term is smaller than that of the second term can be reduced to the statement 1 > 0, which is also true. Thus, Rrr > 0.
p3 3P
C (p, p, p)
D (x, y) Y 3P
B X 3P
A
p1
Fig. 2.
Coordinate system on the boundary
p2
B. Three Users For three users, the sum rate can be written as
p1 p2 1+ + log2 1 + N1 + ap2 + bp3 N2 + cp1 + dp3 p3 + log2 1 + N2 + ep1 + f p2 (3)
R = log2
where letters a to f represent the interference channel coefficients. The optimization problem is to maximize R subject to p1 + p2 + p3 ≤ 3P, pi ≥ 0. The use of 3P instead of P as the power constraint slightly simplifies the algebra. According to Theorem 1, the maximum sum rate occurs on the boundary of the feasible set formed by the plane segment p1 + p2 + p3 = 3P, pi ≥ 0, labeled ABC, in Fig. 2. Consider now the line AB on the edge of this plane. On AB, p3 = 0 and therefore, (3) reduces to a form similar to (2). Hence by Theorem 2, the sum rate is convex on the line AB. The same applies to the edges BC and CA as well, and we arrive at the following theorem. Theorem 3: On the three edges of the plane segment created by the power constraint equality, the sum rate of three transceivers, operating on a Gaussian interference channel, is convex in the powers. Analyzing the convex structure of the sum rate on the entire interior of the plane ABC would be very tedious. In what follows, we prove a theorem that identifies a region in the middle of this plane where the sum rate is concave. This finding is significant because it is in the middle of the plane that the power allocation can be fair to all three users. We begin by motivating the choice of the coordinate system we employ to prove the next theorem. Our purpose is to analyze the convex structure of the sum rate restricted to the plane ABC in Fig. 2. The most straightforward way of parameterizing the points on this plane would be to let p1 = x, p2 = y, and p3 = 3P − x − y. In this parameterization, we found that the evaluation of the second derivative, Rxx , produces an expression consisting of 5 terms; 2 of them negative and 3 of them positive. Establishing the sign of such an expression would be very difficult. We found that a more symmetrical parameterization such as p1 = P + x − y, p2 = P + x − y, and p3 = P − 2x produces 6 terms for Rxx , with 3 negative and 3 positive signs. While this parameterization is more promising than the earlier one, there are many similar parameterizations, and only some of them would enable us to establish the sign relatively easily. Furthermore, there is a more pressing requirement. Familiar 2 > 0 conditions such as Rxx < 0 and Rxx Ryy − Rxy for concavity work only if the coordinate system (x, y) is orthogonal. With this in mind, and after experimenting with several parameterizations, we chose the one illustrated in Fig. 2. Projecting point (x, y) onto the p1 , p2 , and p3 axes respectively, p1 = P + αx − βy, p2 = P + αx + βy, p3 = P − 2αx (4) √ √ where α = 1/ 6 and β = 1/ 2. Armed with this parameterization, we now present and prove the central theorem of our paper. Theorem 4: On the boundary plane formed by the power constraint p1 + p2 + p3 = 3P , there is a region D =
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√ √ {(x, y)| |y| < (1/ 2)P + ( 2/3)x and |x| < (1/ 2)P + ( 2/3)y} (referring to the coordinate system shown in Fig. 2), where the sum rate of three transceivers operating on a Gaussian interference channel is concave in the powers. Proof: We begin by replacing p1 , p2 , and p3 in (3) with the parametric forms in (4). After ignoring variables such as N1 , that are orders of magnitude less than the rest, we arrive at R = log2 [P + αx − βy] − log2 [(a + b)P + (a − 2b)αx + aβy] + log2 [P + αx + βy] − log2 [(c + d)P + (c − 2d)αx − eβy] + log2 [P − 2αx] − log2 [(e + f )P + (e + f )αx + (f − e)βy].
allocation that satisfies this fairness condition by moving the power distribution as much as possible towards the end point with the maximum sum rate (Fig. 3). Two User Protocol 1) Decide which of a11 and a22 is larger (say a11 is larger). 2) Solve the equations R1 = (1 + ρ/100)R2 and p1 + p2 = P for p1 and p2 , using iterated bisection starting from the initial values p1 = p2 = P/2. ( R1 and R2 are rates of users 1 and 2, respectively.)
Representing the 6 terms on the right side above by the letters A, B, C, D, E, F , respectively, we can rewrite R = A − B + C − D + E − F. Twice differentiating with respect to x, Rxx = Axx − Bxx + Cxx − Dxx + Exx − Fxx . It can be shown √ that the statement Axx − Dxx < 0 reduces to y < (3/ 2)P √ . The statement Cxx − Bxx < 0 reduces Exx − Fxx < 0 re(−3/ 2)P . The statement to y > √ y < P + ( 3/2)x. It can be shown that duces to 2 e−f e+f this last statement will be true for any positive e and f , if √ |y| < (1/ 2)P + ( 2/3)x. Since the first two inequalities in y are weaker than the in |y|, we conclude that √ last inequality Rxx < 0, if |y| < (1/ 2)P + ( 2/3)x. 2 Reducing the other inequality Rxx Ryy − Rxy > 0, which is necessary to establish the concavity of R, to simple inequalities in x and y like the above will be extremely tedious. Instead, √ we wereable to prove that the above condition together with the additional |y| < (1/ 2)P +√( 2/3)x condition |x| < (1/ 2)P + ( 2/3)y is sufficient to make 2 Rxx Ryy − Rxy > 0, for all values of the channel coefficients. This part of the proof is lengthy and is omitted here in the interest of space.
It should be noted that Theorem 4 mentions sufficient conditions for concavity of the sum rate. These are not necessary conditions. The inequalities were derived using the easiest upper and lower bounds. Hence it is likely that the actual region in the center of the boundary plane where the sum rate is concave might be larger than D. IV. P OWER A LLOCATION A power allocation protocol and a power allocation algorithm that exploit the findings of the last section to distribute the power with fairness are presented in this section. In the case of two users, Theorem 2 asserts that the sum rate will be convex on the boundary (Fig. 3 and Fig. 4). This implies that the maximum sum rate occurs at one of the end points. At these points one of the users gets all the power while the other gets nothing. This means that at both end points interference from the other user is non existent. Hence, which end point produces the maximum sum rate will be determined by which of the coefficients a11 or a22 is larger. Suppose that we impose a fairness condition by insisting that the users’ rates should be within a certain percentage ρ of each other. We provide a protocol that can be used to find the best power
For the case of three users, since the region D is in the center of plane ABC (Fig. 2), power allocations that stay within D can be considered reasonably fair. The fact that the sum rate is concave here enables us to use the steepest gradient method to find the power allocation that produces the global optimum sum rate within D. We can replace p3 in (3) with 3P − p1 − p2 and ˜ 1 , p2 ) of only write the sum rate of three users as a function R(p ˜p R ˜ p be the partial derivatives two variables p1 and p2 . Let R 1 2 ˜ 1 , p2 ) with respect to p1 and p2 , respectively. We now of R(p present our algorithm that converges to the power distribution that produces the maximum sum rate within D. Three User Algorithm 1) p1 := P , p2 := P . ˜ p , p2 := p2 + tR ˜ p (where t is the step 2) p1 := p1 + tR 1 2 size.) 3) Go to Step 2 unless convergence criterion is satisfied or the boundary of region D is reached.
V. S IMULATION R ESULTS In this section we present simulation results that support the Theorems from Section III and demonstrate the utility of our iterative algorithm from Section IV. To verify Theorem 1, we placed two transceivers at three different configurations. In the first, the transmitter receiver separation for both users are the same but the distances through which interference occurs are different. Fig. 3 shows the individual and sum rates along the line p1 + p2 = 1 Watt, where dij denotes the distance from transmitter j to receiver i. The convex nature of the sum rate is very clear. The point where individual rates are equal represents a minimum in the sum rate. As we move away from this point, on either side, the fairness decreases but the sum rate improves. Fig. 3 also illustrates how we can allocate the power so that the rates of the users are within 50% of each other. In the second configuration, whose simulation results are shown in Fig. 4, the transmitter receiver separations are different as well as the interfering distances. We observe that the sum rate is concave and that the curve is almost symmetrical. This is because the effect of the difference in the transmitter receiver separations is canceled out by the effect of the difference in the interfering distances. The case where the interfering distances are the same but the transmitter receiver separations are different is shown in Fig. 5.
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20
20
d11 = d22 = 10m d12 = 40m, d21 = 20m Optimum Power Region where Allocation with rates are within Fairness 50% of each other Sum Rate
14 12
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Fig. 5.
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m 60
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d21=45
=30
m 60
5 =4
d 23
3 3
m 5
Rate (bits/sec/Hz)
0.7
R3
R2
Path loss coefficient n=3.5
Fig. 6.
Three user simulation details.
work was motivated by our previous work [1], in which we proved that the maximum sum rate occurs at the boundary hyper plane. In this paper we proved that for two transceivers, the sum rate is convex in the powers on the line formed by the power constraint. For three transceivers, we identified a region in the center of the boundary plane formed by the power constraint, where the sum rate is concave. We then provided a protocol for the two user case and an algorithm for the three user case that find the power allocation that maximizes the sum rate while being fair. Simulation results were provided to support the
14 12 10
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2 0
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Fig. 4.
d11=10m
1 =
d =10m, d =20m 11 22 d12=20m, d21=40m
T1
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0.6
Two users, configuration-3.
d2
We checked the effectiveness of our three user algorithm by simulating a range of placements of transceivers summarized in Fig. 6. The convergence of our algorithm to the global maximum sum rate of three users with a total power constraint of 1 Watt for one such placement, is demonstrated in Fig. 7. The step size t used in our algorithm is 0.01. The top dotted line represents the global maximum sum rate obtained using an exhaustive search with a power resolution of 0.01 Watt, over an area larger than D, but avoided the edges the plane with a margin of 0.05 Watts. The bottom doted line represents the Nash equilibrium solution obtained using a payoff function similar to the one used in [21] with individual power constraints of 0.33 Watts. Next, we experimented with the placement of the transceivers. While keeping the transmitter-receiver distance for each user unchanged, we changed the interfering distances at an increment of 2m as shown in Fig. 6. This resulted in more than 250 thousand different placements. For each placement, we ran our three user algorithm, and compared the resulting sum rate to that obtained with equal power allocation. The improvement in the sum rate is shown in Fig. 8.
18
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d
Two users, configuration-1.
Two users, configuration-2.
VI. CONCLUSION This paper analyzed the convex structure of the sum rate of a number of transceivers operating in the same unlicensed band, on the boundary plane created by the power constraint. This
Sum Rate (bits/sec/Hz)
Fig. 3.
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0m
2 0
d11=10m, d22=20m d12=40m, d21=40m
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Nash Equilibrium
11.4 11.3 11.2 0
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Fig. 7.
Convergence of the three user algorithm.
10
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0
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Fig. 8.
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x 10
Improvement in sum rate from equal power allocation.
Theorems developed in the paper as well as to demonstrate the convergence of the algorithm to the global maximum sum rate. The results of the algorithm were also compared with solutions based on Game theory. R EFERENCES [1] K. Illanko, A. Anpalagan, and D. Androutsos, “Dual Methods for Power Allocation for Radios Coexisting in Unlicensed Spectra,” Accepted for Publication on Proc. IEEE Global Communications Conference, Miami, Florida, USA, December 6-10, 2010. [2] Federal Communications Commission (FCC), “Spectrum Policy Task Force,” ET Docket no. 02-135, November 15, 2002. [3] Q. Zhao and B. M. Sadler, “A Survey of Dynamic Spectrum Access,” IEEE Signal Processing Magazine, Vol. 24, Issue 3, May 2007, pp. 7989. [4] C. E. Shannon, “Two-way Communication Channels,” Proc. Fourth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 1 (Univ. of Calif. Press, 1961), pp. 611-644. [5] R. Ahlswede, “The Capacity Region of a Channel with Two Senders and Two Receivers,” The Annals of Probability, 1974, Vol. 2, No. 5, pp. 805-814. [6] T. S. Han and K. Kobayashi, “A New Achievable Rate Region for the Interference Channel,” IEEE Transactions on Information Theory, 1981, Vol. IT-27, No. 1, pp. 49-60. [7] S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, March 2004. [8] W. Yu, and R. Lui, “Dual Methods for Nonconvex Spectrum Optimization of Multicarrier Systems,” IEEE Transactions on Communications, Vol 54, No. 7, July 2006, pp. 1310 - 1322. [9] M. Chiang, C. W. Tang, D. P. Palomar, D. O’Neill, and D. Julian, “Power Control by Geometric Programming,” IEEE Transactions on Wireless Communications, vol. 6, No. 7, July 2007, pp. 2640 - 2651. [10] L. P. Qian, and Y. Jun, “Monotonic Optimization for Non-concave Power Control in Multiuser Multicarrier Network Systems,” Proc. IEEE INFOCOM, Rio De Janeiro, Brazil, April 19-25, 2009, pp. 172 - 180. [11] W. Yu and J. M. Cioffi, “Competitive Equilibrium in the Gaussian Interference Channel,” Proc. IEEE International Symposium on Information Theory, Sorrento, Italy, June 25-30, 2000. [12] R. D. Gibbons, “Game Theory for Applied Economists,” Princeton University Press, 1992. [13] R. Etkin, A. Parekh, and D. Tse, “Spectrum Sharing for Unlicensed Bands,” IEEE Journal on Selected Areas in Communications, Vol. 25, No. 3, April 2007, pp. 517 - 528. [14] Y. Xu, W. Chen, and Z. Chao, “Optimal Power Allocation for Spectrum Sharing in Frequency-Selective Unlicensed Bands,” IEEE Communication Letters, Vol. 12, No. 7, July 2008, pp.511-513. [15] N. Clemens and C. Rose, “Intelligent Power Allocation Strategies in an Unlicensed Spectrum,” Proc. First IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, Nov. 2005, pp. 37-42. [16] A. Leshem and E. Zehavi, “Cooperative Game Theory and the Gaussian Interference Channel,” IEEE Journal on Selected Areas in Communications, Vol.26, No. 7, Sept. 2008, pp.1078-1088.
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