Optimal Power Control for Multiple Access Channel

Transcription

1 2005 International Conference on Wireless Networks, Counications and Mobile Coputing Optial Power Control for Multiple Access Channel with Peak and Average Power Constraints RongRong Chen and Yanzhu Lin Departent of Electrical and Coputer Engineering University of Utah {rchen, yanzhu Abstract In this paper, we study optial power control for ultiple access channel with peak and average power constraints. Perfect channel inforation is assued to be available at both the transitter and the receiver. We characterize the structures of the optial power control policy and show that the optial policy allows ultiple users to transit at peak power and at ost one user to transit at an interediate power level. This is different fro the optial TDMA policy (at ost one user is allowed to transit at any given tie) for the case of average power constraint only. We find closedfor expressions for coputing the optial power control policy and the axial su capacity. Nuerical results indicate that under the peak power constraint, the optial power control policy still achieves very close to the su capacity of the ultiple access channel with average power constraint only. I. INTRODUCTION Due to the tievarying nature of wireless channels, it is shown in [] that for a single user syste, when the channel inforation is available at both the transitter and the receiver, the wireless user should exploit tiediversity to adapt transission power according to the channel state and transit at higher rates when the channel is good. In a ultiuser syste, optial power control policies ([2], [3]) are developed to exploit both ultiuser diversity and tiediversity with the goal of axiizing the su capacity of ultiuser networks. For an uplink cellular syste, [2] shows that tie division ultiple access (TDMA) schees axiize the su capacity. The optial power control policy depends on the channel conditions of all users and it has a structure that, at any given tie, only one user with the best channel condition transits. However, since each user transits only when his channel is the best aong all users, the aount of tie that each user occupies the channel decreases as the nuber of users increases. Under the average power constraint, this iplies that each user would transit at a higher power level when it is his turn to transit. Consequently, as the syste size grows, each users' signal transission becoes increasingly bursty and large peaktoaverage power ratios occur (ore details are included in Section III). In this work, we study optial power control under both peak and average power constraint. The peak power constraint is independent of the nuber of users in the syste, therefore in our syste we have a fixed peak to average power ratio. We characterize the structures of the optial power control policy and provide closefor expressions for coputing the optial su capacity. Under the peak power constraint, we show that the optial policy is no longer a TDMA policy. Instead, it allows ultiple users to transit siultaneously, possibly all with the peak power. Our results show that the peak power constraint does not ipose uch capacity penalties against the case of average power constraint only. In addition, the nuber of siultaneous transissions can be kept sall. In other related work, optial power control with peak power constraint is studied in [4] for a single user syste. In this work, we consider a ultiuser syste. Optial power control for ultiple antenna channels are studied in [5] under the average power constraint. II. UPLIN OPTIMAL POWER CONTROL Consider an uplink cellular network with users. Let yi, i =,., be the channel gain between user i and the base station. Under Rayleigh fading, we assue that Aj_ has a Rayleigh distribution and yi has an exponential distribution with paraeter. Let y = (yr,, ) denote the channel gain vector of all users and ui(y) be the power allocation to user i when the channel gain vector is y. Under the power control policy ui(y), the received signal at the base station can be written as () y= xi+n, i=l where xi is the transitted signal fro user i which satisfies E(xi 2) =, n is white Gaussian noise with zero ean and unit variance. For a fixed y, equation () represents a Gaussian ultiple access channel with a su capacity of 2 log ( + Z= ui(y)'yi) [6]. Assue that channel fading inforation is available at both the transitter and the receiver, we can average over all possible fading realizations to obtain 'E [log ( + Ui(_Y)Yi) as the su capacity for the fading case. Our objective is to find the optial power control policy ui(y),i = to axiize the su capacity under both peak and average power constraints. Let p(y) be the probability density function of y. We forulate the following optiization proble. ax ff... f " log (I + E Ui(_Y)i) p(y)d 'y (2) subjectto ff...fui(y)p(y)dy<i, i= (3) and Ui)< P, i =,. (4) Note that we consider both the average power constraint (3) and the peak power constraint (4), while in [], [2], [3] only the average power constraint is considered /05$ IEEE 407

2 A. Structures of the optial power control policy In order to characterize the structures of the optial power control policy, we first introduce the Lagrange ultipliers, Ai, corresponding to each average power constraint in (3). The Lagrangian function is expressed as L(ui, {>i})= Jf... log ( + )ui(y)i p(7)d y (AiJ...f u i(y)p(y)d ). (5) Due to the convexity of the logarith, the arushuhntucker (T) conditions [7] are necessary and sufficient for optiality. Taking the derivative with respect to ui, we obtain a set of T conditions for every i =,.., and y. 7i > + S uj(q)4yj if 'uiqy) = P (6) Ai Ai X = u5(ay)'yj if O<ui(y)<P (7) ~~J= A < luj(7)'yj if uj(7)=0 (8) Since the suation ters in (6)(8) are the sae for all i.=..,, we ake the following observation: () If s'> aq then we ust have ui (y) >.uiq) (2) At ost one user can be allocated an interediate power level 0 < ui(y) < P. This is true because the probability that the left hand side of (7) is the sae for two different users equals 0. Hence, when there is no peak power constraint (or P = oo), the optial power control policy reduces to the TDMA policy in [2]. (3) Let us reorder the sequence {yi/ai, i =, } such that JXl > JYi2 >X >X A2 Ai It then follows fro () and (2) that, the optial power control policy satisfies the following properties: (a) there exists an integer N (which depends on y) such that the best N users il, in are allocated full power P. (b) User in+ either uses an interediate power level or is inactive (zero power allocation). (c) Users in+2,,i are all inactive. Next, we will show how to deterine the value of N and the optial power allocation for user in+. For notation siplicity, we assue that 7 > MY >,,* A A2 A For every i =,, let ai =iand bi = +P Yj Note that bi equals the suation ter + = uj ('y)yj in (6)(8) if the first i users are allocated peak power and the reaining users are inactive. Since {ai} is an nonincreasing sequence and bi is a nondecreasing sequence, we can find an integer N such that an. bn and an+i < bn+±i There are two cases shown in Figure, in which we let N = 4 as an exaple. 40E avl bli a2q a30 a40 N = 4 b5 y a5 (case ) b4l U5 b2t b3_ P'f4 a5 (case 2).iP'Y3 I Fig.. Optial power control policy with peak power constraint. Case Assue bn < an+ < bn+. In this case, user N + should not be allocated peak power P. Otherwise the T condition (6) will be violated. Since an+ > bn, we can allocate an interediate power level to user N + to ensure that the equality in the T condition (7) holds. The optial power allocation becoes UiX, Qy) = FP, an+bn 0 YN+ if i < N if i = N+ else. Using this power allocation, one can see that the suation ter in (6)(8) equals an+l. Due to the onotone property of the sequence {ai}, the T conditions are clearly satisfied for power control policies of the for (9). Case 2 Assue an+ < bn. In this case, user N + is inactive. The optial power allocation becoes ui(\ =P, if i < N 0 else. (9) (0) The corresponding suation ter in (6)(8) equals bn which also ensures that the T conditions are satisfied. As seen fro (9) and (0), we note a useful fact that the optial power allocation for user i only depends on (y, *, y). B. Coputation of the optial Lagrange ultiplier Given the structures of the optial power control policy (9) and (0), the optial Lagrange ultiplier can be found by choosing {Ai} such that the average power constraint (3) is satisfied for each user i. This, however, involves the evaluation of diensional integrals. In the following, we consider the scenario when the average power constraint Pi = Pa is the sae for all users. By syetry, all Ai ust be equal and we denote the coon value by A. In order to find A that satisfies (3), we need to siplify (3) as uch as possible and reduce the diension of integration.

3 Let {.} denote the indicator function. Since the users are syetric, we have E[ui(y)] = E [Ui(Y)l{user i has the th best channel}] g=i ( ) = E E [EUY)l{user has the th best channel}] = The case when rn = needs to be handled separately. For every rn > 2, we have E [Ur(7){user has the th best channel}]. = (_2) J[j(j...n() Yee(i )'d>dax(y+l.yl <li.. ). = () j [ju/n(i,) i)ea O ' Inorrto c ute(et=ei+ )dry+l(2d) eonsideyr () ~ ~ ~ ~ (2 f U('i = w us hae a.ba) therer d,y,.. daj (e' )fe7 dy. (2) In order to copute the integral in (2), we consider two cases with ub(yi...y) = P or O < ub (y... y) < P. (I) If Zu.(y y) =,we ust have a. > b. and therefore Y E Yi < A vy (3) (2) If 0 < u("ii,,y) < F, then we have b_i < a < b and therefore 2 Y A Y <.Y7j < A (4) It follows fro (9) that the optial power allocation is E (_) [ ~~~~A A,(7) gi (X) = x2ex/( 2)!. The integral in (6) can be siplified as Joe [J ry( d) I g, (x) dx JO ] f,a(y, + X) 9 (X) dx q (,Y) dy (7) where q(y) [exp(y)] [exp(yx)] f,a,(y,x) = /A[+P(x+ (l)"y)]/y r,a(}y) = niax(,(/a)/py) The forula for coputing A is therefore given by Pa = P]q/)d'j + IA i( ) d(y c F, + Z: (MD I ' i(x) dx + fi,,(a, x) 9 (X) dx qq(ry)ddy, (8) where the first two ters correspond to =. C. Coputation of the optial su capacity Once the optial A is found by solving (8), the optial power control policy is uniquely deterined. In this section, we derive analytical expressions for coputing the optial su capacity. Let c(qy) = + Z Uj(y)yj, we have Copt = E[og c(y)] u (t,,'y) = Z3'. (5) (Note that if we let n =, the optial power allocation (5) reduces to the optial waterfilling power control policy in [2] with only average power constraint.) Hence, the integral in (2) equals 00 [jx>.jo ( {. r<f{ A 'n ()( Y in { 2L "Y._ Y3. _ p }J I= *e = dy * d7 ( e7)eedy (6) Let'ai= 3' j =,* * *,and x = Ej ai. Then = 2 L.E 0'(c(')) l there are exactly active users = ) E 2 Ilog(c(7)) { active users. user has th best channel}] = 2 () E [og(c()) active users} 2 S r= lfin(y....y.j)>,ax(y.+j.....y )<Y }] = 2 5 (D) E09(C(7) factive users} = the probability density function of x has a gaa distribution We again consider two cases. 409 l{in(y,...,y) >'yrn,ax(y+i,.,r ). Y } ({U()=P} + {O<U (Y) <P})] (9)

4 () Assue urn (y) = P. To ensure that there are exactly active users, we require that a, > b > a+,. Hence, the following conditions ust be satisfied: su capacity can be written as copt= [j /sl(yl,o) d'7 + j ' l(y )log (i) d,l 7Y+ < A (I+ P Z½ ) z>yj A I < A _p c(?) = b = li + PEZY (20) (2) Note that (2) and (20) iply that y+ <' It follows that the integral ter in (9) that involves {U ()=} equals oo roo OC ro A((I +E') Z=P E a y ~~e j=+ r77?)dy+l ***dw *e(z='yj)d7yl... dy i] ey d7y *(e A (+e P ) (r) e ( Jyjd) dyj dryj ed7t (23) (2) Assue 0 < un(7) < P. We have bi < a < bn which guarantees that there are exactly active users. Hence, we ust have Ey A p _ X _ E l i< A, P tyr c(y) = a = AA (22) where o At + E_ () s[(fyf xg (x) dx dy +fo g (ii\) IT )r9y) ( dx qqty)dayrn] (27) S (Q, ) = log ( + P(Ty + x)) * (e Y) * [ ea\(p(mry+2))]. and the first two ters in (27) correspond to =. III. NUMERICAL RESULTS In this section, we present nuerical results based on the closefor expressions derived in Section II. First, we study the case with only average power constraint to establish our perforance benchark. We assue an average transission power of P, = 0 such that the received signaltonoise power ratio 0 logo Pa equals 0 db. Given a total of users in the syste, the optial power control policy (average power constraint only) allows a axiu transission power Pax() = /A. As shown in Figure 2, the axiu to average power ratio Pax() /Pa increases linearly with respect to due to the fact that as increases, the aount of tie each user transits decreases. When = 6, we have Pax/Pa = 6.03 and the optial su capacity equals bits a 4 X 2 8[ (28) It follows that the integral {o<u(y)<p} equals ter in (9) that involves : nuber of users log i [ Ja ( > ) E { A l ' < I Yj < e e( 3= Yjd7j... d'yrn] ( e e) ne dy (26) Finally, we use the transforation yj = y and x = 2inI y* to arrive at a Gaa distribution and hence reduce the diensions of the integral. After siplifications, the optial Fig. 2. Maxiu to average power ratio versus the total nuber of users in a syste with average power constraint only. Next, we exaine the su capacity under both peak and average power constraints. Copared to the average power constraint only scenario where the axiu transission power Pax() increases linearly with, here we require that at any tie each users' transission power is no ore than P, which is independent of. In Figure 3, we plot the optial su capacity as a function of the peaktoaverage power ratio 0

5 P/Pa For each P/Pa, we first solve (8) nuerically to find the Lagrange ultiplier A. Then we apply (27) to copute the su capacity corresponding to the optial power control policy given by A. Figure 3 shows that as P increases, the su capacity also increases. When P/Pa = 6, the capacity with peak power constraint equals the capacity with average power constraint only ( bits). When P/Pa = 8, the su capacity with peak power constraint attains about / = 97.82% of the su capacity with average power constraint only. Note that by iposing the peak power constraint, we reduce the axiu to average power ratio fro Pax(6)/Pa 6 to P/Pa, = 8, which is a 3 db reduction, with only a 2% loss in su capacity. When P/Pa = 2, which corresponds to a 9 db reduction fro Prax(6)/Pa 6, the su capacity achieved is about 89.3% of the su capacity with average power constraint only. However, as shown in Figure 4, in this case we would require any users to transit siultaneously with large probabilities. 4.5 there is only one active user, hence the resulting optial power control policy is alost TDMA as in the case of average power constraint only. C,) L a) cz co Cu 0 a. Fig [ 0.5[ P/Pa Probability distribution of the nuber of active users *2 *3 *8 * * X 0 7 X* 6 *, 2 * a Ca 0. iu 4 E cx P/Pa Fig. 3. Optial su capacity versus peak to average power ratio. In Figure 4, we study the probability distribution of the nuber of active users under the peak power constraint. Let denote the nuber of active users. Given P, the probability that there are exactly active users can be coputed by evaluating the integral ters in (27) that correspond to active users, and replacing sy(,x) and log () by. In Figure 4, ultiple points are plotted for each P/Pa. The nuber to the right of each point represents, the nuber of active users. The ycoordinate of each point represents the probability that there are exactly active users. The highest point for a given P/Pa represents the ost likely (with the largest probability) nuber of active users. For instance, three points are plotted for P/Pa = 8. Fro top to botto, we see that with probabilities of 0.6, 0.36, and 0.03, there are two, three, and one active users, respectively. The probability that there is no active user is close to 0, hence not shown in the figure. We also observe that when P/Pa is sall, it is likely to have a large nuber of active users. For instance, when P/Pa = 2, with a probability of 0.3, there are 8 active users. However, as P/P,a increases, the nuber of active users reduces. When P/Pa = 8, active users. When P/Pa = 6, there are at ost three with a probability of IV. CONCLUSION In this paper, we study optial power control for ultiple access channel with peak and average power constraints. We obtain a coplete characterization of the optial power control policy which generalizes previous results on the optiality of TDMA type policies for ultiple access channels with only average power constraint. Analytical expressions are derived for coputing the optial policy, the optial su capacity, and the probability distribution of the nuber of active users, assuing that users in the network are syetric. By allowing ore than one user to transit at a given tie, the proposed power control policies under the peak power constraint have the advantage of reducing the axiu to average power ratio and shortening the transission delay due to opportunistic scheduling in the ultiuser network. In addition, the su capacity achieved by the proposed policies are close to that achieved with only the average power constraint. REFERENCES [] A. Goldsith and P. Varaiya, "Capacity of fading channel with channel side inforation," IEEE Trans. InJorn. Theory, vol. 43, pp , Nov [2] R. nopp and P. A. Hublet, "Inforation capacity and power control in singlecell ultiuser counications," IEEE Inter Conf: Connln. (ICC), vol., pp , Jun [3] D. Tse and S. Hanly, "Multiaccess fading channels: part I: Polyatroid structure, optial resource allocation and throughput capacities," IEEE Tranis. InJbrin. Theory, vol. 44, pp , Nov [4] M. A. hojastepour and B. Aazhang, "The capacity of average and peak power constrainted fading channels with channel side inforation," IEEE Wireless Cotnitnnications and Networking Conference (WCNC), vol., pp. 7782, Mar [5] W. Yu, W. Rhee, and J. M. Cioffi, "Optial power control in ultiple access fading channels with ultiple antennas," IEEE Inter ConfJ Cotninn. (ICC), vol. 2, pp , Jun [6] J. M. Cover and J. A. Thoas, Eleents of Inforination Theort. New York, Wiley and Sons, 99. [7] D. P. Bertsekas, Nonlinear Prograning. Belont, Massachusetts: Athena Scientific, 995.