rate in a multiple access channel when there is proper feedback channel information .... we can formulate a power adaptive waterfilling equation for user j, which is .... of user 1 being decoded first, and the dashed line is for the ordering of user ... [7] T. Starr, J. Cioffi, and P. Silverman, Understanding Digital Subscriber. Line ...
Optimum Power Allocation and Control for OFDM in Multiple Access Channels Jisung Oh, Seung-Jean Kim and John M. Cioffi Department of Electrical Engineering Stanford University Stanford, CA 94305-9515 Email: {ofdm, sjkim, cioffi}@stanford.edu Abstract— This paper considers optimal power allocation in a multiple access channel, where multi-user orthogonal frequency division multiplexing is employed. We propose a power allocation algorithm for minimizing the sum of transmitted powers, when each of users in a multiple access cell has a desired data rate. The optimal power allocation can be found very efficiently by an iterative algorithm. The minimum sum power and the convergence behavior turn out to be strongly dependent on the decoding order of the users. We describe the best ordering scheme that provides the global minimum of the sum power. To cope with channel variation in a slowly time-varying channel, we also propose a power control algorithm that modifies the allocated power. In short, the power allocation algorithm assigns optimal power to each user for the initial channel information, and the power control algorithm tracks the channel variation after the initial power allocation.
as a figure of merit the sum of the transmitted powers in the channel, and provide a novel method for allocating minimum sum power while guaranteeing the data rate required by each user. The method minimizes the sum of the transmitted power in a cell, reducing the amount of interference to other cells that might be using the same frequency band at the same time. This paper is organized as follows. Section II shows a multiple access channel model and formulates an optimal sum power problem. Section III solves the optimization problem, and derives an iterative algorithm for optimal power control. Section IV shows a power tracking algorithm in a slowly time-varying channel. Numerical examples for the proposed algorithms are given in section V, and conclusions are drawn in section VI.
I. I NTRODUCTION A multiple access channel, where multiple uncoordinated users send independent information to a common access point, is a typical environment encountered in multi-user communications, such as wireless LAN and cellular systems [1] [2]. In the multiple access channel, power control has been an important mechanism to solve the near-far problem. When successive interference cancelation (SIC) is employed at the receiver, power-control plays a crucial role to maximize the data throughput [3] [4] [5]. In a multiple access channel, orthogonal frequency division multiplexing (OFDM) with SIC (OFDM-SIC) can be employed as an efficient multiplexing scheme in which more than one user share a subchannel and transmit independent data streams through the subchannel. Moreover, OFDM-SIC is known to maximize the sum date rate in a multiple access channel when there is proper feedback channel information from the receiver [6]. In this context, we limit our interest to a multiple access channel where OFDMSIC is employed as a multiplexing scheme. In the OFDM-SIC scheme, the best power allocation for a desired data rate vector in the rate region is not clearly known. Until now, most literature on this topic only addresses the maximum sum rate problem when each user has his own power budget. However, in most practical cases, each user has a desired data rate and likes to achieve it within the available power. Thus, it is also an important problem to guarantee all the users’ desired data rates in a multiple access channel while consuming minimum power. In this paper, we choose 0-7803-8521-7/04/$20.00 © 2004 IEEE
II. S YSTEM M ODEL This paper investigates the power control in a multiple access channel which can be a uplink channel model of a wireless LAN system. The number of users in the channel is K, and each user transmits signals independently to a receiver. The receiver receives signals simultaneously from all the users and decodes them by applying SIC. The channel is assumed frequency-selective, however, it is divided into N independent ISI-free subchannels by employing OFDM in the transmission. This paper also assumes that channel information is perfectly known to the receiver for optimal power control. When OFDM-SIC is employed at the receiver, the received signal contains the signals from all the users and is decoded successively in a pre-determined order. Each user considers the signals of other users with higher decoding orders as interferences. Then, user j, who is the jth decoded one, has the signal to interference-noise ratio (SINR) in subchannel i as sji =
pji h2ji , σi2 + k>j pki h2ki
(1)
where pji and hji are the signal power and the channel response of subchannel i of user j, respectively. Also, σi2 is the channel noise variance of subchannel i of the multiple access channel. Note that σi2 is not dependent on user j. The power control problem is equivalent to finding a power vector {pji } that guarantees the desired data rates of all users.
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Specifically, assuming that all transmitted signals and the noise are Gaussian, for the desired data rate Rj of user j, {pji } should satisfy the following inequality N 1 i=1
2
log2 (1 +
sji ) ≥ Rj , Γ
N
subject to
rji ≥ Rj , j = 1, · · · , K
i=1
III. O PTIMUM P OWER A LLOCATION The power control problem in this paper is also related to quality of service (QoS) of a multiple access system. Each user tries to preserve his data rate independently of the other users, the channel gains and the noise variance, if the data rate is feasible within the power budget. In this sense, the minimum transmitted power can be regarded optimum in the power control problem. In a multiple access channel, minimum sum of transmitted powers is a good measure of optimum power allocation. A. Multiuser Power Optimization This paper derives an optimal power allocation algorithm by solving the following optimization problem: K N
pji
j=1 i=1
subject to
K N 1 2rji 2r (2 − 1) · 2 k>j ki gji j=1 i=1
minimize
(2)
where Γ is the signal-to-noise ratio (SNR) gap that depends on the probability of symbol error, noise margin, and the coding gain [7].
minimize
can be transformed into,
N 1 i=1
2
log2 (1 +
pji ≥ 0.
sji ) ≥ Rj , j = 1, · · · , K Γ
j = 1, · · · , K, i = 1, · · · , N (3)
The optimum power vector {pji } obtained from (3) achieves minimum sum power, which is the least amount of power transmitted in the multiple access channel. The optimization problem cannot be solved directly, since the constraints in (3) are not convex. However, we can convert the problem to a convex optimization by defining the data rate in subchannel i of user j as 1 p g ji ji ), rji = log2 (1 + 2 1 + k>j pki gki
(4)
where the channel state information gji is defined by h2ji /σi2 , and Γ is assumed to be 1 (0 dB) for simplicity. Then, pji can be rewritten as 1 2rji 2r (2 − 1) · 2 k>j ki , (5) pji = gji 1 (22rKi − 1), . . . , p1i = g11i (22r1i − 1) · for example, pKi = gKi 22(r2i +,···,+rKi ) . Using (4) and (5), the optimization problem
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rji ≥ 0.
j = 1, · · · , K, i = 1, · · · , N (6)
Since all the functions in the constraints of (6) are convex, the optimization problem can be easily solved using Lagrange multipliers. Let L({rji }, {λj }) be the Lagrangian: K N 1 2rji 2r L({rji }, {λj }) = (2 − 1) · 2 k>j ki gji j=1 i=1 −
K
N λj ( rji − Rj ).
j=1
(7)
i=1
Deriving its partial derivatives with respect to rji and applying ∂L = 0, Karush-Khun-Tucker (KKT) conditions for user j, ∂r ji we obtain K 2rki j−1 p k=j+1 2 k=1 ki + · 22rji 22rji gji j−1 2r j−1 K 2 ki − 1 1 2r 2rli = + · 2 l=k+1 · 2 m=j mi gji gki k=1
=
λj . i = 1, · · · , N log 2
(8)
The solutions of (8) gives the optimum rate vector {rji } and the Lagrange multipliers {λj }. The optimum power vector is then derived from (5). The KKT conditions in (8) imply that the minimal power allocation for user j is not only dependent on the rates of users with higher decoding orders, but also on the power of users with lower decoding orders. This is quite interesting, since each user in OFDM-SIC considers as interferences only the signals of the users whose decoding orders are higher than his. It also indicates that successive water-filling from the user with the highest decoding order to the user with the lowest decoding order is not a solution to the minimum sum power problem. B. Power Adaptive Iterative Water-filling Unfortunately, there is no closed-form solution for the optimization problem in (6). Thus, we propose an iterative algorithm that converges to the optimum solution. The derivation for the iterative algorithm is similar to [6], where a rate adaptive iterative waterfilling is proposed for sum capacity maximization. From the results of (8), by defining the modified channel state information j−1 2r j−1 K 2 ki − 1 1 1 2r 2rli = + · 2 l=k+1 · 2 m=j+1 mi , g˜ji gji gki k=1 (9)
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we can formulate a power adaptive waterfilling equation for user j, which is given by 22rji g˜ji N rji
=
λj , i = 1, · · · , N. log 2
= Rj .
(10)
users; the user whose channel gain is the highest is given the lowest decoding order (decoded first) and the user who has the lowest channel gain is given the highest decoding order (decoded last). This is not the best ordering to obtain the minimum, but results in the sum power very close to the minimum in most cases.
i=1
User j can allocate power in each subchannel by (10) while achieving the desired data rate Rj . Notice here that g˜ji depends on the other user’s channel gains and data rates. Thus, once user j adapts the power and obtains a new power allocation pji , (i = 1, · · · , N ), the other users’ rates are deviated from their previous rates and need to be updated once again. That is, we have to find a power vector that satisfies (10) simultaneously for all users. The algorithm described below solves this problem by applying water-filling iteratively from user K to user 1 with the modified channel state information. We call this algorithm a power adaptive iterative waterfilling. Algorithm 1: Power Adaptive Iterative Waterfilling initialize pji = 0, j = 1, · · · , K, i = 1, · · · , N repeat from j = K to 1 compute g˜ji by (9) do water-filling for user j with g˜ji until there is no further decrease in the sum power
k=1
The convergence properties of the iterative algorithm depend on the number of users and the channel state information. In most cases, it converges very fast usually taking about 5 to 10 iterations within 0.1% accuracy, which is demonstrated in section V. Until now, we have assumed that the receiver decodes the signals in a pre-determined order. Recall from (8) that the modified channel state information can also be written as K j−1 2rki k=j+1 p 2 1 ki = k=1 + . (11) g˜ji 22rji gji It is obvious from (11) that g˜ji is determined by both the powers of the users who have lower decoding orders and the rates of the users whose decoding orders are higher than that of user j. This indicates that g˜ji may take on different values with different decoding orders, even though the channel gains and the data rates are fixed. Thus, the minimum sum power depends on the decoding order. This is contrasted with the fact that the maximum sum rate in a multiple access channel is independent of the decoding order [6]. On this ground, the optimal power allocation should be performed in two steps; first we have to find the best decoding order among users, and then solve the optimization problem by using the power adaptive iterative algorithm subject to the best decoding order. Finding the best decoding order becomes exponentially complicated as the number of users in the multiple access channel increases. However, we can simply determine the best decoding order by comparing the channel gains of the 0-7803-8521-7/04/$20.00 © 2004 IEEE
IV. T RACKING A LGORITHM Even though it converges very fast, the iterative algorithm needs a considerable amount of operations. Thus, it is preferable to apply the algorithm only when there is severe channel variation. When the channel response is slowly time-varying due to low Doppler frequency, it is more economical to adapt the allocated power to the channel variation. To derive a tracking algorithm that updates the power in the subchannels, we investigate the KKT conditions (8) in detail. Suppose, at the optimal state of (8), user j in the multiple access channel increases the power in the subchannel i to increase the rate by ∆r. From (5) and (8), the amount of power that user j should increase by is 1 2∆r 2r (2 − 1) · 2 k≥j ki ∆pji = gji j−1 = (λj − pki )(22∆r − 1). (12) Since the power change of user j affects the data rates of the other users of lower decoding order as can be seen in (4), the other users should increase the power to prevent the rate reduction due to the power change of user j. User k (k < j) increases the power by an amount of ∆pki = pki (22∆r − 1).
(13)
Thus, the total amount of power required to increase the data rate of user j with the other users’ rates fixed is given by ∆p =
j
∆pki = λj (22∆r − 1).
(14)
k=1
Note that ∆p does not depend on the subchannel, which means that every subchannel requires the same amount of power to increase the data rate by ∆r. This property can be viewed as the equilibrium condition at the optimal state. The tracking algorithm proposed in this paper exploits this property and updates the power to preserve the equilibrium condition for the slowly time-varying channel. Suppose the channel state information of user j has changed from gji to gˆji . Then the updated power pˆki , (k = 1, · · · , j) can be evaluated by applying (10) with updated channel state information gˆji . In this scheme, the higher order users (user j + 1, · · ·, user K) need not to be included, since their power and rate are not affected by those of the lower order users. The tracking algorithm is equivalent to the successive waterfilling from user j to user 1. The successive waterfilling can preserve the equilibrium condition in (14), without applying the iterative waterfilling. Note this simple
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Fig. 1. Optimum power allocation and bit allocation: (a) channel responses (b) power allocations (c) bit allocations.
Fig. 2. Optimum power allocation and bit allocation with the reversed ordering.
tracking algorithm does not necessarily provide the minimum sum power for the varied channel response, but updates the allocated power to the channel variation in a sub-optimal way.
the subchannels to minimize the sum of the transmitted power for the desired data rates. Some of users can share the same subchannel with other users, since SIC is employed at the receiver. Accordingly, we can notice in Fig. 1 that user 1 and user 3 occupy the same subchannels. In Fig. 1, user 1 is decoded first, and then user 2 and user 3 are decoded in this order. We can verify the ordering dependency in the sum power minimization by changing the ordering. Fig. 2 shows the power and bit allocation by the proposed algorithm when the decoding order is reversed for the same channel responses as in Fig. 1. The power and bit allocation is definitely different from the previous one. User 3 at the far distance transmits more power than in the previous case to achieve the same data rate. Also, user 1 transmit little energy in the subchannels in which user 3 allocates the power. This is because user 3 is decoded first, even though he has the smallest channel gain. Consequently, the sum of the power required to achieve the data rates with the reversed ordering is greater than the sum power with the previous ordering. The ordering dependency in sum power minimization becomes more noticeable especially when the channel gains among users differ in magnitude significantly. The best ordering is to decode first the user who has the highest channel gain as is the case of Fig. 1. Fig. 3 shows the convergence of the power adaptive iterative waterfilling for the two orderings we mentioned before. The solid line is for the ordering of user 1 being decoded first, and the dashed line is for the ordering of user 3 being decoded first. The first ordering (solid line) allows the minimum sum power and shows faster convergence than the second one (dashed line). We can notice that more than
V. E XPERIMENTAL R ESULTS In simulations, we evaluate the performance of the proposed algorithm in a multiple access channel and show the dependency of the minimum sum power on the decoding order. The proposed algorithm is applicable to any multiple access channel, once the noise variance and the channel gain of each user are known to the receiver. In the simulations, the number of subchannels for OFDM signals is 64 as in [1], and the SNR gap Γ is assumed 0 dB. First, we consider a 3-user multiple access channel in which the users have the channel responses H1 (D) = 1 + 0.9D + 0.5D2 , H2 (D) = 0.8×[1−0.9D +0.5D2 ], H3 (D) = 0.5×[1+0.9D +0.5D2 ], respectively, and the required data rates are R1 = R2 = R3 = 1 (bit/dim). Note that user 1 and user 3 have the same frequency channel characteristics with difference only in scale. We can see the effect of the decoding order when there are one user near to the receiver and another user at the far distance. The ordering dependency becomes manifest when users share the same frequency band in the multiple access channel like user 1 and user 3. We apply the proposed power adaptive waterfilling by 100 iterations, and obtain the optimum power and bit allocations for the multiple access channel as shown in Fig. 1. Note that the allocated power in each subchannel in Fig. 1(b) is normalized to the noise variance. The power adaptive iterative waterfillling dynamically allocates power in 0-7803-8521-7/04/$20.00 © 2004 IEEE
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40% of power is reduced by choosing the best ordering. VI. C ONCLUSIONS In conclusion, the power allocation algorithm proposed in this paper provides an efficient way of minimizing the sum power of all the users in a multiple access channel, while guaranteeing the desired data rate for each user. In particular, the optimal power allocation algorithm not only economizes the power resources but also reduces the interferences to the adjacent cells. This is achieved by solving the convex optimization problem under the best decoding order. The performance evaluation results on various multi-user channel environments demonstrate both the practical use of the proposed power allocation algorithm and the efficiency in computation. R EFERENCES [1] ”Wireless LAN media access control (MAC) and physical layer (PHY) specification: High-speed physical layer in the 5GHz band,” IEEE std. 802.11a, Sep. 1999. [2] A. J. Viterbi, CDMA-Princeiples of Spread Spectrum Communication. Reading, MA: Addison-Wesley, 1995. [3] D. Warrier and U. Madhow, ”On the capacity of cellular CDMA with successive decoding and controlled power disparities,” Proc. 48th IEEE Vehcular Technology Conf., vol. 3, pp. 1873-1877, May 1998. [4] A. Agrawal, J. Andrews, J. Cioffi, and T. Meng, ”Power control for successive interference cancellation with imperfect cancellation,” Proc. IEEE Int. Conf. Comm. (ICC), pp. 356-360, Apr. 2002. [5] J. Andrews and T. Meng, ”Optimum power control for successive interference cancellation with imperfect channel estimation,” IEEE Trans. Wireless Commun., vol. 2, pp. 375-383, Mar. 2003. [6] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, ”Iterative water-filling for Gaussian vector multiple access channels,” IEEE Trans. on Information Theory, vol. 50, pp. 145-152, Jan. 2004. [7] T. Starr, J. Cioffi, and P. Silverman, Understanding Digital Subscriber Line Technoloy, Upper Saddle River, NJ: Prentice Hall, 1999.
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