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MISO broadcast channels in [5]. Though the scaling laws are the same, there still exists a gap between the sum rates achieved by DPC and opportunistic.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Opportunistic Power Allocation for Random Beamforming in MISO Broadcast Channels Edward W. Jang, Hyukjoon Kwon, and John M. Cioffi STAR Laboratory, Stanford University, Stanford, CA 94305 Email: {ej1130, hjkwon, cioffi}@stanford.edu Abstract— This paper proposes the opportunistic power allocation (OPA) scheme for random beamforming with a limited feedback rate. Without the channel state information, the proposed scheme allocates different amounts of power to different beams. Various low-complexity search algorithms for the proposed OPA scheme are also proposed to find the optimal power allocation that maximizes the sum rate. Computer simulations show that the proposed OPA scheme significantly improves the sum rate, and the optimal power allocation is found with low complexity. In addition, it is shown that the number of beams should increase as the SNR increases in order to maximize the sum rate.

I. I NTRODUCTION Since using multiple antennas is shown to considerably increase the capacity of a channel [1], there has been considerable research not only on point-to-point communications but also on multi-user network systems, such as broadcast channels. Recently, dirty paper coding (DPC) is shown to achieve the sum capacity of a Gaussian broadcast channel when the channel state information (CSI) of the users is known perfectly at the base station (BS) [2]. However, it is impossible to directly implement DPC for practical wireless communication systems for two reasons: First, perfect CSI is hard to obtain because of fading. Second, the computational complexity for a BS to process the CSI feedback from all users is prohibitively high. To overcome these problems, zeroforcing beamforming is proposed [3], resulting in reduced computational complexity. However, it still requires perfect CSI. To maximize the sum rate with only partial CSI feedback from users, opportunistic beamforming, which exploits multiuser diversity, is proposed for multiple-input single-output (MISO) broadcast channels [4]. Since multi-user diversity is best exploited when the dynamic range of channel fluctuation is high, random coefficients are multiplied at each transmit antenna to induce intentional channel variation. This technique supports only the best user based on signal-to-noise power ratio (SNR) feedback from each user. It is shown that if M users are supported simultaneously, the sum rates achieved by DPC and opportunistic beamforming both scale as M log log N for MISO broadcast channels in [5]. Though the scaling laws are the same, there still exists a gap between the sum rates achieved by DPC and opportunistic beamforming. In order to reduce the gap, various suboptimal schemes have been proposed. In [6], the number of beams used during pilot intervals and the number of beams used during data transmission intervals are changed with equal power

allocation across beams. In [7], [8], [9], the BS performs iterative water-filling based on the signal-to-interference-and-noise power ratio (SINR) or the effective channel gain feedback from every user. However, when the feedback rate from users is limited, the use of water-filling is restricted because the waterfilling based on quantized SINR or effective channel gain may be inaccurate. That is, a selected user is not guaranteed to support the data rate determined by the water-filling using quantized partial CSI. Moreover, if a BS changes its power allocation among beams between a training period and a data transmission period, the BS should inform the selected users which modulation and coding scheme (MCS) will be used before transmitting data. This paper proposes a novel scheme, denoted as the opportunistic power allocation (OPA) scheme, to maximize the sum rate for MISO broadcast channels when the feedback rate is limited. In the proposed scheme, the BS opportunistically allocates power across beams without CSI at the beginning and maintains it throughout training periods and data transmission periods, instead of varying the power allocation after receiving feedback as in [6], [7], [8], [9]. Also the power allocation across beams might be unequal whereas equal power allocation is assumed in [5], [6]. This paper is organized as follows: Section II describes the system model, and Section III proposes the OPA scheme and formulates a power allocation search problem for the OPA scheme. Section IV proposes various power allocation search algorithms. Computer simulations in Section IV evaluate the performance of the OPA scheme with various search algorithms. Finally, Section VI provides the conclusion. II. S YSTEM M ODEL This paper considers a MISO broadcast fading channel with a BS equipped with M transmit antennas for N users with only 1 receive antenna. The channel is assumed to be quasi-static, i.e., it is constant during a block, which comprises a training period and a data transmission period. Effectively, there are N MISO channels, y i = hi

M 

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where yi , hi , and ni respectively are the scalar received signal, the 1 × M channel vector, and the scalar additive white circularly symmetric complex Gaussian noise with zero mean and variance N0 for user i. {φm }M m=1 denotes a set

1-4244-0353-7/07/$25.00 ©2007 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

III. P ROBLEM S TATEMENT

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The SINR of user i for beam m is, SIN Ri,m

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of orthonormal random vectors, and xm denotes the symbol transmitted over the beamforming vector φm . The allocated power Mfor each stream xm is Pm with a total power constraint, 1 P m=1 Pm ≤ P , and the average SNR is defined as N0 . M During the training period, the BS broadcasts M pilot sequences known to the users over M randomly generated beams with possibly different power allocations, and each user feeds back its SINR for each beam. The BS selects M best users for each beam from the received SINR feedback and sends data over the beam according to the feedback from the selected M users during the data transmission period. Therefore, the BS maximizes the sum rate by optimizing the distribution of individual users’ SINRs without instantaneous CSIT.

Pj

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where |hi φm |2 ’s are i.i.d over i and m with χ2 (2) distribution for i = 1, · · · , N and m = 1, · · · , M . Here, Zm follows χ2 (2) distribution but Ym is a sum of weighted chi-square random variables, which does not follow χ2 (2M −2) distribution. The distribution of a sum of weighted chi-square random variables lacks a simple closed form expression [10]. Nonetheless, by constructing a random variable with the first three moments equal to the original random variable, a simple N approximate distribution can be obtained [11]. For Y = i=1 di Xi2 , where X iNhave i.i.d. standard normal distribution and di > 0 with i=1 di = 1, the distribution is approximated as,   2d1 −1 − y i y N e 2di 1  2di fY (y)  , y > 0. (5) 2 i=1 Γ( 2d1 i ) As an example, Figure 1 shows the distributions of a sum of weighted chi-square random variables with the degree of freedom 2 with weighting (1, 1, 1, 1) and (3, 1, 0, 0), and their approximate distributions. As shown in the figure, the approximate distributions and the original distributions are very close and therefore the use of approximate distribution is justified. The probability distribution function (PDF) of SIN Ri,m , fm (x), and the cumulative distribution function (CDF) of SIN Ri,m , Fm (x), can be derived by conditioning on y.    ∞ −x(N0 + y) N0 + y fm (x) = exp fYm (y)dy(6) Pm Pm 0    ∞ −x(N0 + y) exp Fm (x) = 1 − fYm (y)dy (7) Pm 0

The BS sends M data streams over M beams with each data stream supporting the selected user with the highest SINR for each beam. Therefore, the sum rate Rsum with the OPA scheme for N users with power allocation {Pm }M i=1 is given by,  M    Rsum = E log 1 + max SIN Ri,m (8) m=1

=

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N −1 log(1 + x)N fm (x)Fm (x)dx. (9)

Hence, the power allocation problem for the proposed OPA scheme to maximize the sum rate is summarized as follows, maximize subject to

Rsum ({Pm }M i=1 ) Pm ≥ 0 M 1 m=1 Pm ≤ P . M

This problem is complex to solve as the objective Rsum is not only strictly non-convex but also hard to transform into convex by relaxation. Therefore, the only way to find the optimal power allocation is a grid-search algorithm, which requires much computation. Although the power allocation is fixed for a given number of users, the number of active users always changes. Therefore, it would be beneficial for the system to be able to easily recalculate the power allocation whenever the number of users changes. The following section proposes several low-complexity power allocation search algorithms that achieve close to the optimal performance with low complexity. IV. P OWER A LLOCATION S EARCH A LGORITHMS This section proposes several power allocation search algorithms for the OPA scheme that maximize the sum rate: a grid-search algorithm, line-search algorithms, and a bisectionsearch algorithm. For expositional simplicity, the power allocation is normalized such that the total power constraint is expressed as sum(P A) ≤ M , where P A denotes the normalized opportunistic power allocation. The normalized incremental power allocation is ∆P .

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

A. Grid-Search Algorithm A grid-search algorithm searches over all the possible combinations of power allocation under the total power constraint and selects the best one. Thus, the power allocation found by the grid-search algorithm is the optimal power allocation that maximizes the sum rate for the proposed OPA scheme. Table 1 shows the proposed grid-search algorithm to find the optimal power allocation. Table 1. Grid-Search Algorithm for all P A such that sum(P A) ≤ M calculate Rsum (P A) Result: P A∗ = argP A max Rsum (P A) ∗ (P A) = Rsum (P A∗ ) Rsum Even though the optimal power allocation can be found by the grid-search algorithm, it suffers from high computational complexity. Therefore, other suboptimal search algorithms with less complexity are proposed in the following subsections. B. Line-Search Algorithm Even though it is suboptimal, a line-search algorithm achieves good performance with fast convergence speed [12]. For the given problem, the line-search algorithm can be implemented with three different starting points: zero power allocation, uniform full power allocation, and full power allocation on only one beam. Table 2 shows the proposed line-search algorithm. Starting from one of the initial power allocations, the line-search algorithm finds another power allocation P A that differs from the current power allocation P A by only ∆P . If P A achieves a higher sum rate compared to P A, the algorithm updates P A with P A and repeats the process until a higher sum rate is not achievable. Table 2. Line Search Algorithm Initialization: P A = [0 0 · · · 0] or [1 1 · · · 1] or [ M 0 · · · 0] calculate Rsum (P A) Recursion: for all P A∆ such that sum(P A∆ − P A) = ±∆P calculate Rsum (P A∆ ) P A = argP A∆ max Rsum (P A∆ ) if Rsum (P A ) ≥ Rsum (P A) P A = P A Rsum (P A) = Rsum (P A ) elseif Rsum (P A ) < Rsum (P A) break Result: P A∗ = P A ∗ (P A) = Rsum (P A) Rsum C. Bisection-Search Algorithm Since using full power maximizes the sum rate, a bisectionsearch algorithm, which is suboptimal but has very fast convergence speed [13], is proposed as a search algorithm. The

bisection-search algorithm starts by sorting all the possible combinations of power allocation that use full power based on their biases. That is, the index of the power allocation that allocates full power on only one beam is 1, while the index for uniform power allocation is K, where K denotes the cardinality of the possible power allocation set. Power allocation that concentrates its power on fewer beams is likely to have a smaller index, whereas power allocation that allocates its power across multiple beams is likely to have a large index. By comparing the rates achievable with the power allocation with the lowest index and the highest index, the bisection-search algorithm updates either the lowest index or the highest index with the middle index between those two. The details of the bisection-search algorithm is shown in Table 3. Table 3. Bisection Search Algorithm Initialization: sort all P A such that sum(P A) = M based on its bias P Ai , i = 1, · · · , K l = 1, u = K Recursion: if Rsum (P Al ) ≤ Rsum (P Au ) l = (l + u)/2 elseif Rsum (P Al ) > Rsum (P Au ) u = (l + u)/2 if u − l ≤ 1 if Rsum (P Al ) > Rsum (P Au ) u=l break Result: P A∗ = P Au ∗ Rsum (P A) = Rsum (P Au ) V. S IMULATION R ESULTS The number of transmit antennas, M , is 4 and the number of users, N , is varied from 1 to 50. The sum rate achieved by the proposed OPA scheme and the number of searches for each N for the proposed power allocation search algorithms are shown in Fig. 2, 3, and 4 for the average SNR equal to 0dB, 10dB, and 20dB, respectively. The grid-search algorithm achieves the maximum sum rate for all SNRs and all possible numbers of users. Nonetheless, it requires the largest number of searches to find the optimal power allocation as expected. For line-search algorithms, the achieved sum rate and the number of searches significantly vary for different starting points. The line-search algorithm starting with using all beams with uniform power allocation performs well for low SNR but poorly for high SNR, while starting with full power allocation for only one beam performs well for high SNR but poorly for low SNR. Therefore, it is conjectured that for low SNR, the optimal power allocation is using all beams with uniform power allocation, while for high SNR, using only one beam with full power is the optimal power allocation. This can

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

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searches for the bisection-search algorithm is the least for all SNRs and for the number of users considered among all search algorithms. The improvement in the sum rate by the proposed OPA scheme over uniform power allocation is larger for a smaller number of users and for higher SNR. When the number of users is 30, the gains are 10% and 30% for 10dB SNR and 20dB SNR, respectively. The scaling law of the proposed OPA scheme is also M log log N as in the DPC case. This is because the optimal power allocation found by the proposed search algorithms for the OPA scheme includes the case of using all beams with uniform power allocation, and it is readily seen in the figures.

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also be verified from observing the cross-over points between uniform power allocation and one beam power allocation. As the SNR increases, the cross-over point moves to the right. This implies that for a fixed number of users, the optimal power allocation is to concentrate the power to fewer beams as the SNR increases. This is reasonable because as the SNR increases, interference also increases. The bisection-search algorithm achieves a sum rate almost equal to the sum rate achieved by the optimal power allocation found by the grid-search algorithm. Because the performance of the bisection-search algorithm is very close to the optimal, it is believed that the strictly non-convex power allocation optimization problem can be cast as a convex problem by sorting possible combinations of power allocation based on their biases as proposed in Section IV-C. The number of

This paper considers a sum rate maximization problem for a MISO broadcast fading channel when users have a limited feedback rate. As the other water-filling-based techniques are not reliable with quantized CSI feedback, the OPA scheme is proposed to maximize the sum rate for a given number of users. It is shown that the optimal power allocation can be found by the bisection-search algorithm with low computational complexity, and the proposed OPA scheme enables significant improvement in the sum rate compared to the case of using all beams with uniform power allocation and the case of using only one beam with full power. The proposed OPA scheme is beneficial mainly in two aspects. First, the transmission protocol is simple and the latency between the training period and the data transmission period is small because the power allocation remains fixed over both training periods and data transmission periods unless the number of users changes. The OPA scheme only needs to notify the selected users that they are selected and then support them with their respective data rate. On the contrary, if the power allocation changes after receiving the CSI feedback,

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

the BS needs verification from the selected users that the data rate with the changed power allocation is possible and this increases the latency of the system. Second, the power allocation only changes when the number of users changes, and recalculating the optimal power allocation for the changed number of users can be done with a low-complexity bisectionsearch algorithm. Thus the computational complexity at the BS is significantly reduced compared to the case when the optimal power allocation is found by water-filling in an iterative way. R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Bell Labs Technical Memorandum, 1995. [2] H. Weingarten, Y. Steinberg, and S. Shamai(Shitz), “The capacity region of the Gaussian MIMO broadcast channel,” Proc. IEEE International Symposium Information Theory, Chicago, p. 174, June 2004. [3] T. Yoo and A. Goldsmith, “On the optimality of multi-antenna broadcast scheduling using zero-forcing beamforming,” IEEE Journal of Selected Areas in Communications, vol. 24, pp. 524-541, Mar. 2006. [4] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Transanctions on Information Theory, vol. 48, no. 6, pp. 1277-1294, Nov. 2002. [5] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channel with partial side information,” IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506-522, Feb. 2005. [6] J. Wagner, Y. C. Liang, and R .Zhang, “On the balance of multiuser diversity and spatial multiplexing gain in random beamforming,” submitted to IEEE Transactions on Wireless Communications, Mar. 2006. [7] J. Chung, C.-S. Hwang, K. Kim, and Y. K. Kim, “A random beamforming technique in MIMO systems exploiting multiuser diversity,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 5, pp. 848-855, 2003. [8] Y. Kim, K. Song, R. Narasimhan, and J. M. Cioffi, “Single user random beamforming in Gaussian MIMO broadcast channels,” Proc. of IEEE International Conference on Communications, Seoul, Korea, vol. 4, pp. 2695-2699, May 2005. [9] M. Kountouris and D. Gesbert, “Robust multi-user opportunistic beamforming for sparse networks,” Proc. of IEEE Workshop on Signal Processing Advances in Wireless Communications, New York, U.S.A., 2005. [10] A. Castano-Martinez and F. Lopez-Blazquez, “Distribution of a sum of weighted central chi-square variables,” Communications in Statistics Theory and Methods, vol. 34, pp. 515-524, 2005. [11] S. Gabler and C. Wolff, “A quick and easy approximation to the distribution of a sum of weighted chi-square variables,” Statistics Hefte, vol. 28, pp. 317-325, 1987. [12] J. M. Cioffi, Advanced Digital Communication, EE379C Stanford University Course Readers, http://www.stanford.edu/class/ee379c. [13] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, 2004.