Average-SNR-Optimized User Selection Algorithm in MIMO BC Systems Jing Wang, Yan Wang, Xiqi Gao and Xiaohu You National Mobile Communications Research Lab Southeast University, Nanjing, P.R.China Email:
[email protected]
Abstract— In this study, a user selection algorithm aiming to optimize the signal-to-noise ratio (SNR) is proposed for MIMO BC systems. After setting a threshold value, only users whose channel gains are above the set threshold are considered. While achieving the full multiplex gain, this method can get a desirable sum capacity and BER performance. Compared with the exhaustive search scheme, the system overheads, such as the amount of feedback and the number of selection in the base station, decrease evidently.
I. I NTRODUCTION Owing to the potential for dramatic improvement in the spectral efficiency, MIMO broadcast (MIMO BC) systems have been received considerable interest in recent years. It has been shown that the sum capacity with full channel state information (CSI) in both the transmitter and all the receivers can be achieved by using dirty-paper coding (DPC) [1]–[5]. This technique is quite sophisticated and challenging to implement in an actual system. Beamforming (BF) is a suboptimal strategy that can serve multiple users at a time, with reduced complexity relative to DPC. And BF has been shown to achieve a fairly large fraction of DPC capacity when the base station has multiple antennas and each user has one antenna. If the BF vectors are chosen optimally, the sumrate of BF approaches that of DPC as the number of users go to infinity [6]. Due to it is a non-convex optimization problem, finding the optimal BF vectors is very difficult. For the case where each user has only a single antenna, zeroforcing beamforming (ZFBF) can be implemented easily [2], [7], [8], which utilizes all available spatial degrees of freedom and performs measurably close to DPC in many scenarios [9]. ZFBF tends to a restriction on the system configuration in terms of the number of antennas. When the number of users is larger than the number of transmit antennas, it is impossible to send data streams to all users simultaneously without interuser interference. In cases that the number of users is equal to the number of transmit antennas, the downlink channel matrix can sometimes be near-singular, and ZFBF can’t offer a good performance [7]. On the other hand, there always be a large number of users in a practical network, and the base station can increase the throughput by selecting the best set of users to communicate with, which is the multiuser diversity gain [10]. As we known, an exhaustive search over all possible user sets will guarantee the optimum multiuser diversity gain. But the complexity is prohibitively large when the number of users
978-1-4244-1645-5/08/$25.00 ©2008 IEEE
becomes large. To reduce the complexity, several suboptimal schemes have been proposed. A greedy user selection approach is presented in [11]. In [12] a semi-orthogonal user selection (SUS) algorithm is given, which shows that as the number of users goes to infinity, the achieved sum capacity is very close to that of DPC. In this method, an orthogonality threshold is used for selecting the users in each step. In [13], a sequential water-filling (SWF) algorithm is investigated, which achieves a comparable maximum sum rate by ZFBF with a lower complexity. In the above mentioned schemes, perfect CSI is assumed at the base station, which tends to involve a prohibitive overhead in the feedback link, especially in frequency division duplex (FDD) systems with a large number of users. [14] and [15] have proposed the transmit strategies in conjunction with limited feedback to implement the beamforming and scheduling for MIMO BC, which reduce the amount of feedback overhead significantly. In [16], a suboptimal algorithm is proposed for selecting a set of users in order to maximize the sum capacity of the system. Through setting a threshold value, the feedback overhead is decreased and the search space of finding the optimal subset of uses becomes small. In this study, a user selection algorithm to optimize the detection signal-to-noise ratio (SNR) is presented. After setting a threshold value, only users whose channel gains are above the set threshold are considered. Then among these candidate users, the transmitter select a set of users whose minimum eigenvalue is maximal among the possible user sets. The amount of feedback and the complexity of search are decreased significantly. The rest of the paper is organized as follows. Section II presents the system model we used. In Section III, the user selection algorithm is presented, and the complexity analysis is considered. Simulations are carried out in Section IV, followed by the conclusions in Section V. II. S YSTEM M ODEL Consider a K-user MIMO BC system in which the transmitter has M antennas, and each of the receivers has one antenna. Assume that K M . The received signal at user k is described as
2101
y k = h k x + nk ,
k = 1 · · · K.
(1)
where x ∈ CM ×1 is the transmitted signal with a power constraint of P . hk ∈ C1×M denotes the channel vector between the base station and user k. The elements of hk are i.i.d complex Gaussian variables with zero mean and unit variance. A block fading condition is assumed for each channel so that the channel realizations stay fixed for the duration of a single frame and change independently frame by frame. {nk } is the additive white Gaussian noise (AWGN) with a distribution CN 0, σn2 . Throughout this study, no CSI is available at the base station, and each user only has the CSI of its own channel. Assume N users, where N M , have been selected for transmission, and channel matrix of these denote the composite H
H users as H = hH , where the superscript H (1) , · · · , h(N ) stands for conjugate transpose. In ZFBF, the BF vectors can be obtained by the pseudo-inverse of the channel downlink and B = t t , · · · , t matrix. Let T = t (1) (N ) (N )
(1) diag b(1) , · · · , b(N ) be the column normalized H† and the allocated power matrix, respectively, where the superscript † denotes the pseudo-inverse operation. Then the transmitted signal can be expressed as
x = TBd,
(2)
T is where the N × 1 symbol vector d = d(1) · · · d(N ) intended to send to users, whose entries are chosen i.i.d. zeromean Gaussian variables with unit power. Then, for the ZFBF system model, the received signal for user (i) is (3) y(i) = b(i) d(i) t(i) + n(i) .
A. Algorithm Description To achieve a good performance by using ZFBF, the selected sub-channels must have high gains and be nearly orthogonal to each other. When the number of users is large, it becomes easier to satisfy these requirements. But the exhaustive search is very complicated, and the amounts of feedback required by providing CSI to the base station is huge. Based on setting a threshold, user selection is performed by maximizing the minimum received SNR in the proposed algorithm. Assume there are L users whose channel
gains are above the L ways in which M prescribed threshold. Then, there are M optimal users out of L may be selected at the base station. Let s and Hs denote the index of selecting and the corresponding
L channel matrix respectively, where s = 1, · · · , . For a M fixed s and Hs , consider equal power allocation, the SNR at the user (i) is γ(i) =
−1
σn2 M (Hs HH s )
In a network with many users, the user selection algorithm should be performed at the beginning of each frame to achieve a high throughput. Meanwhile, in order to achieve a full spatial multiplexing gain, it is required that the base station communicate with M users at each instant simultaneously. In this section, a user selection method is presented, which can acquire the full multiplexing gain and achieve a good performance in terms of the sum capacity and the bit error rate (BER) performance. The proposed algorithm begins with setting a threshold. Then the channel whose gain is higher than the threshold will be fed back to the base station. Among these candidate users, the algorithm chooses a set of size M to serve with, according to the fact that the sum capacity and BER performance are optimal when the minimum SNR of the selected sub-channels is maximized. The amount of feedback and the complexity of search are decreased significantly, especially under a large number of users.
.
(4)
(i),(i)
(i)
gular value and the largest diagonal entry of matrix A, respectively, we have [17], H −1 max Hs Hs λ−2 (5) min (Hs ), (i)
min γ(i)
= min γ(i)
(i)
(i),(i)
as the minimum SNR among
all sub-channels, then, min λ2min (Hs ) γ(i)
III. T HE P ROPOSED U SER S ELECTION A LGORITHM
where [A](i),(i) represents the ((i), (i))th element of matrix A. Let λmin (A) and max [A](i),(i) be the minimum sin-
Denote It can be seen that ZFBF decomposes the MIMO BC channel into a series of subchannels without inter-user interference.
P
P . σn2 M
(6)
Correspondingly, the total system throughput, i.e. the sum capacity of ZFBF MIMO BC system is as follows, sum = CZF
M
min . log2 1 + γ(i) M log2 1 + γ(i)
(7)
i=1
Let Ne denote the average number of nearest neighbors of the constellation used for transmission, and d2min be the minimum squared distance of the constellation. Using the nearest neighbor union bound results from [17], the BER can be bounded by
Pe (Hs ) M Ne Q
min d2 γ(i) min 2
(8)
∞ 2 where Q (x) = √12π x e−t /2 dt is the Gaussian Q-function. According to the analysis above, both the sum capacity min , which and BER performance are functions only of γ(i) shows that they are dominated by the sub-channel with the worst condition. Therefore, the overall performance can be
2102
improved by the following selection criterion, through which the suboptimal set of users will be selected. s˜ = arg max {λmin (Hs )} s
(9)
In brief, the user selection algorithm can be summarized as follows. 2 Step 1: Each user calculate the gain hk of its channel vector,k = 1 · · · K. Step 2: Compare these channel gains with a predetermined threshold t, and feedback the CSI’s of users whose gains are larger than t. Assume there are L users meeting this requirement. Step 3: The transmitter selects M most favorite users from the candidate users to serve according to the selection criterion: s˜ = arg max {λmin (Hs )}. s
B. About the threshold The sum capacity of MIMO BC has been shown to scale as M log log K, as K tends to infinity, which implies that to achieve the optimum sum capacity, the threshold value must behave like log K. According to the asymptotic theory, [16] indicates that the optimal threshold value should be between log K and log log K. 2 As we known, the channel gain η = hk is a chisquare-distributed random variable, and we denote fη (η) as the probability density function of η. Then, the probability that the channel gain of a user is above the threshold is given as ∞ P = Pr {η t} = t fη (η) dη. After a simple deduction, ˜ can be obtained as the average number of feedback users L M −1 ˜ = E {K} = K P = Ke−t tm /m!. L m=0
C. Complexity Analysis In the exhaustive search method, perfect CSI’s of all users are needed in the base station. And an M -dimension channel vector between each user and the transmitter requires 2M real values to describe. So the number of real values required fed back to the base station, i.e. the amount of feedback, is 2M K. The best M users for maximizing the sum capacity can be found by exhaustive
search. In this case, the size of the search K space is equal to . M By introducing the threshold, only the channel vectors whose gain above the threshold should be sent back to the base station in our proposed method. The amount of feedback decreases to 2M L. The size of search space is also decreased from K to L, and
thetotal number of selecting the desired user L . It can be seen in the next simulation set is equal to M results that the complexity of the user selection process is significant reduced. IV. S IMULATIONS This section presents the performance of the proposed user selection algorithm, from three aspects: sum capacity, BER performance and computed complexity. In simulations, the
threshold is set as the lower bound shown in the last sections, i.e. t = log K − log log K. Let {M, K} denote a MIMO BC system with M transmit antennas and K users with one antenna. All channels are i.i.d. Rayleigh fading and distribute according to CN (0, 1). Fig. 1 plots the sum capacity of ZFBF MIMO BC systems versus the number of users. At average received SNR=10dB, two cases are investigated: M =2 and M =3 with equal power allocation and waterfilling strategy, respectively. It can be seen that besides a minor gap between the proposed method and DPC strategy [18], the trend of the sum capacity is similar to DPC as the number of users increases, which show the multiuser diversity can be achieved. Moreover, with the number of users raises, the gap nearly be invariant. For example, for M =2, the gap is less than 1 bps/Hz for the whole range of the number of users. In Fig. 2, with a fixed number of users K=100, the sum capacity versus the average received SNR is given. We also consider the cases of M =2 and M =3. Similar property can be found as in Fig. 1. For comparison, the sum capacity by the exhaustive search is also plotted. And the proposed algorithm can achieve the same sum capacity as the exhaustive one. Fig. 3 and Fig. 4 shows the BER performance of the proposed algorithm with M =2 and M =3, respectively. Binary phase shift keying (BPSK) modulation is employed. Notice that the threshold is only applicable for a large number of users. Meanwhile, the computed complexity is tolerable with a small number of users, even when the exhaustive search is employed. Thus, in this experiment, the exhaustive search is adopted for K=2,3,10 and the proposed algorithm with a predetermined threshold is chosen for K=50,100. It can be seen that, as the number of users increases, selecting an optimal set of users becomes easier and accordingly, the system achieve a better BER performance. For instance, in fig. 3, at BER=10−3 , corresponding to K=10,50,100, the required SNR are 7dB, 3.5dB and 3dB respectively. The following figures give the complexity comparison between the proposed algorithm and the exhaustive search scheme. Fig. 5 depicts the average number of feedback users versus various number of users in a network. For comparison, the case of the exhaustive search is plotted in the figure. Take K=150 and M =2 for example, the average number of feedback users is about 22, which indicates that the amount of feedback will decrease obviously. As the number of users increases, the saving feedback overhead of the proposed algorithm from exhaustive search scheme increases too. In Fig. 6, the number of selection at the base station is depicted, where the proposed algorithm and the exhaustive search scheme are considered. It is shown that number of selection is reduced from 104 to 2.5 × 102 for M =2 and K=150. Define the proportion about the overhead as the ratio of the proposed algorithm and the exhaustive search method, Fig. 7 plots that the proportion of the amount of feedback and the number of selection. Still take M =2 and K=150 for instance, the feedback overhead of our proposed method is about 15% of the exhaustive search. And the number of selection is about
2103
3%. It can be clearly seen that, compared with the exhaustive search, our method has a lower complexity. V. C ONCLUSIONS
[18] N. Jindal, W. Rhee, S. Vishwanath, S. Jafar, and A. Goldsmith, “Sum power iterative water-filling for multi-antenna gaussian broadcast channels,” IEEE Trans. Inform. Theory, vol. 51, no. 4, pp. 1570–1580, April 2005.
In this paper, we investigated the user selection algorithm in MIMO BC systems with many users. In our proposed algorithm, a predetermined threshold was introduced to reduce the amount of feedback. Accordingly, the number of candidates at the base station decreased, and the number of selection reduced too. According to the criterion of maximizing the minimum eigenvalue, this algorithm can achieve the full multiplex gain and multiuser diversity. Simulation results prove the effectiveness of the proposed user selection scheme.
SNR=10dB 20 18 16 M=3
Sum Capacity
14 12
M=2
10 8 6
R EFERENCES
4
[1] M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. 29, no. 3, pp. 439–441, May 1983. [2] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna gaussian broadcast channel,” IEEE Trans. Inform. Theory, vol. 49, no. 7, pp. 1691–1706, July 2003. [3] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector gaissoam broadcast channel and uplink-downlink duality,” IEEE Trans. Inform. Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003. [4] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sum-rate capacity of MIMO broadcast channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2658–2668, Oct. 2003. [5] Y. S. H. Weingarten and S. Shamai(Shitz), “The capacity region of the gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 3936–3964, Sept. 2006. [6] M. Sharif and B. Hassibi, “A comparison of time-sharing, dpc, and beamforming for mimo broadcast channels with many users,” IEEE Trans. Commun., vol. 55, no. 1, pp. 11–15, Jan. 2007. [7] B. M. H. C. B. Peel and A. L. Swindlehurst, “A vector perturbation technique for near-capacity multiantenna multiuser communication - part i: channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195–202, Jan. 2005. [8] ——, “A vector perturbation technique for near-capacity multiantenna multiuser communication - part i: channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 3, pp. 537–544, Mar. 2005. [9] N. Jindal, “High snr analysis of mimo broadcast channels,” in Proc. IEEE Int. Symp. on Info. Theo.(ISIT), Adelaide, Australia, Sept. 2005, pp. 2310–2314. [10] M. Sharif and B. Hassibi, “On the capacity of mimo broadcast channels with partial side information,” IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 506–522, Feb. 2005. [11] G. Dimi´e and N. D. Sidiropoulos, “On donwlink beamforming with greedy user selection: Performance analysis and a simple new algorithm,” IEEE Trans. Signal Process., vol. 53, no. 10, pp. 3857–3868, Oct. 2005. [12] T. Yoo and A. Goldsmith, “On the optimality of multi-antenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528–542, Mar. 2006. [13] D. J. L. J. Wang and M. D. Zoltowski, “User selection for mimo broadcast channel with sequential water-filling,” in in Proc. of 44th Annual Allerton Conf. on Commun., Control, and Computing, Monticello, IL, Sept. 2006, pp. 27–34. [14] T. Yoo and A. Goldsmith, “Multi-antenna broadcast channels with limited feedback and user selection,” Submitted to IEEE J. Sel. Areas Commun., [Online]. Available: http://systems.stanford.edu. [15] D. J. L. P. Ding and M. D. Zoltowski, “Multiple antenna broadcast channels with shape feedback and limited feedback,” Accepted to IEEE Trans. Singal Process. [Online]. Available: http://cobweb.ecn.purdue.edu/ djlove/papers/26. [16] A. Bayesteh and A. K. Khandani, “On the user selection for mimo broadcast channels,” in Proc. IEEE Int. Symp. on Info. Theo.(ISIT), Adelaide, Australia, Sept. 2005, pp. 2325–2329. [17] S. S. R.W. Heath and A. Paulraj, “Antenna selection for spatial multiplexing systems with linear receivers,” IEEE Commun. Lett., vol. 5, no. 4, pp. 142–144, April 2001.
2
DPC Proposed,equal power Proposed,waterfilling
0 50
Fig. 1.
100
150 200 Number of users
250
300
Sum Capacity versus the number of users with SNR=10dB. K=100 25
20
DPC Proposed,equal power Proposed,waterfilling Exhaustive,equal power Exhaustive,waterfilling
Sum Capacity
M=3 15 M=2 10
5
0 0
5
10 15 Average received SNR(dB)
20
Sum Capacity versus the average received SNR with K=100.
Fig. 2.
M=2
0
10
−1
Average BER
10
−2
10
−3
10
K=2 K=3 K=10 K=50 K=100
−4
10
Fig. 3. M =2.
2104
−4
−2
0 2 4 Average received SNR(dB)
6
8
Average BER performance versus the average received SNR with
M=3
0
7
10
10
M=2,Exhaustive M=2,Proposed M=3,Exhaustive M=3,Proposed
6
10 −1
Number of selection
Average BER
10
−2
10
−3
10
K=3 K=10 K=50 K=100
5
10
4
10
3
10
2
10
−4
10
−2
0 2 4 Average received SNR(dB)
6
8
1
10
50
100
150 200 Number of users
250
300
Average BER performance versus the average received SNR with Fig. 6.
The number of selection at the base station.
3
10
0.7
M=2,Proposed M=3,Proposed Exhaustive
Proportion about the overhead
Average number of feedback users
Fig. 4. M =3.
−4
2
10
M=2,Amount of feedback M=2,Number of selection M=3,Amount of feedback M=3,Number of selection
0.6 0.5 0.4 0.3 0.2 0.1
1
10
50
100
Fig. 5.
150 200 Number of users
250
0 50
300
Average number of feedback users.
100
Fig. 7.
2105
150 200 Number of users
250
The proportion about the overhead.
300
