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Orthogonal beamforming using GramSchmidt orthogonalization for multiuser MIMO downlink system
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 41 (2011)
Abstract
Simultaneous transmission to multiple users using orthogonal beamforming, known as spacedivision multipleaccess (SDMA), is capable of achieving very high throughput in multipleinput multipleoutput (MIMO) broadcast channel. In this paper, we propose a new orthogonal beamforming algorithm to achieve high capacity performance in MIMO broadcast channel. In the proposed method, the base station generates a unitary beamforming vector set using GramSchmidt orthogonalization. We extend the algorithm of opportunistic SDMA with limited feedback (LFOSDMA) to guarantee that the system never loses multiplexing gain for fair comparison with the proposed method by informing unallocated beams. We show that the proposed method can achieve a significantly higher sum capacity than LFOSDMA and the extended LFOSDMA without a large increase in the amount of feedback bits and latency.
1 Introduction
In multipleinput multipleoutput (MIMO) broadcast (downlink) systems, simultaneous transmission to multiple users, known as spacedivision multipleaccess (SDMA), is capable of achieving very high capacity. In general, the capacity of SDMA can be considerably improved in comparison with timedivision multipleaccess [1] because of multiuser diversity gain, which refers to the selection of users with good channels for transmission [2, 3]. The optimal SDMA performance can be achieved by dirty paper coding (DPC) [4], however, implementation of DPC is infeasible since it requires complete channel state information (CSI) and high computational complexity. More practical SDMA algorithms are based on transmit beamforming, including zero forcing [5], minimum mean square error [6], and channel decomposition [7].
Various algorithms for limited feedback SDMA schemes have been proposed recently. When the number of users exceeds the number of antennas at the base station, a user scheduling algorithm should be jointly designed with limited feedback multiuser precoding. For the opportunistic SDMA (OSDMA) algorithm proposed in [8], the feedback of each user is reduced to a few bits by constraining the choice of beamforming vector to a set of orthonormal vectors. In OSDMA, base station sends orthogonal beams, and each user reports the best beam and their signaltointerferenceplusnoise ratio (SINR) to the base station. The base station then schedules transmissions to some users based on the received SINR. For a large number of users, OSDMA ensures that the sum capacity increases with the number of users. However, the sum capacity of the OSDMA is limited if there are not a sufficient number of users.
To solve this problem, an extension of OSDMA, called OSDMA with beam selection (OSDMAS), is proposed in [9]. OSDMAS improves on OSDMA using beam selection to get capacity gain for any number of users in the system. However, multiple broadcast and feedback are required for implementing OSDMAS, which incurs downlink overhead and feedback delay.
An alternative SDMA algorithm with orthogonal beamforming and limited feedback is proposed [10], called OSDMA with LFOSDMA. LFOSDMA results from the joint design of limited feedback, beamforming and scheduling under the orthogonal beamforming constraint. In LFOSDMA, each user selects the preferred beamforming vector with their normalized channel vector, called the Channel shape, using a codebook made up of multiple orthonormal vector sets. Then, each user sends back the index of the preferred beam vector as well as SINR to the base station. Using multiuser feedback and a criterion of maximum capacity, the base station schedules a set of simultaneous users with the beamforming vectors. More details of LFOSDMA algorithm are stated in Section 3.
The simulation in [10] shows that LFOSDMA can achieve significant gains in sum capacity with respect to OSDMA. However, LFOSDMA does not guarantee the existence of N_{ t } (the number of transmit antennas) simultaneous users whose beam vectors belong to same orthonormal vector set, since each user selects a beamforming vector. This can result in the loss of multiplexing gain and hence the sum capacity of LFOSDMA decreases for an increase of the number of subcodebooks.
In this paper, we propose a new orthogonal beamforming algorithm using GramSchmidt orthogonalization for achieving high capacity in MIMO broadcast channel. In this algorithm, the base station initially selects one or more users, and let them feed their full CSI back. Among the feedback users, the base station selects the one having highest channel gain. Using full CSI information, the base station generates beamforming vector for the selected user, and using GramSchmidt orthogonalization, the base station can generate a unitary orthogonal vector set. On the other hand, each user can generate the same unitary orthogonal vector set in the same way for the base station using CSI of the selected user from the base station. Each user selects the preferred beam from the generated beamforming vector set, and feeds the index of the preferred beam and quantized SINR back. Among feedback users, the base station schedules users using the criterion of maximizing sum capacity. More details of the proposed method are shown in Section 4. Because the base station generates the beamforming vector for the selected user and schedules the one, the proposed method is expected to achieve high sum capacity, though the number of feedback bits and the amount of latency increase in our system. For fair comparison of the amount of the latency, we extend the algorithm of LFOSDMA to guarantee that the system never loses multiplexing gain in Section 5. Section 6 presents the analysis of the proposed method in terms of encoding, the effect of changing the number of initially selected users and the complexity at mobile terminal. In Section 7, we compare the number of feedback bits, the amount of latency, and the sum capacity of the proposed beamforming algorithm with LFOSDMA and the extended LFOSDMA. In the result, we show that the proposed method can achieve a significantly higher sum capacity than LFOSDMA and the extended LFOSDMA without a large increase in the amount of feedback bits and latency.
2 System Model
We consider a downlink multiuser multipleantenna communication system, made up by a base station and K active users. The base station is equipped with N_{ t } transmit antennas, and each user terminal is equipped with a single receive antenna. The base station can separate the multiuser data streams by beamforming, assigning a weight vector to each of N_{ t } active users. The weight vectors are unitary orthogonal vectors, where is a beamforming vector with  w_{ n } ^{2} = 1. We assume that the equal power allocation over scheduled users. The received signal of the user k is represented as
where is a channel gain vector of user k with i.i.d. complex Gaussian entries , x_{ b } is the transmitted symbol with x_{ b } = 1 and E [x_{ b }] = 1, B is the index set of scheduled users, and n_{ k } is complex Gaussian noise with zero mean and unit variance of user k. The superscript T denotes the vector transpose. It is assumed that the user k has perfect CSI h _{k}.
3 Conventional Orthogonal Beamforming
An orthogonal beamforming and limited feedback algorithm were proposed in [10], called LFOSDMA, which results from the joint design of limited feedback, beamforming and scheduling under the orthogonal beamforming constraint.
The CSI h_{ k } can be decomposed into two components: gain and shape. Hence, h_{ k } = g_{ k }s_{ k } where g_{ k } = h _{k} is the gain and s_{ k } = h_{ k }/h _{k} is the shape. The channel shape is used for choosing weight vector, and the channel gain is used for computing SINR value. The user k quantizes and sends back to the base station two quantities: the index of a selecting weight vector and the quantized SINR. We assume that a codebook is created using the method in IEEE 802.20 [11], which can be expressed as , where the subcodebook F_{ i } is the unitary matrix and M is the number of subcodebooks. By expressing each unitary matrix as , the preferred beam q_{ k } selected by the user k, as a function of CSI's shape s _{k}, is given by
where · ^{T} means transposition. To compute SINR, we define the quantization error as
It is clear that the quantization error is zero if s_{ k } = q _{k}. The SINR for the user k is a function of channel power ρ_{ k } = h _{k}^{2} and the quantization error δ_{ k }
where γ is the input SNR. Each user feeds back its SINR along with the index of the preferred beam. Only the index of q_{ k } needs to be sent back, because the quantization codebook can be known a priori to both the base station and the users. We assume that the SINR _{ k } is perfectly known to the base station by feedback processing. The same assumption is used in [8], [10]. Let the required number of bits for quantizing SINR be Q_{SINR}, and the total amount of required feedback per user becomes log_{2}(N_{ t }M ) + Q_{SINR} bits.
Among feedback users, the base station schedules a subset of users using the criterion of maximizing sum capacity. Using the algorithm discussed in [10], [12], we group feedback users according to their quantized channel shapes as follows.
where is the i th beam vector in the j th subcodebook. Among these subgroups, the one having the maximum sum capacity is scheduled, and base station selects the subcodebook having the maximum sum capacity for transmission. The resultant sum capacity can be written as
If is empty, we set SINR _{ k } = 0.
In the situation that there is a large number of active users, LFOSDMA can achieve high capacity. However, in the situation that there is a small number of active users, its capacity is limited because LFOSDMA does not guarantee the existence of N_{ t } simultaneous users whose beam vectors belong to the same orthogonal vector set, in other words, there is an unallocated beam vector in the selected subcodebook. This can result in the loss of multiplexing gain and hence the sum capacity of LFOSDMA decreases for an increase of the number of subcodebooks where there is a small number of active users.
4 Proposed Orthogonal Beamforming Algorithm
In this section, we propose a new orthogonal beamforming algorithm using GramSchmidt orthogonalization. The proposed method is described from Steps I to VI as follows.
Step I
The base station initially selects S users, and sends pilot signals to let all users estimate CSI, where S is the number of users selected by the base station. In this paper, we assume that all users have perfect CSI. We denote the latency, until pilot signals are received by all users in the cell, by δ_{ BC }
Step II
Users who are initially selected by the base station feed back their full CSI, analog CSI. In this paper, we randomly selected the initial users who feed their full CSIs back, because at the initial step the base station does not have users' CSI and the proposed method does not want to increase the amount of feedback. We denote the latency, until selected users' feedback information are received by the base station, by δ_{select}, and the number of feedback bits is SQ_{CSI} bits, where Q_{CSI} is the number of feedback bits of the full CSI.
Step III
Among the feedback users, the base station picks up the one having the highest channel gain from the initially selected users, which is defined as user u that has CSI h_{ u } and we refer to this user as the pivotal user. Using full CSI of user u, the base station generates a unitary orthogonal vector set, as follows.
Where · ^{H} means Hermitian transposition. We assume X is (N_{ t } × N_{ t } ) unit matrix, which is used for generating orthogonal weight vectors. Using GramSchmidt algorithm with w_{1}, we generate orthogonal beams to w_{1}. The vector w_{1} is the beamforming vector for user u, and the vector set of represents generated orthogonal beamforming vectors.
Step IV
The base station informs all users about information of w_{1}. We denote the latency, until the information of w_{1} is received by all users in the cell, by δ_{ad}, and the number of information bits is Q_{CSI} bits which is the number of feedback bits of full CSI
Step V
Using information from the base station about w_{1}, each user can generate the same unitary orthogonal vector set for the base station using (8) and (9). We assume that the algorithm for getting the unitary vector set is known a priori to both the base station and users. Then, each user selects the preferred beam which is given by
The quantization error and SINR for the user k is defined as
Each user feeds the quantized SINR' and the index of the preferred beam vector back. We denote the latency, until all users' feedback information are received by base station, by δ_{all}. The number of feedback bits is log_{2}N_{ t } + Q_{SINR} bits.
Step VI
Among feedback users, the base station schedules users using the criterion of maximizing sum capacity. Certainly, the beam w_{1} is assigned by the user u, the pivotal user, because this beam is the beamforming vector for the user u.
5 Extended Conventional Orthogonal Beamforming
In this section, we extend the algorithm of conventional orthogonal beamforming to guarantee that there is no unallocated beam in the selected subcodebook. The proposed method always supports N_{ t } users, while the conventional LFOSDMA cannot always support N_{ t } users, though its latency is smaller than that of the proposed method. Therefore, to compare the performance of those algorithms under more similar condition, we allow LFOSDMA to support always N_{ t } users but with higher latency, which is the extended LFOSDMA. The scheduling algorithm with the extended LFOSDMA is described from Step 1 to Step 6 as follows.
Step 1
A base station sends pilot signals to let users estimate CSI. In this paper, we assume that all users have perfect CSI h _{k}. We denote the latency, until pilot signals are received by all users in the cell, by δ_{BC}
Step 2
Using CSI, each user chooses the preferred beam vector from codebook and calculates the receive SINR. Then, each user feeds back indexes of the preferred beam vector and quantized SINR_{k}. We denote the latency, until all users' feedback information are received by base station, by δ_{all}.
Step 3
Among feedback users, the base station schedules a subset of users, and selects the subcodebook having the maximum sum capacity.
So far, during Step 1 and Step 3, the algorithm is same as that of LFOSDMA, and the extended part begins from Step 4 to Step 6.
Step 4
If the selected subcodebook has an unallocated beam vector, the base station informs all users about indexes of the selected subcodebook and the unallocated beam vector. We denote the latency, until the information of the unallocated beam vector is received by all users in the cell, by δ_{ad}, and the number of informed bits is log_{2}M + N_{ t } bits.
Step 5
Using information from the base station about the unallocated beam vector, each user can generate the unallocated beam vector set F_{ m }= {f_{m,n}, ...}, n ∈{1,2,..., N_{ t } }, and selects the preferred beam which can be given by
The quantization error and SINR for the user k is defined as
Each user feeds back the quantized and the index of the preferred beam vector. In this step, the latency is same as that of Step 2, and the number of feedback bits is log_{2}N_{ t } + Q_{SINR} bits.
Step 6
Among feedback users, the base station assigns a user to the unallocated beam vector of the selected subcodebook using the criterion of maximizing sum capacity.
The extended algorithm can guarantee the existence of N_{ t } simultaneous users, so even if there is a small number of users, and the extended LFOSDMA can achieve high capacity. However, the extended LFOSDMA leads to the large increase in the number of feedback bits, and worsens system latency. We make comparisons of the number of the feedback bits and a system latency in Sect. 7.
6 Analysis of the Proposed Method
In this section, we analyze the proposed method in terms of encoding, the effect of changing the number of initially selected users S and the complexity at mobile terminal.
6.1 Encoding of the proposed method
In this subsection, we evaluate the capacity performance of the proposed method when CSI is quantized by a random vector quantization codebook, because the feedback of the full CSI results in a large amount. The size of the codebook is 2 ^{QCSI} where Q_{CSI} is the number of feedback bits of the CSI. Figure 1 shows the sum capacity of the proposed method for different codebook sizes, Q_{CSI} = {5, 10, 15, 20, analog CSI}, for an increase of users. The number of transmit antennas is N_{ t } = 4, SNR is 5 dB and the number of the initially selected user is S = 1. We come up with the results based on Monte Carlo simulation.
As the codebook size becomes larger, the sum capacity of the proposed method increases. This is because the quantization error of CSI becomes smaller, as the codebook size becomes larger. As observed from Figure 1, 15 bits of the CSI feedback causes only marginal loss in sum capacity with respect to the analog CSI feedback. Such loss is negligible for 20bits feedback. Therefore, the feedback by the codebook of Q_{CSI} = 20 from the initially selected users is as good as the analog CSI case. Thus, in this paper, we assume that the number of the feedback bits of the full CSI is Q_{CSI} = 20 when we evaluate the feedback bits. Actually, the codebook of Q = 20 is not preferable in practice because of the large complexity at the mobile terminal side.
6.2 Effect of changing the number of initially selected users
In this section, we show the capacity result and the number of feedback bits of the proposed method with the increase of the number of initially selected users by the base station. Note that S affects the amount of feedback, but is not dependent on the number of transmit antennas N_{ t } .
Figure 2 shows the sum capacity of the proposed method for different number of initially selected users, S = 1,3,5, all active users, for an increase of users. The number of transmit antennas is N_{ t } = 4 and the SNR is 5 dB. We came up with the results of the capacity based on Monte Carlo simulation. By Monte Carlo simulation, we generate each user's flat Rayleigh fading channel and AWGN. Based on these values, we calculate each user's SINR using (4), (12) or (15). Using the SINR and (6), we calculate sum capacity. For the increase of the number of initially selected users, the sum capacity of the proposed method increases, however, the rate of improvement of the sum capacity decreases. The difference of the sum capacity between S = 1 and S = 2 is about 0.4 bits/Hz at K = 100, but there is little difference between S = 2 and S = 3. Therefore, S = 1 or S = 2 are practical.
Figure 3 shows the number of feedback bits of the proposed method with the increase of the number of initially selected users by the base station. We calculate the number of feedback bits based on the analytic formula, and there are two times for the base station to receive feedback from active users. First time, the initially selected S users feed their full CSIs back to the base station, and the number of the firstfeedback bits is SQ_{CSI} bits, where Q_{CSI} is the number of feedback bits of the full CSI per user. Thus, the number of the initially selected users, S, affects the number of firstfeedback bits, but is not dependent on the number of the active users, K nor the number of transmit antennas N_{ t } . Second time, the base station receives feedback about the selected beamforming vector from all active users other than the pivotal user, and the number of the secondfeedback bits is (K  1)(log_{2}N_{ t } + Q_{SINR}), where Q_{SINR} is the number of feedback bits of quantizing SINR. Thus, the number of active users, K, affects the number of secondfeedback bits, but is not dependent on the number of the initially selected users, S.
When S = all active users, the proposed method produces explosive growth of the number of feedback bits, because all users in the cell feed back their full CSI. When S ≠ all active users, the difference of the number of feedback bits is constant, which represents that of the full CSI from initially selected users. If we increase the number of initially selected users S by 1, the number of feedback bits is increased by Q_{CSI} = 20 bits.
6.3 Complexity at the mobile terminal
In this section, we show the complexity of the proposed method at the mobile terminal side in comparison with LFOSDMA. We evaluate the complexity by the number of scalar multiplications and square roots. Table 1 shows the complexity of LFOSDMA at users. In LFOSDMA, each user has N_{ t }M (4N_{ t } + 2) multiplications and N_{ t }M square roots when selects the beamforming vector from the codebook, where N_{ t } is the number of transmit antennas and M is the number of subcodebooks. Table 2 shows the complexity of the proposed method. In the proposed method, the implementation of each user consists of two stages: generation of the same unitary orthogonal vector set for the base station using (8) and (9), and the selection of the beamforming vector. We neglect the complexity of the initially selected users, because they feed the analog CSI back. In the former, (8) has (N_{ t }  1)!8N_{ t } + (N_{ t } +1) multiplications and one square root. In the latter, each user has (N_{ t }  1)(4N_{ t } + 2) multiplications and (N_{ t }  1) square roots.
7 Performance Comparison
7.1 Feedback comparison
In this subsection, we compare the number of feedback bits among the proposed method, LFOSDMA and the extended LFOSDMA. We calculate the number of the feedback bits based on the analytic formula, and summarize them in Table 3. Actually, the feedback bits of the extended LFOSDMA in Step 5 cannot be calculated by the analytic formula, and we assume it K(log_{2}N_{ t } + Q_{SINR}) this time. Figure 4 shows the number of feedback bits for an increase of the number of users until K = 20. To compare the extended LFOSDMA with the proposed method in terms of latency, we assume the extended LFOSDMA always informs all users about the index of the unallocated beam vector. Thus, every system in this paper has the linearlyincreasing number of feedback bits. We assume that the number of transmit antennas is N_{ t } = 4, the number of feedback bits of the full channel information is Q_{CSI} = 20 bits and that of quantizing SINR is Q_{SINR} = 3 bits [10].
Figure 4 shows that the proposed method needs fewer number of feedback bits than the extended LFOSDMA, and needs almost the same number of feedback bits as LFOSDMA. We can also observe from Figure 4 that the difference of the number of feedback bits between the proposed method and LFOSDMA for M = 1 is constant, which represents the number of feedback bits of the full CSI from initially selected users. If there is a large number of users, e.g. K = 100, the proposed method needs much fewer number of feedback bits than the extended LFOSDMA and LFOSDMA with M = 8. Therefore, the increase of the number of the feedback bits for the proposed method against that of LFOSDMA with M = 1 is not large compared with that of LFOSDMA with M = 8 and extended LFOSDMA.
7.2 Latency comparison
In this section, we compare the latency among the proposed method, LFOSDMA, and the extended LFOSDMA. Table 4 lists the comparison of system latency. δ_{BC} is the latency that is the amount of time from the sending pilot signals of the base station to the receiving of all users in the cell; δ_{all} is the latency that is the amount of time from the sending feedback information of all users to the receiving of the base station; δ_{ad} is the latency that is the amount of time from the sending the information of unallocated beam vector of the base station to the receiving of all users in the cell; and δ_{select} is the latency that is the amount of time from the sending the feedback information of the initially selected users to receiving of the base station.
Table 4 shows that the extended LFOSDMA and the proposed method have to tolerate higher latency than that of LFOSDMA. In practical systems, δ_{ BC } and δ_{ad} are much lower than δ_{all} or δ_{selec}, because δ_{BC} and δ_{ad} use a downlink broadcast channel. In addition, if there is a large number of users in the cell, δ_{selec} is much smaller than δ_{ all } . Therefore, the increase of the latency for the proposed method against LFOSDMA is not large. However, the increase of the latency affects the capacity of the proposed method, particularly in case of high mobility.
7.3 Capacity comparison
In this section, we show the capacity result of the proposed beamforming algorithm. Figure 5 compares the sum capacity of the proposed method with that of LFOSDMA and the extended LFOSDMA for an increase of the number of users. The number of transmit antennas is N_{ t } = 4 and SNR is 5 dB. Moreover, the number of subcodebooks is M = {1, 8} for LFOSDMA and the extended LFOSDMA. The number of initially selected users by the base station is S = {1, 2} for the proposed method. We came up with the results of the capacity based on Monte Carlo simulation. By Monte Carlo simulation, we generate each user's flat Rayleigh fading channel and AWGN. Based on these values, we calculate each user's SINR using (4), (12) or (15). Using the SINR and (6), we calculate sum capacity.
Firstly, the proposed method achieves a significantly higher sum capacity than LFOSDMA and the extended LFOSDMA for any number of users. This is because in the proposed method, the base station generates the beamforming vector for the initially selected user using full CSI, and allocates other users to the vectors that do not cause interference to the beamforming vector for the initially selected user. The sum capacity of LFOSDMA decreases for an increase of the number of subcodebooks where there is a small number of active users. On the other hand, the extended LFOSDMA improves the sum capacity on that of LFOSDMA for the small number of users, because the extended LFOSDMA guarantees that there is no unallocated beam in the selected subcodebook. However, for a large number of users, there is little difference in the sum capacity between LFOSDMA and the extended LFOSDMA, because LFOSDMA can sufficiently get the multiplexing gain since there is a large number of users. At K = 20, the capacity gain of the proposed method with respect to LFOSDMA with M = 1 is 2 bps/Hz and with respect to the extended LFOSDMA with M = 8 is 1 bits/Hz. At K = 100, the proposed method also improves the sum capacity of LFOSDMA and the extended LFOSDMA by 0.5 bps/Hz. In the result, the proposed method can achieve a significantly higher sum capacity than LFOSDMA and the extended LFOSDMA without a large increase in the amount of feedback bits and latency.
7.4 Cumulative distribution function
In this section, we show the cumulative distribution function (CDF) of the capacity on a peruser basis, because it is important for a system designer to consider this performance. We come up with the results based on the Monte Carlo simulation. Figure 6 compares the CDF of the proposed method with that of LFOSDMA and the extended LFOSDMA. The number of transmit antennas is N_{ t } = 4, SNR is 5 dB and the number of users is K = 50. In this simulation, we also randomly selected the initially selected users who feed their full CSIs back, and S affects the amount of feedback, but is not dependent on the number of transmit antenna N_{ t } .
Figure 6 shows that the proposed method has a higher variance of the capacity on a peruser basis than LFOSDMA and the extended LFOSDMA. All users of LFOSDMA and the extended LFOSDMA achieve the capacity between 1 and 3 bps/Hz/User. On the other hand, in the proposed method, the users achieve the capacity higher than or equal to those in LFOSDMA. In addition, the variance of the capacity in the proposed method is larger as well. These are because in the proposed method, the pivotal user can have a much higher capacity than the users in LFOSDMA and the extended LFOSDMA. In addition, for the selected users other than the pivotal user, the amount of mismatch between each user's channel and the selected beamforming vector is about the same as that in the conventional algorithms. Therefore, the proposed method achieves the improvement of the capacity for the whole of the system compared with LFOSDMA and the extended LFOSDMA without the loss of the capacity on a peruser basis, though the variance of the capacity on a peruser basis becomes large.
8 Conclusion
In this paper, we proposed a new orthogonal beamforming algorithm for the MIMO BC aiming to achieve high capacity performance for any number of users. In this algorithm, we do not use codebook, and the base station generates a unitary beamforming vector set using GramSchmidt orthgonalization using the beamforming vector for the pivotal user. Then, the pivotal user can use the optimal beamforming vector because of using analog value of the actual CSI. The proposed method increases the number of feedback bits and the amount of latency. For fair comparison about the amount of latency, we extend the algorithm of LFOSDMA to guarantee that the system never loses multiplexing gain. Finally, we compare the number of feedback bits, the amount of latency, and the sum capacity of the proposed beamforming algorithm with LFOSDMA and the extended LFOSDMA. We showed that the proposed method can achieve a significantly higher sum capacity than LFOSDMA and the extended LFOSDMA without a large increase in the amount of feedback bits and latency. In this paper, we adopt IEEE 802.20 codebook for the LFOSDMA, but there may exist optimal codebook for LFOSDMA. In addition, the high correlation among the users's channels may affect the capacity of the proposed method largely. We want to examine these point in our future research.
Abbreviations
 CDF:

cumulative distribution function
 CSI:

channel state information
 DPC:

dirty paper coding
 LFOSDMA:

SDMA with limited feedback
 MIMO:

multipleinput multipleoutput
 OSDMA:

opportunistic SDMA
 SDMA:

spacedivision multipleaccess
 SINR:

signaltointerferenceplusnoise ratio.
References
 1.
Viswanath P, Tse D: Sum capacity of the vector Gaussian broadcast channel and uplinkdownlink duality. IEEE Trans Inf Theory 2003, 49(8):191221. 10.1109/TIT.2003.814483
 2.
Yoo T, Goldsmith A: On the optimality of multiantenna broadcast scheduling using zeroforcing beamforming. IEEE J Sel Areas Commun 2006, 24(3):528541.
 3.
Shen Z, Chen R, Andrews JG, Heath RW Jr, Evans BL: Low complexity user selection algorithms for multiuser MIMO systems with block diagonalization. IEEE Trans Signal Process 2006, 54(9):36583663.
 4.
Costa M: Writing on dirty paper. IEEE Trans Inf Theory 1983, 29(3):439441. 10.1109/TIT.1983.1056659
 5.
Dimic G, Sidiropoulos ND: On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm. IEEE Trans Signal Process 2005, 53(10):38573868.
 6.
Serbetlli S, Yener A: Transceiver optimization for multiuser MIMO systems. IEEE Trans Signal Process 2004, 52(1):214226. 10.1109/TSP.2003.819988
 7.
Choi YS, Alamouti S, Tarokh V: Complementary beamforming: new approaches. IEEE Trans Commun 2006, 54(1):4150.
 8.
Sharif M, Hassibi B: On the capacity of MIMO broadcast channels with partial side information. IEEE Trans Inf Theory 2005, 51(2):506522. 10.1109/TIT.2004.840897
 9.
Choi W, Frenza A, Andrews JG, Heath RW Jr: Opportunistic space division multiple access with beam selection. IEEE Trans Commun 2007, 55(12):23712380.
 10.
Huang KB, Heath RW, Andrews JG: Performance of orthogonal beamforming for SDMA with limited feedback. IEEE Trans Veh Technol 2009, 58(1):152164.
 11.
IEEE 802.20 C802.200604, Part 12: Precoding and SDMA codebooks 2006.
 12.
Huang K, Heath RW, Andrews JG: Space division multiple access with a sum feedback rate constraint. IEEE Trans Signal Process 2007, 55(7):38793891.
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Matsumura, K., Ohtsuki, T. Orthogonal beamforming using GramSchmidt orthogonalization for multiuser MIMO downlink system. J Wireless Com Network 2011, 41 (2011). https://doi.org/10.1186/16871499201141
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Keywords
 Multiuser MIMO
 GramSchmidt orthogonalization
 Spacedivision multipleaccess (SDMA)