Opportunistic round robin scheduling for VBLAST systems over multiuser MIMO channels
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Abstract
In this paper, we study opportunistic round robin (ORR) uplink scheduling for vertical Bell Labs layered spacetime architecture (VBLAST) systems over multiuser multipleinput multipleoutput (MIMO) channels. The proposed ORR scheduling method is compared to greedy scheduling. In greedy scheduling, the base station selects the best user based on a certain criterion without any consideration for fairness. On the other hand, ORR scheduling guarantees full fairness and each user will be served by excluding the previous selected users from the competition in the next round. The selected user spatially multiplexes his data over the transmit antennas. This spatial multiplexing (SM) scheme provides high data rates, while multiuser diversity obtained from scheduling improves the performance of the uplink system. The results show the performance and capacity gains obtained by scheduling. The greedy scheduler captures full multiuser diversity. In contrast, the ORR scheduler provides substantial signaltonoise ratio (SNR) gains compared to round robin while guaranteeing full fairness to all users.
Keywords
Opportunistic round robin scheduling MIMO multiuser uplink scheduling VBLAST Spatial multiplexingIntroduction
In multiuser multipleinput multipleoutput (MIMO) wireless systems, optimizing the physical layer for each user does not necessarily optimize system performance nor takes advantage of statistical independence of fading channels for different users. Furthermore, users have different needs in terms of data rates, power constraints, and quality of service (QoS). These requirements make scheduling an important technique for optimizing the performance of communication systems and utilizing system resources efficiently. The wireless fading channels of users are usually independent. Thus, scheduling transmission to best users leads to a form of selection diversity known as multiuser selection diversity. In general, schedulers are designed to maximize system throughput and capacity or to minimize error rates. However, they should also provide fairness to users and minimize packet delays.
In singleinput singleoutput (SISO) systems, where the base station and each mobile have one antenna, it has been shown that selecting the user who has the maximum signaltonoise ratio (MaxSNR) maximizes the total information capacity of uplink systems [1]. Similar results are also found for downlink systems from the base station to the mobile unit [2]. This selection criterion is known as MaxSNR.
For multiuser MIMO channels, most of the studies are based on theoretical information capacity and on the downlink [3, 4, 5, 6], which is the broadcast channel from the base station to the mobile unit. Furthermore, it has been shown in [7] that spacetime block coding (STBC) and scheduling are not a good match. In fact, scheduling to a user with a single antenna outperforms scheduling using STBC. The reason is that STBC averages the fades, while scheduling tends to benefit from high peaks in the fading channel. In addition, multiuser diversity obtained from scheduling is much higher than the spatial diversity of STBC. Therefore, STBC inherent diversity does not add much benefits. On the other hand, spatial multiplexing (SM) schemes match perfectly with scheduling since they provide high data rates while scheduling provides multiuser selection diversity.
In addition, for MIMO systems, scheduling could be done to a single user or multiple users. Scheduling to multiple users, i.e., allowing more than one user to transmit or receive at the same time, is optimal in terms of maximizing system capacity and throughput [3, 4, 8]. That is because the degrees of freedom provided by the multiuser MIMO system are much larger than those of the single user selection case. In [3], downlink scheduling to multiple users improved the average throughput compared to a single user scheduling. Furthermore, the authors in [8] showed that the optimal scheduler should allocate all power to at most M_{R} users, where M_{R} is the number of receive antennas at the base station. Also, they found that the optimal power resource allocation is waterfilling in space and time. In [4], the authors found that multiuser scheduling reduces average delays experienced by users compared to a singleuser scheduling.
For spatially multiplexed systems, greedy multiuser scheduling over MIMO systems was studied in [9], where we proposed and compared the performance of several user selection criteria for uplink vertical Bell Labs layered spacetime architecture (VBLAST) systems. Later in [10], the authors investigated user selection criteria that minimized the pairwise error performance for optimal and suboptimal lattice reduction (LR)based MIMO detectors. In addition, a lowcomplexity user selection scheme with an iterative lattice reduction algorithm was proposed in [11], where the authors showed that their proposed algorithm provided comparable performance with the combinatorial approaches with much lower complexity. The effect of imperfect channel state information (CSI) was investigated in [12], where they showed that their proposed robust multiuser MIMO scheduling improved the system average throughput significantly.
To overcome the drawback of the greedy scheduler in [9] and to guarantee full fairness to all users, we propose in this paper to use opportunistic round robin (ORR) scheduling. In addition, we compare the performance of greedy and ORR scheduling schemes. We demonstrate the fundamental tradeoff between performance and fairness in multiuser scheduling. The greedy scheme selects the best user without considering fairness among users. On the other hand, the ORR algorithm selects the best user first based on a scheduling criterion, then this selected user will be excluded from the search in the next round until all users are served. The result of this work shows that the ORR scheduler provides SNR gains compared to round robin scheduling while still providing full fairness to all users.
System model
In this paper, the base station compares the MIMO channels of all users and selects the best user one at a time based on a certain criterion. In general, for MIMO multiuser scheduling, the best set of transmit antennas could be selected, and this set might belong to more than one user. However, this approach requires more feedback and synchronization than a single user selection constraint. In addition, for MIMO multiuser scheduling, the scheduler should select the best set of M_{T} transmit antennas out of K M_{T} antennas. Thus, the search space will be huge, and suboptimal search algorithms should be proposed. However, this is out of the scope of this paper where we are focusing on analyzing and comparing the performance of user selection criteria.
where y_{ k } is an M_{R}×1 received vector, H_{ k } is an M_{R}×M_{T} MIMO channel matrix for the k th user, x_{ k } is an M_{T}×1 transmitted symbols from user k, and η_{ k } is an M_{R}×1 i.i.d complex AWGN vector of zero mean and variance N_{0}/2 per dimension.
Channel state information (CSI) is assumed to be known only at the receiver (base station) for all users. Based on a selection criterion, the receiver compares all users and selects the best user at that time. Then, it informs the best user to transmit through a feedback channel.
Scheduling schemes
We compare in this study two scheduling schemes. They are greedy and opportunistic round robin. The greedy scheduler selects the best user based on a user selection criterion. This scheduler does not guarantee fairness in the sense that users with weak channel conditions will not be served. However, if all users have same channel statistics and strict power control is applied, then greedy scheduling will be fair on average. On the other hand, the ORR scheduler guarantees fair scheduling to all users. It selects the best user first based on a scheduling criterion. In the next round, this selected user is excluded from the search and only the remaining users are considered. This procedure is repeated until all users are served.
Optimal MIMO user selection criteria
VBLAST user selection criteria
VBLAST [14] is a practical MIMO architecture that spatially multiplexes transmitted data over multiple transmit antennas. Data transmitted from each antenna is called a layer of information. At the receiver, a serial interference nulling and cancellation algorithm is used to detect each layer. Although VBLAST is a full spatial multiplexing scheme, it has poor energy performance because of the lack of spatial diversity. The diversity order of VBLAST is M_{R}−M_{T}+1 [15]. Thus, when the number of receive and transmit antennas is equal, there will be no spatial diversity. Therefore, VBLAST makes a good match with scheduling since multiuser diversity will improve the system performance significantly.
VBLAST applies successive nulling and cancellation algorithm to detect the spatially multiplexed data. The nulling part could be done by zero forcing (ZF) or minimum mean squared error (MMSE). For a single user system, the authors in [15] showed that both ZF and MMSE provide the same spatial diversity order. However, MMSE provides SNR gains compared to ZF. We investigate in this paper the effect of ZF and MMSE nulling matrices on multiuser systems with scheduling. As will be shown in the ‘Simulation Results’ section, the SNR gains provided by MMSE diminishes with multiuser selection diversity.
Therefore, the weakest layer that determines the capacity of VBLAST is the one with the largest norm of the ZF projection row. Let ${w}_{k}={\text{max}}_{n=1,2,\dots ,{M}_{\text{T}}}\parallel {W}_{\text{ZF},n}^{k}{\parallel}^{2}$ be the largest projection value for user k, then the scheduler that maximizes VBLAST capacity will select the user with minimum w_{ k }. In other words, the capacity maximization scheduling for VBLAST selects the user with largest postprocessing SNR of its weakest layer.
where λ_{max} and λ_{min} are the largest and smallest eigenvalues of ${\mathbf{H}}_{k}{\mathbf{H}}_{k}^{H}$, respectively. The eigenspread gives insight into the orthogonality of the channel. The smaller the value of s, the closer the matrix is to be orthogonal. The minimum value of s is 1, and it occurs when the channel matrix is orthogonal.
Thus, selecting the largest ρ_{min} means selecting a large ρ_{max}, which measures the norm of H_{ k } and hence the power, and a small eigenspread (s). We will refer to this scheduler as MaxMinSV.
In addition, we compare the above user selection criteria with the product of eigenvalues (PEV) of ${\mathbf{H}}_{k}{\mathbf{H}}_{k}^{H}$, as proposed in [17] for multiuser downlink MIMO scheduling. We refer to this criterion as MaxPEV.

MaxVBLASTCapc: select the user with mink=1,2,…,K{w_{ k }}, where ${w}_{k}={\text{max}}_{n=1,2,\dots ,{M}_{\text{T}}}\phantom{\rule{0.3em}{0ex}}\left\{\parallel {W}_{\text{ZF},n}^{k}{\parallel}^{2}\right\}$ and W_{ZF,n} is defined in (8).

MaxMinSV: select the user with maximum minimum singular value of H_{ k }.

MaxPEV [17]: select the user with maximum product of eigenvalues of ${\mathbf{H}}_{k}{\mathbf{H}}_{k}^{H}$.

MinES: select the user with minimum eigenspread of ${\mathbf{H}}_{k}{\mathbf{H}}_{k}^{H}$ as defined in (10).

MaxSNR [1]: select the user with maximum Frobenius norm of ${\mathbf{H}}_{k}\phantom{\rule{0.3em}{0ex}}\left(\text{trace}\left({\mathbf{H}}_{k}{\mathbf{H}}_{k}^{H}\right)\right)$.

RR: round robin scheduling allows each user to transmit in a time division fashion.
Capacity bounds on scheduling criteria
where $\underset{i}{\text{min}}\parallel {W}_{\text{ZF},n,i}^{k}{\parallel}^{2}$ is the minimum i th diagonal value of ${\left({\mathbf{H}}_{k,n}^{H}{\mathbf{H}}_{k,n}\right)}^{1}$ of the n th layer of the k th user. In this section, we will find bounds that are independent of n and i for each scheduling technique.
MaxMinSV
where U is a unitary matrix with orthonormal eigenvectors and Λ_{k,n} is the diagonal matrix of eigenvalues $\left[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{{M}_{T}}\right]$. Another approach is to use QR decomposition as done in [18].
According to the inclusion principle for matrices, the minimum value of λ_{min} occurs at n=1 and the minimum vale of λ_{max} occurs at n=M_{T}.
We will use this inequality to establish lower bounds on MaxMinSV.
MinES
where α>0 and β>0.
MaxSNR
Again from the inclusion principle, the minimum of the trace occurs at n=M_{T}.
Simulation results
BER performance
Effect of ZF and MMSE nulling matrices for VBLAST detection
Outage capacity
Conclusions
This paper proposed an opportunistic round robin (ORR) scheduling algorithm for uplink VBLAST users over multiuser MIMO channels. The proposed algorithm overcomes the drawbacks of greedy scheduling algorithm and provides full fairness to all users. In addition, error rate performance and outage capacities for user selection criteria were analyzed and compared for both ORR and greedy scheduling. User selection criteria for ORR perform very close to each other, while there are substantial differences in the greedy scheduling algorithm. This suggests that for ORR scheduling, it is sufficient to use suboptimal user selection criteria such as MaxMinSV.
The main results of this paper show the SNR and capacity gains obtained by ORR scheduling while providing full fairness to all users. Compared to round robin scheduling, the ORR scheduler provides around 12dB gain at BER = 10^{−3} at 10 users. In addition, there is a fundamental tradeoff between performance and fairness for ORR and greedy scheduling algorithms. The ORR scheduler does not capture the whole available multiuser diversity, but it provides full fairness to all users. On the other hand, greedy scheduling provides substantial performance improvements while there is no consideration for user fairness. Thus, ORR scheduling is an excellent candidate for nextgeneration high data rate system to satisfy certain quality of experience for all users.
Notes
Acknowledgements
The author would like to acknowledge the support provided by King Fahd University of Petroleum and Minerals (KFUPM) and King Abdulaziz City for Science and Technology (KACST) through the Science and Technology Unit at KFUPM for funding this work through project number 09ELE7814 as part of the National Science, Technology and Innovation Plan.
Supplementary material
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.