 Research
 Open Access
Feedback channel designs for fair scheduling in MISO–OFDMA systems
 Berna Özbek^{1}Email author and
 Didier Le Ruyet^{1}
https://doi.org/10.1186/168714992012220
© Özbek and Le Ruyet; licensee Springer. 2012
 Received: 1 December 2011
 Accepted: 14 July 2012
 Published: 17 July 2012
Abstract
In the next generation of wireless communication, adaptive resource allocation techniques will play an important role to improve quality of service and spectral efficiency. In order to employ adaptive fairness scheduling for multipleinput singleoutput (MISO) orthogonal frequencydivision multiple access (OFDMA), the channel state information (CSI) belonging to all users is required at the base station. However, the amount of feedback increases with the number of users, transmit antennas and subchannels. Therefore, it is important to perform a user selection at the receiver side without cooperation among the users and to quantize the CSI. In this article, the reduced feedback channel designs are examined for MISO–OFDMA systems while providing fairness between the users. In order to reduce the feedback rate, we choose the users considering their norm and orthogonality properties as well as their location in the cell. In order to limit the feedback rate, channel direction information is quantized by designing a specific codebook thanks to the proposed criterion. We obtain an expression to determine the amount of required feedback information for MISO–OFDMA systems to support more than one beam per subchannel for a given number of users, subchannels and transmit antennas. The performance results of the reduced feedback channel designs are evaluated for fair scheduling in wireless channels.
Keywords
 Orthogonal Frequency Division Multiplex
 Channel State Information
 OFDMA System
 Feedback Channel
 Multiuser Diversity
Introduction
In future wireless radio networks, adaptive resource allocation can have a significant role to improve the spectral efficiency for multiuser orthogonal frequency division multiplexing (OFDM) systems with multiple transmit and/or receive antennas by exploiting a large number of degrees of freedom in space, frequency, and time. OFDM inherits its superiority of mitigating multipath fading to maximize throughput. Besides, in a multipleantenna system with the help of precoding techniques, it is possible to increase the spectral efficiency by multiplexing multiple users on the same subchannel. For orthogonal frequencydivision multiple access (OFDMA) systems, the sum rate can be maximized by assigning each subcarrier to the user having the best channel gain. However, for multiuser multipleinput singleoutput (MISO)–OFDM systems, the sum rate is maximized by choosing the optimal set of cochannel users for each subchannels[1, 2]. It is also important to satisfy all users’ requirements even if the users are far from the base station (BS) by allocating the clusters and beams fairly. Therefore, not only the optimal set but also the fairness issue is considered by employing proportional fair scheduling algorithms (PFS) for MISO–OFDMA systems[3].
In order to achieve the gain of MISO–OFDMA systems, the channel state information (CSI) of all subchannels from all users for all antennas is required at the transmitter side. This causes a high feedback load and a sophisticated resource allocation algorithm at the BS. In order to design an efficient MISO–OFDMA system, the selection of users and the quantization of their CSI using efficient codebooks at the receiver side is needed to fed back the users’ CSI to the BS through a reduced rate feedback channel.
In order to design reduced feedback channels, user selection algorithms at the receiver side are performed by deciding the number of users that are fed back their quantized CSI by using codebooks known at both the transmitter and the receiver sides[4]. For multiuser MISO singlecarrier systems with linear precoding, the semiorthogonal user selection algorithm to reduce the feedback load by combining the classical norm criterion and with a criterion based on the orthogonality between the cochannel users has been presented in[5, 6]. The codebook design for narrowband single user communication system has been well studied in[7]. The extension to the OFDM case has been considered in[8, 9]. For multiuser multipleinput multiple output (MIMO) with zeroforcing (ZF) beamforming, the codebook design and quantization algorithm have been proposed in[10] to enable high resolution quantization. In[11], an adaptive limited feedback linear precoding technique for temporally correlated MIMO channels has been presented by performing a differential feedback where a perturbation added to the previous precoder for adaptation to the time correlation structure. Limited feedback using a polarcap differential codebook which utilizes the temporal correlation in MISO channels has been presented in[12].
A twostage feedback scheme has been presented in[13] where in the first step a coarse estimates of all user channels are feedback to the BS then in the second step only N_{ t } users are selected to feed back more accurate channel quantization. In[14], it has been underscored the tradeoff between getting coarse channel feedback from large number of users and providing multiuser diversity gain versus getting highquality channel feedback from a low number of users. In[15], two stages for scheduling process based on partial probing of the users has been presented, in order to reduce the feedback, at the second stage the probing process stops and only the remaining users are requested to feedback their channel quality.
A scheduler at the BS usually does not schedule users on their weakest clusters. Hence, the amount of feedback information can be reduced by letting each user feed back information only about its strongest clusters. Following this phenomenon, in order to achieve reduction on the feedback load without sacrificing performance too much, the socalled clustered Sbest criterion has been proposed for OFDMA systems in[16]. This criterion is based on the clusters where adjacent subcarriers are grouped and only the CSI related to the strongest S clusters of each user are fed back to the transmitter. In[17], a quantization method for OFDMA systems has been proposed to reduce the amount of feedback bits by determining subchannel block size and feedback periodicity according to users’ channel conditions. This scheme has been extended to MISO–OFDMA systems in[18] by using spacefrequency matrix. For OFDMA systems, a feedback reduction algorithm that considers feedback efficiency as a feedback decision metric instead of the received signaltonoise ratio (SNR) has been examined in[19] to increase the fairness between the users.
In this article, we propose reduced feedback channel designs for fair scheduling where the users are uniformly distributed in the cell for MISO–OFDMA systems. Both user selection and quantization algorithms dedicated to the selected users are applied to solve the major problem which arises from the fact that the total feedback load increases with the number of users, subchannels and antennas. The clustered Sbest and the combined user selection algorithms with a codebook design thanks to the semiorthogonal criterion are presented by choosing adaptively the number of feedback clusters depending on the location of the users and/or the number of users in the cell. These algorithms allow a precoding scheme creating more than one beam to schedule more than one user for each cluster by providing fairness between the users. We also provide the amount of required feedback information to create more than one beam for a given number of users, subchannels and transmit antennas.
This article is organized as follows: In Section “System model”, the system model for MISO–OFDMA over wireless channels is described. Then, the reduced feedback channel designs are presented for fair scheduling in Section “The proposed designs”. The performance results are demonstrated in Section “Performance results” considering perfect and quantized CSI in wireless channels. Section “Conclusion” draws the conclusion of the article.
System model
where${H}_{k,q,t}^{n}$ is the channel coefficient from the t th transmit antenna to the k th user for the q th cluster at frame n.
where N_{ Q } is the number of subcarriers in one cluster and calculated as N_{ Q }=M/Q with M is the total number of subcarriers in a OFDM symbol.
where${h}_{k,\ell ,t}^{n}$ is channel coefficient in time domain belonging to k th user, t th antenna and ℓ th path and L_{ t } is the number of multipath components.
The channel coefficients belonging to subcarriers for each user and each antenna are obtained by applying the Fourier Transform to the channel vector in Equation (4).
where${A}_{k,q,b}^{n}$ is a binary variable that indicates cluster q is allocated to user k for beam b.
which includes N_{ t }elements as${\mathit{W}}^{n}\left({\mathbb{S}}_{q}\right)={\left[{\mathbf{W}}_{1}^{n}\left({\mathbb{S}}_{q}\right){\mathbf{W}}_{2}^{n}\left({\mathbb{S}}_{q}\right)\phantom{\rule{0.3em}{0ex}}\dots \phantom{\rule{0.3em}{0ex}}{\mathbf{W}}_{{N}_{t}}^{n}\left({\mathbb{S}}_{q}\right)\right]}^{T}$ where${\mathbf{W}}_{b}^{n}\left({\mathbb{S}}_{q}\right)$ is the precoding vector for b th beam and q th cluster with the dimension of N_{ t }×1.
where${\mathit{P}}_{q}=\text{diag}\left(\sqrt{{P}_{T}/\left(Q{N}_{t}\right)},\dots ,\sqrt{{P}_{T}/\left(Q{N}_{t}\right)}\right)$ denoting that the total transmit power P_{ T } is equally shared between the clusters and beams and${\mathbf{X}}^{n}\left({\mathbb{S}}_{q}\right)\in {\mathcal{C}}^{{N}_{t}\times 1}$ is the transmitted vector from the BS.
Our objective is to maximize the average sum rate while keeping fairness between the users by optimizing both the cluster and beam allocation[3, 22],${\mathit{A}}^{n}=[{\mathit{A}}_{1}^{n},{\mathit{A}}_{2}^{n},\dots ,{\mathit{A}}_{Q}^{n}]$. In order to construct${\mathit{A}}_{q}^{n}$, all vectors of${\mathbf{A}}_{k,q}^{n}={[{A}_{k,q,1}^{n},{A}_{k,q,2}^{n},\dots ,{A}_{k,q,{N}_{t}}^{n}]}^{T}$ are stacked column by column.
where N_{0} is the power spectral density of additive white Gaussian noise (AWGN) and B is the total available bandwidth.
where t_{ c } is the average window size.
The proposed designs
In order to exploit the multiuser diversity, the users need to feedback their CSI to the BS. The feedback rate can be reduced by letting each user send their CSI associated only to a subset of clusters after applying quantization. We present four different algorithms to perform user selection at the receiver side and a quantization method thanks to the properties of the semiorthogonal criterion.
P1: The clustered Sbest criterion
For the clustered Sbest criterion, each user selects independently a set of${\mathbb{S}}_{k}^{n}$ composed of the S clusters with the highest channel norm$\parallel {\mathbf{H}}_{k,q}^{n}\parallel $. Then, each user fed backs only its CSI associated to the selected clusters to the BS.
Then, for each cluster q, the BS selects the set${\mathbb{S}}_{q}^{n}$ from the set${\mathbb{T}}_{q}^{n}$ to maximize the sum rate in Equation (14).
Since the total feedback rate is proportional to KS, it is reasonable to adjust S in function of K according to a desired function S=f(K).
The case of N_{ t }=1 has been examined in[16] for OFDMA systems with single antenna. In this article, we extend it to the MISO–OFDMA systems considering the case of N_{ t }=2 to allocate more than one beam for each cluster.
The calculation of P_{ K }(i) in Equation (18) is given in the Appendix.
P2: The combined criterion
The clusters can be selected based not only on channel norm/quality but also on channel direction information (CDI)[23]. Therefore, for each cluster, N_{ t } random orthonormal vectors ϕ_{b,q}(N_{ t }×1), b=1,…,N_{ t } are generated.
where${\stackrel{~}{\mathbf{H}}}_{k,q}^{n}=\frac{{\mathbf{H}}_{k,q}^{n}}{\u2225{\mathbf{H}}_{k,q}^{n}\u2225}$ is the normalized channel vector of the user k and cluster q.
Following the definition of spherical cap with the center ϕ_{b,q} and square radius ϵ, the constructed open set corresponds to semiorthogonal neighborhood set of any orthonormal vector. Then, it is possible to calculate that the normalized channel vector is in the open set or not according to chordal distance metric.
where${{\mathbb{T}}^{\prime}}_{q}^{n}$ is the set of semiorthogonal users for cluster q.
According to Mukkavilli et al.[24], the number of clusters which satisfy ϵ criterion for each user is calculated approximately as${N}_{t}{\u03f5}_{\mathrm{th}}^{{N}_{t}1}$. Therefore, the choice of ϵ is critical since it is directly relative to the total number of clusters per user. Consequently, it should be guaranteed that at least S clusters which satisfy the semiorthogonal criterion for each user are selected by adjusting the ϵ parameter properly.
Then, each user selects S best clusters in terms of channel quality/norm from the set${{\mathbb{T}}^{\prime}}_{k}^{n}$ and constructs a set${\mathbb{S}}_{k}^{n}$. From this set, for each q, we can obtain the set${\mathbb{T}}_{q}^{n}$ using Equation (16).
Adaptive P1: Adaptive clustered Sbest criterion
In order to further reduce the feedback rate, the number of feedback clusters,$\left{\mathbb{S}}_{k}^{n}\right={S}_{k}^{n}$, is adjusted adaptively according to the location of the users in the cell. Since the users that are far from BS have lowest rate per cluster, they require more clusters for fair scheduling. Therefore, the purpose of the adaptive feedback algorithm is to give more scheduling opportunities to the users far from the BS by providing more feedback clusters.
${Z}_{k}^{n}$ is proportionally changed according to the location of the users where the farthest user will required more clusters than the nearest user for proportionally fair scheduling. Since${p}_{k}^{n}$ is proportionally to${Z}_{k}^{n}$, the farthest user will feedback more clusters than the nearest user.
After that, each user generates a binomial random variable with parameters Q and${p}_{k}^{n}$ to determine the number of feedback clusters,${S}_{k}^{n}$. Then, as described in P1 algorithm, the set${\mathbb{S}}_{k}^{n}$ which includes the best${S}_{k}^{n}$ clusters of user k is constructed.
Adaptive P2: adaptive combined criterion
The adaptive combined criterion is described as follows:
For each user k:

As described in Adaptive P1 algorithm, the required number of feedback clusters,${S}_{k}^{n}$, is calculated depending on the instantaneous number of allocated clusters.

Choose the corresponding epsilon value,${{\u03f5}^{\prime}}_{k}^{n}$ as well as the number of feedback cluster,${{S}^{\prime}}_{k}^{n}$.

The set of the predefined number of clusters and the corresponding epsilon values are given, respectively:$\begin{array}{ll}\mathbf{C}\phantom{\rule{0.3em}{0ex}}& =\phantom{\rule{0.3em}{0ex}}\{{C}_{1},{C}_{2},\dots ,{C}_{T}\}\phantom{\rule{2em}{0ex}}\\ \mathbf{E}\phantom{\rule{0.3em}{0ex}}& =\phantom{\rule{0.3em}{0ex}}\{{e}_{1},{e}_{2},\dots ,{e}_{T}\}\phantom{\rule{2em}{0ex}}\end{array}$

Find the index of the closest value:${t}^{\ast}=\text{arg}min{C}_{t}{S}_{k}^{n},\phantom{\rule{1em}{0ex}}t=1,2,\dots ,T$(25)

Set the parameters:$\begin{array}{ll}{{S}^{\prime}}_{k}^{n}\phantom{\rule{0.3em}{0ex}}& =\phantom{\rule{0.3em}{0ex}}{C}_{{t}^{\ast}}\phantom{\rule{2em}{0ex}}\\ {{\u03f5}^{\prime}}_{k}^{n}\phantom{\rule{0.3em}{0ex}}& =\phantom{\rule{0.3em}{0ex}}{e}_{{t}^{\ast}}\phantom{\rule{2em}{0ex}}\end{array}$(26)

According to these parameters, the semiorthogonal criterion described in P2 algorithm is performed and the set${\mathbb{S}}_{k}^{n}$ which includes${{S}^{\prime}}_{k}^{n}$ best clusters for user k is obtained.
The codebook design
is maximized. As in the i.i.d., case for CDI, we use a practical vector quantization scheme, namely the generalized LloydMax algorithm to design these local packings.
where U_{rot} is the unitary rotation matrix.
The users selected by the combined criterion feed back log_{2}(N_{ c }) bits corresponding to the codebook index. In addition to that, it is necessary to feedback log_{2}(N_{ t }) bits for the index of the vector ϕ_{b,q}. Consequently, for a codebook size of N_{ c }, log_{2}(N_{ c }×N_{ t }) bits are necessary to quantify the CDI for P2 algorithm while log_{2}(N_{ c }) bits are required for P1 algorithm for each cluster. In terms of complexity, predefined tables can be used for generation of N_{ t }random orthogonal vectors in practical systems.
PFS algorithm
The FI ranges between 0 (no fairness) and 1 (perfect fairness) in which all users would achieve the same data rate.
After applying user selection algorithms and quantization at the receiver side as presented in the previous sections, the CSI of the selected users is feedback through the feedback channel. Then, the users’ CDI and CQI are available at the BS to perform PFS by choosing the user set for each cluster. Since the search space becomes${2}^{\left{\mathbb{T}}_{q}\right}$ instead of 2^{ K }with the advantage of the user selection algorithms, the complexity of the resource allocation is significantly reduced at the BS.

For each cluster q:

Set${A}_{k,q,b}^{n}=0$ for k = 1,2,…,K and b=1,2,…,N_{ t }.

Construct$\left(\begin{array}{l}\left{T}_{q}\right\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{N}_{t}\end{array}\phantom{\rule{0.3em}{0ex}}\right)$ sets as${\mathcal{S}}_{q}^{b,i};b=1,2,\dots ,{N}_{t}$ from the set of${\mathcal{T}}_{q}$.For i=1,2,…T_{ q }:

Calculate the user rate for ${k}^{\prime}\in {\mathcal{S}}_{q}^{b,i};{k}^{\prime}=1,2,\dots ,K;b=1,2,\dots ,{N}_{t}$:${\left[{R}_{{k}^{\prime},q}^{n}\right]}^{b,i}=\underset{2}{log}(1+{\mathbf{SINR}}_{{k}^{\prime},q,b}^{n})$(29)

Choose the allocation pattern that maximizes the sum weighted rate as:${i}^{\ast}=\text{arg}\underset{i}{max}\sum _{b=1}^{{N}_{t}}\frac{{\left[{R}_{{k}^{\prime},q}^{n}\right]}^{b,i}}{{\left[\overline{{T}_{{k}^{\prime}}^{n}}\right]}^{\alpha}}$(30)

Update ${A}_{{k}^{\prime},q,{b}^{\prime}}^{n}=1$ for ${k}^{\prime}\in {\mathcal{S}}_{q}^{{b}^{\prime},{i}^{\ast}};{b}^{\prime}=1,2,\dots ,{N}_{t}$.

Update the weighted average data rate according to Equation (15).

End.
Performance results
We obtain the performance results to illustrate the benefits of the reduced rate feedback channels in a singlecell MISO–OFDMA system with two transmit and one receive antennas through wireless channels. The users are uniformly distributed in a cell with a diameter of 750 m. The transmitted power and the noise density power are set at 43.10 dBm and −174 dBm/Hz, respectively. The path loss model is L_{ p }= 128.1 + 37.6log_{10}(d(km)) dB and the wireless channel is modeled using 3GPPTU. The bandwidth, the carrier frequency and the number of clusters are selected 10 MHz, 2.4 GHz and 48 with a velocity of 30 km/h. The clusters are grouped into 18 subcarriers. Assuming the slot duration is 100 ms, the feedback information is provided every 1 ms. The parameter α is chosen as 2 with t_{ c }=100 for the PFS algorithm.
The parameters for the case of L=N_{ t }=2, η=0.95, P_{obj}=0.1 at the spectral usage=0.99
K  10  20  30  40  50 

S  30  15  10  8  6 
ϵ  0.4  0.2  0.15  0.125  0.1 
e  2.2  2.4  2.6  2.8  3 
The average number of feedback users per cluster
K  10  20  30  40  50 

Full  10  20  30  40  50 
Fixed  6.25  6.25  6.25  6.67  6.25 
Adaptive  4.15  4.48  4.74  5.12  5.51 
Conclusion
In this article, we have examined efficient algorithms to design reduced rate feedback channel for MISO–OFDMA systems. We have presented fixed and adaptive cluster Sbest and the combined semiorthogonal criterion by employing more than one user for each cluster. According to the number of active users in cell and a given target, we have obtained the calculations of the number of feedback clusters for OFDMA with multiple antennas. The quantization error on the channel direction has been reduced since the CDI codebooks are designed using a local packing by taking into account the users’ direction. It has been illustrated that the proposed algorithms in a quantized reduced feedback link improve sum capacity significantly while providing fairness among the users in wireless channels. It has been also shown that the adaptive algorithm adjusts the number of selected clusters according to the location of the users and achieve the same fairness performance while further reducing the feedback link load. The proposed reduced feedback designs will be extent to multicell MISO–OFDMA networks as a future work.
Appendix
We define respectively U_{ k }and V_{ k } as the number of clusters having CSI of only one user and more than one user when k users are fed back their CSI to the BS. For the case N_{ t } = 2, we compute the probability that v different clusters are sending back less than N_{ t } users. It is assumed that each user feds back S clusters among the Q clusters. The fraction of clusters that have at least N_{ t } users’ CSI is$\u016a=({U}^{\prime}/Q)$ where${U}^{\prime}=\left\bigcap _{q=1}^{Q}{\mathbb{S}}_{q}\right$.
Otherwise, Pr(U_{ k }= A;V_{ k }= CU_{k−1} = B;V_{k−1}= D) =0.
where P_{ K }(i) denotes the i th element of P_{ K }.
Declarations
Acknowledgements
This research was supported by the Marie Curie Intra European Fellowship within the 7th European Community Framework Programme as a part of the INTERCELL project under the contract number PIEFGA2009255128.
Authors’ Affiliations
References
 Zhang YJ, Letaif KB: An efficient resource allocation scheme for spatial multiuser access in MIMO/OFDM system. IEEE Trans. Commun 2005, 53: 107116. 10.1109/TCOMM.2004.840666View ArticleGoogle Scholar
 Maciel TF, Klein A: A resource allocation strategy for SDMA/OFDMA systems. Proc. of 16th IST Mobile and Wireless Communications Summit 2007.Google Scholar
 Maciel TF, Klein A: On the performance, complexity and fairness of suboptimal resource allocation for multiuser MIMOOFDMA systems. IEEE Trans. Veh. Technol 2010, 59: 406419.View ArticleGoogle Scholar
 Love DJ, Heath RW, Lau VKN, Gesbert D, Rao BD, Andrews M: An overview of limited feedback in wireless communication systems. IEEE J. Sel. Areas Commun. (JSAC) 2008, 26: 13411365.View ArticleGoogle Scholar
 Swannack C, UysalBiyikoglu E, Wornell GW: Finding NEMO: near mutually orthogonal sets and applications to MIMO broadcast scheduling. Proc. of Int. Conf. Wireless Networks, Communnications, Mobile Computing (WIRELESSCOM) 2005.Google Scholar
 Le Ruyet D, Khanfir H, Ozbek B: Performance of semiorthogonal user selection for multiuser MISO systems. Proc. of IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) 2008.Google Scholar
 Zhou S, Wang Z, Giannakis G: Performance analysis for transmitbeamforming with finiterate feedback. Proc. of 38th Conf. on Info. Sciences and Systems 2004.Google Scholar
 Choi J, Heath RW: Interpolation based transmit beamforming for MIMOOFDM with limited feedback. IEEE Trans. Signal Process 2005, 53: 41254135.MathSciNetView ArticleGoogle Scholar
 Pande T, Love DJ, Krogmeier JV: Reduced feedback MIMO–OFDM precoding and antenna selection. IEEE Trans. Signal Process 2007, 55: 22842293.MathSciNetView ArticleGoogle Scholar
 Heath RW, Wu T, Soong ACK: Progressive refinement of beamforming vectors for highresolution limited feedback. EURASIP J. Adv. Signal Process 2009, 2009: 13 (Article ID 463823).Google Scholar
 Kim T, Love DJ, Clerckx B: MIMO systems with limited rate differential feedback in slowly varying channels. IEEE Trans. Commun 2011, 59: 11751189.MathSciNetView ArticleGoogle Scholar
 Choi J, Clerckx B, Lee N, Kim G: A new design of polarcap differential codebook for temporally/spatially correlated MISO channels. IEEE Trans. Commun 2012, 11: 703711.Google Scholar
 Zakhour R, Gesbert D: A twostage approach to feedback design in MUMIMO channels with limited channel state information. Proc. IEEE Personal, Indoor and Mobile Radio Commun. Symp (PIMRC) 2008.Google Scholar
 Ravindran N, Jindal N: Multiuser diversity vs. accurate channel feedback for MIMO broadcast channels. Proc. IEEE Personal, Indoor and Mobile Radio Commun. Symp (PIMRC) 2008.Google Scholar
 AlHarthi Y: Opportunistic multiuser scheduling with reduced feedback load. Eur. Trans. Telecommun 2010, 21: 299311. 10.1002/ett.1403View ArticleGoogle Scholar
 Svedman P, Wilson Sk, Cimini LJ, Ottersten B: Opportunistic beamforming and scheduling for OFDMA systems. IEEE Trans. Commun 2007, 55: 941952.View ArticleGoogle Scholar
 Majjigi V, Agarwal R, Cioffi J: User profiling: a method for limited feedback in OFDMA systems. Proc. of IEEE International Communications Conference (ICC) 2009.Google Scholar
 Trivellato M, Tomasin S, Benvenuto N: On channel quantization and feedback strategies for multiuser MIMOOFDM downlink systems. IEEE Trans. Commun 2009, 57: 26452654.View ArticleGoogle Scholar
 Jeon J, Son K, Lee HW, Chong S: IEEE Trans. Veh. Technol. 2010, 59: 160169.Google Scholar
 Spencer QH, Swindlehurst AL, Haardt M: Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. Signal Process 2004, 52: 461471. 10.1109/TSP.2003.821107MathSciNetView ArticleGoogle Scholar
 Yoo T, Goldsmith A: On the optimality of multiantenna broadcast scheduling using zeroforcing beamforming. IEEE J. Sel. Areas Commun. (JSAC) 2006, 24: 528541.View ArticleGoogle Scholar
 Papoutsis VD, Fraimis IG, Kotsopoulos SA: Fairnessaware user selection and resource allocation in MISOOFDMA. Eur. Trans. Telecommun 2010, 21: 568574.View ArticleGoogle Scholar
 Ozbek B, LeRuyet D: Reduced feedback designs for SDMAOFDMA systems. Proc. of IEEE International Communication Conference (ICC) 2010.Google Scholar
 Mukkavilli KK, Sabharwal A, Erkip E, Aazhang B: On beamforming with finite rate feedback in multiple antenna systems. IEEE Trans. Inf. Theory 2003, 49: 25622579. 10.1109/TIT.2003.817433MathSciNetView ArticleGoogle Scholar
 Jain R, Chiu DM, Hawe WR: A quantitative measure of fairness and discrimination for resource allocation shared computer systems. Digital Equipment Corporation technical report TR301 1984.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.