Research Article | Open Access

Feng Zhao, Wen Wang, Hongbin Chen, "Interference Alignment and Fairness Algorithms for MIMO Cognitive Radio Systems", *Mobile Information Systems*, vol. 2015, Article ID 907142, 8 pages, 2015. https://doi.org/10.1155/2015/907142

# Interference Alignment and Fairness Algorithms for MIMO Cognitive Radio Systems

**Academic Editor:**Qilian Liang

#### Abstract

Interference alignment (IA) is an effective technique to eliminate the interference among wireless nodes. In a multiinput multi-output (MIMO) cognitive radio system, multiple secondary users can coexist with the primary user without generating any interference by using the IA technology. However, few works have considered the fairness of secondary users. In this paper, not only is the interference eliminated by IA, but also the fairness of secondary users is considered by two kinds of algorithms. Without losing generality, one primary user and secondary users are considered in the network. Assuming perfect channel knowledge at the primary user, the interference from secondary users to the primary user is aligned into the unused spatial dimension which is obtained by water-filling among primary user. Also, the interference between secondary users can be eliminated by a modified maximum signal-to-interference-plus-noise algorithm using channel reciprocity. In addition, two kinds of fairness algorithms, max-min fairness and proportional fairness, among secondary users are proposed. Simulation results show the effectiveness of the proposed algorithms in terms of suppressed interference and fairness of secondary nodes. What is more, the performances of the two fairness algorithms are compared.

#### 1. Introduction

With the rapid deployment of various wireless systems, the limited radio spectrum is becoming increasingly crowded. What is worse is that it is evident that most of the allocated spectrum experiences low utilization. Cognitive radio has been recently proposed as a potential solution for efficient utilization of the scarce radio resources [1]. The key feature of the cognitive radio is to allow a class of radio devices, called secondary users, to opportunistically access a licensed spectrum that is left unused by the primary user as long as the secondary users will not affect the operation of the primary user adversely. This improves the spectral efficiency greatly for allowing more users to coexist in the same frequency band. Due to the significantly increased channel capacity in MIMO systems, it has become a dominating technique in next generation wireless systems. It is thus quite natural to combine the MIMO and the cognitive radio techniques to achieve higher spectral efficiency. This technological combination results in the so-called cognitive MIMO radio [2].

Interference alignment (IA), a recently emerging idea for wireless networks, is an effective approach to manage interference. IA is a technique of signal construction that the interference casts overlapping shadows at the unintended receivers, while the desired signals can still be distinguished at the intended receivers free of interference [3]. Hence, a suitable precoding matrix can be found by which all the interference can be constrained into one-half of the signal spaces at each receiver, leaving the other half not interfering with the desired signal; then, the expected signal can be obtained using the simple zero-forcing method. The concept of IA has been introduced independently in [4–6]. What is more, the idea of IA for the -user interference channel has been introduced by Gomadam et al. in [6]. They have showed the degrees of freedom (DoF); that is, the interference-free signaling dimensions of this channel are much more than previously thought. For the sake of achieving IA, two kinds of iterative distributed algorithms, that is, minimum weighted leakage interference (MWLI) and maximum signal-to-interference-plus-noise (MSINR), were presented in [7] by exploiting the channel reciprocity. In [8], the authors have analyzed these two kinds of algorithms and compared the bit error rate performance of them. In cognitive MIMO radio systems, interference alignment is a new research direction in recent years. In [9], the interference alignment techniques in cognitive radio systems were introduced for the first time. This study is under cognitive MIMO interference network equipped with one primary user and one secondary user who have the same antennas in the transmitting and receiving nodes. Opportunistic interference alignment (OIA), a technique proposed to allow a single secondary user to exploit the unused spatial directions by the primary user, has received great attention recently. In [10], OIA for one primary user and multisecondary user network was proposed. The relation between the DoF and the number of antennas in cognitive radio systems was studied by author in [11]. Moreover, an imposed minimum weighted leakage interference (MWLI) algorithm in cognitive radio networks with multiprimary user and multisecondary user was proposed in [12].

In cognitive MIMO networks, some of works tried to maximize the throughput of users under the premise of sacrificing the fairness which uses some algorithms like [13]. In [13], game theory is introduced to maximize the throughput of users. The researches above did not consider the fairness of secondary users. However, fairness is an essential problem in cognitive MIMO systems. In recent years, two kinds of fairness are taken into consideration when the researchers study the throughput of users. On one hand, max-min fairness is a hotpot problem. The authors in [14, 15] have considered the max-min signal-to-interference-plus-noise ratio (SINR) for a MIMO downlink system. In [16], the max-min fairness has been considered in a multipair two-way relay system. In [17, 18], the authors have achieved fairness by allocating the resource in MIMO networks. In cognitive radio networks, the max-min fairness has also been considered in [19]. On the other hand, the proportional fairness was also studied by many authors. Max-min fairness aimed to maximize the fairness while proportional fairness was considered to achieve tradeoff between system throughput and fairness [20]. In [20, 21], proportional fairness has been considered in OFDM systems. In [22], the authors have proposed a Quality of Experience- (QoE-) based proportional fair scheduling considering network-wide users’ QoE maximization as well as fairness among users in the multicell OFDMA networks. Also, in [23], proportional fairness has been studied in virtual MIMO networks which have combined the advantages of proportional fairness and maximum rate rules. Moreover, proportional fairness has been studied in wireless mesh networks in [24] which derived the joint allocation of flow airtimes and coding rates that achieves the proportionally fair throughput allocation. In [25], the resource allocation problem with the proportional fair constraint condition based on quantised feedback for multiuser orthogonal frequency division multiplexing access system has been proposed. Besides, the proportional fair scheduling has been studied under imperfect channel state information (CSI) in [26]. However, few works have thought about the IA and fairness together. Next, we will solve this fairness problem in a cognitive MIMO network.

In this paper, we consider the case of multiple secondary users opportunistically exploiting the same frequency band utilized by a primary user. IA-based cognitive transmission schemes are developed that secondary users can exploit the unused spatial dimensions left by the primary user so that no interference is generated at the primary receiver. What is more, no interference from the primary transmitter or other secondary transmitters is generated to each of the secondary receiver [9]. IA is performed by us for the secondary links by a modified version of MSINR to find the precoders and reception filters [13]. After the interference alignment, we consider the max-min fairness and proportional fairness among secondary users. The rest of the paper is organized as follows. In Section 2, the system model and the IA problem for the primary user are introduced. In Section 3, we present an iterative distributed IA algorithm. Section 4 presents the fairness for secondary users in the case of a -user symmetric secondary system. Simulation results are presented in Section 5, and the paper is concluded in Section 6 finally.

*Notation*. We use lower-case bold symbols for vectors and upper-case bold font to denote matrices. represents the identity matrix and the null matrix is represented by . , , , , , and denote the trace, rank, range, nullspace, norm, and Hermitian transpose of the matrix, respectively, and denotes the th column of the matrix. Besides, denotes the statistical expectation and .

#### 2. System Model

As shown in Figure 1, a ()-user MIMO interference channel in a CR MIMO network is considered by us. Assume that the secondary users share the same spectrum with the primary user. The primary user is denoted by index 0 while the secondary users are denoted by index . The symbols and denote the number of antennas at the th transmitter and receiver, respectively. We assume that the th user sends independent streams meaning the DoF are . And denotes the transmit signal of th user, and the power matrix of the th user is denoted by . Let denote the precoding matrix of the th user. So, the received signal at the th receiver of dimension can be written as [11]where is the matrix that represents the channel gain between the transmitter and the receiver . The channels are assumed to experience block fading and the elements of the channels are constant.

It must be noted that the entries of all channel matrices are drawn from a continuous distribution independently. Accordingly, all channel matrices are surely full rank [12]. () denotes the additive white Gaussian noise vector with zero mean, whose covariance matrix is denoted by . The received signal at the th receiver is postprocessed by the postprocessing matrix to extract the transmitted symbols. Therefore, the postprocessed output of the th receiver is given by [11].

Also, we assume that the primary transmitter is oblivious to the presence of the secondary users. And the primary channel matrix is known at the primary transmitter and receiver. By choosing the precoding matrix in the transmitter side and selecting the postprocessing matrix in the receiver side, the primary link channel can be transferred into a diagonal matrix. The singular value decomposition of the matrix is given by , where is diagonal matrix whose diagonal elements are the nonzero singular values of the matrix . The columns of the precoding matrix and the postprocessing matrix correspond to a nonzero power allocation [11]. Then, the achievable rate of the primary user is maximized by the power allocation matrix which is the solution to the following optimization problem [9]:

The solution to (2) is the well-known water-filling algorithm. According to this method, the optimal power allocation matrix is a diagonal matrix with entries [9]:where the constant is the Lagrangian multiplier that is determined to satisfy the power constraint in (2).

Noting the received signal at the th receiver due to the th transmitter lies in the subspace spanned by the columns of . For the purpose of ensuring that the secondary transmitters do not generate any interference to the primary receiver and the secondary users are orthogonal to the received signal from the primary link, we have the following conditions [11]:The feasibility of the cognitive interference alignment problem in (4) was proved in [11].

It is obvious that the interference between the primary link and the secondary links can affect the performance of cognitive MIMO systems adversely. Now, this interference is eliminated by (4). Thus, an iterative IA algorithm can be performed to eliminate the interference between secondary users in Section 3 and achieve the fairness of secondary users in Section 5 without considering the interference above.

#### 3. IA Scheme for the Secondary Link

In this section, a -user secondary system with constant channel coefficients is considered by us. We will start by providing an achievable scheme for the above system and then present an iterative IA algorithm for the system where channel reciprocity holds.

##### 3.1. Achievability of IA for Symmetric MIMO Cognitive System

In order to achieve IA, the following constraint must be satisfied on the precoding and postprocessing matrices of the secondary system [6]:The constraint in (5) ensures that no interference is generated from other undesired secondary transmitters. Moreover, the constraint in (6) guarantees that the dimension of the desired signal space at the th receiver is .

Using the idea of decomposition in [11], we define the matrices and the matrices such that [11]where () and () span the nullspace of the matrices and , respectively.

Using the decomposition in (7), we can convert (5) and (6) into the following equations [11]:The modified problem in (8) is a standard IA problem in the variables and [11].

##### 3.2. Iterative IA Algorithm

In this subsection, we propose a modified version of the iterative MSINR algorithm in [7] for IA of the secondary users in the presence of the primary user whose interference is aligned with such a similar algorithm. A reciprocal channel in which the roles of the transmitters and receivers are switched is defined by us. The reciprocal channel between the transmitter and the receiver in the reciprocal network is denoted by and due to the channel reciprocity. It is shown in [6] that if the DoF allocation is feasible on the original system, it is also feasible on the reciprocal system (and vice versa).

Note that, by listening to the primary transmitter and estimating the subspaces spanned by , the th secondary receiver can estimate . Similarly, by listening to the transmission from the primary receiver and estimating the subspace spanned by , the th secondary transmitter can estimate [11]. Therefore, the distributed IA algorithm in [7] can be used to estimate the th modified precoding matrix and the th postprocessing matrix by secondary transmitters and receivers, respectively. We must note that, in this algorithm, the secondary users do not need the whole channel state information, and local channel state information is sufficient for the secondary users [13].

*Algorithm 1 (Max-SINR Algorithm). *We have the following:(1)Start with any matrix : ; the columns of are linearly independent unit vectors.(2)Iteration is beginning.(3)Compute interference plus noise covariance matrix for stream at the receiver :(4)Compute receiver combining vectors: (5)Reverse the communication direction and use the receiver combining vectors as precoding vectors: , .(6)Compute interference plus noise covariance matrix in the reciprocal system:(7)Calculate the receiver combining vectors in the reciprocal system:(8)Reverse the communication direction and use the receiver combining vectors as precoding vectors: , (9)Repeat until convergence or the number of iterations reaches a limit defined earlier.

#### 4. Fairness for the Secondary Users

After the dispose in Section 3, the received signal at the th secondary user can be denoted bywhere denotes the additive noise matrix and is assumed to be complex Gaussian, with zero mean and variance . Then, the signal-to-interference-plus-noise ratio (SINR) of the th secondary user iswhere denotes the interference from the th secondary user to the primary user, denotes the desired signal, and denotes the interference from other secondary users. Also, the utility of users can be denoted as

##### 4.1. Max-Min Fairness

In this subsection, we will realize the fairness by maximizing the minimum SINR of secondary users. In cognitive MIMO systems, the secondary users share the same spectrum with the primary user and achieve the best throughput. Therefore, we should investigate appropriate power weights to distribute them among the users so that the minimum throughput of secondary users is maximized. From (15), we know that we can try to maximize the minimum SINR instead of the throughput in order to simplify this problem. Actually, maximizing the minimum SINR amounts to equalizing the SINR performance of all users [14]. Thus, the optimization problem can be formally stated as follows:where and are the given values with respect to the total transmit power and interference threshold, respectively. Our objective is to consider the optimization of power allocation in order to maximize the minimum SINR of the secondary system with the following three constraints. The first constraint restricts the total transmission power from the secondary transmitters. The second constraint is to guarantee the interference power to the primary receiver is bounded by a certain limit. In the third constraint, each power is limited in secondary users.

By introducing a new variable , which is , we can rewrite this optimization problem in (16) as follows:

###### 4.1.1. Max-Min Algorithm

In this subsection, a kind of intelligent algorithm, called particle swarm optimization (PSO), is presented to allocate the power to maximize the minimum SINR. The particle swarm optimization (PSO) algorithm is a kind of evolutionary computation, which is derived from the study on the behavior of the birds swarm and is similar to the genetic algorithm, and is a kind of iterative optimization tool.

The algorithm is summarized as follows.

*Algorithm 2 (Max-Min SINR Algorithm). *
We have the following:(1)Initialize particle position ; speed is denoted by power allocation matrix , .(2)Compute the value of , , for , calculate, compare, and find the minimum SINR which is indicated by .(3)Compute the fitness function: (4)Compute the optimal solutions of population and individual, denoted by gbest and pbest, respectively: If pbest = gbest, output the optimal result. Or else, update the particle position and power allocation matrix ; then, update the fitness function by Steps (2) and (3).(5)Repeat Steps (2), (3), and (4) until convergence.

In the algorithm above, our goal is to minimize the fitness function of PSO by changing the norm of power matrix. When the power is updated, the value of , , will be changed afterwards; then, we should reorder the value of , . Therefore, the fitness function will be updated too.

##### 4.2. Proportional Fairness

In order to obtain a balance between maximizing the system throughput and maintaining user fairness, proportional fairness was proposed in [23]. In this subsection, we let denote the average throughput of user at th time slot and denotes the estimated supportable data rate of user at the th time slot. The user to be scheduled at the th time slot is [23]Then, is updated using [27]:where is a constant parameter which determines the effective memory of the throughput averaging window [23].

The algorithm is summarized as follows.

*Algorithm 3 (Proportional Fairness Algorithm). *
We have the following:(1)Initialize power allocation matrix , : For , Compute the throughput of user by (10) and (11): End.(2)Compute by (14):Update according to (15).(3)Update the power allocation matrix, and repeat Steps (1) and (2) until reaching the maximum number of iterations.

Max-Min and PF algorithms are two effective and common algorithms to study the fairness problem. In our network, we want to show their effectiveness and compare these two algorithms in different aspects in the simulation results.

#### 5. Simulation Results

In this section, a 3-secondary user and 1-primary user symmetric cognitive MIMO system with constant channel coefficients are simulated. Each node is equipped with antennas. Furthermore, the channel coefficients are selected as zero mean unit-variance circularly symmetric complex Gaussian random variables. On the other hand, the primary transmitter and receiver are also equipped with antennas [13]. We select the channel such that only two eigenmodes are active at the primary transmitter; that is, [11]. From [11], we get that for a symmetric system in which each secondary transmitter and receiver have and antennas, respectively, having the same number of DoF , the system is proper if ; thus, the maximum number of achievable DoF per secondary user is , if is selected to be 2. In the following simulation, we choose noise power W and maximum sum transmit power of secondary users W, the primary user transmit power is W for the sake of fairness [13], and the interference threshold is chosen to be 100 [28]. In the PSO, the number of particles is chosen to be 40.

Figure 2 shows the minimum SINR calculated by max-min algorithm and the throughput computed by PF algorithm among secondary users, respectively. Great advantage of the IA algorithm can be found. From the simulation results above, we know that the result above with IA performed better than that without IA because the interference between secondary users has been eliminated by IA. Besides, the convergence of max-min algorithm is also shown in Figure 2.

In Figure 3, we compare the sum throughput of the secondary users with max-min, PF, and game-theoretic algorithm in [13] which maximize the sum utility and ignore the fairness among users. From this curve, we know that the authors in [13] who have not considered the fairness problem get the largest sum throughput. In addition, in the algorithms which have considered the fairness problem, the max-min algorithm achieves the low throughput while the PF algorithm performs better than it in terms of throughput.

In Figure 4, we compute the fairness index by using Jain’s fairness index [29] which is defined asAs shown in Figure 4, the algorithms which have considered the fairness problem perform better than the algorithm in [13]. In detail, the max-min algorithm has the higher fairness index than the PF algorithm while the game-theoretic algorithm which ignores the fairness has the lowest fairness index. Combined with the results in Figure 3, we know that the PF algorithm achieves the tradeoff of throughput and fairness, and the max-min fairness is better in achieving fairness of secondary users.

#### 6. Conclusion

In this paper, we have presented a cognitive IA scheme, allowing multiple secondary users to access the free spatial dimensions of a primary user, which protects the transmission of the primary user while providing interference-free communication for the secondary users. Also, we presented an iterative algorithm that utilizes channel reciprocity to achieve the proposed cognitive IA scheme for secondary user MIMO cognitive radio systems. Moreover, we have thought about the fairness among secondary users; thus, two kinds of algorithms are proposed for the cognitive MIMO system which consists of single primary user and multiple secondary users sharing the same spectrum. Finally, from the simulation results, we get that the algorithms with IA are much more effective than those without IA. Besides, the fairness and sum throughput among different algorithms are also compared. To sum up, in this paper, we not only eliminate the interference by IA but also achieve the fairness among secondary users.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (61162008, 61172055, and 61471135), the Guangxi Natural Science Foundation (2013GXNSFGA019004), and the Director Fund of Key Laboratory of Cognitive Radio and Information Processing (Guilin University of Electronic Technology), Ministry of Education, China (2013ZR02).

#### References

- S. Haykin, “Cognitive radio: brain-empowered wireless communications,”
*IEEE Journal on Selected Areas in Communications*, vol. 23, no. 2, pp. 201–220, 2005. View at: Publisher Site | Google Scholar - G. Scutari, D. P. Palomar, and S. Barbarossa, “Cognitive MIMO radio,”
*IEEE Signal Processing Magazine*, vol. 25, no. 6, pp. 46–59, 2008. View at: Publisher Site | Google Scholar - A. S. Motahari, S. Oveis-Gharan, M.-A. Maddah-Ali, and A. K. Khandani, “Real interference alignment: exploiting the potential of single antenna systems,”
*IEEE Transactions on Information Theory*, vol. 60, no. 8, pp. 4799–4810, 2014. View at: Publisher Site | Google Scholar - M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, “Communication over MIMO X channels: interference alignment, decomposition, and performance analysis,”
*IEEE Transactions on Information Theory*, vol. 54, no. 8, pp. 3457–3470, 2008. View at: Publisher Site | Google Scholar | MathSciNet - H. Weingarten, S. Shama, and G. Kramer, “On the compound MIMO broadcast channel,” in
*Proceedings of the Annual Information Theory and Applications Workshop*, Tel Aviv, Israel, January 2007. View at: Google Scholar - K. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching the capacity of wireless networks through distributed interference alignment,” in
*Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '08)*, pp. 4260–4265, IEEE, New Orleans, La, USA, December 2008. View at: Publisher Site | Google Scholar - K. Gomadam, V. R. Cadambe, and S. A. Jafar, “A distributed numerical approach to interference alignment and applications to wireless interference networks,”
*IEEE Transactions on Information Theory*, vol. 57, no. 6, pp. 3309–3322, 2011. View at: Publisher Site | Google Scholar - X. Baile, L. Yang, H. Minn, and A. Nosratinia, “Adaptive interference alignment with CSI uncertainty,”
*IEEE Transactions on Communications*, vol. 61, no. 2, pp. 792–801, 2013. View at: Publisher Site | Google Scholar - S. M. Perlaza, M. Debbah, S. Lasaulce, and J.-M. Chaufray, “Opportunistic interference alignment in MIMO interference channels,” in
*Proceedings of the IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '08)*, pp. 1–5, IEEE, Cannes, France, September 2008. View at: Publisher Site | Google Scholar - B. Abdelhamid, M. Elsabrouty, and S. Elramly, “Novel interference alignment in multi-secondary users cognitive radio system,” in
*Proceedings of the 17th IEEE Symposium on Computers and Communication (ISCC '12)*, pp. 785–789, Cappadocia, Turkey, July 2012. View at: Publisher Site | Google Scholar - M. Amir, A. El-Keyi, and M. Nafie, “Constrained interference alignment and the spatial degrees of freedom of MIMO cognitive networks,”
*IEEE Transactions on Information Theory*, vol. 57, no. 5, pp. 2994–3004, 2011. View at: Publisher Site | Google Scholar - A. Tadaion and F. Rezaei, “Interference alignment in cognitive radio networks,”
*IET Communications*, vol. 8, no. 10, pp. 1769–1777, 2014. View at: Publisher Site | Google Scholar - F. Zhao, W. Wang, H. Chen, and Q. Zhang, “Interference alignment and game-theoretic power allocation in MIMO Heterogeneous Sensor Networks communications,”
*Signal Processing*, 2015. View at: Publisher Site | Google Scholar - D. W. H. Cai, T. Q. S. Quek, and W. T. Chee, “A unified analysis of max-min weighted SINR for MIMO downlink system,”
*IEEE Transactions on Signal Processing*, vol. 59, no. 8, pp. 3850–3862, 2011. View at: Publisher Site | Google Scholar - Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, “Max-min fairness linear transceiver design for a multi-user MIMO interference channel,”
*IEEE Transactions on Signal Processing*, vol. 61, no. 9, pp. 2413–2423, 2013. View at: Publisher Site | Google Scholar - M. Tao and R. Wang, “Linear precoding for multi-pair two-way MIMO relay systems with max-min fairness,”
*IEEE Transactions on Signal Processing*, vol. 60, no. 10, pp. 5361–5370, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Chou, C. Tsao, J. Wu, J. Hsu, and P. Ting, “Fair resource allocation for multiuser MIMO communications network,” in
*Proceedings of the IEEE Global Communication Conference (GLOBECOM '14)*, pp. 3910–3915, Austin, Tex, USA, December 2014. View at: Google Scholar - X. Li, X. Ge, F. Li, and V. C. Leung, “Max-Min fair resource allocation for min-rate guaranteed services in distributed antenna systems,” in
*Proceedings of the IEEE 80th Vehicular Technology Conference (VTC Fall '14)*, pp. 1–5, Vancouver, Canada, September 2014. View at: Publisher Site | Google Scholar - L. Tang, H. Wang, and Q. Chen, “Power allocation with max-min fairness for cognitive radio networks,” in
*Proceedings of the Global Mobile Congress (GMC '10)*, pp. 1–5, IEEE, Shanghai, China, October 2010. View at: Publisher Site | Google Scholar - T.-D. Nguyen and Y. Han, “A proportional fairness algorithm with QoS provision in downlink OFDMA systems,”
*IEEE Communications Letters*, vol. 10, no. 11, pp. 760–762, 2006. View at: Publisher Site | Google Scholar - S. M. Maung and S. Iwao, “Efficient resource allocation for multiuser MIMO-OFDM uplink system to guarantee the proportional data rate fairness among users in a system,” in
*Proceedings of the 1st International Symposium on Access Spaces (ISAS '11)*, pp. 132–137, June 2011. View at: Publisher Site | Google Scholar - Y. H. Cho, H. Kim, S.-H. Lee, and H. S. Lee, “A QoE-aware proportional fair resource allocation for multi-cell OFDMA networks,”
*IEEE Communications Letters*, vol. 19, no. 1, pp. 82–85, 2015. View at: Publisher Site | Google Scholar - L. Yong, W. Wang, X. Zhang, and M. Peng, “Combined proportional fair and maximum rate scheduling for virtual MIMO,” in
*Proceedings of the IEEE 68th Vehicular Technology Conference*, pp. 1–4, Calgary, Canada, September 2008. View at: Publisher Site | Google Scholar - K. Premkumar, X. Chen, and D. J. Leith, “Proportional fair coding for wireless mesh networks,”
*IEEE/ACM Transactions on Networking*, vol. 23, no. 1, pp. 269–281, 2015. View at: Publisher Site | Google Scholar - L. Xu, Y.-P. Li, Y.-W. Yang, X. Zhang, Z.-M. Tang, and S.-H. Lan, “Proportional fairness resource allocation scheme based on quantised feedback for multiuser orthogonal frequency division multiplexing system,”
*IET Communications*, vol. 8, no. 16, pp. 2925–2932, 2014. View at: Publisher Site | Google Scholar - R. Fritzsche, P. Rost, and G. P. Fettweis, “Robust rate adaptation and proportional fair scheduling with imperfect CSI,”
*IEEE Transactions on Wireless Communications*, 2015. View at: Publisher Site | Google Scholar - A. Goldsmith,
*Wireless Communications*, Standford University Press, 2005. - F. Zhao, B. Li, and H. Chen, “Joint beamforming and power allocation algorithm for cognitive MIMO systems via game theory,” in
*Wireless Algorithms, Systems, and Applications*, vol. 7405 of*Lecture Notes in Computer Science*, pp. 166–177, Springer, Berlin, Germany, 2012. View at: Publisher Site | Google Scholar - R. Jain, D. W. Chiu, and W. R. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in shared systems,”
*DEC Research Report*TR-301, 1984. View at: Google Scholar

#### Copyright

Copyright © 2015 Feng Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.