Interference Alignment and Fairness Algorithms for MIMO Cognitive Radio Systems

Interference alignment (IA) is an effective technique to eliminate the interference among wireless nodes. In a multiinput multioutput (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 andK 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.


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][5][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 interferencefree 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 signalto-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-interferenceplus-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 networkwide 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.I  represents the × identity matrix and the  ×  null matrix is represented by 0 × .tr{A}, rank{A}, span{A}, {A}, ‖A‖, and A  denote the trace, rank, range, nullspace, norm, and Hermitian transpose of the matrix, respectively, and A *  denotes the th column of the matrix.Besides, {⋅} denotes the statistical expectation and () + = max{0, }.

System Model
As shown in Figure 1, a ( + 1)-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 1, . . ., .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 x  = [ 1 , . . .,    ]  denotes the transmit signal of th user, and the power matrix of the th user is denoted by P  = {x  x   } = diag(√ 1 , . . ., √    ).Let V  denote the   ×  precoding matrix of the th user.So, the received signal at the th receiver of dimension   can be written as [11] where H  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].z  (  × 1) denotes the additive white Gaussian noise vector with zero mean, whose covariance matrix is denoted by  2 I   .The received signal at the th receiver is postprocessed by the   ×   postprocessing matrix U  to extract the   transmitted symbols.Therefore, the postprocessed output of the th receiver is given by U     [11].Also, we assume that the primary transmitter is oblivious to the presence of the secondary users.And the primary channel matrix H 00 is known at the primary transmitter and receiver.By choosing the precoding matrix V 0 in the transmitter side and selecting the postprocessing matrix U 0 in the receiver side, the primary link channel can be transferred into a diagonal matrix.The singular value decomposition of the matrix H 00 is given by H 00 = U  0 Λ 0 V 0 , where Λ 0 is  0 ×  0 diagonal matrix whose diagonal elements are the min{ 0 ,  0 } nonzero singular values of the matrix H 00 .The columns of the precoding matrix V 0 and the postprocessing matrix U 0 correspond to a nonzero power allocation [11].Then, the achievable rate of the primary user is maximized by the power allocation matrix P 0 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  0 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 H  V  .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.

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.

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]: rank 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 (  −  0 ) ×   matrices { Ṽ }  =1 and the (  −  0 ) ×   matrices { Ũ }  =1 such that [11] V  = G  Ṽ , ∀ = 1, . . ., , where G  (  × (  −  0 )) and B  (  × (  −  0 )) span the nullspace of the matrices U  0 H 0 and H 0 V 0 , 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 { Ṽ }  =1 and { Ũ }  =1 [11].
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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 ⃖ H  and ⃖ H  = H   due to the channel reciprocity.It is shown in [6] that if the DoF allocation {  }  =1 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 H 0 V 0 , the th secondary receiver can estimate B  .Similarly, by listening to the transmission from the primary receiver and estimating the subspace spanned by ⃖ H 0 U 0 , the th secondary transmitter can estimate G  [11].Therefore, the distributed IA algorithm in [7] can be used to estimate the th modified precoding matrix Ṽ and the th postprocessing matrix Ũ 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].
(9) Repeat until convergence or the number of iterations reaches a limit defined earlier.

Fairness for the Secondary Users
After the dispose in Section 3, the received signal at the th secondary user can be denoted by where n denotes the additive noise matrix and is assumed to be complex Gaussian, with zero mean and variance  2 .Then, the signal-to-interference-plus-noise ratio (SINR) of the th secondary user is where ‖U   H 0 V 0 P 0 ‖ denotes the interference from the th secondary user to the primary user, ‖U   H  V  P  ‖ denotes the desired signal, and ‖ ∑  =1, ̸ = U   H  V  P  ‖ 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: max where   and  th 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  = min =1,..., SINR  , we can rewrite this optimization problem in (16) as follows: min 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.
(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 V  and power allocation matrix P  ; then, update the fitness function by Steps ( 2) and ( 3).
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 SINR  ,  = 1, . . ., , will be changed afterwards; then, we should reorder the value of SINR  ,  = 1, . . ., .Therefore, the fitness function will be updated too.

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.
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.

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   =   = 6 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  0 =  0 = 6 antennas [13].We select the channel H 00 such that only two eigenmodes are active at the primary transmitter; that is,  0 = 2 [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  +  − 2 0 − ( + 1) ≥ 0; thus, the maximum number of achievable DoF per secondary user is  = 2, if  0 is selected to be 2.In the following simulation, we choose noise power  2 = 1 W and maximum sum transmit power of secondary users  max = 20 W, the primary user transmit power is  0 = 20 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 maxmin 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 as As 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.

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.

Figure 2 :
Figure 2: SINR and throughput comparison with/without IA.

Figure 3 :Figure 4 :
Figure 3: The sum throughput of SUs with three algorithms.