Mei Rong＊

Department of Electronic and Information Engineering, Chang’an University, Xi’an 710064, China

I. INTRODUCTION

Interference alignment (IA) is an interference cancellation technique for multi-user multi-antenna systems. The main idea of IA is to process signals at each transmitter and receiver,so as to align interference to null-space of each receiver’s desired signal. There are several implementations of IA for different networks. In centralized systems, IA has the same form with null-space projection[1]. However,degrees of freedom (DOF) of system are restricted by number of base station’s antennas.Alternative minimization is an effective way to achieve IA in distributed networks, whose convergence has already been proved[2].Compared to non-interfering case, each user can acquire half DOF through IA, as long as the number of users satisfies specific conditions[3].

IA is suitable for spectrum sharing (SS) in cognitive radio networks (CRNs) owing to sensing ability[4, 5] of cognitive users (CUs)and their tolerance of interference due to the“best-effort” QoS demand. Besides, distributed construction and flexible reconfiguration of CRNs prefer IA to deal with signals distributively and independently. However, the biggest challenge of IA based SS in CRNs is to deal with the absence of primary users (PUs)in IA process and guarantee the transmissions of PUs. There have been several ways to apply IA in MIMO CRNs: 1. PUs adopt water filling power allocation, and some unused spatial directions exist. CUs can project their signals on these unused sub-channels, so as to eliminate interference between cognitive radio system(CRS) and primary system (PRS), and then actualize IA inside CRS[6-14]; 2. PUs only use some of the spatial sub-channels, and leave other subspaces to CUs. IA in CRS can be implemented in these subspaces. This virtual cooperation between CRS and PRS can improve spectrum efficiency[15]; 3. IA is adopted in both CRS and PRS, and interference between the two systems can be kept below a threshold by designing proper precoding and equalizing matrices for each transmitter and receiver, respectively[16, 17]; 4. In MIMO-OFDM CRN,Lu etc. developed an IA based spatial-frequency signal alignment SS scheme. Spatial water filling power allocation and cyclic-prefix(CP) of PRS bring idle spatial and frequency sub-channels, which are used for IA among CUs[18, 19]. Moreover, In Ref. [20, 21], CUs realize IA on idle spectrums, and guarantee that adjacent-channel interference is under a threshold through power control; 5. CUs have sufficient antennas to provide idle spatial sub-channels for their transmission, as well as to cancel the interference to PRS[22-24];6. Primary receiver (PRR) has channel state information (CSI) from each cognitive radio transmitter (CRT) to itself, so it can guarantee its transmission rate by designing shaping matrices and feeding them back to CRTs.Then IA can be exploited inside CRS with the restriction of shaping matrices[25]; 7. Both PUs and CUs participate in an IA process, so as to eliminate interference among them perfectly[26-28]. These methods are valuable for application of IA SS in CRNs. However, all the methods above are not suitable for general CRNs, where PUs can neither be aware of the coexistence of CUs nor cooperate with CRS.Furthermore, water filling power allocation and idle spatial or frequency sub-channels in PRS cannot be guaranteed. Motivated by all these reasons, we investigate SS method for general CRNs to increase sum rate of CUs through adaptive IA. Considering interference constraint (IC) of PU, we should decrease transmit power of CRTs. Unfortunately, this may push the CRS into low SNR regime, where power gain, instead of DOF, has dominant effect to sum rate of CUs, so IA may jeopardize the sum rate. To solve this Conflict, we propose a new IA based SS strategy. Alternative minimization based IA[2] is performed among CUs.In the meantime, access control, adaptive spatial projection and power control are assistant steps in different cases of scenarios to improve sum rate of CUs as well as satisfy IC.

This paper focuses on how to introduce IA into general CRNs as well as guarantee PU’s transmission. The authors propose an adaptive IA SS method for general distributed multi-user multi-antenna CRNs.

The main contributions of this paper are summarized as follows: firstly, we break the restriction that CUs can only transmit on idle sub-channels of PRS, and provide an adaptive IA based SS method for general CRNs, which can not only satisfy IC of PU but also improve sum rate of CUs by scenario classification,adaptive spatial projection and access/power control. Secondly, the proposed method does not require any cooperation between PRS and CRS. Furthermore, PU is unaware of the coexistence of CUs. Thirdly, because of the iteration in IA, it is hard to determine whether IC is satisfied without completion of IA.As a consequence, scenario classification is impossible. To solve the problem, we utilize Rayleigh quotients of channel matrices to predict the maximum interference from each CRT to PRR. On the basis of predictions, scenario classification can be achieved.

The rest of this paper is organized as follows: Section II provides system model of CRN. Section III describes the proposed adaptive IA based spectrum sharing (AIASS)method. Performance analysis is shown in Section IV, including convergence, achievable sum rate of CUs, outage probability of PU and complexity. Section V provides the simulation results and discussions. Finally, we conclude the paper in Section VI.

II. SYSTEM MODEL

2.1 System Description

We consider a CRN withKpairs of CUs,each including a CRT withMantennas and a cognitive radio receiver (CRR) withNantennas. The CRS coexists with a primary transmitter (PRT) and a PRR withMandNantennas, respectively. The CRN is shown in figure 1.

Channels between users are all quasistatic Rayleigh flat. Hi,kand Gkdenote channel matrices from CRTkto CRRiand PRR, respectively. Dkis channel matrix between PRT and CRRk. Transmitted signal from CRTkis xk,and signal from PRT is s. Receiving signal at CRRkis

Fig. 1. Block diagram of cognitive radio network.

where zkis complex Gaussian white noise with power σ02. Every CU is attainable to Hi,k, Gkand Dkby sensing[6-8].

2.2 Mathematics model of spectrum sharing problem

Spectrum sharing problem aims at maximizing sum rate of CRS on the premise of satisfying IC of PRR. Based on IA, we design pre-processing matrices for xkand post-processing matrices for ykas Fkand Wk, which subjectIn this paper, A?represents the conjugate transpose of matrix A. Signal model at CRRkis

The set of active CUs is A. Transmit power of CRTkand PRT arePc,kandPpwith equal power on each antenna. Interference power at PRR is

wheredkis the rank of Fk, andmeans F-norm of A. Accordingly, mathematics model of SS problem is in Eq. (12) to (14),wherePis transmit power budget of each CRT, and γ0is the IC of PRR. The problem can be solved by searching optimal A and designing Fk, Wk,dkandPc,kproperly. For simplification, we suppose

Due to complexity, we prefer to look for suboptimal solution through IA. It can be noticed that there is neither specific requirement on transmission of PUs, nor cooperation between PRS and CRS, so the proposed method is more general than methods in Ref. [6-28].

III. ADAPTIVE IA BASED SPECTRUM SHARING

3.1 Scenario classification

The relationship between interference and noise (including white noise and the interference from PRT) power at each CRR is influenced byP, γ0and stochastic channel matrices. In this paper, the set of these time-varying relationships at all CRRs are refered to as instantaneous scenarios of CRN, which can be described byP, γ0and channel matrices.

IA has different effect on the sum rate of CUs in different types of scenarios, e.g.,whenPand γ0are high, and most CUs are influenced mainly by interference from other CRTs, IA is preferable for CUs to cancel the interference so as to aquire large DOF; contrarily, if CUs are basically affected by noise,CU access control is suitable for reserving transmit power of CRTs. Therefore, different IA based SS algorithms should be properly designed for different types of scenarios with the aim of increasing sum rate of CUs. Obviously, scenario classification is the basis of SS algorithms. Based onP, γ0and stochastic channel matrices, we provide a scenario classification scheme.

Firstly, we analyze effect ofPand set its threshold without considering γ0. IfPis high, interference from undesired CRTs is larger than noise on each CRR, so DOF is the major factor affecting sum rate of CUs; on the contrary, ifPis low, noise is larger, and transmit power of CRTs mainly effects the sum rate. According to above analysis, we suppose that the number of CUs satisfying (4) isSc,

Pis considered to be high, otherwise low.

Secondly, we determine whether γ0is high.

The number of CUs whose channel matrices satisfy (6) is denoted asSi,

γ0is high; otherwise γ0is low.

At last, we should check whether IC is always satisfied. Substitute Eq. (3) into (13), we can get

If (8) is true, IC condition is satisfied; otherwise it is not. However, the pre-processing matrices Fk,1≤k≤Kare not attainable before IA, so here the maxima ofis used for decision.

According to Theorem 1, the maxima of

where λi,kis theith maximum eigenvalue ofGk. Therefore, we revise (8) to

Theorem 1 Both G and F are complex matrices, with dimensionsN×MandM×d,respectively.is

where λiis theith maximum eigenvalue of G?G.

ProofSee AppendixA.

Based on above three steps, CRN scenarios

can be classified into 23=8 types. Combination and processing of different scenarios will be given in the following subsections.

3.2 Decoupling of SS problem

There are three kinds of controllable variables in SS problem: the set of accessing CUs(A), transmit power of each CRT (Pc), and processing matrices (Fkand Wk). These variables affect optimization together, and complicate the problem. Hence, in this subsection,we look for key parameters in different type of scenarios, so as to offer suitable transmission mode.

In light of the analysis in last subsection,we combine 8 types of scenarios into 4 cases:

1) Both ofPand λ0are high, which means that both (5) and (7) are true. Contrarily, (10) is not true. In this case, spatial DOF has dominant effect on sum rate of CUs[29],so we should decreasePcafter IA, in order to meet IC condition as well as keep spatial DOF for each pair of CUs.

2)Pis high, but λ0is low. In addition,(10) is not true. IfPcis reduced, it is very likely CRS will be in low SNR regime, where transmit power of CRTs is the major influence on sum rate[29], so power reduction will harm sum rate seriously. Due to these reasons, we permit only one pair of CUs to access each time, so as to not only meet IC but also alleviate power reduction.

3)Pis low, and (10) is not true. Since CRS is in low SNR regime, where transmit power is the major factor,Pcshould be as large as possible. Differing from case 2), CRS is already in low SNR regime, so interference between CUs is less than noise. As a result, the number of accessing CUs should be as large as possible. To meet IC and increase the number of accessing CUs simultaneously, we adopt spatial projection adjustment, which will be shown in details shortly.

4) IC is always satisfied (Inequality (10) is true). In this case, we apply IA in CRS directly.

Obviously, case 1), 2), 3) and 4) include 1, 1, 2, and 4 types of scenarios defined in subsection 3.1, respectively. So these 4 cases contain all the 8 types of scenarios completely.

3.3 Adjusted spatial projection scheme

Considering the structure of CRN, we propose the following adjusted spatial projection scheme for IA in CRN.

On one hand, we look for an interference subspace for each CRR, which contains interference to the CRR as much as possible. Interference subspace for CRRkcan be represented by Uk, with orthonormal basis Uoptk. So

where(A) denotes theddominant eigenvectors of A. Consequently,

On the other hand, interference from CRTkto other CRRs, e.g. CRRl,l≠k, is expected to lie in the interference subspace of CRTl,which means that the distance betweenandshould be minimizing. Moreover, it is necessary to minimize the interference from CRTkto PRR. Therefore,

where(A) denotes thedleast dominant eigenvectors of A.

Based on the above schemes, we propose adaptive IA based spectrum sharing method in Algorithm 1, where δ is the accuracy of convergence, andNmaxis the maximum number of iteration.

IV. PERFORMANCE ANALYSIS

4.1 Convergence analysis

In this subsection, we discuss convergence of AIASS method in different cases.If CRN is in case 2), there is no iteration.If CRN is in case 1) or 4), β=0. When α=0, iterations in AIASS method are the same as in Ref. [2]. Since the object functionwill never increase in iterative steps 8 and 9[2], and it is nonnegative, AIASS method is convergent.When α=1, pre-processing matrix Fkcan also project the interference from CRTkto other CRRs on their interference subspaces Ul,l≠k, so IA is convergent.

On the contrary, if CRN is in case 3),β=1. Accordingly, Fkshould not only project interference from CRTkto other CRRs onto their interference subspaces, but also minimize interference from CRTkto PRR.Under these circumstances, the object functioncannot be minimized, so IA is nonconvergent. In this case,IA process ends when the maximum iteration numberNmaxis reached.

4.2 Achievable sum rate of CRS

Achievable sum rate of CRS is sum rate of all accessing CUs. When AIASS method is adopted, achievable sum rate can be described as(19) at page 104. Since Fkand Wkare calculated through iterative IA process, we substitute optimized results in each random channel realization into (19) to obtain average sum rate from a large amount of simulations as ergodic sum rate of CRS.

4.3 Outage probability of PU

Obviously, outage of PU caused by CRS happens only when IC condition is not satisfied.In AIASS method, IC is always guaranteed by transmit power adjustment in step 12, so CRS will not cause outage of PU.

Algorithm 1. adaptive IA based spectrum sharing method.

4.4 Complexity

We analyze complexity of AIASS method in terms of an addition or a multiplication, and compare it with Interference Alignment Spectrum Sharing (IASS) method, which adopts alternative minimizing IA[2] inside CRS, and meets IC condition by transmit power adjustment. It is an intuitive application of IA in general CRNs. The complexity is shown in Table 1, whereTdenotes the average number of iterations for each convergent IA process,usually in 102order[2].

Here ρ denotes the probability of case 3).From the comparison, it can be noted that our method has the same order of complexity with IASS when ρNmaxandTare in the same order.

V. SIMULATION RESULTS AND DISCUSSIONS

We consider PRS as a single-cell scene,and the cell is a round region with radius asD=500m. As in figure 2, PRT locates in the center of the cell. Polar coordinates of PRR distribute uniformly in the cell, i.e., radial coordinate is ρ0～U(0,D] and angular coordinate is θ0～U[0,2π] in each simulation independently. CRS is a distributed system. There areK=3 pairs of CUs situated uniformly in a round region with radius ofD1inside the cell. The center of this region has radial coordinate η0～U(0,D] and angular coordinateφ0～U[0,2π]. Every terminal has 4 antennas,i.e.,M=N=4. Distance between PRT and CRRk, CRTkand PRR, CRTkand CRRlare denoted bydcp,k,dpc,kanddcc,l,k, respectively. Large scale fading model of PRS is Urban Macro NLoS model[31]:

Table I. The complexity of two methods.

Fig. 2. Cognitive radio network.

Lpc,krepresents the large scale fading between CRTkand PRR. In the same way,we can getLcp,kas the large scale fading between PRT and CRRkfromdcp,k. HereW=20m,h=20m,hBS=50m,hUT=1.6m,fc=2.6GHz. For the CRS, we adopt Indoor HotSpot NLoS mode[31] as

Noise power is=6.3246× 10?13w. The elements of small scale fading channel matrices are assumed to be independent complex Gaussian random variables with mean 0 and unit variance[2, 6, 7, 10]. Furthermore, we chooseNmax=1000 to keep the complexity of AIASS and IASS method in the same order.Convergence accuracy is δ=10?13.

Achievable sum rate of CRS brought by six SS methods are shown in figure 3. In the simulations, we suppose thatPp=100Pand γ0=10-13P. There are 4 different radii of CRS, includingD1=10m,D1=20m,D1=50m andD1=100m. Figure 3 shows that since average distance between CUs increases withD1, sum rate provided by every method decreases. Alternative minimization based IA algorithm can bring larger spatial

DOF to MIMO CRS in high SNR regime,which improves sum rate of CUs. However, ifPis low, or CRRs are close to PRT, CRS is in low SNR regime and IA is not suitable. From the curves in figure 3, we can notice that the proposed AIASS method can not only provide high DOF through IA in high SNR regime,but also alleviate harm from IA in low SNR regime by scenario classification and adaptive IA. That is to say, besides IA and power control, AIASS method applies adjusted spatial projection and user access control in low SNR regime, which are beneficial to the sum rate.Therefore, AIASS method can achieve larger sum rates than other methods in all the four situations. Max-SINR SS and IASS2 are intuitive expansions of IA methods in [32]. IA based SS methods (including IASS, IASS2 and Max-SINR SS) restrain interference to PRR only by power control, which leads to a reduction of sum rate, especially whenD1is large, such asD1=50m andD1=100m.In these scenarios, average distance between PR terminals and CR terminals is small, so inter-system interference is serious. In order to satisfy IC, we should decrease transmit power of each CRT, so sum rate of CRS decreases.Nevertheless, IA based SS methods are superior to the other two methods due to the application of IA, especially in high SNR regime.Orthogonal spectrum sharing (ORTSS) allows only 1 pair of CUs to access, so as to keep transmit power of CRT as much as possible while satisfying IC. IfPis low, noise is the predominant influence in reducing sum rate of CRS, so ORTSS is preferable. in figure 3,the curve of ORTSS is getting closer to IASS curve whenD1is increasing. Isotropic SS(ISOSS) method treats all the interference as noise. Without consideration of interactions between CUs, ISOSS brings the lowest sum rate.

Figure 4 shows sum rate of CRS with different γ0. HereD1=10m,Pp=1000P, and sub figure (a) to (d) correspond to γ0=10-10w,γ0=10-13w, γ0=10-15w and γ0=10-18w respectively. From the figure, we can conclude that sum rate increases with γ0. Meanwhile,AIASS method can provide the largest sum rate of CRS in each situation. Arched curves reflect different effects ofPand γ0on sum rate of CRS. Before vertexes of the curves,sum rates are constrained byP, so they increase withP. On the contrary, γ0constrains sum rate after vertexes. Under the latter circumstance, whenPis growing,Pccannot increase withP, butPpincreases, so interference from PRT to each CRR keeps growing,which leads to decrease of sum rate. Besides,if γ0is smaller, the vertex appears earlier. IA based SS methods have superiority before the vertexes because of application of IA. Nevertheless, owing to IC, IA based SS methods cannot acquire larger DOF after vertexes, so sum rate provided by IA based SS methods decrease more seriously. Obviously, AIASS method provides larger sum rate than other methods by means of adaptive IA.

Fig. 3. Achievable sum rate of CUs.

Fig. 4. Achievable sum rate of CUs with different γ0.

Sum rates of CRS with differentD1are illustrated in figure 5, where γ0=10-13w andPp=100w. Sub figure (a) to (d) correspond toD1=10m,D1=20m,D1=50m andD1=100m circumstances, respectively. It is notable that sum rate decreases asD1increases. The main reason is that average distance between CRRs and their corresponding CRTs increases whenD1grows, as a result, sum rate of CRS decreases. However, AIASS method can provide the largest sum rates in all the three situations because of the adaptation.

Fig. 5. Achievable sum rate of CUs with different D1.

Fig. 6. Achievable sum rate of CUs with different Pp.

At last, figure 6 shows the sum rate of CRS with γ0=10-13PandD1=10m. Subfigure (a) to (d) representPp=1w,Pp=10w,Pp=100w andPp=1000w circumstances.Along with growth ofPp, each CRR suffers more interference from PRT, so sum rate of CUs brought by every method decreases. Nevertheless, the proposed AIASS method can provide larger sum rate than other methods in all situations. Furthermore, it is worth noting that AIASS has more advantages whenPpis larger, which shows the robustness of the proposed method.

VI. CONCLUSIONS

This paper focuses on how to introduce IA into general CRNs as well as guarantee PU’s transmission. In the proposed AIASS method,a scenario classification mechanism is formulated according to the analysis of influence on CUs’ sum rate, including spatial DOF and transmit power of CRT. Through access/power control and adjusted spatial projection, we keep spatial DOF and transmit power for CUs in high and low SNR regime, respectively. We prove that maximum interference from CRS to PU can be predicted by eigenvalues of channel matrices. Based on these results, an iterative AIASS method is designed to maximize achievable sum rate of CRS on the premise of satisfying IC, which solves the problem of the absence of PU in IA, and eases the restriction that CUs can only transmit on idle sub-channels. Simulation results show that achievable sum rate of CRS can be enlarged in different CRN scenarios. In future, we will expand the ideas to situations with multiple PUs.

ACKNOWLEDGEMENTS

This work was supported by National Natu-ral Science Foundation of China under Grant 61201233; 61271262 and 61701043.

Appendix A Proof of Theorem 1

Proof of Theorem 1We suppose fiis theith column of F. Since F?F = I ,=1. According to properties of trace in a matrix[33],we have

For brevity, we suppose G?G = A , so A should be an Hermitian matrix, whose dimension isM×M, and

On the other hand,

Consequently,

From Eq. (20) to (23),

Besides, on the basis of the Rayleigh quotient property of matrix[34], we can conclude that

Where λiis theith maximum eigenvalue of G?G. Moreover, if and only if fiis theith dominant eigenvector of G?G,reaches the maxima.

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