ZHANG Hui-jie (張慧杰)，LI Li (李 莉) ，QIU Hao (邱 昊)，XIA Lin (夏 林)

College of Information，Machanical and Electrical Engineering，Shanghai Normal University，Shanghai 200234，China

Introduction

The soaring demand for wireless spectrum resource has narrowed the access to the resource that is applied to wireless communication.Cognitive radio (CR)can improve the usage of spectrum according to dynamic spectrum access and intelligent spectrum allocation.Multiple input multiple output(MIMO)system is also considered as an alternative technique to enhance the spectrum efficiency.

Though the fundamental barriers in the field ofCR can be cleared both in time and frequency domains，space domain just presents a more distinctive capability of advancing the spectrum efficiency，enabling the overlap upon time as well as frequency of the primary users (PUs)and cognitive users (CUs).For further improvement on the spectrum efficiency，a series of advantages of MIMO technology， including array gain，diversity gain，and the multiplexing gain are employed in CR networks while technology such as beam forming，pre-coding technology［1］focuses on counteracting various interferences in the networks［2］.The cognitive MIMO system have been proposed in order to efficiently utilize radio spectrum which is a limited natural resource［3-4］and satisfy the demand for high data rate multimedia wireless services［5-6］.PU and CU coexist in MIMO CR system where data transmission is transmitted through MIMO antenna.As the increasing of the number of CUs and the enlargement of the system's throughput in cognitive MIMO system， an effective management method for interference is badly needed.Interference alignment (IA)［7-12］has been widely used in MIMO technique to restrain and even eliminate the interference between different users.

The basic principle of IA is to align multi-cell interference to a limited sub-space，then leave more interference-free space［13］for data transmission by designing a pre-processing matrix to signal transmission，and obtain the useful signal by post-processing matrix at the receiver.Reference［14］designed the pre-coding matrix of CU's transmitter to reduce the interference to PU.Although Ref.［14］allowed a simultaneous PU and CU access by keeping the orthogonality between them，it only discussed the PU's interference reduction and the degree of freedom that CU could attain and，it did not consider the transmission performance of the PU and CU.Reference［15］discussed the CR system concluding single PU and multiple CUs.It computes the gradient of rate formula of the CUs and acquires the CU's power allocation matrix and pre-coding matrix to align the interference caused by CUs to a sub-channel space.In Ref.［16］，the pre-coding matrices of CU's transmitter and receiver interference subspaces are jointly designed to minimize the interference outside the interference subspace so that the PU's receiver eliminates some interference，and it requires joint optimization on CU's transmitter and CU's receiver，where both the transmitter and the receiver are required to update their designs periodically within the iterative algorithm.What's more，they still fail to take into account the transmission performance of the CUs.

In this paper，a new interference alignment algorithm based on cognitive MIMO networks is proposed.The proposed IA algorithm is realized by designing two-level pre-coding，the first-level pre-coding aligns the interference generated by the CUs to unused sub-channels of the PU，thereby eliminating the interference of CUs to primary user;the second-level pre-coding is used to improve the throughput of CUs.Simulation results show the throughput of CUs improves by using the proposed algorithm.

1 System Model

Cognitive MIMO communication system is shown in Fig.1.We consider a PU and multiple CUs.CUs communicate by opportunistically accessing the sub-channes space［15］of PU.The CUs can cause interference to the PU and mutual interference among other CUs.We assume in this paper，that all terminals can get the perfect channel information.

Fig.1 Transmission model ofcognitive MIMO system

As shown in Fig.1，the transmitter and the receiver of PU are equipped with M antennas while all transmitters and receivers of CUs have N antennas.All of the transmission channels are Rayleigh flat-fading channels.The received signal of the primary user is represented as Eq.(1)，

where the M × M matrix H00represents the PU's transmission channel matrix，the M ×N matrix H0jrepresents the interference channel matrix，the M-dimensional vector x0represents the transmitted symbols of the PU，and the N-dimensional vectors xjrepresent the transmitted symbols of the jth CUs，K represents the number of CUs，and N0is the complex Gaussian noise，with zero mean and variance σ2.

2 The Proposed IA Algorithm

In cognitive MIMO network，PU has priority in the process of using frequency resources，the CUs communicate by taking the way of waiting for opportunity to get access to PU's licensed spectrum.A basic challenge is to maximize the throughput of CU while ensuring the transmission performance of the PU［1］.

Theproposed IA is achieved by designing the transmitter's pre-coding.The transmitter's pre-coding of the CUs not only considers the interference generated by themselves to PU，but also takes into account guaranteeing the performance of CUs.In PU's transmitter，a pre-coding matrix is designed to improve the transmission performance of the PU.In CUs' transmitter，a precoding matrix is designed to align the interference caused by CUs to the unused space sub-channel，which ensuring the CU's own transmission performance at the same time.

For H00，we can use singular value decomposition scheme，then get M parallel space sub-channel of PUs.

where the singular values of transmission channel matrix H00are λ01，λ02，…，λ0M，and meet λ01＞λ02＞… ＞λ0M;the M ×M diagonal matrix Λ0= diag(λ01，λ02，…，λ0M);V0is the conjugate transpose matrix of the right singular matrix;and U0is the left singular matrix of channel transfer matrix H00，where v01v02…v0Mand u01u02…u0Mare singular vectors corresponding to V0and U0.

In the transmitter of the PU，the power allocation for M parallel space sub-channel according to water-filling scheme，then the power allocation matrix P is achieved as Eq.(3)，

where the M × M matrix P is a diagonal matrix，pi=i represents the subscript of PU's space sub-channel，i = 1，2，…，M，σ2is the noise power，λiis the ith singular value of the transmission channel matrix H00decomposition，β is the Lagrange multipliers within water-filling scheme，function(x)+=max(x，0)means take the maximum of real numbers x and 0.

According to the results of the power distribution construct diagonal matrix Q，

In the power allocation matrix P，the elements pi≠0，i =1，2，…，M，corresponding to the space sub-channels are called space sub-channels which PU has used;the elements pi= 0，i = 1，2，…，M，corresponding to the space sub-channels are called space sub-channels which PU has unused.

Only considering PU subject to the interference caused by CUs，and taking V0as a pre-coding matrix in PU's transmitter，the pre-coding matrices of the jth CU's transmitters are designed as Vj，from Eq.(1)，the received signal y0by the PU is represented by Eq.(5)，

In terms of PUs，the essence of our design is an optimization problem，that is，in the case of a fixed total transmit power，how to design power distribution and pre-coding matrices，making the maximum throughput of the PUs.For CUs，in order to reduce PU's interference caused by CUs，take the way of waiting for opportunity to get access to PU's space sub-channel.With paying attention to improving the throughput of CUs，a two-level pre-coding matrix is designed in the CU's transmitter.The first-level pre-coding is used to achieve IA，and the secondlevel pre-coding is used to improve throughput of CU.For the jth CU，the pre-coding matrices Vjare designed with the method as follows.

First，for the jth CU，design the first-level pre-coding matrixin the transmitter to ensure the interference on the PU caused by the jth CU align to the space sub-channel which has not been used by PU in the process of information transmission，in order to avoid the interference channel matrix of unfilled rank，takingas the pseudo-inverse matrix of interference channel matrix H0j，and the pseudo-inverse matrix satisfies Eq.(6)，

Solving the first-level pre-coding matrixas Eq.(7)shows below:

Design the second-level pre-coding matrix in the transmitter of the jth CU to ensure CU's own transmission performance.Make singular value decomposition for the CU's own transmission channel matrix Hjj:

where the singular values of transmission channel matrix Hjjis λj1，λj2，…，λjN，and meet λj1＞λj2＞… ＞λjN，the M × M diagonal matrix Λj= diag(λj1，λj2，…，λjN)，and Vjis the conjugate transpose matrix of the right singular matrix，Ujis the left singular matrix of channel transfer matrix Hjj，where vj1vj2… vjNand uj1uj2… ujNare the singular vectors corresponding to Vjand Uj，respectively.

Taking Ujas the second-level pre-coding matrix in the jth CU's transmitter，we get the pre-coding matrix of the jth CU's transmitter:

3 Simulations Results

In this section，simulation results illustrate the efficiency of the proposed IA scheme.In our simulation，the channel system model is mentioned in Section 2.There is a PU (marked 0)and two CUs (marked 1 and 2)carrying on the transmission task.Suppose the numbers of antennas of PU and CU at the transmitter and the receiver are M and N，separately.All of the transmission channels are Rayleigh flat-fading channels.To simplify the problem，M and N are defined the value of 3 and the noise variance σ2is assumed to be 1;both the PU and CUs are in an identical signal to noise ratio (SNR)condition.

In the cognitive MIMO communication system， the throughput C'kof the kth user which doesn't use IA can be expressed as Eq.(10)，

The throughput Ckof the kth user which uses IA is expressed as Eq.(11)，

where the subscripts like k and r represent the sequence of users.It is necessary to notify here that k，r ∈{0，1，2 }，Pkis the power allocation matrix of the kth user as Prto the rth user.Vkand Vrare the pre-coding matrices of the kth and the rth，respectively.Hkkshows the transmission channel matrix of the kth user and Hkris introduced to note the interference channel matrix that is passed from the rth user to the kth user.What's more，Nkrepresents the noise matrix of channel of the kth user in the course of transmission.

In the simulation environment above，in order to highlight the superiority of the IA algorithm，the transmission capacity of the CUs is considered.The IA algorithm developed in this paper which considers CU's throughput is referred as method 1，the comparative traditional IA algorithm is referred as method 2.The pre-coding matrix is solved by aligning interference to the space expanded by an array of channel considering just the interference from CU to PU with method 2［7］.The users'throughput is calculated based on Eq.(11)，by replacing matrix V in Eq.(11)with pre-coding matrix of methods 1 and 2，respectively.

The simulations of the throughput for CUs calculated with methods 1 and 2 respectively，the simulation of the CU 1 and CU 2's throughput are shown in Figs.3 and 4，respectively，and the throughput of the CU system is also simulated in Fig.5.Here，the throughput calculation of CU system is expressed as Eq.(12)，

It is shown in Fig.2，with the increase of SNR，the throughput of the PU enlarges.The IA can improve PU throughput more effectively than without IA method，and the PU's throughput obtained by the proposed IA method and the traditional IA method are almost the same in Fig.2，that is because both methods allow for the PU's performance.Considering method 1 needs more channel information，method 1 is a little bit better than method 2.

Fig.2 Comparison of PU's throughput

Fig.3 Comparison of CU1's throughput

Fig.4 Comparison of CU2's throughput

Fig.5 Comparison of CU system's throughput

Figures 3-5 demonstrate that the throughput of CU grows along with the increasing of SNR.The capacity peaks when the SNR changes from 7 to 10 dB.However，a descending tendency appears if there is a continuous increasing on SNR.The reasonable explanation is that，as the SNR grows in CR，the channel quality of PU improves，and the probability of CU's opportunistic spectrum access to the PU's channel reduces，which leads to a decline on the CU's throughput.In Figs.3 and 4，although the two CUs play the same role，and CU1 and CU2 show different capacity on our illustration，for the channel matrix is random， it is reasonable that the throughput performances of CU1 and CU2 have difference.

4 Conclusions

The achievement of the paper can be briefly concluded as follows.In the cognitive MIMO system，the IA can be realized by the design of pre-coding at the PU's transmitter as well as the two-level pre-coding at the CU's transmitter and it weakens the interference that CU produces on the PU during transmission，guaranteeing the CU's own transferring performance at the same time.The result suggests that the proposed scheme is of effectiveness on cutting down the PU's interference caused by CU，prompting not only the transmission capability of PU，but also the throughput of the CU by applying on the two-level precoding.But in terms of the complexity，we can see，when the number of users is K，the number of antennas is N.Because the proposed method in this paper has a singular value decomposition，its complexity is expressed as O (KN3).Compared with the traditional IA algorithm with the complexity of O(KN).The proposed method will bring more complexity，but it improves the throughput of the CU.In next stage，we will take into consideration the interference that the PU produces on CU and emerges inside the CU group，besides，we will study into the circumstance where channel state information is partially attained.

［1］Jung M，Hwang K，Choi S.Interference Minimization Approach to Precoding Scheme in MIMO-Based Cognitive Radio Networks［J］.Communications Letters，IEEE，2011，15(8):789-791.

［2］Li M M.Research on Key Techniques in Cognitive MIMO Wireless Networks［D］.Beijing:Beijing University of Posts and Telecommunications，2011.(in Chinese)

［3］Akyildiz I F，Lee W Y，Vuran M C，et al.A Survey on Spectrum Management in Cognitive Radio Networks［J］.Communications Magazine，IEEE，2008，46(4):40-48.

［4］Haykin S.Cognitive Radio: Brain-Empowered Wireless Communications［J］.Selected Areas in Communications，IEEE Journal on，2005，23(2):201-220.

［5］Zhang R， Liang Y C.Exploiting Multi-antennas for Opportunistic Spectrum Sharing in Cognitive Radio Networks［J］.Selected Topics in Signal Processing，IEEE Journal of，2008，2(1):88-102.

［6］Sridharan S，Vishwanath S.On the Capacity of a Class of MIMO Cognitive Radios［J］.Selected Topics in Signal Processing，IEEE Journal of，2008，2(1):103-117.

［7］Cadambe V R，Jafar S A.Interference Alignment and Degrees of Freedom of the-User Interference Channel［J］.Information Theory，IEEE Transactions on，2008，54(8):3425-3441.

［8］Zhou H，Ratnarajah T，Liang Y C.On Secondary Network Interference Alignment in Cognitive Radio［C］.New Frontiers in Dynamic Spectrum Access Networks (DySPAN)，2011 IEEE Symposium on，Aachen，Germany，2011:637-641.

［9］Maddah-Ali M A， Motahari A S， Khandani A K.Communication over MIMO X Channels:Interference Alignment，Decomposition，and Performance Analysis［J］.Information Theory，IEEE Transactions on，2008，54(8):3457-3470.

［10］Abdelhamid B，Elsabrouty M，Elramly S.Novel Interference Alignment in Multi-secondary Users Cognitive Radio System［C］.Computers and Communications (ISCC)，2012 IEEE Symposium on，Cappadocia，Turkey，2012:000785-000789.

［11］Farhadi H， Zaidi A A， Fischione C， et al.Distributed Interference Alignment and Power Control for Wireless MIMO Interference Networks with Noisy Channel State Information［C］.Communications and Networking (BlackSeaCom)，2013 First International Black Sea Conference on，Batumi，Georgia，2013:23-27.

［12］Zhao L，Biao S，Li J D.Interference Alignment and Cancellation Based Concurrent Transmission and Scheduling Scheme for Multiuser CR-MIMO System［J］.China Communications，2013，10(8):36-43.

［13］Amir M， El-Keyi A， Nafie M.Constrained Interference Alignment and the Spatial Degrees of Freedom of MIMO Cognitive Networks［J］.Information Theory，IEEE Transactions on，2011，57(5):2994-3004.

［14］Perlaza S M，F(xiàn)awaz N，Lasaulce S，et al.From Spectrum Pooling to Space Pooling:Opportunistic Interference Alignment in MIMO Cognitive Networks［J］.Signal Processing，IEEE Transactions on，2010，58(7):3728-3741.

［15］Sun Q Y，Tian H，Dong K.Dynamic Resource Allocation with QoS Guaranteed in MIMO-OFDM System［J］.Journal of Electronics amp Information Technology，2009，31(1):41-44.(in Chinese)

［16］Du H Q，Ratnarajah T，Zhou H C，et al.Interference Alignment for Peer-to-Peer Underlay MIMO Cognitive Radio Network［C］.Signals， Systems and Computers (ASILOMAR)， 2011 Conference Record of the Forty Fifth Asilomar Conference on，Pacific Grove，CA，USA，2011:349-353.