Iterative Transceiver Design for Opportunistic Interference Alignment in MIMO Interfering Multiple-Access Channels

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1 Journa of Communications Vo. 0 No. February 0 Iterative Transceiver Design for Opportunistic Interference Aignment in MIMO Interfering Mutipe-Access Channes Weipeng Jiang ai Niu and Zhiqiang e Schoo of Information and Communication Engineering Beiing University of Posts and Teecommunications Beiing 0087 China Emai: wpqty@gmai.com; {niuai hezq}@bupt.edu.cn Abstract In this paper we propose an iterative transceiver design agorithm for opportunistic interference aignment in MIMO interfering mutipe-access channes. The proposed agorithm iterativey optimizes the transmit beamforming vectors the receive matrices and the user seection set which digs into the mutiuser diversity gain. Specificay data transmission can be operated in each iteration using the updated transceiver and the seected user set which means that no additiona processing deay is introduced. Simuation resuts demonstrate the improved average rate per ce performance of the proposed agorithm compared to the conventiona ones. Index Terms Interference aignment OIA user scheduing iterative I. INTRODUCTION In the wireess communication networs how to share the imited bandwidth among mutipe users is the primary chaenge. Interference aignment (IA) as a promising technique to eiminate the interference and achieve the optima degrees of freedom (DoF) aso nown as mutipexing gain has shown the bandwidth avaiabe to each user can be significanty improved. Different interference aignment agorithms are studied for various scenarios. For the user interfering channe it is proved in [] that each user can get haf of the interference free channe capacity even though the number of users can be arbitrariy arge. For the mutipe-input mutipe-output (MIMO) X networ [] the DoF outer bound is derived where every transmitter has an independent message for every receiver. In [] subspace interference aignment is proposed for ceuar networs. owever the optima DoF gain is usuay achieved with goba channe state information (CSI) which is not practica. Thus imited-feedbac interference aignment agorithms ie [] and bind interference aignment agorithms ie [] have been proposed. Nevertheess these agorithms either need the feedbac bits to scae fast or require many time frequency or space domain extensions. Manuscript received November 9 0; revised February 0. This wor is supported by Nationa igh-tech R&D Program of China (8 Program) (SS0AA00) Nationa Natura Science Foundation of China ( ) uawei Technoogies Co. Ltd. Proect (YB0008). Corresponding author emai: wpqty@gmai.com. doi:0.70/cm In []-[8] opportunistic interference aignment (OIA) is proposed for MIMO interfering mutipe-access channes (IMAC). Different from the above interference aignment methods OIA taes advantage of the mutiuser diversity (MUD). Ony partia CSI is needed and there is no requirement for time frequency or space domain extensions. Each user obtains its beamforming vector by minimizing the eaage of interference (LIF) and the optima DoF can be achieved through user seection ony if the user number scaes fast enough. Simiar OIA schemes have been proposed in [9] and [0] for the downin transmission. Athough OIA is DoF optima as SNR (signa to noise ratio) approaches infinity it is not optima in the sense of sum rate especiay at intermediate SNR vaues. In [] the sum rate is enhanced by considering the efficient signa in based on the signa to eaage and noise ratio (SLNR). In [] an energy efficient agorithm with power constraints at the transmitter is proposed. Instead of optimizing each ce's achievabe DoF [] gives priority to increasing the possibiity for perfect IA in one ce (named as active OIA). When the number of users does not meet the scaing condition a the ces are interference-imited for the other schemes above. owever the average rate per ce (ARPC) for the active OIA scheme can sti have inear growth with the SNR because one ce is perfecty interference aigned with no interference eft. Obviousy fairness among ces is sacrificed. The ey idea of OIA is to get the mutiuser diversity gain. owever the conventiona OIA schemes mentioned above have potentia shortcomings in user seection. Theoreticay the user seection shoud be optimized ointy with the transmit beamforming vectors and the receive matrices. owever user seection in a the above methods depends on the receive matrices which are randomy generated. Athough [] attempts to optimize the receive matrices after user seection the user set is sti not changed. ence the seected users are actuay those who match the given signa space which is not optima for achieving the MUD gain. In []-[] iterative interference aignment agorithms are proposed for -user interference channe which achieve much gain in terms of sum rate. owever these iteration agorithms cannot be used for OIA in MIMO IMAC because no user scheduing is considered. 0 Engineering and Technoogy Pubishing 0

2 Journa of Communications Vo. 0 No. February 0 In both agorithms the user set shoud be decided before the cacuation of both the transmit beamforming vectors and the receive matrices. In this situation ony the beamforming vectors of the seected users can be updated. Thus even with the updated transmit beamforming vectors the seected user set wi not change in the iteration. In this paper we propose an iterative transceiver design and user seection agorithm for OIA in MIMO IMAC channes. Firsty at the transmitter side SLNR maximization is adopted as the design criterion and both the transmit beamforming vectors and the seected user set are optimized. Secondy at the receiver side SINR maximization is adopted as the performance criterion and the receive matrices are optimized. The two steps are iterativey executed unti converging or reaching the maximum number of iterations. Extensive simuations show the improved performance in terms of ARPC compared with the conventiona schemes. The rest of this etter is organized as foows. In Section the mode of the -ce upin MIMO IMAC system is introduced. The proposed iterative transceiver design and user scheduing agorithm is deveoped in Section. In Section numerica exampes are provided to show the improved performance of the proposed agorithm over the conventiona schemes. Concusions are drawn in Section. II. SYSTEM MODEL We consider a ce MIMO IMAC woring in TDD mode as shown in Fig.. Every ce has one base station (BS) equipped with M antennas and N users each equipped with L antennas. In each ce ony S users ( S M ) out of the N users are seected to transmit signas simutaneousy. The seected user set of ce is defined as I { S} where i { N} denotes the i th seected user in ce. It is assumed that every seected user transmits a singe spatia stream. Each BS ony decodes the signas from the seected users in its own ce and treats the signas from other ces as interference. In Fig. the interfering ins and the effective signa ins are denoted using dotted and soid ines respectivey. BS Ce M Rx S seected users/ce L Tx N users/ce Fig.. ce MIMO IMAC mode. L Tx M Rx BS S seected users/ce N users/ce Ce We use i which is an M L Gaussian random matrix with i.i.d. zero mean and unit variance entries to denote the fading channe matrix between the th user in the i th ce and the BS of the th ce. v i is the beamforming vector with unit energy for user in the ith ce. The transmit message of user in the i th ce is denoted by m i. We assume that the average power of the transmit message at each transmitter is where E( m m ) P * i i * () stands for the compex conugate. The seected users in different ces transmit in the same time using the same frequency band. Thus the received signa vector at the th BS can be written as S S m i i mi i i () y v v n where n denotes the M additive noise vector each eement of which is an i.i.d. compex Gaussian random variabe with zero mean and unit variance. Thus the SNR at each transmitter is defined identicay as SNR P / P. The expression v m is the S desired signa part from the S seected users in ce. S The expression i vi mi is the interference from i i the seected users of other ces. Using a inear receiver the estimated signa vector is given by r [ r r ] U y () T S where U is the M S receive matrix for BS. T () and () denote matrix (vector) transpose and ermitian transpose respectivey. U [ U U S ] is fu coumn ran in other words the signa space has S dimensions. Each coumn U i I has unit energy and corresponds i to the receive vector of the seected user i. The conventiona OIA scheme is proposed for achieving the optima DOF but its ARPC performance can be further improved. A gobay optima agorithm shoud ointy optimize the receive matrices the beamforming vectors and the seected user set together to get fu MUD gain. The sum rate optimization probem can be formuated as arg max og( SINR ) U v I I where SINR is the signa to interference pus noise power ratio of the th user in ce which can be written as Eq. ()-() U D U () U C U SNR SINR () C v v ( ) () M i i i ( i ) i D I v v SNR I () v v ( ) SNR 0 Engineering and Technoogy Pubishing 0

3 Journa of Communications Vo. 0 No. February 0 where I M denotes the M M identity matrix. D is the correation matrix of the interference and the noise. III. TE PROPOSED ITERATIVE TRANSCEIVER DESIGN AND USER SCEDULING ALGORITM In this section we present our iterative transceiver design and user scheduing agorithm. The probem Eq. () is nonconvex and the coection variabes I mae it more difficut to sove. Moreover the receive matrices beamforming vectors and the seected user set of different ces or users are highy couped which further maes the gobay optima soution intractabe. To get a nearoptima soution the iteration scheme is worth considering. The critica question is how to decoupe the users' beamforming vectors and the BSs' receive matrices. The proposed agorithm is a SLNR-SINR bi- criteria combined iteration agorithm. Compared with the agorithms in [] and [] the proposed agorithm has better performance for the oint optimization of the beamforming vectors the receive matrices and the user seection. Compared with the iteration agorithms using ony SINR criteria the proposed agorithm has ower compexity because each user can optimize its own beamforming vectors using SLNR criteria without need of iterations among different users. Besides the seected user set can be easiy updated in the iteration process. At the transmitter side given the receive matrices of each ce each user cacuates the optima beamforming vector that maximizes its SLNR. At the receiver side each BS seects the users of its ce according to the reported SLNRs. Given the beamforming vectors of each user each BS aso optimizes the receive matrices by maximizing the SINR. Iterations are carried out unti convergence or unti the maximum number of iteration. The seected user set the beamforming vectors and the receive matrices are a updated in each iteration. This way every BS can optimize its U and I independenty. Every user can optimize its v without nowing the beamforming vectors of the other users. In the foowing we give the agorithm steps as foows: Step. Each BS randomy generates and broadcasts the M S receive matrix U the coumns of which are ineary independent unit vectors. S and M are defined in Section II. Set SumRate 0 0 which is the initiaized sum rate of a the ces; The broadcast of the receive matrix is ony performed once for the initiaization. Start iteration. Step. Each user cacuates the SLNR according to Eq. (7)-(9) and gets the optima transmit beamforming vector by optimizing Eq. (0). Each user sends the computed SLNR to its ce s BS. Detais are as foows: The th user in the i th ce can cacuate its SLNR independenty ony using its own transmit beamforming vector v i. The SLNR is written as Eq. (7)-(9) v A v SNR SLNR (7) i i i i vi Bi vi A ( ) U U (8) i i i i i i i i L ( i ) i i B I U U SNR (9) where I L denotes the L L identity matrix. i i is the effective data in between the th user in the i th ce and the i th BS. i i. is the eaage signa in between the th user in the i th ce and the th BS. The overa procedure of our protoco is based on the channe reciprocity of TDD systems. Due to the channe reciprocity the receive matrices can be obtained using downin piot signaing. Because the optima receive matrices in the upin are aso the optima transmit matrices in the downin the expression U i i i can be i fuy estimated as equivaent channe ( i ) Ui in the downin transmission. So according to Eq. (7)-(9) and the equivaent channe estimation the optimization obective can be written as arg max SLNR (0) v i which is a generaized Rayeigh quotient maximization probem []. The maxima vaue is the argest generaized eigenvaue of A B i i and the optima v i is the corresponding eigenvector. Step. Each BS seects the S users who have the S argest SLNRs. Using Eq. ()-() each BS cacuates the SINRs for the seected users and gets the new receive vector U i I for the corresponding seected user by i optimizing Eq. (). Detais are as foows: In Eq. ()-() is the effective data in between the th user in the th ce and the th BS. i i i is the interference in from the th user in the i th ce to the th BS. The beamforming vectors can be obtained by the BS in the same way during upin transmission that is v is fuy estimated as the equivaent channe. So according to Eq. ()-() and the equivaent channe estimation the optimization obective can be written as U i arg max SINR () which is aso a generaized Rayeigh quotient maximization probem. The maxima vaue is the argest generaized eigenvaue of C D and the optima U is the corresponding eigenvector. Step. Compute the new sum rate SumRate if SumRate SumRate 0 update the seected user set et SumRate 0 SumRate and go to step (); otherwise iteration ends. 0 Engineering and Technoogy Pubishing 0

4 Journa of Communications Vo. 0 No. February 0 Detais are as foows: Sum rate of a the ces is cacuated using Eq. () S S () R R og( SINR ) sum Some sort of coaboration is needed for the BSs to cacuate the sum rate (though the overhead of coaboration is minima). For exampe the quantified rate information can be transmitted using the X interface in LTE systems. Remar: Precise vaues of the SLNR are not required to feed bac for comparison which eaves much space for feedbac compression. It is noted that a configurations of the transceiver such as the receive matrices the beamforming vectors and the seected user set shoud be updated in each iteration. The iterative agorithm is an onine process that is a the optima soutions in the current iteration can be utiized to configure the transceiver in the next transmission. In the iteration process given the seected users and the beamforming vectors the updated receive matrices wi achieve higher SINRs which mean better ARPC performance. Given the receive matrices the updated seected users and beamforming vectors can ensure higher SLNRs but it's not aways better for ARPC performance because the SLNR metric is not optima in terms of capacity. Nevertheess using the SLNR metric is hepfu to decoupe the beamforming vectors of the different users effectivey which simpifies the iteration progress. Moreover it has better performance than other metrics such as the LIF metric. The average power between every user and every base station is assumed to be equa. Simiar with the conventiona OIA agorithms we assume perfect oca CSI estimation and that the channes remain constant throughout the duration of the operation. Once the agorithm has converged each base station woud be in contact with ony S seected users per ce. If the channe matrices are constant the eft N S users wi not be served unti channes change. IV. NUMERICAL RESULTS In this section we study the performance of the proposed iterative transceiver design and the user scheduing agorithm through numerica simuations. For the convenience of comparison with the conventiona agorithms we choose the same simuation parameters as those in [] or [] as foows: We consider a ce MIMO IMAC woring in TDD mode as shown in section II. is or for different degrees of interference. There are M antennas at each BS and L antennas at each user. We set the noise power to and the transmit SNR changes with the signa power. For the sae of simpicity we choose S in the simuation. ARPC is used as the overa performance metric which is defined S as R E( R / ) where R og( SINR ). We compare the performance of our proposed agorithm with the foowing three agorithms: ) the SLNR metric based agorithm in [] with ZF receiver (referred to as the SLNR-ZF agorithm); ) the SLNR metric based agorithm with optima receiver (referred to as the SLNR-Opt agorithm); ) the agorithm named CATB-Sb-Opt Rx in [] (referred to as the active OIA agorithm). A simuation resuts are averaged over 0 independent channe reaizations. Average Rate per Ce (bps/z) Average Rate per Ce (bps/z) = SNR(dB) (a) = proposed agorithm active-oia agorithm SLNR-Opt agorithm SLNR-ZF agorithm proposed agorithm active-oia agorithm SLNR-Opt agorithm SLNR-ZF agorithm SNR(dB) (b) Fig.. ARPC vs SNR for different numbers of ces. Number of users N=0. In the first simuation exampe of Fig. we compare the ARPC performance versus the SNR of the four agorithms. Fig. shows that the proposed agorithm has better performance than the conventiona agorithms at a SNRs for the mid interference case with. For strong interference case with the proposed agorithm is sti better at ow-to-medium SNR ( SNR 0 db ) whie the active OIA agorithm performs better at SNR higher than 0dB. This can be expained as foows. At SNR 0dB the interference instead of noise is the ey factor which affects the performance. The interference can be aigned simutaneousy in the three ces when and then the interference can be fuy 0 Engineering and Technoogy Pubishing 0

5 Journa of Communications Vo. 0 No. February 0 eiminated. So SINR is equivaent to SNR in this situation and the rate increases with SNR at high SNR region. owever there is no soution to aign the interference in four ces simutaneousy which means that the interference cannot be fuy eiminated. In this situation at high SNR the noise is negigibe compared with the interference. Thus the ARPC performance of the proposed agorithm reaches a foor at high SNR. owever the ARPC of the active OIA agorithm can sti have inear growth with the SNR because one of the four ces is given priority to aign interference perfecty. The average rate of this ce increases which maes the ARPC of the four ces increase. SNR=dB Average Rate per Ce (bps/z) Average Rate per Ce (bps/z) Number of Users (a) SNR=0dB proposed agorithm active-oia agorithm SLNR-Opt agorithm SLNR-ZF agorithm Number of Users (b) proposed agorithm active-oia agorithm SLNR-Opt agorithm SLNR-ZF agorithm Fig.. ARPC vs number of users for different SNR. Number of ces =. In the second simuation exampe shown in Fig. we estimate the ARPC performance versus the number of users with mid interference ( ). It is demonstrated that the proposed agorithm performs better than its counterparts under a numbers of users for both medium SNR db and high SNR 0 db which transates to arger MUD gain. The third simuation exampe of Fig. shows the convergence performance of the proposed agorithm. The ARPC performance for the first iteration is equivaent to the SLNR-Opt agorithm which aso proves the reiabiity of the simuation resuts. It is seen that the ARPC improves rapidy at the beginning which demonstrates the gain from iteration. More iterations are needed at higher SNR and the agorithm approximatey approaches the fina performance after seven iterations in most cases. Average Rate per Ce (bps/z) Average Rate per Ce (bps/z) dB 0dB 0dB = dB 0dB 0dB Iteration Number (a) = Iteration Number (b) Fig.. ARPC vs number of iterations for different number of ces. Number of users N=0. Through the simuation resuts above we can concude that the proposed agorithm outperforms the conventiona agorithms in most cases. Athough the active OIA agorithm performs better at SNR 0 db with which means better DoF gain the proposed agorithm is more practica because the SNR is usuay sma than 0dB under the wireess fading channe. The signaing overheads of the four agorithms are amost the same except that the proposed agorithm and the active OIA agorithm need some sort of minor coaboration between the BSs. Finay we compare the computationa compexity of the agorithms. For the sae of notationa simpicity we assume the antenna number at a nodes equas to N. In each of the simuated agorithms the compexity order of a 0 Engineering and Technoogy Pubishing 0

6 Journa of Communications Vo. 0 No. February 0 one basic cacuation is O(( N ) ) which mainy invoves matrix mutipication and matrix eigen vaue decomposition. The compexity of the proposed agorithm is higher than the other three agorithms for the need of iteration. In Fig. and Fig. the iteration number is eight. But the average compexity of the agorithm is not ie eight times high because the optimized beamforming vectors receive matrices and user seection set in each iteration can be used for the data transmission in the next iteration. V. CONCLUSIONS In this paper an iterative transceiver design and user scheduing agorithm is proposed which further digs into the mutiuser diversity. The proposed agorithm is a SLNR-SINR bi-criteria combined iteration agorithm which ointy optimizes the transmit beamforming vectors the receive matrices and the user seection set. Athough iteration is needed between the users and the BSs no extra processing deay is introduced. The optimization resuts in each iteration can be used for the transmission of the next iteration which is important for practica appications. Simuation resuts show performance gains to the conventiona agorithms. ACNOWLEDGMENT The authors woud ie to than the editor and anonymous reviewers for their vauabe comments and suggestions that improved the quaity of the paper. a REFERENCES [] V. Cadambe and S. A. Jafar Interference aignment and degrees of freedom of the -user interference channe IEEE Trans. Inf. Theory vo. pp. - Aug [] V. Cadambe and S. A. Jafar Interference aignment and the degrees of freedom of wireess X networs IEEE Trans. Inf. Theory vo. no. 9 pp Sept [] C. Suh and D. Tse Interference aignment for ceuar networs in Proc. Aerton Conf. Commun. Contro and Computing Sept. 008 pp [] X. Rao L. Ruan and V.. N. Lau Limited feedbac design for interference aignment on MIMO interference networs with heterogeneous path oss and spatia correations IEEE Trans. Signa Process. vo. no. 0 pp May. 0. [] T. G. Gou C. W. Wang and S. A. Jafar Aiming perfecty in the dar-bind interference aignment through staggered antenna switching IEEE Trans. Signa Process. vo. 9 no. pp. 7-7 June 0. [] B. C. Jung and W. Y. Shin Opportunistic interference aignment for interference-imited ceuar TDD upin IEEE Commun. Lett. vo. pp. 8-0 Feb. 0. [7] B. C. Jung D. Par and W. Y. Shin Opportunistic interference mitigation achieves optima degrees-of-freedom in wireess muti-ce upin networs IEEE Trans. Commun. vo. 0 no. 7 pp. 9-9 Juy. 0. [8]. J. Yang W. Y. Shin B. C. Jung and A. Paura Opportunistic interference aignment for MIMO interfering mutipe-access channes IEEE Trans. Wire. Commun. vo. no. pp May 0. [9] T. M. Nguyen T. Q. S. Que and. Shin Opportunistic interference aignment in MIMO femtoce networs in Proc. IEEE Internationa Symposium on Information Theory Proceedings Juy 0 pp. -. [0]. J. Yang W. Y. Shin B. C. Jung and C. Suh Opportunistic interference aignment for MIMO interfering broadcast channes in Proc. IEEE Internationa Conference on Acoustics Speech and Signa Processing May 0 pp. -9. [] S.. ur B. C. Jung and B. Rao Sum rate enhancement by maximizing SGINR in an opportunistic interference aignment scheme in Proc. Asiomar Conf. Pacific Grove CA Nov. 0 pp. -8. [] J. Yoon W. Y. Shin and. S. Lee Energy-efficient opportunistic interference aignment IEEE Commun. Lett. vo. 8 no. pp. 0- Jan. 0. []. Gao J. Leithon C. Yuen and. A. Suraweera New upin opportunistic interference aignment: An active aignment approach in Proc. Wireess Communications and Networing Conference Apr. 0 pp []. R. umar and F. Xue An iterative agorithm for oint signa and interference aignment in Proc. IEEE Internationa Symposium on Information Theory Proceedings Jun. 00 pp []. Gomadam V. R. Cadambe and S. A. Jafar A distributed numerica approach to interference aignment and appications to wireess interference networs IEEE Trans. Inf. Theory vo. 7. no. pp. 09- Jun. 0. [] R. orn and C. Johnson Matrix Anaysis New Yor: Cambridge University Press 990. Weipeng Jiang received the B.Eng. degree from Beiing University of Posts and Teecommunications Beiing China in 00. e is currenty woring towards the Ph.D. degree in the Schoo of Information and Communication Engineering Beiing University of Posts and Teecommunications Beiing China. is current research interests incude wireess communications cooperative communication systems and interference aignment. ai Niu received a B.S. degree in information engineering and a Ph.D. in signa and information processing from Beiing University of Posts and Teecommunications (BUPT) Beiing China in 998 in 00 respectivey. Currenty he is an associate professor in the Schoo of Information and Communication Engineering of BUPT. is research interests are in the area of channe coding and broadband wireess communication particuary on the practica design of poar codes and study of poar decoding agorithms. Zhiqiang e received the B.E. degree and Ph.D. degree (with distinction) from Beiing University of Posts and Teecommunications China a in signa and information processing in 999 and 00 respectivey. Since Juy 00 e has been with the Schoo of Information and Communication Engineering Beiing University of Posts and Teecommunications where he is currenty an Associate Professor and the director of the Center of Information Theory and Technoogy. is research interests incude signa and information processing in wireess communications networing architecture and protoco design and underwater acoustic communications. 0 Engineering and Technoogy Pubishing 0