Cognitive MIMO Relaying with Multiple Primary Transceivers

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Cognitive MIMO Relaying with Multiple Primary Transceivers Yeoh P.L.; Elkashlan M.; Kim K.J.; Duong T..; Karagiannidis G.K. TR23-8 December 23 Abstract We examine the impact of clusters of primary transceivers in cognitive multiple-input multipleoutput MIMO relay networks with underlay spectrum sharing. In such a network we propose antenna selection as an interference-aware design to satisfy the power constraints in the primary and secondary networks. To demonstrate this we consider transmit antenna selection with maximal ratio combining TAS/MRC in the primary and secondary networks. With this in mind we derive new closed-form asymptotic expressions for the outage probability and the symbol error rate SER over independent Nakagami-m fading channels. Our results lead to several new fundamental insights. In particular we highlight that TAS/MRC achieves a full diversity gain when the maximum transmit power in the secondary network is proportional to the peak interference temperature in the primary network. IEEE Global Communicatiions Conference GLOBECOM This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying reproduction or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories Inc Broadway Cambridge Massachusetts 239

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3 Cognitive MIMO Relaying with Multiple Primary Transceivers Phee Lep Yeoh Maged Elkashlan Kyeong Jin Kim Trung. Duong and George K. Karagiannidis Department of Electrical and Electronic Engineering University of Melbourne Australia ueen Mary University of London London United Kingdom Mitsubishi Electric Research Laboratories 2 Broadway Cambridge MA Blekinge Institute of Technology Karlskrona Sweden Aristotle University of Thessaloniki Thessaloniki Greece phee.yeoh@unimelb.edu.au maged.elkashlan@eecs.qmul.ac.uk kkim@merl.com quang.trung.duong@bth.se geokarag@auth.gr Abstract We examine the impact of clusters of primary transceivers in cognitive multiple-input multiple-output MIMO relay networks with underlay spectrum sharing. In such a network we propose antenna selection as an interference-aware design to satisfy the power constraints in the primary and secondary networks. To demonstrate this we consider transmit antenna selection with maximal ratio combining TAS/MRC in the primary and secondary networks. With this in mind we derive new closed-form asymptotic expressions for the outage probability and the symbol error rate SER over independent Nakagami-m fading channels. Our results lead to several new fundamental insights. In particular we highlight that TAS/MRC achieves a full diversity gain when the maximum transmit power in the secondary network is proportional to the peak interference temperature in the primary network. I. INTRODUCTION Cognitive spectrum sharing in cooperative networks is a revolutionary paradigm to support the proliferation of wireless devices and combat spectrum scarcity []. From a power perspective cognitive spectrum sharing with network cooperation addresses fundamental constraints on the transmit power at the secondary users SUs while keeping the interference temperature at the primary users PUs to a minimum [2]. Against this background how to manage the transmit power relative to the interference temperature in cognitive spectrum sharing with multiple antennas at the primary and secondary users remains an open question. A common approach to cognitive spectrum sharing is the underlay model in which the SUs are permitted to transmit over the same spectrum as the PUs [3]. In underlay spectrum sharing the transmit power at the SUs must be managed under a peak interference temperature to guarantee reliable communication between the PUs. Under this paradigm majority of works in the literature have considered single antennas at the primary and secondary users. Among them the exact outage probability of cognitive spectrum sharing was derived in [4] for decode-and-forward DF relaying in Nakagami-m fading. Considering amplify-and-forward AF relaying in Rayleigh fading the exact outage probability was examined in [5]. In [6] AF relaying was found to achieve a full diversity gain in Nakagami-m fading when the transmit power at the SU transmitters SU-Tx is managed according to the peak interference temperature of a single PU receiver PU-Rx. More recently in [7] the impact of multiple PU-Rx on the outage probability was addressed. To do so the transmit power at the SU transmitters SU-Tx was adapted to the maximum interference constraint arising from the multiple PU-Rx. In [8] [9] the impact of interference from PU transmitters PU- Tx on the SU receivers SU-Rx was considered. In [] several relay selection and power allocation constraints were proposed. In this paper different from [4] [] we investigate cognitive spectrum sharing from the viewpoint of multiple-input multiple-output MIMO in the primary and secondary networks. In the primary network we consider a cluster of L PU-Tx transmitting to a cluster of L PU-Rx with N antennas at each of the PUs. In the secondary network we assume N S N R and N D antennas at the source relay and destination respectively. To harness the diversity benefits of MIMO we consider transmit antenna selection with receive maximal-ratio combining TAS/MRC in both the primary and secondary networks. In TAS/MRC a single antenna at the transmitter is selected for transmission and all the antennas at the receiver are combined using MRC. In this setting we seek to address the impact of three cognitive design parameters namely maximum transmit power at the SU- Tx P T 2 peak interference temperature at the PU-Rx and 3 interference power at the SU-Rx due to the PU-Tx P I. In doing so we derive new closed-form expressions for the outage probability with Nakagami-m fading in the primary and secondary networks. Based on this we derive the symbol error rate SER under the higher order constellations of M- ary quadrature amplitude modulation M-AM and M-ary phase-shift keying M-PSK. Our results are given in terms of concise asymptotic expressions which accurately characterize the network performance when P T grows large. II. NETWORK AND CHANNEL DESCRIPTION We consider a cognitive MIMO relay network as illustrated in Fig.. The primary network comprises L PU-Tx transmitting to L PU-Rx each equipped with N antennas.

4 Fig.. Cognitive spectrum sharing with MIMO in the primary and secondary networks. The secondary network consists of a secondary source S a secondary relay R and a secondary destination D equipped with N S N R and N D antennas respectively. In the secondary network G denotes the N S N R channel for the S R link and G 2 denotes the N R N D channel for the R D link. The interference channels from the SU- Tx to the PU-Rx are represented by: H l which denotes the N S N channel from S to the l-th PU-Rx and 2 H 2l which denotes the N R N channel from R to the l-th PU-Rx. The interference channels from the PU-Tx to the SU-Rx are represented by: H 3l which denotes the N N R channel from the l-th PU-Tx to R and 2 H 4l which denotes the N N D channel from the l-th PU-Tx to D. The channel coefficients in G G 2 H l H 2l H 3l and H 4l follow a Nakagami-m distribution with independent fading parameters m g m g2 m h m h2 m h3 m h4 and channel powers Ω g Ω g2 Ω h2 Ω h3 Ω h4 respectively. In the following is the Euclidean norm is the absolute value and E[ ] is the expectation. We now detail the interference-limited scenario with TAS/MRC in the primary and secondary networks. In the S R link a single antenna at S that maximizes the signalto-interference ratio SIR at R is selected to transmit whereas all the antennas at R are combined using MRC. In the R D link the signal received at R is decoded and forwarded to D. In doing so a single antenna at R that maximizes the SIR at D is selected to transmit whereas all the antennas at D are combined using MRC. In the primary network a single transmit antenna is selected at each PU-Tx and all the antennas at each PU-Rx are combined using MRC. According to underlay spectrum sharing the interference at the PU-Rx inflicted by S and R should not exceed a given We focus on the interference-limited scenario where the interference power is dominant relative to the noise and therefore the effects of noise is neglected. maximum tolerable level. Further to satisfying S and R are power-limited terminals such that the maximum allowable transmit power is P T. As such the transmit powers at S and R are constrained as [] and P S = min P T P R = min P T h l i 2 h 2l j 2 2 respectively. In due to MRC at the PU-Rx h l i = max l { [h li...h li N ] } is the largest channel vector from S to the L PU-Rx where h liq are channel coefficients in H l with l {...L} i {...N S } q {...N }. In 2 h 2l j = max l{ [h 2lj...h 2lj N ] } is the largest channel vector from R to thelpu-rx whereh 2ljq are channel coefficients in H 2l with l {...L} i {j...n R } q {...N }. Given the co-existence of transmissions in the primary and secondary networks the received signals at S and R are impacted by interference from the PU-Tx denoted by P I. As such the instantaneous received SIR in the S R link can be written as γ = P S g i 2 L l= P 3 I h 3lq 2. In 3 we define g i = max i { [g i...g inr ] } as the largest channel vector from S to R where g ij are channel coefficients in G with i {...N S } j {...N R }. We also define h 3lq = [h 3lq...h 3lq N R ] as the channel vector from a single transmit antenna at the l-th PU- Tx to R where h 3lqj are channel coefficients in H 3l with q {...N } and j {...N R }. In the R D link the instantaneous received SIR is given by γ 2 = P R g 2j 2 L l= P 4 I h 4lq 2. In 4 we define g 2j = max j { [g 2j...g 2jND ] } as the largest channel vector from R to D where g 2jk are channel coefficients in G 2 with j {...N R } k {...N D }. We also define h 4lq = [h 4lq...h 4lq N D ] as the channel vector from a single transmit antenna at the l-th PU- Tx to D where h 4lqk are channel coefficients in H 4l with q {...N } and k {...N D }. Finally the instantaneous end-to-end SIR with TAS/MRC and DF relaying is defined as [2] γ D = min γ γ 2. 5 III. OUTAGE PROBABILITY In this section we seek to address the impacts of the maximum transmit power at the SU-Tx P T the peak interference temperature at the PU-Rx and the interference power at the SU-Rx P I on the outage probability of cognitive MIMO relaying. The cognitive network is considered to be in outage

5 when γ D falls below a minimum threshold γ th. As such the outage probability is P out = Pr{γ D γ th } = F γd γ th 6 where F γd γ th is the cumulative distribution function cdf of γ D. Based on 5 the cdf of γ D is evaluated as F γd γ = F γ γ+f γ2 γ F γ γf γ2 γ 7 where F γ γ is the cdf of γ and F γ2 γ is the cdf of γ 2. We now present the asymptotic cdf of γ in the following theorem. Theorem : The asymptotic cdf of γ as P T is given by 8. Note that 8 contains easy-to-evaluate finite summations of the gamma function Γ [3 eq. 8.3.] and the upper incomplete gamma function Γ [3 eq ]. Proof: See Appendix A. The asymptotic cdf of γ 2 follows directly from 8 by exchanging the parameters ǫ ǫ 2 m h m h2 m h3 m h4 m g m g2 Ω h2 Ω h3 Ω h4 Ω g Ω g2 N S N R and N R N D. Applying the asymptotic cdf s of γ and γ 2 the asymptotic outage probability for the end-to-end SIR is presented as P out where mg N SN R mg N SN R Θ γpi γpi P + T m g N S N R < m g2 N R N D mg2 N RN D mg2 N RN D Θ γpi γpi 2 P + 2 T m g N S N R > m g2 N R N D Θ +Θ 2 γpi mg N SN R γpi m g N S N R = m g2 N R N D P T mg N SN R mg Ω mg N ǫ h3 SN R Ω g m Γmg h3 N S N R +m h3 N R L Θ = Γm g N R + NS Γm h3 N R L mg2 Ω mg2 N ǫ h4 RN D 2 Ω g2 m Γmg2 h4 N R N D +m h4 N D L Θ 2 = Γm g2 N D + NR Γm h4 N D L mg Ω L h Ω mg N h3 SN R m h Ω g m Γmg h3 N S N R +m h3 N R L = Γm h N Γm g N R + NS Γm h3 N R L L m h N [ L ar ] ar a r+ l l l= a r= l+ mg NSNR m h N m h N a r Γ m g N S N R +m h N + m h N a r l+m h P T 9 2 and mg2 Ω L h2 Ω mg2 N h4 RN D m h2 Ω g2 m Γmg2 h4 N R N D +m h4 N D L 2 = Γm h2 N Γm g2 N D + NR Γm h4 N D L L m h2 N [ L ar ] ar a r+ l l l= a r= l+ mg 2 NRND m h 2 N m h N 2 a r Γ m g2 N R N D +m h2 N + m h2 N a r l+m h 2 P T Ω h2. 3 Our result in 9 clearly highlights that the outage probability of TAS/MRC increases with the interference power at the SU-Rx P I. We also see that the outage probability decreases with increasing P T and. IV. SYMBOL ERROR RATE In this section we derive new asymptotic expressions for the SER of TAS/MRC under the higher order constellations of M-AM and M-PSK. In our closed-form expressions we define the ratio of the peak interference temperature to the maximum transmit power as µ = /P T. The asymptotic SER with M-AM is evaluated using [4] 6 Pe M = πm 2 6 M πm π 4 π 2 e γ a MAM e γ a MAM F γ D γdγdθ F γ D γdγdθ 4 where a MAM = 3/2M and M represents the constellation size. We substitute 9 into 4 and solve the nested integrals which results in P e Ξ TAS/MRC Γq +B M 2 3 M P q I P q T πm aq+ 2 +q 2 π Γ 2 +q 5 MAM where q = minm g N S N R m g2 N R N D Θ + m µ mg N S N R g N S N R < m g2 N R N D Θ Ξ TAS/MRC = m µ mg 2 N R N D g N S N R > m g2 N R N D Θ +Θ µ mg N S N R m g N S N R = m g2 N R N D. 6 and B z ab = z ta t b dt is the incomplete beta function. The asymptotic SER with M-PSK can be derived as [4] P e = a MPSK π τ e γ a MPSK F γ D γdγdθ 7

6 F γ γ PT L + l= L l ǫ Γm g N S N R +m h3 N R L γmg P I Ω h3 Γm g N R + NS Γm h3 N R L P T Ω g m h3 l LΓm g N S N R +m h3 N R L γmg P I Ω h3 Γm h N Γm g N R + NS Γm h3 N R L l+ mg NSNR m h N m h N a r Γ mg N SN R m h Ω g m h3 m g N S N R +m h N + mg N SN R m h N m h N [ ar a r= ] ar a r+ a r l+m h P T. 8 Fig. 2. Outage probability with N S = N D = L = N = 2 = 2P T and P I = 5 db. Fig. 3. SER of 6-AM with N S = N D = N R = 2 L = N = 2 = 2P T and P I = 3 db. where τ = πm /M and a MPSK = sin 2 π/m. Substituting 9 into 7 and solving the nested integrals results in P e Ξ TAS/MRC Pq I Γq + πγ P q 2 +q T πaq MPSK 2Γ+q + a MPSK 2 F 2 2 q; 3 2 ; a MPSK. 8 Our new closed-form expression for the SER of TAS/MRC is given in terms of the Gauss hypergeometric function 2F ab;c;z [3 eq. 9.] which is straightforward to evaluate with numerical software such as Matlab. Our asymptotic solutions can be re-expressed as P e G A P T GD +o P GD T 9 where G D is the diversity gain which determines the asymptotic slope of the SER curve and G A is the array gain which represents the shift of the SER curve with respect to a reference curve of P GD T. Based on 5 and 8 we see that the diversity gain of TAS/MRC is G D = minm g N S N R m g2 N R N D. V. NUMERICAL EXAMPLES Numerical examples are provided to highlight the impact of TAS/MRC on the outage probability and the SER of cognitive MIMO networks. As shown in Fig. we consider a cluster of PU-Tx nodes communicating with a cluster of PU-Rx nodes in the primary network. In the secondary network we consider a simple collinear topology where S R and D are placed along the same straight line with R located halfway between S and D. In the following examples we assume a two dimensional network topology where the three nodes in the secondary network are placed along the x-axis with S located at R located at 2 D located at and the channel mean power of the link from S to D is normalized to unity. We further assume an exponential decaying path loss where the channel mean power is proportional to d ν with d denoting the distance between the transceivers and ν = 3 denoting the path loss coefficient. As such the channel mean powers for the links in the secondary network are Ω g = Ω g2 = 8. The channel mean powers for the links from the primary network to the secondary network are defined by the the locations of the PU-Tx and PU-Rx clusters for example Ω h3 = d Tx /2 2 +d 2 T y 3/2 and = d 2 R x +d 2 R y 3/2

7 where d Tx d Ty and d Rx d Ry are the coordinates of the PU-Tx and PU-Rx clusters respectively. Fig. 2 plots the outage probability versus the maximum transmit power in the secondary network P T. We set the outage threshold as γ th = 3 db and the coordinates of the PU-Tx and PU-Rx clusters are and respectively. The dashed lines represent the asymptotic outage probability derived in 9 and + denotes the simulation points. We see in the figure that there is a good agreement between our analytical results and the simulations. We further see that the outage probability of TAS/MRC decreases with increasing number of antennas at the relay N R. Fig. 3 plots the SER with 6-AM versus P T. The dashed lines represent the asymptotic SER derived in 5. We see that the asymptotic SER provides an accurate approximation of the simulation points at medium to high P T. Furthermore we observe that the SER decreases as the fading parameter in the secondary network m g increases. We note that TAS/MRC achieves a full diversity gain of G D = minm g N S N R m g2 N R N D which serves to verify our analytical results. VI. CONCLUSION We proposed TAS/MRC with DF relaying in underlay spectrum sharing with cognitive MIMO relaying. For such networks we derived new closed-form expressions for the asymptotic outage probability and symbol error rate with L primary transceivers equipped with N antennas and a secondary source relay and destination equipped with N S N R and N D antennas respectively. Our results are valid for Nakagamim fading with distinct fading parameters in the primary and secondary networks. We show that TAS/MRC achieves a full diversity gain of G D = minm g N S N R m g2 N R N D for both M-PSK and M-AM modulations. APPENDIX A PROOF OF THEOREM The cdf of γ conditioned on Z is derived as { gi 2 F γ Z γ = Pr min P T h l i 2 Z { = Pr g i 2 γz & h l i P 2 } T P T }{{} I Z } γ { + Pr g i 2 γ h l i 2 Z & h l i 2 } P T }{{} I 2Z where we re-express γ as γ = min 2 P T Y X Z 2 with Y = h l i 2 X = g i 2 and Z = L l= P I h 3lq 2. The first term I Z can be evaluated as I Z = Pr g i 2 γz Pr h l i P 2 PT T γz = F gi F 2 hl i P 2 22 T PT where F gi 2 is the cdf of g i 2 and F hl i 2 is the cdf of h l i 2. The cdf of g i 2 is given by p m g N R F gi 2x = e xmg x mg Ω g Ωg NS 23 p! p= where the channel gains in g i follow a Gamma distribution with Nakagami-m fading parameter m g and channel power Ω g. The cdf of h l i 2 can be written as F hl i 2x = m h N e xm h x m h r= r L 24 where the channel gains in h l i 2 follow a Gamma distribution with Nakagami-m fading parameter m h and channel power. At high P T 22 can be approximated as I Z PT ǫ γzm g P TΩ g mg N SN R where the asymptotic cdf of g i 2 is and we define x + F gi 2 x ǫ = F hl i 2 P T = e m m h N h P T Γm g N R + NS 25 mg xmg N SN R Ω g Γm g N R + NS 26 r= mh P T r Integrating 25 with respect to the pdf of Z given by f Z x = yields E Z {I Z } mh3 P I Ω h3 mh3 N RL x m h 3 N RL e x m h 3 P I Ω h3 Γm h3 N R L L ǫ γmg P IΩ h3 P TΩ g m h3 mg N SN R Γmg N S N R +m h3 N R L Γm g N R + NS Γm h3 N R L where we solve the integral according to [ ] 29 x m exp βxdx = Γm+ β m+ 3

8 The second term I 2 Z can be evaluated as I 2 Z = = P T yγz f hl i 2y f gi 2xdxdy yγz f hl i 2yF g i 2 P T dy 3 where f hl i 2 is the probability density function pdf of h l i 2 given by f hl i 2x = L mh m h N e xm h x m h r= mh N x m h N e xm h r Γm h N L 32 where the channel coefficients in h l i 2 follow a Gamma distribution with fading severity parameter m h and channel power. We perform a change of variables y = P T t in 3 which results in γz I 2 Z = f hl i 2 t F gi P 2 t dt T P T P T 33 where dy = P T dt. Applying the Taylor series expansion as P T we approximate the cdf of g i 2 as γzm mg N g SN R γz P TΩ g t F gi 2 t 34 P T Γm g N R + NS Substituting 32 and 34 into 33 results in I 2 Z PT m h N L l= [ ar a r= L l l L γzm g m h Ω g mg N SN R Γm h N Γm g N R + NS ] ar a r+ l+ mg NSNR m h N m h N a r Γ m g N S N R +m h N + m h N a r l+m h P T where we solve the resulting integral using [3 3.35/2] 35 = x n e µx dx = µ n Γn+µ. 36 with Γ denoting the upper incomplete gamma function [3 eq ]. We expand the power sum according to [ N n= ] l x n = n! N n= [ an a n= an a n ] an a n+ x N n= an n! 37 where N > a = l and a N =. Integrating 35 with respect to 28 results in E Z {I 2 Z } PT L L l l L Γm h N Γm g N R + NS γmg P I Ω h3 m h Ω g m h3 l= mg N SN R Γm g N S N R +m h3 N R L Γm h3 N R L m h N [ ar a r= l+ mg NSNR m h N m h N a r Γ m g N S N R +m h N + m h N ] ar a r+ a r l+m h P T 38 Finally the asymptotic cdf of γ is the sum of 29 and 38 as presented in 8 which completes the proof. REFERENCES [] L. Musavian S. Aïssa and S. Lambotharan Effective capacity for interference and delay constrained cognitive radio relay channels IEEE Trans. Wireless Commun. vol. 9 no. 5 pp May 2. [2] A. Alshamrani X. S. Shen and L.-L. Xie os provisioning for heterogeneous services in cooperative cognitive radio networks IEEE J. Sel. Areas Commun. vol. 29 no. 4 pp Apr. 2. [3] H. A. Suraweera P. J. Smith and M. Shafi Capacity limits and performance analysis of cognitive radio with imperfect channel knowledge IEEE Trans. Veh. Technol. vol. 59 pp May 2. [4] C. Zhong T. Ratnarajah and K.-K. Wong Outage analysis of decodeand-forward cognitive dual-hop systems with the interference constraint in Nakagami-m fading channels IEEE Trans. Veh. Technol. vol. 6 no. 6 pp Jul. 2. [5] T.. Duong V. N.. Bao and H.-J. Zepernick Exact outage probability of cognitive AF relaying with underlay spectrum sharing Electronics Letters vol. 47 no. 7 pp. 2 Aug. 2. [6] T.. Duong D. B. da Costa T. A. Tsiftsis C. Zhong and A. Nallanathan Outage and diversity of cognitive relaying systems under spectrum sharing environments in Nakagami-m fading IEEE Commun. Lett. vol. 6 no. 2 pp Dec. 22. [7] D. B. da Costa M. Elkashlan P. L. Yeoh N. Yang and M. D. Yacoub Dual-hop cooperative spectrum sharing systems with multi-primary users and multi-secondary destinations over Nakagami-m fading in IEEE International Symposium on Personal Indoor and Mobile Radio Communications PIMRC Sydney Australia Sep. 22. [8] T.. Duong P. L. Yeoh V. N.. Bao M. Elkashlan and N. Yang Cognitive relay networks with multiple primary transceivers under spectrum sharing IEEE Signal Process. Lett. vol. 2 no. pp Nov. 22. [9] K. J. Kim T.. Duong H. V. Poor and L. Shu Performance analysis of cyclic prefixed single-carrier spectrum sharing relay systems in primary user interference IEEE Trans. Signal Process. vol. 6 pp Dec. 22. [] K. J. Kim T.. Duong and X.-N. Tran Performance analysis of cognitive spectrum-sharing single-carrier systems with relay selection IEEE Trans. Signal Process. vol. 6 pp Dec. 22. [] A. Ghasemi and E. S. Sousa Fundamental limits of spectrum-sharing in fading environments IEEE Trans. Wireless Commun. vol. 6 no. 2 pp Feb. 27. [2] T. Wang A. Cano G. B. Giannakis and J. N. Laneman Highperformance cooperative demodulation with decode-and-forward relays IEEE Trans. Commun. vol. 55 pp Jul. 27. [3] I. S. Gradshteyn and I. M. Ryzhik Table of Integrals Series and Products 6th ed. New York NY USA: Academic Press 2. [4] M. K. Simon and M.-S. Alouini A unified approach to the performance analysis of digital communications over generalized fading channels Proc. IEEE vol. 86 pp Sep. 998.