Achievable rate region for the Gaussian MIMO MAC with ...

G. Fraidenraich and J. PortugheisThe rest of paper is organised as follows. In Section 2,we describe the communication system model and makesome definitions. In Section 3, we characterise an achievablerate region for the model of Section 2. Then, inSection 4, we give some examples of cooperating systemsand obtain their achievable regions. Finally, in Section 5,we make some final comments and conclusions.2. SYSTEM MODELWe consider a two-user Gaussian vector memoryless MACwhere users can cooperate when transmitting their messages.This is illustrated in Figure 1. The BS receiver hasn r0 antennas, and user i, i D 1; 2, transmits and receiveswith n ti and n ri antennas, respectively.Users 1 and 2 generate their messages W 1 and W 2independently and uniformly distributed over the setsf1;2;:::;2 NR 1g and f1;2;:::;2 NR 2g. Lety i 2 R nri1 ,i D 0; 1; 2, be the received signals at the BS, user 1and user 2, respectively. Encoder E i , i D 1; 2, mapsthe message W i and a sequence of previous received signalsy 1 i ; y2 i ;:::;yk i1 into the next channel input x k i 2Rn ti, ky 1 0 ; y2 0 ;:::;yN 0D 1;2;:::;N. Upon receiving the sequence, the BS receiver’s decoder obtains messageestimates OW 1 and OW 2 . An error occurs whenever. OW 1 ; OW 2 / ¤ .W 1 ;W 2 /.A.2 NR 1;2 NR 2;N/code consistsof two sets of N encoding functions and a BS receiver’sdecoding function.The discrete time-invariant channel model is given by(where time index k is dropped)y 0 D H 10 x 1 C H 20 x 2 C z 0 (1)W 1 x 1E 1xx H 12+Ŵ 1W 2z 2H 21z 1H 20H 10z 0+xE 2 xx 2Figure 1. MIMO MAC cooperation channel model.+Ŵ 2y 1 D H 21 x 2 C z 1 (2)y 2 D H 12 x 1 C z 2 (3)where z i N 0; N i I nri , i D 0; 1; 2, are independentadditive white Gaussian noises at the BS, user 1 and user2, respectively. The matrix H i0 2 R n r0n ti models thechannel between user i, i D 1; 2, and the BS, whereasmatrices H 12 2 R n r2n t1 and H 21 2 R n r1n t2 modelthe inter-user channels. Each channel matrix is assumedto be known to their corresponding transmitter andreceiver. Additionally, we assume the power constraintstr.E X i X T i / 6 Pi ;i D 1; 2; where EŒ stands for theaverage operator, ./ T stands for transpose and tr./ is thetrace operator.Theratepair.R 1 ;R 2 / is achievable for the Gaussianvector MAC with cooperation, if for any ">0 andfor sufficiently large N there exists a sequence of.2 NR 1;2 NR 2;N/ codes such that the error probability isless than ". The set of all achievable rate pairs is thecapacity region for the multiple access system describedin this section. In the next section, we will characterise anachievable rate region for this system.3. ACHIEVABLE RATE REGIONIn [5], an achievable rate region for the discretememoryless MAC with generalised feedback wasdescribed. As the result of [5] is very general, we arguein the Appendix A that the result of [5] can also be appliedto the Gaussian vector memoryless MAC considered in thelast section.User 1 divides its message W 1 into two parts, W 10 2f1; 2; : : : ; 2 NR 10g and W 12 2f1;2;:::;2 NR 12g, andthenuses signals x 10 2 R n t11 to send the first part directly tothe BS and signals x 12 2 R n r11 to send the second partvia user 2.User2 also divides its message in a similar form:W 20 2f1;2;:::;2 NR 20g and W 21 2f1;2;:::;2 NR 21g.Note that R 1 D R 10 C R 12 and R 2 D R 20 C R 21 .User2 has estimated the second part of user 1’s message andvice versa. Based on these previous estimated messages,they cooperate by simultaneously defining two vectors,u 1 2 R n t11 and u 2 2 R n t21 . We call the definition ofthese vectors as a cooperation strategy. The users’ signalscan thus be decomposed asx 1 D x 10 C x 12 C u 1 (4)x 2 D x 20 C x 21 C u 2 (5)Now define the covariance matrices Q i D E X i X T i ,Q i0 D E X i0 X T i0, QUi D E U i U T i , i D 1; 2, Q12 DE X 12 X T 12and Q21 D E X 21 X T 21. We assume that allvectors x i0 ; u i ;i D 1; 2; x 12 and x 21 have a Gaussiandistribution. We show in the Appendix A that the rateTrans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett

G. Fraidenraich and J. Portugheispair .R 1 ;R 2 / is achievable if the inequalities (6)–(11) aresimultaneously satisfied.R 10 6 1 2 log 2 det I n r0C H !10Q 10 H T 10N 0R 20 6 1 2 log 2 det I n r0C H !20Q 20 H T 20N 0Following the same reasoning of [8,10], the boundary ofthe achievable rate region can be characterised by solvingR 12 6 1 12 log 2I det nr2 C N 2 I nr2 C H 12 Q 10 H12 T H12 Q 12 H T 12(6)(7)(8)R 21 6 1 12 log 2I det nr1 C N 1 I nr1 C H 21 Q 20 H21 T H21 Q 21 H T 21R 10 C R 20 6 1 2 log 2 det I n r0C H 10Q 10 H T 10 C H !20Q 20 H T 20N 0R 10 C R 12 C R 20 C R 21 6 1 2 log 2 det I nr0 C HQHTN 0(9)(10)(11)In these inequalities,!H D ŒH 10 H 20 and Q DQ 1 Q 3, with Q 3 D E U 1 U T 2. Note that matricesQ T 3 Q 2Q 1 and Q 2 can be expressed asQ 1 D Q 10 C Q 12 C Q U1 (12)Q 2 D Q 20 C Q 21 C Q U2 (13)By using Lemma 7.1 of [5], it is possible to show thatan equivalent set of inequalities defining achievable ratepairs isR 1 6 F 10 C F 12 (14)R 2 6 F 20 C F 21 (15)R 1 C R 2 6 minf.F 0 C F 12 C F 21 /; F g (16)where F 10 , F 12 , F 20 , F 21 , F 0 and F are the righthandsides of inequalities (6), (8), (7), (9), (10) and (11),respectively.Let I 1 D F 10 C F 12 , I 2 D F 20 C F 21 and I 3 Dminf.F 0 C F 12 C F 21 /; F g. We show in Appendix B thatI 3 6 I 1 C I 2 holds true. This implies that for a fixed inputQ i0 ;i D 1; 2, Q 12 , Q 21 , and cooperation strategy Q, theachievable rate pairs define in general a pentagon. The convexhull of the union of all possible pentagons is an achievableregion. In order to obtain this region, a brute-forceapproach is to generate all pentagons, store their union andthen apply a convex hull algorithm. This may be computationallyhard. An easier way to obtain the boundary ofthe region is to maximise a weighted sum of the rates R 1and R 2 [8, 10], that is, to maximise a 1 R 1 C a 2 R 2 , witha 1 C a 2 D 1.the following optimisation problem:maxŒ.a 2 a 1 /.F 20 CF 21 /Ca 1 min..F 0 CF 12 CF 21 /; F /(17)subject toQ 10 ; Q 20 ; Q 12 ; Q 21 ; Q U1 ; Q U2 > 0 (18)tr .Q 1 / 6 P 1 (19)tr .Q 2 / 6 P 2 (20)with a 1 6 a 2 ;a 1 C a 2 D 1 and Q 3 is a cross-correlationmatrix. A > 0 denotes that A is a positive semi-definitematrix. The region obtained using Equation (17) will bedenoted, hereinafter, as the achievable rate region.The functions F 12 , F 21 and F are not concave in thespace of matrices given in Equation (18), and Q 3 is nota positive semi-definite matrix. Hence, the above optimisationproblem is not in the class of convex programmingproblems. The MATLAB (MathWorks, Natick, MA, USA)used fmincon with a large number of initial conditions scatteredaround the domain. After an exhaustive search, wehave chosen the best result. Although there is no certaintythat the function reaches the global solution, the appropriatechoice of the initial conditions makes it very likely thatthe global solution is found. Indeed, we have comparedsome of the results obtained with the fmincon MATLABcommand to an exhaustive search over a very large set ofcovariance matrices, and we found that the final result wasbasically the same.Now suppose that we find the optimal Q 10 and Q 12that maximise F 10 C F 12 . Then, using matrices Q 10and Q 12 , we find the other optimal strategies that maximiseEquation (16). This defines an achievable pentagon.Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett

G. Fraidenraich and J. PortugheisThe same procedure can be carried out to find a secondachievable pentagon by first obtaining Q 20 and Q 21 thatmaximise F 20 C F 21 . And finally, a third pentagon isobtained by maximising Equation (16) and using theseoptimal strategies to find R1 and R 2 . The union of thesethree pentagons is achievable. We can enlarge even furtherthis region if we allow time sharing combinationof the corner points [10, Fig. 5]. This last region is alower bound for the achievable region of our MIMO MACwith cooperation.In the next section, we will show results for ourachievable rate region and the lower bound.4. RESULTSWe have obtained results for two different systems. In thefollowing, all signals x ij and u i are zero mean and unitvariance Gaussian variables, respectively.System 1: Consider the case where the transmitters haveone antenna each and the BS has two antennas, that is,n ti D n ri D 1, i D 1; 2, andn r0 D 2. This systemis denoted as system 1 1 2. Setx i0 D p P i0 x i0,i D 1; 2, u 1 D . p P 1 u 1 / and u 2 D p P 2 u 2.Setalso x 12 D p P 12 x 12and x21 D p P 21 x 21. Then,x i D .x i /, i D 1; 2, can be expressed asx 1 D p P 10 x 10 C p P 12 x 12 C p P 1 u 1x 2 D p P 20 x 20 C p P 21 x 21 C p P 2 u 2In the superposition block Markov encoding process,cooperation is based on previously estimated messages.This implies that components x ij are independent ofu i ;i D 1; 2. Then, Q 1 D ŒP 10 C P 12 C P 1 ,Q 2 D ŒP 20 C P 21 C P 2 and Q 3 D p P 1 P 2 , withjj 6 1.System 2: Consider the case where both transmittershave two antennas and the BS has four antennas,that is, n ti D n ri D 2, i D 1; 2, and n r0 D 4.This system is denoted as system 2 2 4. Assumethe following:q q x T i0 D P .1/i0 x.1/ i0 ; P .2/i0 x.2/ i0 ;i D 1; 2q q u T 1 D P .1/1 u 1; P .2/1 u 2q q u T 2 D P .1/2 u 3; P .2/2 u 4q q x T 12 D P .1/12 x.1/ 12 ; P .2/12 x.2/ 12q q x T 21 D P .1/21 x.1/ 21 ; P .2/21 x.2/ 21Then, x T i D .x i1;x i2 /, i D 1; 2, can be expressed asq q qx 11 D P .1/10 x.1/ 10 C P .1/12 x.1/ 12 C P .1/1 u 1q q qx 12 D P .2/10 x.2/ 10 C P .2/12 x.2/ 12 C P .2/1 u 2q q qx 21 D P .1/20 x.1/ 20 C P .1/21 x.1/ 21 C P .1/2 u 3q q qx 22 D P .2/20 x.2/ 20 C P .2/21 x.2/ 21 C P .2/2 u 4Again, the components u j , j D 1; 2; 3; 4, are independentof components x .k/ij . Then, Q i , i D 1; 2; 3, aregiven by0BQ 1 D @BC @C0BQ 2 D @qP .1/10 10 P .1/10 P .2/10q 10 P .1/10 P .2/10 P .2/100qP .1/12 12 P .1/12 P .2/12q 12 P .1/12 P .2/12 P .2/1210@ P .1/1 0BC @C0BQ 3 D @0 P .2/1AqP .1/20 20 P .1/20 P .2/20q 20 P .1/20 P .2/20 P .2/200qP .1/21 21 P .1/21 P .2/21q 21 P .1/21 P .2/21 P .2/2110@ P .1/2 00 P .2/2qP .1/1 P .1/2 11qP .2/1 P .1/2 21AqP .1/1 P .2/2 12qP .2/1 P .2/2 221CA C11CA CCA C1CA1CA Cwhere j ij j 6 1 and j ij j 6 1.The achievable rate region using cooperation has beencompared with a non-cooperation case in which the capacityregion is given in [10].Besides the non-cooperation case, an outer region wasgenerated by maximising the three inequalities (14)–(16)independently.The total cooperation line is obtained by evaluating apoint-to-point MIMO with n r0 receiving and .n t1 C n t2 /transmitting antennas [11].Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett

G. Fraidenraich and J. PortugheisFigures 2 and 3 show results for systems 1 1 2and 2 2 4, respectively. For the system of Figure 2,the achievable rate region and the lower bound are notshown because they are almost indistinguishable from oneanother and coincide with the outer region. For the systemof Figure 3, the achievable rate region, the lowerbound, the outer region, the total cooperation line and theMIMO MAC without cooperation are shown. Note that theachievable rate region and the lower bound are very close,showing the tightness of this bound.As can be observed in both figures, cooperation enlargessignificantly the rate region. It is worthwhile to note thatthis enlargement is only possible when the inter-user channelmatrices H 12 and H 21 represent better channels thanthe direct links channels H 10 and H 20 . In other words, theusers are close to each other.The point where user 2 acts as a relay to user 1,(R1 ;0), where R 1 is the maximum of the right-hand sideR232.521.510.5System 2x2x4, P1=2, P2=2Cooperation outer regionTotal Cooperation LineMIMO MAC without CooperationAchievable rate region0 0.5 1 1.5 2 2.5R1Figure 4. Achievable rate region for a 2 2 4 systemwith H 10 D Œ0:5 0:45I 0:55 0:5I 0:4 0:6I 0:6 0:55; H 20 DŒ0:9 0:89I 0:85 0:95I 0:98 0:94I 0:94 0:97; H 12 D H 21 DŒ0:95 0:93I 0:93 0:99.R21.61.41.210.80.60.40.2System 1x1x2, P1=2, P2=2Cooperation outer regionWithout CooperationTotal Cooperation Lineof Equation (14), is an achievable rate for a MIMO relaychannel [12]. The fact that R 2 D 0 does not mean thatP 2 D 0, because user 2 is cooperating with user 1. Asimilar conclusion holds for the case where user 1 acts as arelaytouser2.In order to illustrate a case where the direct channelis very dissimilar, Figure 4 shows a rate region when thedirect channel of user 2 is stronger than channel of user 1.Note that the achievable rate region and the outer region arecoincident. Note also that the achievable region for user 1,the weak user, has been enlarged considerably.00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6R1Figure 2. Achievable rate region for a 1 1 2systemwith(H 10 D Œ0:65 0:67; H 20 D Œ0:67 0:61; H 12 D H 21 D 0:985).R22.521.510.5System 2x2x4, P1=2, P2=2Cooperation outer regionTotal Cooperation LineMIMO MAC without CooperationAchievable rate regionCooperation lower bound00 0.5 1 1.5 2 2.5R1Figure 3. Achievable rate region for a 2 2 4 systemwith H 10 D Œ0:5 0:45I 0:55 0:5I 0:4 0:6I 0:6 0:55; H 20 DŒ0:3 0:6I 0:45 0:7I 0:6 0:24I 0:34 0:7; H 12 D H 21 DŒ0:95 0:93I 0:93 0:99.5. CONCLUSIONSAn achievable rate region for the two-user Gaussian MIMOMAC with cooperating encoders was presented. The regiongenerally enlarges the capacity region of a MIMO MACwithout cooperation. Although the achievable rate boundsare described by non-concave functions, the boundaryof the achievable rate region could be generated bymaximising a weighted sum of the rates.APPENDIX AWillems gave in [5, Theorem 7.1, Lemma 7.1] an achievablerate region for a discrete memoryless MAC withgeneralised feedback. The achievability proof is basedon superposition block Markov encoding and backwarddecoding. The set of inequalities defining achievable ratepairs can be expressed asR 1 6 I.V 1 I Y 2 jX 2 ;U/C I.X 1 I Y jX 2 ;V 1 ;U/ (21)R 2 6 I.V 2 I Y 1 jX 1 ;U/C I.X 2 I Y jX 1 ;V 2 ;U/ (22)Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett

G. Fraidenraich and J. PortugheisR 1 C R 2 6 min fI .V 1 I Y 2 jX 2 ;U/C I.V 2 I Y 1 jX 1 ;U/C I.X 1 ;X 2 I Y jV 1 ;V 2 ;U/;I.X 1 ;X 2 I Y/g (23)over the joint probabilityP.u;v 1 ;v 2 ;x 1 ;x 2 ;y 1 ;y 2 ;y/DD P.u/P.v 1 ju/P .v 2 ju/P .x 1 jv 1 ;u/ P.x 2 jv 2 ; u/P .y; y 1 ;y 2 jx 1 ;x 2 /:By considering an appropriate definition of the jointAEP property [13] for jointly typical decoding forcontinuous random variables and making the substitutions,U .U 1 ; U 2 / U, V 1 X 12 , V 2 X 21 ,X 1 X 1 ,X 2 X 2 , Y 1 Y 1 , Y 2 Y 2and Y Y 0 , inequalities (14), (15) and (16) can bederived. Note that in our case, P.y;y 1 ;y 2 jx 1 ;x 2 / DP.yjx 1 ;x 2 /P .y 1 jx 1 ;x 2 /P .y 2 jx 1 ;x 2 / because of thenoise independency.For example, the first mutual information inEquation (14), that is, F 10 , can be evaluated using the chainrule asI. X 1 I YjX 2 ; X 12 ; U/ Dh.YjX 2 ; X 12 ; U/ h.YjX 1 ; X 2 ; X 12 ; U/D 1 2 log 2e det N 0 I nr0 C H 10 Q 10 H T 1012 log 2 det e N 0I nr0D 1 2 log 2 det I n r0C H !10Q 10 H T 10N 0All the other evaluations of mutual information followthe same rationale and will not be shown here.APPENDIX BConsider inequalities (21), (22) and (23). LetI 1 D I.V 1 I Y 2 jX 2 ;U/C I.X 1 I Y jX 2 ;V 1 ;U/I 2 D I.V 2 I Y 1 jX 1 ;U/C I.X 2 I Y jX 1 ;V 2 ;U/I 3 D minfI 3A ;I 3B gD minfI .V 1 I Y 2 jX 2 ;U/C I.V 2 I Y 1 jX 1 ;U/C I.X 1 ;X 2 I Y jV 1 ;V 2 ;U/;I .X 1 ;X 2 I Y/gShowing that I 3 6 I 1 C I 2 is equivalent to showing thatI 3A 6 I 1 C I 2 , which in turn is equivalent to showingthat I.X 1 ;X 2 I Y jV 1 ;V 2 ;U/ 6 I.X 1 I Y jX 2 ;V 1 ;U/ CI.X 2 I Y jX 1 ;V 2 ;U/.Because of the independence of .X 1 ;Y/ with respectto V 2 and of .X 2 ;Y/ with respect to V 1 , we can writethat I.X 1 I Y jX 2 ;V 1 ;U/ D I.X 1 I Y jX 2 ;V 1 ;V 2 ;U/ andI.X 2 I Y jX 1 ;V 2 ;U/ D I.X 2 I Y jX 1 ;V 1 ;V 2 ;U/.However,conditioned on V 1 ;V 2 ,andU , X 2 and X 1 are independent.Therefore, we can apply the same reasoningused in [13] for a two-user MAC without cooperation, toconclude that I 3 6 I 1 C I 2 .ACKNOWLEDGEMENTSThe authors would like to thank Prof. Paulo A. V. Ferreiraand Prof. Wei Yu for the helpful discussions.REFERENCES1. Boche H, Jorswieck EA. On the performance optimizationin multiuser MIMO systems. European Transactionson Telecommunications 2007; 18(3): 287–304.2. Maric I, Goldsmith A, Kramer G, (Shitz) SS. On thecapacity of interference channels with one cooperatingtransmitter. European Transactions on TelecommunicationsApril 2008; 19(19): 405–420.3. Sendonaris A, Erkip E, Aazhang B. User cooperationdiversity. Part I. System description. IEEE Transactionson Communications November 2003; 51(11):1927–1938.4. Sendonaris A, Erkip E, Aazhang B. User cooperationdiversity—part II: implementation aspects and performanceanalysis. IEEE Transactions on Communications2003; 51: 1939–1948.5. Willems F. Information theoretical results for the discretememoryless multiple access channel, PhD Thesis,1982. 109–126.6. Kaya O, Ulukus S. Power control for fading cooperativemultiple access channels. IEEE Transactions onWireless Communications 2007; 6(8): 2915–2923.7. Edemen C, Kaya O. Achievable rates for the three usercooperative multiple access channel. In IEEE WirelessCommunications and Networking Conference, WCNC2008, 2008; 1507–1512.8. Tse D, Viswanath P. Fundamentals of Wireless Communication.Cambridge University Press Ed. 2004.9. Goldsmith A, Jafar SA, Jindal N, Vishwanath S. Capacitylimits of MIMO channels. IEEE Journal on SelectedAreas in Communications 2003; 21(5): 684–702.10. Yu W, Rhee W, Boyd S, Cioffi JM. Iterative waterfillingfor Gaussian vector multiple-access channels.IEEE Transactions on Information Theory 2004; 50(1):145–152.11. Telatar IE. Capacity of multi-antenna Gaussian channels.European Transactions on TelecommunicationsNovember 1999; 10(6): 585–595.12. Wang B, Zhang J, Host-Madsen A. On the capacityof MIMO relay channels. IEEE Trans. Inform. Theory2005; 51(1): 29–43.13. Cover TM, Thomas JA. Elements of Information Theory.Wiley, Ed. 2006.Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett