Noncooperative Eigencoding for MIMO Ad hoc Networks

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 2, FEBRUARY Noncooperatve Egencodng for IO Ad hoc Networks Duong Hoang, Student ember, IEEE, and Ronald A. Ilts, Senor ember, IEEE Abstract A new noncooperatve egencodng algorthm s ntroduced for multple-nput multple-output (IO) ad hoc networks. The algorthm performs generalzed waterfllng wth respect to ts transmt covarance matrx and the receve node covarance matrx, both of whch can be estmated locally. A cooperatve algorthm s also developed as a benchmark n smulatons. Smulaton results show that the noncooperatve algorthm performance s close to that of the cooperatve method and s much better than greedy optmzaton. A game theoretc nterpretaton of the algorthm s also provded. Index Terms Ad hoc networks, game theory, generalzed egencodng, multple-nput multple-output (IO), waterfllng. I. INTRODUCTION The problem of mnmzng total transmt power n an ad hoc network under capacty constrants s consdered. Each node employs multple antennas so as to ncrease the channel capacty by explotng multpath scatterng [1], [2]. The jont transmtter recever optmzaton nvolves multple-nput multple-output (IO) precodng and decodng at the transmtter and the recever. The problem of precodng/decodng under mnmum mean square error (SE) optmzaton crtera for a sngle pont-to-pont lnk has been extensvely studed n [3] [8]. The applcaton of SE technques to multuser optmzaton for a sngle-cell IO network has also been addressed n [9] [11]. In the context of ad hoc networks, the precoder optmzaton was consdered n [12]. In [12], the authors formulate the problem as maxmzng the total capacty subject to power constrants at each node and also ntroduce a centralzed optmzaton procedure based on gradent projecton. Though a global optmum s not guaranteed, t was shown by smulaton n [12] that the centralzed algorthm performs best. The dea of usng local nformaton to enable dstrbuted optmzaton n a IO network s presented n [13]. A tme-dvson duplexng (TDD) network s consdered n [13], where the objectve s to maxmze the mnmum capacty of each substream n the network. For TDD, the transmt vector proportonal to the complex conjugate of the receve vector s shown to be optmal n [13]. Another approach to fndng the transmt vector wthout explctly estmatng the channel matrx s gven n [14]. The technque used n [14] s based on the power algorthm to fnd the domnant egenvector and s then extended to full egencodng. We prevously consdered dstrbuted beamformng for ad hoc networks n [15]. Ths paper extends the results of [15] to full egencodng. The man results of ths paper can be summarzed as follows: anuscrpt receved October 24, 2006; revsed June 20, The assocate edtor coordnatng the revew of ths manuscrpt and approvng t for publcaton was Dr. Hongbn L. Ths work was supported n part by NSF Grant No. CCF and a grant from the Internatonal Foundaton for Telemetern. The authors are wth the Department of Electrcal and Computer Engneerng Unversty of Calforna, Santa Barbara, CA (e-mal: duong@ece. ucsb.edu; lts@ece.ucsb.edu). Dgtal Object Identfer /TSP a) the concept of a total nterference functon (TIF) prevously developed for spatal beamformng s extended here to full egencodng; b) a new algorthm, Noncooperatve Generalzed Egencodng wth Taxaton (NGET), s developed to mnmze the TIF whle mantanng the qualty of servce (QoS) for every par of nodes; c) the NGET algorthm s nterpreted usng a game theoretc approach; d) the exstence of Nash equlbrum ponts for NGET s proven; and (e) a centralzed optmzaton algorthm s developed as a benchmark for NGET. Secton II presents the narrowband IO ad hoc network model, the optmzaton problem, and the TIF functon. The NGET algorthm s presented n Secton III. Secton IV presents numercal smulaton results for the NGET, greedy, and centralzed optmzaton algorthms. II. NARROWBAND NETWORK ODEL Consder an ad hoc network of N nodes wth each uncastng node communcatng wth node l(); = 1...N. Each node uses transmt and receve antennas. The channels are assumed to be flat fadng. Let H 2 ;j 2 be the channel response from node j to node. Channel recprocty s assumed [13] such that for every ; j we have H ;j = H T j;. Let G 2 2 and P 2 2 be the normalzed transmt precoder and power matrx of node. The unnormalzed transmt precoder of node s then G ~ = G P 1=2, where P s a dagonal matrx wth nonnegatve entres and G has unt norm columns. Let b 2 be the symbol vector sent from node to node l(). The sgnal receved at node l() s r l() = H ~ l(); G b + H l();k G ~ k b k + n l() : (1) k6=;l() In (1), the frst term represents the desred sgnal from node, whle the second term s multuser nterference, and the thrd term s crcular whte Gaussan nose. It s assumed w.l.o.g. when all recevers have the same nose fgure that Efn l() n H l() g = I. The decoupled capacty of the channel from node and node l() s then [12] c l() =log I + ~ G H H H l();r 01 l() H l(); ~ G (2) where R l() s the nterference plus nose covarance at node l() and can be computed as follows: R l() = I + H l();k Gk ~ ~ H Gk Hl();k: H (3) k6=;l() The objectve s to mnmze the total transmt power n the network, whle mantanng channel capacty c l() r at all l(). The network optmzaton problem s then N mnmze Tr( G ~ G ~ H ); =1 subject to log I + G ~ H Hl();R H 01 l() H l(); G ~ r; =1;...N: (4) Optmzng the total network transmt power n (4) drectly s a nonconvex problem and does not have a closed-form soluton. The greedy algorthm n [12] mnmzes Tr( G ~ G ~ H ) ndependently at each node whle consderng the nterference from other users as colored Gaussan nose. Ths approach results n a waterfllng procedure for each par of nodes as n [1] X/$ IEEE

2 866 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 2, FEBRUARY 2008 In the followng, we wll present an alternatve objectve functon called the total nterference functon (TIF). For each transmsson lnk, nstead of mnmzng the transmt power drectly, each node mnmzes the TIF. The TIF s the total nterference power of the substreams of users n the network. TIF = Tr G ~ H H 3 ;k G ~ 3 k G ~ T k H;k T G ~ +2 k6=;l() Tr( ~ G ~G H ): The TIF functon defned here s a generalzed verson of that defned n [15] and s smlar to total squared correlaton n [16]. The nterference nterpretaton follows from the matrces ~ G H H 3 ;k ~ G 3 k ~ G T k H T ;k ~ G. These can be nterpreted as the covarance of the nterference vector at node k as the result of transmsson at node when we set the decoder at node k; ~ W k, equal to the complex conjugate of the precoder at that node: ~W k = ~ G 3 k. It s emphaszed that when channel recprocty holds, settng the decoder proportonal to the complex conjugate of the precoder at each node s a standard technque that has been wdely appled n wreless communcatons for cellular networks [17], [18] and for ad hoc networks [19], [13], [15]. The term Tr( ~ G ~G H ) n (5) represents the thermal nose contrbuton at node when the decoder s ~W = ~ G 3. The TIF functon can be decomposed as (5) TIF =2Tr ~G H R 3 ~ G + f ( ~ G 0 ) (6) where here the game theoretc notaton f ( ~ G 0 ) mples functonal dependence on all ~ G l ;l 6=. Equaton (6) s very mportant n the sense that t s decomposable. To mnmze TIF wth respect to ~ G, we only need to mnmze the frst term n (6), and the only knowledge requred s the nterference covarance matrx, whch can be estmated locally. III. NONCOOPERATIVE GENERALIZED EIGENCODING FOR IO NETWORKS At each node, the proposed noncooperatve optmzaton algorthm solves the local problem nmze Tr ~G H R 3 ~ G ; subject to log I + ~ G H H H l();r 01 l()h l(); ~G r; = 1...N where r s a predefned transmsson rate and R l() s held fxed. Note that (7) corresponds to mnmzaton of TIF wth regard to ~ G Proposton 1: At the optmal pont, the constrants n (7) are actve, that s the nequaltes n (7) become equaltes. Proof: We generalze [18]. Suppose that at the optmum pont there exsts a user such that the nequalty n (7) s strct, then we can reduce the transmt power of that user by replacng ts transmt matrx by (1 0 ) ~ G, where >0 s arbtrarly small. The transmt powers of other users do not change, but the transmt power of user decreases; therefore, both the total transmt power and the objectve functon n (7) decrease. Ths contradcts the optmalty of the current pont. Theorem 1: A (nonunque) soluton ~ G to the local optmzaton problem (7) s the generalzed egenmatrx of A = Hl();R H 01 l() H l(); and B = R 3 ; wth A ~G = B ~G 3 AB. The elements of the dagonal matrx 3 AB 2 2 are the generalzed egenvalues of A and B. (7) and trans- ~G. The optmzaton problem (7) s Proof: Defne Cholesky decomposton b = L L H formed precoder G = L H rewrtten as mnmze Tr( G H G ) subject to log I + G H L 01 A (L 01 ) H G r: (8) ) H. Let U be the egen- Let Q = G G H, and K = L 01 A (L 01 matrx of K. The spectral decomposton of K s K = U 3 K U H where 3 K s dagonal. Applyng matrx propertes: ji + ABj = ji + BAj and Tr(AA H )=Tr(A H A), the optmzaton n (8) becomes mnmze Tr(Q ) subject to log I + U H Q U 3 K r: (9) Denote ~ Q = U H Q U. Because U s untary Tr( ~ Q )=Tr(Q ). For fxed Tr( ~ Q ), by the Hadamard nequalty [20] log I + U H Q U 3 K = log I + ~ Q 3 K log(1 + Q ~ (k; k)3 K (k; k)) (10) where Q ~ (k; k) and 3 K (k; k) are the kth dagonal elements of Q ~ and 3 K. The equalty occurs ff Q ~ s dagonal (because 3 K s dagonal). Note that the condton that Tr( Q ~ ) s fxed nvolves only dagonal elements of Q ~ and the rght-hand sde (RHS) of (10) also nvolves only dagonal elements of Q ~. Ths means that f the optmum Q ~ s not dagonal, we can set all the nondagonal elements of Q ~ to zero and thereby make the constrant n (8) nactve wthout volatng the condton that Tr( Q ~ ) s fxed. But an nactve constrant contradcts Proposton 1, and therefore Q ~ must be dagonal. Therefore ~Q (01)=(2) U H Q U ~Q (01)=(2) = Q ~ 0 U H G G H U ~Q 0 = I: Ths means that T = G H U ~Q 0 G = U ~Q T H : s a untary matrx or The optmal soluton only depends on Q. Thus, f G s an optmal soluton, then the product of G wth an arbtrary untary matrx s also optmal. Therefore, G can be replaced by G T yeldng G = U ~Q : Because ~ Q s dagonal and U s an egenmatrx of K ; G s an egenmatrx of K wth unnormalzed columns. Thus where 3 AB L H ~G,wehave L 01 A (L 01 ) H G = G 3 AB s dagonal and G s orthogonal. Substtutng back G = A ~G = L G 3 AB = L L H ~G 3 AB = B ~G 3 AB : (11) Thus, ~ G s the generalzed egenmatrx of A and B. Because ~ G s the generalzed egenmatrx of A and B ; G s also the generalzed egenmatrx of A and B. Note that G s not an or-

3 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 2, FEBRUARY thonormal matrx unless B = I for some 2. However, the generalzed egenmatrx G has the followng dagonalzaton propertes [21]: TABLE I NGET ALGORITH FOR IO AD HOC NETWORKS G H A G = 3 A ; G H b G = 3 B (12) where 3 A and 3 B are dagonal. Substtute G ~ = G P (1)=(2) nto the objectve and constrant functons to obtan a new local optmzaton problem, as follows: mnmze Tr(P 1=2 G H B G P 1=2 ) B = P (k; k)3 (k; k); subject to log ji + P 1=2 3 A P 1=2 j = log(1 + P (k; k)3 A (k; k)) r (13) where P (k; k); 3 A (k; k) and 3 B (k; k) are the kth elements of the respectve dagonal matrces. The objectve functon (13) s lnear n P (k; k) whle the constrant sets n (13) are convex. Ths combned wth the constrants that P (k; k) 0 whch s also convex makes the optmzaton problem convex. Any local soluton s thus global. The problem can be solved by the generalzed waterfllng algorthm. The Lagrangan s L(P ; )= P (k; k)3 B (k; k) + r 0 log(1 + P (k; k)3 A (k; k)) : (14) Takng the dervatve of the Lagrangan wth respect to P (k; k), we (k; k) = 3 A 3B (k; k) (k; k) A(k; k)p (k;k) : (15) At a statonary pont, (15) equals zero; therefore P (k; k) = 3 AB + (k; k) A (16) (k; k) where 3 AB (k; k) s the kth generalzed egenvalue of (A ; B ) and equals 3 AB (k; k) =3 A (k; k)=3 B (k; k). The waterfllng level s =exp r 0 k:p (k;k)>0 log 3 AB (k; k) N m : (17) N m s the number of modes for whch P (k; k) > 0. The NGET algorthm for a IO ad hoc network s summarzed n Table I. Theorem 2: The NGET algorthm presented n Table I s a noncooperatve game n whch the utlty functon of each user s concave. Therefore, there exsts at least one fxed pont. Proof: Let us defne the strategy of each user as a par fg ; r g, where G 2 2 determnes drectons of transmsson of the precoder and r 2 + s the rate allocaton vector of user. The kth element of r s r (k) = log 1+P (k; k)3 A (k; k) : (18) For clarty, defne ^G as the generalzed egenmatrx of A and B wth unt norm columns n (12). At node, the optmzaton mnmzng (13) can be reformulated usng game theoretc notatons as where fr ; G g = arg max u(r; G; r0; G0) r ;G u (r ; G ; r 0 ; G 0 )=0 3 B (k; k)(e r (k) 0 1) 3 A (k; k) 0Tr (G 0 ^G )(G 0 ^G ) H (19) s the utlty functon of that node. The strategy set s fr 2 + ; G 2 Sj1 T r = rg, where S s any convex set n 2 contanng the set of matrces wth unt norm columns. The strategy set s convex and compact. The frst term n the utlty functon n (19) forces (13) to be mnmzed whle the second term n (19) forces G = ^G as the optmal strategy for user. Note that the frst term s concave n r. The second term s concave n G. Because the frst term depends only on r and the second term depends only on G, the utlty functon (19) s a concave functon n fg ; r g. Therefore, the noncooperatve algorthm has at least one fxed pont [22], [23]. Smulaton results n Secton IV show that NGET always outperforms the greedy algorthm and gves the same result as a centralzed algorthm based on the augmented Lagrangan technque. A game theoretc nterpretaton of the algorthm s presented next to suggest why NGET works better than the greedy counterpart. At each step, node s utlty functon can be expressed as follows: where and ~G = arg max u( 0 G ~ ; G ~ 0) (20) ~G 2f G ~ 2 jr ( G ~ )rg u 0 ( ~ G ; ~ G 0 )=0Tr ~R 0 = = 0Tr( ~ G H k6=;l() ~ G H R 3 ~ G ~G H ) 0 Tr( ~ G H H ;k ~ G k ~ G H k H H ;k ~R 03 ~G ) (21) s the covarance of the nterference from other nodes n the network. At each teraton at node the greedy algorthm mnmzes the power Tr( G ~ H ~G ) only. Thus, the greedy utlty functon s the frst term on the RHS of (21). Due to the channel recprocty n the network, the second term n (21) represents the total nterference power from node

4 868 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 2, FEBRUARY 2008 Fg. 2. Total transmt power of nodes n the network. Fg. 1. Node locatons and beampatterns correspondng to the largest egenmode of the network. TABLE II AVERAGE TOTAL TRANSIT POWER OF NGET, GREEDY, AND CENTRALIZED ALGORITHS to other nodes n the network when the decoder s set equal to the complex conjugate of the precoder ~ W = G ~ 3. Ths term can also be nterpreted as a tax to the utlty functon mposed on node. Ths makes node whle mantanng ts QoS, attempt to mnmze ts nterference to other nodes. In the greedy algorthm, each node acts selfshly by mnmzng power alone, and thus the equlbrum pont s not necessarly the optmum from the socal vewpont. By approprately prcng (.e., va taxaton) the system resource, the NGET algorthm thus gudes user behavor toward a more socally effcent pont [24]. IV. NUERICAL RESULTS AND CONCLUSION Smulatons were carred out to fnd the total transmt power of the nodes n the network. The frst network consdered has 14 nodes n a 1000 m m area as n Fg. 1. Each node has = 8 transmt and receve antennas. The channels are assumed to be full rank wth angle of arrval unformly dstrbuted over [0=4;=4]. Each multpath has a Raylegh fadng magntude. Transmt power s normalzed to the nose power such that n the absence of fadng, a transmt power of unty on an omndrectonal antenna creates a sgnal-to-nose rato (SNR) of 10 db at the receve antenna. The pathloss exponent s 4. The requred channel capacty for each lnk s r = 2 b/s/hz. Fg. 2 shows the total transmt power of the followng algorthms: NGET, the greedy algorthm, and the centralzed algorthm usng an augmented Lagrangan algorthm. The fgure shows that NGET outperforms the greedy algorthm and ts performance s comparable to that of the cooperatve algorthm. To further evaluate the performance of NGET, for each value of the number of nodes n the network N 2 f10; 14; 18; 22; 26; 30g, 200 random networks were generated on a square of 1000 m m. The dstance between the transmtter and the recever s unformly dstrbuted between 0 and 200 m. Other parameters are the same as n Fgs. 1 and 2. The average total transmt power of nodes over all the network nstances of NGET, the greedy algorthm, and the centralzed algorthm are shown n Table II. The results show that performance of NGET s better than that of the greedy algorthm and comparable to that of the centralzed algorthm. To conclude, a noncooperatve egencodng algorthm wth a rate constrant was developed ncorporatng taxaton. Smulaton results for an ad hoc network demonstrated that the resultng NGET algorthm outperforms greedy optmzaton from the standpont of mnmzng total transmt power, and performs almost dentcally to a cooperatve augmented Lagrangan approach. REFERENCES [1] I. E. Telatar, Capacty of mult-antenna {G}aussan channels, Eur. Trans. Telecommun., vol. 10, no. 6, pp , Nov./Dec [2] G. J. Foschn and. J. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Personal Commun., vol. 6, no. 3, pp , ar [3] J. Yang and S. Roy, On jont transmtter and recever optmzaton for mult-nput-mult-output (IO) transmsson systems, IEEE Trans. Commun., vol. 42, no. 12, pp , Dec [4] A. Scaglone, G. B. Gannaks, and S. Barbarossa, Redundant flterbank precoder and equalzers part I: Unfcaton and optmal desgns, IEEE Trans. Sgnal Process., vol. 47, no. 7, pp , Jul [5] A. Scaglone, P. Stoca, S. Barbarossa, G. B. Gannaks, and H. Sampath, Optmal desgns for space-tme lnear precoders and decoders, IEEE Trans. Sgnal Process., vol. 50, no. 5, pp , ay [6] H. Sampath, P. Stoca, and A. Paulraj, Generalzed lnear precoder and decoder desgn for IO channels usng the weghted SE crteron, IEEE Trans. Commun., vol. 49, no. 12, pp , Dec [7] D. P. Palomar, J.. Coff, and. A. Lagunas, Unform power allocaton n IO channels: A game-theoretc approach, IEEE Trans. Inf. Theory, vol. 49, no. 7, pp , Jul [8] D. P. Palomar,. A. Lagunas, and J.. Coff, Optmum lnear jont transmt-receve processng for IO channels wth QoS constrants, IEEE Trans. Sgnal Process., vol. 52, no. 5, pp , ay [9] E. A. Jorsweck and H. Boche, Transmsson strateges for IO AC wth SE recever: Average {SE} optmzaton and achevable ndvdual SE regon, IEEE Trans. Sgnal Process., vol. 51, no. 11, pp , Nov [10] S. Serbetl and A. Yener, Transcever optmzaton for multuser IO systems, IEEE Trans. Sgnal Process., vol. 52, no. 1, pp , Jan [11] Z. Luo, T. N. Davdson, G. B. Gannaks, and K.. Wong, Trancever optmzaton for block-based multple access through ISI channels, IEEE Trans. Sgnal Process., vol. 52, no. 4, pp , Apr

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