Power Allocation/Beamforming for DF MIMO Two-Way Relaying: Relay and Network Optimization

Transcription

1 Power Allocaton/Beamformng for DF MIMO Two-Way Relayng: Relay and Network Optmzaton Je Gao, Janshu Zhang, Sergy A. Vorobyov, Ha Jang, and Martn Haardt Dept. of Electrcal & Computer Engneerng, Unversty of Alberta, Edmonton, Alberta, Canada, T6G2V4 Emals: {gao3, vorobyov, Communcatons Research Lab, Ilmenau Unversty of Technology, Ilmenau, Germany, Emals: {Janshu.zhang, Abstract The problem of ont sum-rate optmzaton and power mnmzaton s studed for a decode-and-forward DF) multple-nput multple-output MIMO) two-way relayng system consstng of two source nodes and one relay. Two scenaros are nvestgated. In the frst scenaro, the relay optmzes ts own power allocaton/beamformng strategy gven that the source nodes strateges are optmal n terms of sum-rate mzaton n the multple-access channel MAC) phase. In the second scenaro, the relay and the source nodes ontly optmze ther power allocaton/beamformng strateges to acheve network optmzaton over both the MAC and the broadcastng BC) phases. The problem of ont sum-rate optmzaton and power mnmzaton s shown to be nonconvex n both scenaros. For the frst scenaro, an algorthm s proposed to effcently derve the relay s optmal strategy. For the second scenaro, the problem s consdered n dfferent cases based on an ntal power allocaton. It s shown that the orgnal nonconvex problem can be transferred nto convex problems n all but one cases. For the rest case, an algorthm s proposed to derve the optmal strateges for the sources and the relay. Smulatons demonstrate the performance of the proposed algorthms. I. INTRODUCTION Two-way relayng has recently attracted sgnfcant nterests [1]- [5]. A typcal two-way relay network nvolves two source nodes whch need to exchange nformaton wth each other va one relay. As the source nodes subtract self-nterference whle the relay performs smultaneous b-drectonal communcaton, two-way relayng acheves hgher spectral effcency as compared to one-way relayng. The performance of two-way relayng depends on the transmsson strateges of both the sources and the relay. Power allocaton and related problems, e.g., beamformng, have been one of the research nterests of two-way relayng. In [2], the optmal power allocaton for two-way relayng s studed under an equal rate constrant for both amplfy-and-forward AF) and decode-and-forward DF) relayng. Whle ths work assumes sngle antenna at both the sources and the relay, the case wth multple antennas s nvestgated n [3], [4]. Assumng that the number of antennas for the relay and the source nodes are the same, the problem of ont sum-rate mzaton and transmsson power mnmzaton for AF MIMO two-way relayng s nvestgated n [3]. The relay s optmal power allocaton for MIMO DF two-way relayng consderng farness and channel estmaton error s derved n [4] assumng that the nformaton rate on each drecton s domnated by the rate of the transmsson from the relay to the source nodes. The achevable rate regon of MIMO DF two-way relayng s characterzed n [5]. The problem of ont sum-rate mzaton and transmsson power mnmzaton for MIMO DF two-way relayng s consdered n ths work. The obectve s to derve the optmal power allocaton/beamformng strateges for ont sum-rate mzaton and transmsson power mnmzaton n two scenaros. In the frst scenaro, the relay optmzes ts own power allocaton/beamformng strategy gven that the source nodes mze the sum-rate of the multple access channel MAC) phase. In the second scenaro, the relay and the source nodes ontly mze ther power allocaton/beamformng strateges to acheve ont sum-rate mzaton and transmsson power mnmzaton on the system level over both the MAC and the broadcastng channel BC) phase. The contrbuton of ths work s that the optmal soluton of the consdered nonconvex ont optmzaton problem s derved effcently ether usng proposed algorthms or through smplfyng the orgnal problem nto convex problems n dfferent cases. II. SYSTEM MODEL Consder a two-way relay system wth two source nodes and one relay, where source node = 1,2) and the relay have n and n r antennas, respectvely. In the frst phase MAC phase), source node transmts the sgnal W s to the relay, where W s the beamformng matrx of source and s s the nformaton symbol vector of source satsfyng E{s s H } = I and E{s s H } = 0 wth the superscrpt H standng for conugate transpose. The channels from source to the relay and from the relay to source are denoted as H r and H r, respectvely. It s assumed that the relay has all channel nformaton. The receved sgnal at the relay n the MAC phase s y r = H 1r W 1 s 1 + H 2r W 2 s 2 + n r 1) where n r s the nose at the relay wth covarance 2 ri. Source node has a power budget lmt formulated as the constrant

2 Tr{W W H } P. The sum-rate of the MAC phase s bounded by [7] R mac W 1,W 2 ) = 1 2 log I+ 1 r 2 H 1r W 1 W1 H H H 1r+H 2r W 2 W2 H H H 2r) 2) In the BC phase, the relay decodes s from the receved sgnal, re-encodes messages usng superposton codng and transmts the sgnal x r = T r2 s 1 + T r1 s 2 3) where T r s the relay s beamformng matrx of sze n r n r for relayng the sgnal from source node to node. 1 The relay s power budget s formulated as Tr{T r1 T H r1 + T r2 T H r2} P r. Note that the relay needs to know W n order to decode s. It s not dffcult for the relay to calculate the optmal beamformng for both of the sources snce t has all channel nformaton. Moreover, n order to acheve sum-rate optmzaton and power consumpton mnmzaton on the system level n a two-way relayng system, t s necessary for the relay to coordnate the sources transmsson strateges through nformng the sources of ther optmal beamformng matrces. Such coordnaton s requred especally when the sources are n lack of certan channel nformaton. For the same reason, T r should be sent to source for self-nterference cancelaton. The receved sgnal at source node s y = H r x r + n 4) where n s the nose at the relay wth covarance 2 I. Wth the knowledge of H r and T r, source subtracts the self-nterference H r T r s from the receved sgnal and the equvalent receved sgnal at source s y = H r T r s + n. 5) The nformaton rate for the communcaton from relay to source n ths phase s [1] where R bc rt r ) = 1 2 mn{ ˆR r T r ), R r W )} 6) ˆR r T r ) = log I + H r T r T H rh H r) 2 ) 1 7) R r W ) = log I + H r W W H H H r) 2 r) 1. 8) The sum-rate for the DF two-way relayng n the BC phase s then R bc T r1,t r2 ) = R bc r1t r1 ) + R bc r2t r2 ). 9) The sum-rate for the communcaton over both phases for the consdered DF two-way relayng s then R tw = mn{r mac W 1,W 2 ),R bc T r1,t r2 )}. 10) In the followng two sectons, the problem of sum-rate optmzaton wth transmsson power mnmzaton s consdered frst for the relay only and then for the network of the two sources and the relay. 1 It s assumed as default throughout the paper that the user ndex and satsfy. III. RELAY OPTIMIZATION In the scenaro of relay optmzaton n ths secton, the relay does not coordnate the sources. Instead t ams at optmzng ts own power allocaton/beamformng strategy to acheve sum-rate mzaton n the BC phase usng mnmum power. The sources n ths scenaro fnd ther beamformng matrces by solvng the followng convex problem {W Rmac W 1,W 2 ) 11a) 1,W 2} s.t. Tr{W W H } P,, 11b) whch s a basc power allocaton problem on MAC studed n [6]. Denote the soluton of the above problem as W 0 1,W 0 2. For the sum-rate optmzaton, the relay needs to solve the followng problem {T r1,t r2} mn{rmac W 0 1,W 0 2),R bc T r1,t r2 )} 12a) s.t. Tr{T r1 T H r1 + T r2 T H r2} P r. 12b) The problem 12a)-12b) s convex. However, n order to fnd the par {T r1,t r2 } wth mnmum Tr{T r1 T H r1 +T r2 T H r2} among all possble {T r1,t r2 } that acheve the same mum of the obectve functon 12a), the relay needs to solve the problem {T mn{rmac W1,W 0 2),R 0 bc T r1,t r2 )} 13a) r1,t r2} s.t. Tr{T r1 T H r1 + T r2 T H r2} P r 13b) ˆR r T r ) R r W), 0 13c) ˆR r1 T r1 ) + ˆR r2 T r2 ) R mac W1,W 0 2). 0 13d) The above problem s nonconvex due to constrants 13c) and 13d). In order to solve ths problem, we develop an effcent algorthm based on waterfllng. Note that waterfllng cannot be drectly appled here due to the fact that the relay has to dstrbute power for the communcatons on two drectons and that Rr bct r) s the mnmum of two tems as shown n 6). Denote the rank of H r as r r and the sngular value decomposton SVD) of H r as U r Ω r Vr H. Assume that the frst r r dagonal elements of Ω r are non-zero, sorted n the descendng order and denoted as ω r 1),...,ω r r r ) whle the last mn{n,n r } r r dagonal elements are zeros. Then the algorthm for relay optmzaton s summarzed as n Table I, where the rates R mac W1,W 0 2), 0 ˆRr T r ) and R r W 0) are brefly denoted as Rmac, ˆRr and R r, respectvely. Theorem 1: The algorthm n Table I acheves optmal relay power allocaton. Based on ths power allocaton, the followng beamformers T r = V r P r, 14)

3 TABLE I THE ALGORITHM FOR RELAY OPTIMIZATION. 1. Intal waterfllng: allocate P r on ω r k) 2, k {1,..., r r }, = 1, 2 usng waterfllng. Denote the ntal water level as 1. The power λ 0 allocated to ω r k) 2 s then p r k) = λ 0 ω r k) Set 2 M + r = {k p rk) > 0}, = 1, Check f log1 + p rk) ω r k) 2 2 ) R r for both = 1, 2. M + r If yes, output p r k), k {1,..., r r }, = 1, 2 and break. If there exsts such that log1+ p rk) ω r k) 2 2 ) > R r, denote the set M + r M + r of such as I and proceed. 3. Search for λ λ > λ 0 ) such that M + r log λ = log ω rk) 2 2 R r, where p r k) = 1 2 +, λ ω r k) 2 M + r = {k p rk) > 0} and M + r s the cardnalty of set M+ r. log1 + p rk) ω r k) 2 2 ). Check f I,. If Set ˆRr = M + r yes, proceed to Step 4. Otherwse, proceed to Step Search for λ λ > λ 0 ) such that M + r log λ = M + r log ω rk) 2 2 R r, where p r k) = 1 λ 2 +, ω r k) 2 M + r = {k p rk) > 0}. Set ˆRr = log1 + p rk) ω r k) 2 M + 2 ). r Proceed to Step Calculate P r = Pr k p rk). Allocate P r on ω rk) 2, k {1,..., r r } va waterfllng. Obtan water level 1 λ. Calculate p r k) = λ ω r k) and M + 2 r = {k p rk) > 0}. Set ˆR r = M + r Step 6. Otherwse, go to Step 4. log1 + p rk) ω r k) 2 2 ). If ˆRr R r, proceed to 6. Check f ˆRr + ˆR r R mac. If yes, output p r k), k, and break. Otherwse, fnd out λ m r r such that log ω r k) 2 + λ m 2 1 = R mac. If λ > λm and λ > λm, set λ = λ = λm. If λ < λm and λ > λm, update λ, p rk), k and M + r such that M+ r log λ = log ω rk) 2 M + 2 R mac + ˆR r. r Update p r k), k, accordng to new λ and/or λ. where P r s a dagonal matrx of sze n r n r gven as P r = pr 1)... prrr) 0, 15) are the optmal soluton of the rate mzaton and power mnmzaton problem 13a)-13d). Proof: Proof s omtted due to space lmtaton and wll be avalable n the correspondng ournal paper. The algorthm for relay optmzaton can be brefly understood as follows. Step 1 obtans ntal power allocaton P 0 r1,p 0 r2 and correspondng BC phase rates ˆR r1 T 0 r1) and ˆR r2 T 0 r2). The beamformng matrces T 0 r1,t 0 r2 mze ˆR r1 T r1 )+ ˆR r2 T r2 ) among all possble T r1 and T r2 subect to the relay s power lmt. Step 2 checks f Rr bct r) n 6) s upper-bounded by R r W 0 ), 1,2. If t s true,.e., ˆR r T 0 r ) > R r W 0 ), the relay reduces ts power allocated for relayng the sgnal from source to source such that ˆR r T r ) = R r W 0 ) n Step 3. It can be proved that ˆR r T r ) = R r W 0) at the pont when M+ r log λ = log ωrk) 2 k) R 2 r. If ˆRr T r ) s reduced n Step 3, M + r more power s avalable for relayng the sgnal from source to source. In ths case, f ˆR r T 0 r ) > R r W 0 ), the relay reduces ts power allocated for relayng the sgnal from source to source untl ˆR r T r ) = R r W 0 ) n Step 4. Otherwse, ˆR r T 0 r ) < R r W 0 ) and more power should be allocated for relayng the sgnal from source to source untl the power lmt s reached or untl ˆRr T r ) = R r W 0 ), and that s Step 5. It can be shown that f P r s below certan threshold, the algorthm proceeds through Steps 1-2 only. Increasng P r to a certan pont, the algorthm proceeds through Steps For further ncreased P r, the algorthm proceeds through Steps or Steps As a result, the algorthm n Table I provdes the optmal soluton to the nonconvex problem 13a)-13d) n at most sx steps wthout teratons. IV. NETWORK OPTIMIZATION In ths secton, the relay and the sources ontly optmze ther power allocaton/beamformng strateges to acheve sumrate mzaton for the two-way relayng over two phases. The sum-rate mzaton part can be formulated as the followng optmzaton problem {W mn{rmac W 1,W 2 ),R bc T r1,t r2 )} 1,W 2,T r1,t r2} 16a) s.t. Tr{W W H } P, 16b) Tr{T r1 T H r1 + T r2 T H r2} P r. 16c) The above problem s a convex problem whch can be rewrtten nto the standard form by ntroducng varables t,t 1,t 2 as follows {t,t 1,t 2,W 1,W 2,T r1,t r2} t 17a) s.t. t R mac W 1,W 2 ) 17b) t t 1 + t 2 t ˆR r, t R r, Tr{W W H } P, Tr{T r1 T H r1 + T r2 T H r2} P r. 17c) 17d) 17e) 17f) 17g) Smlar to the scenaro of relay optmzaton n Secton III, extra condtons are ntroduced f transmsson power mnmzaton needs to be taken nto account at the same tme. For network optmzaton, these constrants are R mac W 1,W 2 ) = R bc T r1,t r2 ) ˆR r T r ) R r W ),. 18a) 18b)

4 Consderng the above constrants, problem 17a)-18b) becomes nonconvex. Consderng that the optmal T r2,t r1 and W 1,W 2 n problem 17a)-18b) depend on each other, ont sum-rate mzaton and power mnmzaton for network optmzaton generally nvolves teratve optmzaton of T r2, T r1, W 1, and W 2. Such process provdes no nsght and can be hghly neffcent. Next we show that the optmzaton of T r2, T r1, W 1, and W 2 can be decoupled n all but one cases. In these cases, the orgnal problem s smplfed to dfferent convex problems. For the case when the optmzaton of the aforementoned beamformng matrces cannot be decoupled, an algorthm s proposed for solvng the problem effcently. In order to decouple the optmzaton varables of the orgnal problem, an ntal power allocaton s requred. Obtan 1/λ 0, p r k) and M + r, va the waterfllng procedure as descrbed n Step 1 of Table I. Denote T 0 r as the correspondng beamformng matrces obtaned from p r k), usng 14) and 15), and defne ˆR r T 0 r) = M + r log 1 + p rk) ω r k) 2 ),. 19) k) 2 Use ˆRr T 0 r ), as the ntal BC phase rates. Denote R bc T 0 r1,t 0 r2) = ˆR r1 T 0 r1) + ˆR r2 T 0 r2). Next, accordng to R mac W1,W 0 2) 0 and R bc T 0 r1,t 0 r2), we consder the orgnal problem n two cases, each of whch s dvded to four subcases. A. The case that R mac W 0 1,W 0 2) R bc T 0 r1,t 0 r2) In ths case, the sum-rate R tw n 10) s upper-bounded by the sum-rate n the BC phase. The relay consumes all avalable power and uses the optmal beamformers T 0 r1 and T 0 r2 for mzng R bc T r1,t r2 ), and the sources adust ther power allocaton/beamformng accordngly. If ˆR r1 T 0 r1) R 2r W 0 2) and ˆR r2 T 0 r2) R 1r W 0 1), the sources smply solve the followng convex problem mn Tr{W 1W1 H + W 2 W2 H } 20a) {W 1,W 2} s.t. R mac W 1,W 2 ) ˆR r1 T 0 r1) + ˆR r2 T 0 r2) 20b) R 1r W 1 ) ˆR r2 T 0 r2) 20c) R 2r W 2 ) ˆR r1 T 0 r1). 20d) However, f ˆRr1 T 0 r1) > R 2r W 0 2) and ˆR r2 T 0 r2) R 1r W 0 1), t s not straghtforward to see whether 20a)-20d) s feasble. Therefore, we present the followng result. Theorem 2: Gven that R mac W 1,W 2 ) R bc T 0 r2,t 0 r1), the optmal soluton to the problem R 2r W 2 ) {W 1,W 2} 21a) s.t. R1r W 1 ) ˆR r2 T 0 r2), 21b) denoted as W2, always satsfes R 2r W2) > ˆR r1 T 0 r1). Proof: The proof s omtted due to space lmtaton and wll be avalable n the correspondng ournal paper. TABLE II ALGORITHM FOR DERIVING OPTIMAL SOLUTION FOR SUBCASE IV WHEN R bc T 0 r1,t0 r2 ) > Rmac W 0 1,W0 2 ) 1. Source decreases ts transmsson power from P to P < P. Solve 11a)-11b) wth P substtuted by P. Obtan Rmac W 1,W 2 ), R r W ) and R r W ). If R r W ) > ˆR r T 0 r ) or R r W ) < ˆR r T 0 r ), P = P + P )/2. r r log 1+ 1 ω r k) 2 λ k) 2 2. Search for λ such that ˆR r T r ) = 1) + s equal to R r W ). Set p r k) = 1 k) 2 λ ω r k) 2 ) +. Allo- cate P r p r k) on M + r usng waterfllng and obtan ˆR r T r ). k If ˆRr T r ) > R r W ), search for λ such that ˆRr T r ) = r r log 1 + 1λ ω r k) 2 k) 2 1) + s equal to R r W ). Set p r k) = 1 k) 2 λ ω r k) 2 ) If ˆR r T r ) + R r W ) R mac W 1,W 2 ) < ǫ, where ǫ s the tolerance, output W and T r, and break. If ˆRr T r ) + R r W ) R mac W 1,W 2 ) < ǫ, set P = P /2 and go back to Step 1. If ˆRr T r ) + R r W ) R mac W 1,W 2 ) > ǫ, set P = P + P )/2 and go back to Step 1. Based on Theorem 2, t can be seen that the problem 20a)-20d) s stll feasble n the subcase that ˆRr1 T 0 r1) > R 2r W 0 2) and ˆR r2 T 0 r2) R 1r W 0 1). Therefore, n ths subcase the sources also solve 20a)-20d) for power optmzaton. Smlar result can be derved f ˆR r1 T 0 r1) R 2r W 0 2) and ˆR r2 T 0 r2) > R 1r W 0 1), whch s omtted here. Gven that R mac W 0 1,W 0 2) R bc T 0 r1,t 0 r2), the subcase that ˆRr1 T 0 r1) > R 2r W 0 2) and ˆR r2 T 0 r2) > R 1r W 0 1) s mpossble. B. The case that R bc T 0 r1,t 0 r2) > R mac W 0 1,W 0 2) In ths case, defne three varables µ 1, µ 2 and µ m such that ) r r2 log ω r2k) 2 µ 1 2k) 1) + = R 2 1r W1), 0 22) ) r r1 log ω r1k) 2 µ 2 1k) 1) + = R 2 2r W2), 0 23) r r log 1+ 1 ω rk) 2 µ m k) 1) )=R + mac W1,W 0 2). 0 24) 2 It can be shown that 1/µ m < {1/µ 1,1/µ 2 }. Snce R bc T 0 r2,t 0 r1) > R mac W1,W 0 2), 0 t s necessary that 1/λ 0 > 1/µ m where 1/λ 0 s obtaned by performng the procedure n Step 1 of Table I. The followng four subcases are possble. Subcase I: 1/µ m mn{1/µ 1,1/µ 2 }. In ths subcase, the optmal beamformer for source s W 0. The relay fnds ts optmal strategy through solvng the followng convex problem mn Tr{T r1t H r1 + T r2 T H r2} 25a) {T r1,t r2} s.t. ˆRr1 T r1 ) + ˆR r2 T r2 ) R mac W 0 1,W 0 2). 25b)

5 The soluton s n closed-form and can be found usng 14) and 15) wth 1 p r k) = k) 2 ) + µ m ω r k) 2,k = 1,...,r r,, 26) where µ m satsfes 24). Subcase II: 1/µ 1/µ m < 1/µ 1/λ 0. In ths subcase, the optmal beamformer for source s also W 0. The relay fnds ts optmal strategy by solvng the problem mn Tr{T r1t H r1 + T r2 T H r2} 27a) {T r1,t r2} s.t. ˆRr1 T r1 ) + ˆR r2 T r2 ) R mac W 0 1,W 0 2) 27b) ˆR r T r ) = R r W 0 ). 27c) The soluton s n closed-form. The optmal T r can be also found usng 14) and 15) wth 1 p r k) = k) 2 ) + µ ω r k) 2,k = 1,...,r r, 28) where µ satsfes 22) f = 1 and µ satsfes 23) f = 2. The optmal T r can be shown as 14) and 15) wth 1 p r k) = k) 2 ) + ω r k) 2,k = 1,...,r r, 29) µ where µ satsfes r r log 1+ 1 ω r k) 2 ) µ k) 2 1)+ + R r W )=R 0 mac W1,W 0 2). 0 30) Subcase III: 1/µ 1/µ m < 1/λ 0 < 1/µ and there exsts µ such that r r log 1+ 1 ω r k) 2 ) µ k) 2 1)+ R mac W1,W 0 2) 0 R r W ) 0 31) r r 1 k) 2 ) + r r 1 µ ω r k) 2 Pr k) 2 ) +. 32) µ ω r k) 2 In ths subcase, the optmal solutons for p r k) and p r k) are the same as those gven by 28), 29) and 30). Subcase IV: 1/µ 1/µ m < 1/λ 0 < 1/µ and there whch satsfes condtons 31) and 32). In ths subcase, the optmal sum-rate of the BC phase cannot acheve R mac W1,W 0 2) 0 although R mac W1,W 0 2) 0 < ˆR r1 T 0 r1) + ˆR r2 T 0 r2). Ths subcase becomes complcated snce the strateges of the sources and the relay cannot be decoupled. The algorthm n Table II s proposed to derve the optmal soluton n ths case. The algorthm gradually reduces R mac W 1,W 2 ) untl the mum of R bc T r1,t r2 ) acheves R mac W 1,W 2 ). Detaled explanaton s omtted due to space lmtaton. Theorem 3: The optmal soluton of problem 17a)-18b) acheves the same sum-rate and power consumpton as that acheved by the optmal soluton derved, n the respectve s no µ TABLE III SUMMARY OF THE ALGORITHM FOR NETWORK OPTIMIZATION. 1. The source nodes solve the MAC sum-rate mzaton problem 11a)-11b). Obtan R 1r W1 0) and R 2r W2 0). 2. The relay performs Step 1 of Table I. Obtan p r k) and M + r, = 1, 2. Calculate ˆR r 0 T0 r ) accordng to 19). 3. Check f R mac W1 0,W0 2 ) > Rbc T 0 r2,t0 r1 ). If yes, proceed to Step 4. Otherwse, proceed to Step Relay s optmal strategy s to use T 0 r1 and T0 r2. Source nodes solve problem 20a)-20d) for transmsson power mnmzaton. 5. Determne the subcase based on µ 1, µ 2, µ m, and λ 0. For subcases I, II, and III, the optmal beamformer for source s W 0 and the relay mnmzes ts transmsson power va solvng the problems 20a)-20d), 25a)-25b), and 27a)-27c), respectvely. For subcase IV, perform the algorthm n Table II for dervng the optmal strateges for both the source nodes and the relay. subcase, by solvng the problems 20a)-20d), 25a)-25b), and 27a)-27c), or by performng the algorthm n Table II. Proof: Proof s omtted due to space lmtaton and wll be avalable n the correspondng ournal paper. The complete procedure of dervng the optmal soluton of the ont sum-rate mzaton and transmsson power mnmzaton problem for the scenaro of network optmzaton s summarzed n Table III. V. SIMULATIONS In ths secton, we show the performance of the proposed algorthms for relay optmzaton n Table I and for network optmzaton n Table II. The general setup s as follows. The elements n channels H r and H r, are generated from complex Gaussan dstrbutons wth zero mean and unt varance. Nose powers 2, and 2 r are set to 1. The rates R mac W 1,W 2 ), R bc T r1,t r2 ), R r W ), and ˆR r T r ) are denoted as R mac, R bc, Rr and ˆR r, respectvely, n ths secton. Fg. 1 shows the BC phase rates for the optmal soluton of the problems 12a)-12b) and 13a)-13d) versus the relay s power lmt P r for one channel realzaton. The specfc setup for ths smulaton s as follows. The number of antennas n 1,n 2, and n r are set to be 6,5 and 8, respectvely. Power lmts for the source nodes are P 1 = P 2 = 3. The MAC phase rates for ths channel realzaton are for R mac, for R1r, and for R2r. In ths fgure, ˆR r s the rate from the relay to the source node for the optmal soluton of the convex problem 12a)-12b) obtaned usng CVX semdefnte programmng SDP) and ˆR r s the rate from the relay to the source node for the optmal soluton of the problem 13a)-13d) obtaned usng the algorthm n Table I. It can be seen from the fgure that the algorthm n Table I generates the same rates as those obtaned from solvng the problem 12a)-12b) when P r s small. The reason s that ˆR r s small when P r s below certan threshold and as a result 13c) and 13d) are always satsfed. As P r ncreases, the obectve functon 12a) s bounded by R mac whle the relay s transmsson power s not necessarly mnmzed n the soluton of the problem 12a)-12b). In Fg. 1, the optmal soluton of the problem 12a)-12b) always uses all avalable

6 13 a few teratons ˆR r1 ˆR r ˆR r1 ˆR r R tw R mac R bc Pr Number of teratons Fg. 2. Convergence of R tw, R mac and R bc. Fg. 1. ˆRr n the optmal soluton of problems 12a)-12b) and 13a)-13d) power whle the optmal soluton of the problem 13a)-13d) obtaned usng the proposed algorthm does not use any more power after P r exceeds 4.16, at whch pont R bc acheves R mac. Fg. 2 shows nstantaneous R tw, R mac and R bc versus the number of teratons when the algorthm n Table II s performed for network optmzaton Subcase IV when R bc T 0 r1,t 0 r2) > R mac W 0 1,W 0 2). The specfc setup for ths smulaton s as follows. Number of antennas at the source 1, source 2, and the relay are 6, 4, and 8, respectvely. Power lmts are set as P 1 = 2,P 2 = 8,P r = 8. The channel realzaton leads to the result that 1/λ 0 = 1.12, 1/µ 1 = 1.83, 1/µ 2 = 0.61 and 1/µ m = It can be seen from the fgure that the algorthm searches for the mum R mac such that t can be acheved by R bc. Durng ths search, R tw gradually acheves ts mum value. Fg. 3 shows the nstantaneous R r and ˆR r, and the power usage of the relay denoted as P c r and the source node 2 denoted as P c 2 source node 1 always uses all power). Two observatons can be drawn from Fg. 3. Frst, ˆRr2 = R 1r and ˆR r1 < R 2r for the optmal soluton because the sum-rate s bounded by R bc = R mac < R 1r + R 2r. Second, the source node 2 does not use all avalable power n the optmal soluton snce P c 2 < P 2. VI. CONCLUSION In ths work, we have effcently solved the ont problem of sum-rate mzaton and transmsson power mnmzaton for DF two-way relayng n two scenaros. For scenaro of relay optmzaton, the proposed algorthm allows the relay to decde ts optmal power allocaton/beamformng n at most 6 steps. For the scenaro of network optmzaton, for all except one subcases, we have shown that the problem can be smplfed to a convex problem, whch has a closed-form soluton n some subcases; and for the excepton subcase, our proposed algorthm fnds the optmal soluton numercally n Fg Number of teratons Number of teratons Convergence of the relay s and the source node 2 s power usage. REFERENCES [1] B. Rankov and A. Wttneben, Spectral effcent protocols for half-duplex fadng relay channels, IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp , Feb [2] M. Pschella and D. Le Ruyet, Optmal power allocaton for the twoway relay channel wth data rate farness, IEEE Commun. Lett., vol.15, no. 9, pp , Sept [3] C. Y. Leow, Z. Dng, and K. K. Leung, Jont beamformng and power management for nonregeneratve MIMO two-way relayng channels, IEEE Trans. Veh. Technol., vol.60, no. 9, pp , Nov [4] I. Krkds and J. S. Thompson, MIMO two-way relay channel wth superposton codng and mperfect channel estmaton, n Proc. Global Telecommun. Conf. 2010, Dec. 2010, Mam, Florda, USA. [5] T. J. Oechterng, H. Boche, Optmal Transmt Strateges n Mult- Antenna Bdrectonal Relayng, n Proc. IEEE Int. Conf. Acoust., Speech, Sgnal Process. 2007, vol. 3, pp. III-145-III-148, Apr. 2007, Honolulu, HI, USA. [6] Y. We, R. Wonong, S. Boyd, and J. M. Coff, Iteratve water-fllng for Gaussan vector multple-access channels, IEEE Trans. Inf. Theory, vol.50, no.1, pp , Jan [7] A. Goldsmth, S. A. Jafar, N. Jndal, and S. Vshwanath, Capacty lmts of MIMO channels, IEEE J. Sel. Areas Commun., vol.21, no.5, pp , June R1r R2r ˆRr1 ˆRr2 P c r P c 2