Capacity of Gaussian MIMO Bidirectional Broadcast Channels

Capacity of Gaussian MIMO Bidirecti onal Broadcast Channels Rafael F . W yrembelski ∗ , T obias J. Oechtering † , Igor Bjelakovi ´ c ∗ , Clemens Sch nurr † , and Holger Boche ∗ † ∗ Heinrich-Her tz-Chair for Mobile Communicatio ns, T ech nical Uni versity of Berlin † Fraunho fer German -Sino Lab f or Mobile Communication s Einsteinufer 25/37, D-1058 7 Berlin, Germany Email: { rafael.wyre mbelski, igor .bjelakovic, holger . boche } @mk .tu-ber lin.de, { tobias.oec htering, clemen s.schnurr } @hhi.fraunhofer .de Abstract — W e consider the b roadcast phase of a th ree-node network, where a relay node establishes a bidirectional commu- nication between two nodes using a spectrally efficient tw o-phase decode-and-forwar d protocol. In the first ph ase th e two n odes transmit their messages to the relay node. Then th e relay node decodes the messages and b roadca sts a re-encoded composition of them in the second phase. W e consider Gaussian MIM O channels and d etermine the capacity region fo r the second phase which we call the Gaussian M IMO bid irectional broadcast channel. I . I N T R O D U C T I O N Future wireless communication systems should provide high data rates r eliably in a cer tain area, even if the direct link does not have the desire d quality due to path loss or shad owing. T o face this challenge there h as been growing interest in cooper ati ve p rotocols where some nodes act as relay nodes to guaran tee a clo sed coverage by multi-ho p co mmunicatio n. In this work, we con sider a th ree-no de n etwork, w here one relay node establishes a bid irectional commu nication between the two o ther nodes. The prob lem of the two-way commu nication without a relay node is first studied in [1 ]. Since it is difficult to isolate simultan eously tran smitted and received wireless sig nals within the same fr equency band, we assume h alf-dup lex nodes and th erefore allocate o rthogo nal resources in tim e for or thogon al tr ansmission and recep tion. According ly , the whole transmission is separated into two phases as dep icted in Figure 1. In the fir st phase of a deco de- and-fo rward pro tocol both nodes tr ansmit their informa tion to the relay node an d in th e second phase the relay node d ecodes the messages an d bro adcasts a re-enco ded composition of them. Since we do no t allow any coop eration between no des 1 and 2, this can be seen as a restricted two-way relay c hannel. W e assume mu ltiple antennas at a ll nodes since they can increase the c apacity of a system significantly [2]. The op timal coding strategy for the Gaussian multiple access (MAC) phase is well known [3], [4] a nd extends to the Gaussian MIMO case, see f or in stance [2]. W e can assume that the r elay node can successfully d ecode the messages w 1 ∈ W 1 and w 2 ∈ W 2 from no des 1 an d 2 if we choose the correspon ding rate pair within the cap acity r egion. In the following bidirectio nal broadc ast phase w 1 is known at th e relay node and n ode 1 and w 2 is known at the re lay node an d node 2 . It rem ains for the Node 1 Node 2 Relay W 2 Source 1 Source 2 W 1 X Node 1 Node 2 Relay Y 2 Source 1 Source 2 1 MAC BC W W X 1 X 2 Y 1 Y 2 Fig. 1. Multipl e access and broadcast phase of the bi directi onal relay channel. relay node to broadcast a message wh ich a llows both nodes to recover the unknown message. The bidirectional broadcast phase is an alyzed for the d is- crete memoryless channel with finite alp habets in [5]. An achiev able rate region of a co mpress-and -forward approach, where the relay no de broadcasts a comp ressed version o f the MA C o utput to both n odes, can be found in [6]. In this work , we extend the pro tocol of [5] to the Gau ssian MIMO bid irectional bro adcast ch annel. The capacity r egion for this case canno t be given in closed f orm becau se of its complicated stru cture. The refore we use conv ex op timization methods to ch aracterize the b ound ary of the capacity region. 1 I I . P R E L I M I N A R I E S Let N R be the numb er of transmit antenn as at th e relay node and N k be the number of recei ve anten nas at node k , k = 1 , 2 . W e define the discrete-time , memor yless Gaussian MIMO channe ls between the relay no de an d nodes 1 and 2 respectively a s linear time-inv ariant m ultiplicative chann els with additive white Gaussian noise. The vector-valued linear input-o utput relations at o ne time in stant can b e exp ressed as y k = H k x + n k , k = 1 , 2 , (1) where y k ∈ C N k × 1 denotes th e ou tput, H k ∈ C N k × N R the channel ma trix, x ∈ C N R × 1 the inp ut, and n k ∈ C N k × 1 the complex circula r symm etric distributed Ga ussian noise o f the channel according to C N ( 0 , σ 2 I N k ) . W e assum e that the inp ut alp habet is contin uous so that it is reasonable to consider an inpu t constrain t. A comm on 1 Notati on : Matrice s and random variabl es are denoted by bold capital lette rs, vectors by bold lo wer case letters, and sets by calligra phic letters; R + denotes the set of non-negati v e real numbers and M ( N, C ) the space of N × N matri ces with complex entries; ( · ) − 1 and ( · ) H denote in ve rse and Hermitia n transpose; E [ · ] is the expecta tion; Q  0 means Q is positi v e semidefinit e; lim inf and li m sup denote the limit inferio r and limit superior . and physically mea ningfu l constraint is an average p ower constraint. This means any sequen ce x 1 , x 2 , ..., x n of length n must satisfy 1 n n X i =1 x H i x i ≤ P . (2) Definition 1: The Gaussian MIMO bidir ectional br oadcast channel with average power limitation consists of two ch an- nels between the relay node and no des 1 and 2 as d efined in (1 ) with x ∈ X ⊂ C N R × 1 , where X describes the set of po ssible input sequen ces wh ich satisfy the a verage power constraint (2), i.e., f or a sequence o f length n we h av e X n := { ( x 1 , x 2 , ..., x n ) ∈ C N R × n : 1 n P n i =1 x H i x i ≤ P } . Let W 1 and W 2 be the in depend ent inform ation sour ces at nodes 1 and 2, which are also k nown at th e relay node. W e assume th at W k is u niform ly distributed on the message set W k := { 1 , 2 , ..., M ( n ) k } with n the leng th of the blo ck code. Further, we use th e abbreviation V := W 1 × W 2 . Definition 2: A ( M ( n ) 1 , M ( n ) 2 , n ) -co de fo r the Gaussian MIMO bid irectional br oadcast chann el with av erage p ower limitation consists of one encoder at the re lay node f : V → X n and decoder s at nodes 1 and 2 g 1 : C N 1 × n × W 1 → W 2 ∪ { 0 } , g 2 : C N 2 × n × W 2 → W 1 ∪ { 0 } . The e lement 0 in the deco der plays the role o f an erasure symbol and is included in th e definition for con venience on ly . When the relay node sends the message v = [ w 1 , w 2 ] , the receiver o f nod e 1 is in er ror if g 1 ( y n 1 , w 1 ) 6 = w 2 . W e denote th e p robab ility of this ev ent by λ 1 ( v ) := P [ g 1 ( y n 1 , w 1 ) 6 = w 2 | f ( v ) has been sent ] . According ly , we denote the pro bability tha t the r eceiver of n ode 2 is in error by λ 2 ( v ) := P [ g 2 ( y n 2 , w 2 ) 6 = w 1 | f ( v ) has been sent ] . Now , we are able to introduce th e no tation fo r the average pro bability of error for the k -th node µ ( n ) k := 1 |V | X v ∈ V λ k ( v ) . Definition 3: A rate pair [ R 1 , R 2 ] is said to be achievable for the Gaussian MIMO bidir ectional broadcast chann el with av erage power limitation if for any δ > 0 there is an n ( δ ) ∈ N and a sequen ce of ( M ( n ) 1 , M ( n ) 2 , n ) -co des satisfying the power constraint such that for all n ≥ n ( δ ) we have log M ( n ) 1 n ≥ R 2 − δ and log M ( n ) 2 n ≥ R 1 − δ wh ile µ ( n ) 1 , µ ( n ) 2 → 0 as n → ∞ . The capacity region is the set of all achiev ab le rate p airs wh ich is defined as C BDBC := { [ R 1 , R 2 ] ∈ R 2 + : [ R 1 , R 2 ] achiev ab le } . I I I . C A PAC I T Y R E G I O N In th is section we pre sent and p rove our main result which is th e capac ity region of the Gaussian MIMO b idirectional broadc ast chann el. Theor em 1 : For gi ven cov arian ce matrix Q with tr ( Q ) ≤ P satisfyin g the power co nstraint the cor respond ing rate pair [ C 1 ( Q ) , C 2 ( Q )] is giv en by C k ( Q ) := log det( I N k + 1 σ 2 H k QH H k ) , k = 1 , 2 . Then th e capacity region C BDBC of the Gaussian MIMO bidirection al bro adcast ch annel is given b y C BDBC := [ Q : tr ( Q ) ≤ P , Q  0 dpch  [ C 1 ( Q ) , C 2 ( Q )]  where dp ch  ·  denotes th e downward po siti ve co mprehe nsiv e hull which is defined for the vector x ∈ R 2 + by the set dpch  x  := { y ∈ R 2 + : y i ≤ x i , i = 1 , 2 } . A. Pr o of of Achievability W e follow [5] a nd adap t the ran dom codin g proof fo r the degraded b roadcast channel o f [7] to our context. For a gi ven covariance matrix Q with tr ( Q ) ≤ P satisfying the power constrain t we have to show th at all r ate pairs [ R 1 , R 2 ] are achiev able which satisfy R k ≤ lo g det( I N k + 1 σ 2 H k QH H k ) , k = 1 , 2 . W e denote the achiev able rate region as R BDBC . 1) Rand om co deboo k generation and encodin g: For any δ > 0 we have to ensure that the prob ability tha t a co dew ord does no t satisfy the power con straint g oes to zero as the blo ck length n go es to infinity . Therefor e, we define the covariance matrix ˆ Q := ˆ P P Q with ˆ P := P − ǫ P , ǫ P ∈ (0 , P ] , wh ere ǫ P allows us to get the rate ˆ R k correspo nding to th e tran smit strategy ˆ Q arbitrarily close to R k . Then we define for any ˆ R k the rate of the code R ⋆ k := ma x { 1 n ⌊ n ( ˆ R k − δ 2 ) ⌋ , 0 } , k = 1 , 2 . 2 W e gen erate M ( n ) 1 M ( n ) 2 codewords of len gth n with M ( n ) 1 := 2 nR ⋆ 2 and M ( n ) 2 := 2 nR ⋆ 1 , where for each v = [ w 1 , w 2 ] ∈ V each entr y of the co rrespon ding code- word f ( v ) = x n ( v ) is indep endently chosen accor ding to C N ( 0 , ˆ Q ) . W e first bou nd the error p robab ility with respect to the codeb ook wh ich mig ht violate the power c onstraint. Therefo re let ˆ λ k ( v ) an d ˆ µ ( n ) k , k = 1 , 2 , be the correspo nding error probabilities. In the fo llowing the random variable X denotes an entry of the codew o rd X n and the random variable Y k an en try of the output Y n k , k = 1 , 2 . 2) Decodin g: The receiving nodes use typical set dec oding. Let I ( X ; Y k ) := E X n , Y n k [ i ( X n ; Y n k )] , k = 1 , 2 , d enote the av erage m utual informa tion with i ( x n ; y n k ) := 1 n log p ( y n k | x n ) p ( y n k ) for realization s x n , y n k of the random variables X n , Y n k . At each receiving nod e k we have the decod ing sets S ( y n k ) :=  x n ∈ X n : i ( x n ; y n k ) > R ⋆ k + I ( X ; Y k ) 2  and the indicator f unction d ( x n , y n k ) := ( 1 , if x n / ∈ S ( y n k ) 0 , else . 2 If R ⋆ k = 0 , the error probabil ity is zero by definit ion so that we al ways assume R ⋆ k > 0 in the follo wing. When x n has been sent, and y n 1 and y n 2 have been receiv ed at nodes 1 and 2, two d ifferent events of er ror m ay occur at the d ecoder: the codew o rd x n is not in S ( y n k ) (occurring with probab ility P (1) e,k ( v ) ) and at no de one x n ( w 1 , ˆ w 2 ) with ˆ w 2 6 = w 2 is in S ( y n 1 ) or at nod e two x n ( ˆ w 1 , w 2 ) with ˆ w 1 6 = w 1 is in S ( y n 2 ) (o ccurring with probab ility P (2) e,k ( v ) ). If the re is no o r more th an one codeword x n ( w 1 , · ) ∈ S ( y n 1 ) or x n ( · , w 2 ) ∈ S ( y n 2 ) , the decoder s map on the erasure symbol 0 . 3) An alysis of the pr obability o f decod ing err or: From the union bound we have ˆ λ k ( v ) ≤ P (1) e,k ( v ) + P (2) e,k ( v ) with P (1) e,k ( v ) := Z C n p  y n k | x n ( v )  d  x n ( v ) , y n k  d y n k for k = 1 , 2 , P (2) e, 1 ( v ) := Z C n p  y n 1 | x n ( v )  |W 2 | X ˆ w 2 =1 ˆ w 2 6 = w 2  1 − d  x n ( w 1 , ˆ w 2 ) , y n 1  d y n 1 . The error event P (2) e, 2 ( v ) is d efined similarly . For un iformly distributed messages W 1 and W 2 we d efine P ( m ) e,k := 1 |V | P v ∈ V P ( m ) e,k ( v ) , m = 1 , 2 , so that ˆ µ ( n ) k ≤ P (1) e,k + P (2) e,k . For app lying the weak law of large nu mbers we have to ensure that the first two m oments a re finite [8, Section 7.3]. Lemma 1: The mean and variance of i ( X n ; Y n k ) , k = 1 , 2 , are finite. Pr oof: The pr oof is a generalizatio n of [ 9, Theorem 8.2.2] to the vector-valued case and is om itted here for brevity . Next, we average ov er all codeb ooks and show that E X n [ P (1) e,k ] , E X n [ P (2) e,k ] → 0 a s n → ∞ if ˆ R k ≤ I ( X ; Y k ) , k = 1 , 2 . Recall that R ⋆ k ≤ ˆ R k − δ 2 holds so that we ha ve E X n [ P (1) e,k ] = 1 |V | X v ∈ V E X n [ P (1) e,k ( v )] = E X n  Z C n p  y n k | X n ( v )  d  X n ( v ) , y n k  d y n k  = Z C n Z C n p  x n  p  y n k | x n  d  x n , y n k  d y n k d x n = E X n , Y n k  d  X n , Y n k  = P  d  X n , Y n k  = 1  = P  i ( x n ; y n k ) ≤ R ⋆ k + I ( X ; Y k ) 2  ≤ P  i ( x n ; y n k ) ≤ I ( X ; Y k ) − δ 4  → n →∞ 0 by the law of large numb ers since Lemma 1 holds. T he fou rth equality follows from Fubini’ s theo rem. For the calculation of E X n [ P (2) e,k ] we h av e to distinguish betwe en the nod es. W e present the analysis for k = 1 , th e case k = 2 f ollows similarly . W e use the fact tha t fo r v = [ w 1 , w 2 ] 6 = [ w 1 , ˆ w 2 ] the random variables in p ( y n 1 | X n ( v )) and d ( X n ( w 1 , ˆ w 2 ) , y n 1 ) are indepen dent for each choice of y n 1 ∈ C n . E X n [ P (2) e, 1 ] = 1 |V | X v ∈ V E X n [ P (2) e, 1 ( v )] = E X n " Z C n p  y n 1 | X n ( v )  |W 2 | X ˆ w 2 =1 ˆ w 2 6 = w 2  1 − d  X n ( w 1 , ˆ w 2 ) , y n 1  d y n 1 # = Z C n |W 2 | X ˆ w 2 =1 ˆ w 2 6 = w 2 E X n  p  y n 1 | X n ( v )   E X n  1 − d  X n ( w 1 , ˆ w 2 ) , y n 1   d y n 1 = Z C n |W 2 | X ˆ w 2 =1 ˆ w 2 6 = w 2 p  y n 1  E X n  1 − d  X n ( w 1 , ˆ w 2 ) , y n 1  d y n 1 = Z C n |W 2 | X ˆ w 2 =1 ˆ w 2 6 = w 2 p  y n 1  Z C n p  x n  1 − d  x n ( w 1 , ˆ w 2 ) , y n 1  d x n d y n 1 =( |W 2 | − 1) Z C n Z S ( y n 1 ) p  x n  p  y n 1  d x n d y n 1 , where in the third eq uality the change of the order of integration fo llows ag ain f rom Fubini’ s theorem . Whenever x n ∈ S ( y n 1 ) , we have i ( x n ; y n 1 ) = 1 n log p ( y n 1 | x n ) p ( y n 1 ) > 1 2 ( R ⋆ 1 + I ( X ; Y 1 )) or p ( y n 1 ) < p ( y n 1 | x n )2 − n 2 ( R ⋆ 1 + I ( X ; Y 1 )) . Consequently , E X n [ P (2) e, 1 ] < |W 2 | Z C n Z S ( y n 1 ) p  x n  p  y n 1 | x n  × 2 − n 2 ( R ⋆ 1 + I ( X ; Y 1 )) d x n d y n 1 ≤ 2 nR ⋆ 1 2 − n 2 ( R ⋆ 1 + I ( X ; Y 1 )) = 2 n 2 ( R ⋆ 1 − I ( X ; Y 1 )) ≤ 2 n 2 ( ˆ R 1 − δ 2 − I ( X ; Y 1 )) ≤ 2 − nδ 4 → n →∞ 0 if ˆ R 1 ≤ I ( X , Y 1 ) . Th e case k = 2 fo llows immed iately so that if ˆ R k ≤ I ( X ; Y k ) , k = 1 , 2 , th e average probab ility of error gets arbitrar ily small for suf ficiently large block length n . 4) Codebo ok that satisfies the power constraint: Up to n ow some co dew ords f ( v ) may v iolate the power constrain t. The probab ility of this e vent is given b y P (0) e ( v ) := P  1 n k X n ( v ) k 2 > P  . Next, for each rando mly gene rated codeb ook we con struct a new codebo ok wh ere we c hoose for all codewords f ( v ) , which d o no t satisfy the p ower constraint, the zero sequen ce instead which obviou sly satisfies the power co nstraint. W e upper bound the prob ability of a decoding error o f th e zero sequence with 1 . Then it easily follows λ k ( v ) ≤ P (0) e ( v ) + P (1) e,k ( v ) + P (2) e,k ( v ) , k = 1 , 2 . Since W 1 and W 2 are unifo rmly d istributed, we have P (0) e := 1 |V | P v ∈ V P (0) e ( v ) and P ( m ) e,k := 1 |V | P v ∈ V P ( m ) e,k ( v ) , m = 1 , 2 , so that µ ( n ) k ≤ P (0) e + P (1) e,k + P (2) e,k . A veraging over all codebo ok realization s, we get E [ µ ( n ) k ] ≤ E [ P (0) e ] + E [ P (1) e,k ] + E [ P (2) e,k ] . The first term describe s th e probability of violating the power constraint. Thereby 1 n k X n k 2 is the ar ithmetic av e rage of n indepen dent, iden tically distributed r andom variables with E [ k X k 2 ] = ˆ P . By the we ak law of large nu mbers, the arithmetic av erage c on verges in probability to ˆ P . Since ˆ P < P , we have E [ P (0) e ] → 0 as n → ∞ . Since E X n [ P (1) e,k ] , E X n [ P (2) e,k ] → 0 as n → ∞ as well, we have µ ( n ) k → 0 as n → ∞ , k = 1 , 2 . 5) Achievable rates: Since Y ki = H k X i + N ki , k = 1 , 2 , and X i ∼ C N ( 0 , ˆ Q ) with tr ( ˆ Q ) = ˆ P and N ki ∼ C N ( 0 , σ 2 I N k ) are indep endent and multiv ariate normal distributed, it f ollows that I ( X ; Y k ) = log det( I N k + 1 σ 2 H k ˆ QH H k ) with tr ( ˆ Q ) = ˆ P . It exists a n ǫ P > 0 such that ˆ R k = log det( I N k + 1 σ 2 H k ˆ QH H k ) > log det( I N k + 1 σ 2 H k QH H k ) − δ 2 = R k − δ 2 . Finally , we hav e R ⋆ k > log det( I N k + 1 σ 2 H k QH H k ) − δ = R k − δ while µ ( n ) k → 0 as n → ∞ , k = 1 , 2 , which pr oves the achiev ab ility . B. Pr o of of W eak Converse W e h av e to show that a ny given sequence of ( M ( n ) 1 , M ( n ) 2 , n ) -co des with µ ( n ) 1 , µ ( n ) 2 → 0 there exists a covariance matrix Q with tr ( Q ) ≤ P satisfying the p ower constraint such that R 1 := lim inf n →∞ 1 n log M ( n ) 2 ≤ log det( I N 1 + 1 σ 2 H 1 QH H 1 ) + o ( n 0 ) R 2 := lim inf n →∞ 1 n log M ( n ) 1 ≤ log det( I N 2 + 1 σ 2 H 2 QH H 2 ) + o ( n 0 ) are satisfied. That means we hav e C BDBC ⊆ R BDBC . Lemma 2: For our context we have Fano’ s inequality H ( W 2 | Y n 1 , W 1 ) ≤ µ ( n ) 1 log M ( n ) 2 + 1 = nǫ ( n ) 1 with ǫ ( n ) 1 = log M ( n ) 2 n µ ( n ) 1 + 1 n → 0 for n → ∞ as µ ( n ) 1 → 0 . Pr oof: From Y n 1 and W 1 node 1 estimates the in dex W 2 from the sent co dew ord X n ( W 1 , W 2 ) . W e define the event of an error at node 1 as E 1 := ( 1 , if g 1 ( Y n 1 , W 1 ) 6 = W 2 0 , if g 1 ( Y n 1 , W 1 ) = W 2 so that we ha ve for the a verage p robab ility of error µ ( n ) 1 = P [ E 1 = 1 ] . From the c hain rule for entro pies [ 9, Lemma 8.3.2] we have H ( E 1 , W 2 | Y n 1 , W 1 ) = H ( W 2 | Y n 1 , W 1 ) + H ( E 1 | Y n 1 , W 1 , W 2 ) = H ( E 1 | Y n 1 , W 1 ) + H ( W 2 | E 1 , Y n 1 , W 1 ) . Since E 1 is a function o f W 1 , W 2 , an d Y n 1 , we hav e H ( E 1 | Y n 1 , W 1 , W 2 ) = 0 . Further, since E 1 is a binary- valued random v ariable, we get H ( E 1 | Y n 1 , W 1 ) ≤ H ( E 1 ) ≤ 1 . So that finally with the next ineq uality H ( W 2 | Y n 1 , W 1 , E 1 ) = P [ E 1 = 0] H ( W 2 | Y n 1 , W 1 , E 1 = 0 )+ P [ E 1 = 1] H ( W 2 | Y n 1 , W 1 , E 1 = 1 ) ≤ (1 − µ ( n ) 1 )0 + µ ( n ) 1 log( M ( n ) 2 − 1) ≤ µ ( n ) 1 log M ( n ) 2 we get Fano’ s inequ ality fo r our co ntext. W ith a similar d eriv atio n we get H ( W 1 | Y n 2 , W 2 ) ≤ µ ( n ) 2 log M ( n ) 1 + 1 = nǫ ( n ) 2 with ǫ ( n ) 2 = log M ( n ) 1 n µ ( n ) 2 + 1 n → 0 for n → ∞ as µ ( n ) 2 → 0 . Lemma 3: The rate 1 n H ( W 2 ) can be bounde d as follows 1 n H ( W 2 ) ≤ log det  I N 1 + 1 σ 2 H 1  1 n n X i =1 Q i  H H 1  + ǫ ( n ) 1 with ǫ ( n ) 1 = log M ( n ) 2 n µ ( n ) 1 + 1 n → 0 for n → ∞ as µ ( n ) 1 → 0 . Pr oof: First, we boun d the entropy H ( W 2 ) as follows H ( W 2 ) = H ( W 2 | W 1 ) = I ( W 2 ; Y n 1 | W 1 ) + H ( W 2 | Y n 1 , W 1 ) ≤ I ( W 2 ; Y n 1 | W 1 ) + nǫ ( n ) 1 ≤ I ( W 1 , W 2 ; Y n 1 ) + nǫ ( n ) 1 ≤ I ( X n ; Y n 1 ) + nǫ ( n ) 1 where the equ alities and inequalities fo llow f rom the indepen - dence of W 1 and W 2 , the definition of mutual inf ormation , Lemma 2, and the cha in rule for mutual inform ation. Sinc e ( W 1 , W 2 ) , X n , Y n 1 form a Markov chain, it can be shown that the last ineq uality holds. If we use the d efinition of mutual informa tion and the mem oryless prop erty of the channel, we can express the inequ ality as H ( W 2 ) ≤  h ( Y n 1 ) − h ( Y n 1 | X n )  + nǫ ( n ) 1 = n X i =1  h ( Y 1 i ) − h ( Y 1 i | X i )  + nǫ ( n ) 1 = n X i =1 I ( Y 1 i ; X i ) + nǫ ( n ) 1 . If we divide the ine quality by n and use a gain the definition of mutual informatio n we get 1 n H ( W 2 ) ≤ 1 n n X i =1  h ( Y 1 i ) − h ( N 1 i )  + ǫ ( n ) 1 with Y 1 i = H 1 X i + N 1 i . The ran dom variables X i and N 1 i with h ( N 1 i ) = log det( π e σ 2 I N 1 ) are ind ependen t. It follows from the entropy maximization theorem that h ( Y 1 i ) ≤ log det  π e ( σ 2 I N 1 + H 1 Q i H H 1 )  with equ ality if we have Gaussian input, i.e., X i ∼ C N ( 0 , Q i ) . Therewith we have h ( Y 1 i ) − h ( N 1 i ) ≤ log det  I N 1 + 1 σ 2 H 1 Q i H H 1  . Finally , we get 1 n H ( W 2 ) ≤ 1 n n X i =1 log det  I N 1 + 1 σ 2 H 1 Q i H H 1  + ǫ ( n ) 1 ≤ lo g det  I N 1 + 1 σ 2 H 1  1 n n X i =1 Q i  H H 1  + ǫ ( n ) 1 where the second inequality fo llows fr om the concavity of the log det function [10, T heorem 7.6.7] . W ith a similar d eriv ation we get 1 n H ( W 1 ) ≤ log det  I N 2 + 1 σ 2 H 2  1 n P n i =1 Q i  H H 2  + ǫ ( n ) 2 with ǫ ( n ) 2 = log M ( n ) 1 n µ ( n ) 2 + 1 n → 0 for n → ∞ as µ ( n ) 2 → 0 . It is clear that R 1 = lim inf n →∞ 1 n log M ( n ) 2 ≤ lim sup n →∞ 1 n log M ( n ) 2 holds. Since W 2 is un iformly dis- tributed, we have 1 n log M ( n ) 2 = 1 n H ( W 2 ) and obtain w ith Lemma 3 R 1 ≤ lim sup n →∞ h log det  I N 1 + 1 σ 2 H 1  1 n n X i =1 Q i  H H 1  + ǫ ( n ) 1 i . (3) Next, we defin e the comp act set G := { Q ∈ M ( N R , C ) : tr ( Q ) ≤ P , Q  0 } with 1 n P n i =1 Q i ∈ G since 1 n P n i =1 Q i  0 and 1 n P n i =1 tr ( Q i ) = tr ( 1 n P n i =1 Q i ) ≤ P hold. This implies that the re exists a subseq uence ( n l ) l ∈ N such th at 1 n l P n l i =1 Q i → Q as n l → ∞ with Q ∈ G . Ther ewith and with the continuity of the log det we have lim sup n l →∞  log det  I N 1 + 1 σ 2 H 1  1 n l n l X i =1 Q i  H H 1  + ǫ ( n l ) 1  = lo g det  I N 1 + 1 σ 2 H 1 QH H 1  . (4) Combining (3) and (4) we obtain R 1 ≤ log det  I N 1 + 1 σ 2 H 1 QH H 1  . Using the sam e sub sequence ( n l ) l ∈ N and arguments we get R 2 ≤ log det  I N 2 + 1 σ 2 H 2 QH H 2  which proves the conv erse. I V . D I S C U S S I O N Since the capacity regio n is conve x, we can co mpletely characterize C BDBC by its bound ary which co rrespond s to the weighted rate sum optimal rate p airs. Therefore, we introduce a weight vecto r q = [ q 1 , q 2 ] ∈ R 2 + \{ 0 } and expr ess the weighted rate sum fo r given q as R Σ ( Q ) = q 1 R 1 ( Q ) + q 2 R 2 ( Q ) with R 1 ( Q ) := log det( I N 1 + 1 σ 2 H 1 QH H 1 ) and R 2 ( Q ) := log det( I N 2 + 1 σ 2 H 2 QH H 2 ) . The a im is now to find the o ptimal covariance matrix Q ∗ ( q ) with tr ( Q ∗ ( q )) ≤ P satisfying the power constraint which maximizes the weigh ted rate sum. This c an be expressed as the following o ptimization problem max Q q 1 R 1 ( Q ) + q 2 R 2 ( Q ) s.t. tr ( Q ) ≤ P , Q  0 . (5) The Lagrangia n for this optimization problem is giv en by L ( Q , µ, Ψ ) = − q 1 R 1 ( Q ) − q 2 R 2 ( Q ) − µ  P − tr ( Q )  − tr ( Q Ψ ) . Therewith, the cov ariance matrix maximizing (5) for gi ven q is uniquely ch aracterized by th e Karush- Kuhn-T u cker conditions − µ I N R + Ψ = − q 1 H H 1 ( σ 2 I N 1 + H 1 QH H 1 ) − 1 H 1 − q 2 H H 2 ( σ 2 I N 2 + H 2 QH H 2 ) − 1 H 2 , (6a) Q  0 , P ≥ tr ( Q ) , (6b) Ψ  0 , µ ≥ 0 , (6c) tr ( Q Ψ ) = 0 , µ  P − tr ( Q )  = 0 , (6d) with prim al, dual, and c omplemen tary slackness con ditions (6b), (6c), and (6d) respectiv ely . The weighted ra te sum o ptimal rate pa irs de scribe the curved section o f th e boun dary an d are uniquely ch aracterized by (6a)-(6d). The two endpoints correspond to the cases, where q is chosen to optimize one unidire ctional rate. More precisely , the case q 1 > 0 , q 2 = 0 means we want to maxim ize the rate R 1 , for which we can achieve the single-user capacity R (1) 1 := max R 1 ( Q ) = lo g det( I N 1 + 1 σ 2 H 1 Q (1) H H 1 ) , where the su perscript (1) indicates that the weight vector q is chosen to optimize the rate R 1 . For R 2 this leads to an achievable rate R (1) 2 = log det ( I N 2 + 1 σ 2 H 2 Q (1) H H 2 ) . Similar, for 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 rate R 1 [bit/channel use] rate R 2 [bit/channel use] B A C superposition XOR optimal H 1 =  1 − 0 . 5  1 − 0 . 5  − 0 . 5 1 . 3  H 2 =  1 . 2 − 0 . 2  0 . 7 1 − 0 . 2   Fig. 2. Achie vable rate regions for N 1 = N 2 = N R = 2 . q 1 = 0 , q 2 > 0 we can ach iev e R (2) 2 = log det( I N 2 + 1 σ 2 H 2 Q (2) H H 2 ) an d R (2) 1 = log det( I N 1 + 1 σ 2 H 1 Q (2) H H 1 ) . Figure 2 ex emplarily dep icts th e capacity region in compar- ison to the achiev ab le rate regions of the superposition cod ing [11] and th e XOR codin g a pproach [12], whe re th e optimal rate pair is gi ven by R ∗ 1 = R ∗ 2 := max min { R 1 ( Q ) , R 2 ( Q ) } . The o ptimal un idirectional rate pairs corre spond to the points A and B in Figu re 2. Point C in Figure 2 describ es th e maxmin op timal rate pair , which is the on ly rate pair where the XOR coding approach achiev es the capacity . V . C O N C L U S I O N In this w ork, we extended the b idirectional broadcast p hase for the discrete memory less channel with finite alphabets of [5] to the Gaussian M IMO case and der i ved the capacity region. W e showed that there is not a uniqu e rate sum optimal transmit strategy . Similar to the Gaussian M IMO MAC the weighted rate sum optimal transmit strategy fo r th e bidirection al broad - cast phase depends now on the weights of the two rates. R E F E R E N C E S [1] C. E . Shannon, “T wo-W ay Communication Channel, ” Pr oc. 4th Berkele y Symp. Math Stat. and Pr ob . , vol. 1, pp. 611–644, 1961. [2] E. Biglieri, R. Calderbank, A. Constant inides, A. Goldsmith, A. Paul raj, and H. V . Poor , MIMO W ireless Communicat ions . Cambr . Univ . Press, 2007. [3] R. Ahlswede, “Multi-W ay Communication Channels, ” in 2nd Int. Symp. on Inf. Theory , Tsahkadsor , Armenian, USSR, Sep. 1971, pp. 23–52. [4] H. Liao, “Multiple Access Channels, ” P h.D. dissertatio n, Univ ersity of Haw aii, Honolulu, HI, 1972. [5] T . J. Oech tering, C. Schnurr , I. Bjelak ovic , and H. Boche, “Broadca st Capaci ty Re gion of T wo-Phase Bidirecti onal Relaying , ” IE EE T ransac- tions on Information Theory , vol. 54, no. 1, pp. 454–458, Jan. 2008. [6] C. Schnurr , T . J . 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