On the Capacity of the Diamond Half-Duplex Relay Channel

On the Capaci ty of the Diamond Half-Duple x Rela y Channel Hossein Bagheri , Abolfazl S. Motahari, and Amir K. Khandani Dept. of Electrical Engineer ing University of W aterloo W aterloo, ON, Canada N2L3G1 { hbag heri, ab olfazl, khandani } @cst.uwaterloo.ca Abstract W e consider a diamond-sh aped dual-ho p commu nication system con sisting a source, two pa r- allel ha lf-duplex relays an d a destination . In a sing le a ntenna con figuration, it has been p reviously shown that a two-phase node- scheduling algorithm, along with the d ecode and fo rward strate gy can achieve the capacity of the diamo nd cha nnel fo r a cer tain sy mmetric channel gains [1]. In this pa per , we o btain a mo re general condition for th e optimality o f th e scheme in terms of power resources and chann el gain s. I n particular, it is proved that if the prod uct of th e capacity of the simultaneou sly activ e link s are equal in both tran smission p hases, the scheme achieves the capacity of the chann el. I . I N T R O D U C T I O N Half-duplex relays i.e., relays t hat can not transmi t and recei ve at the same ti me, hav e rece ntly attracted enormous attention due to their simplicity and cost efficiency . Some capacity results for the case of a single half-duplex relay are presented in [2]. T o realize multiple-antenna benefits wit hout increasing th e w eight and s ize of t he equipments, m ultiple relays come i nto play . A s imple model for in vestigating the p otential benefits of the multip le relays is depicted in Fig. 1. The end-to-end capacity of th e relay channel h as been studied in [1], [3]–[7] and is referred to as diamond r elay channel in [1]. Schein in [3] and [4] Financial supp ort pro vided by Nortel and t he corresponding matching funds by the Natural Sciences and E ngineering Research Council of Canada (NSERC), and Ontario Centres of Excellence (OCE) are gratefully acknowledg ed. established upper and lo wer bounds on the ca pacity of the full-duplex diamond channel. Half- duplex relays ha ve b een considered in [1], [5]–[7]. Xue, and Sandhu in [1] prop osed sev eral communication schemes including mult ihop wit h spatial reuse, scale-forward, broadcast- multiaccess with common message, compress-forward, as well as hybrid ones. They assumed that there i s no interference between the transm itting and receiving relays. Howe ve r [5]–[7] considered s uch int erference and used di f ferent n ames of two-way [5], [6] and s uccessive [7] relaying, respective ly for th e multihop with spatial r euse prot ocol. In th is work, we follow the set-up of [1] and assume t here i s no s uch int erference. In t his paper , we are interested in s ituations where t he s imple st rategy o f successive relaying achieves the capacity of the diamond channel. Surprisingly , we prove t hat when the product of the capacity of the sim ultaneously active links in both transmission phases are equal, the scheme achieves the capacity . Note that t his condition includes the result indicated in [1] as a special case. A. Notations Throughout this paper , boldface letters are used to denote vectors. A → B represents the l ink from node A to node B . The notation also means: appr oaches to when the right hand si de of → is a number . A circularly symmetric com plex Gaussian rando m variable Z with mean 0 and variance σ 2 , is represented by Z ∼ C N (0 , σ 2 ) . I I . T R A N S M I S S I O N M O D E L OV E R D I A M O N D R E L AY C H A N N E L In t his work, a dual -hop wireless system depicted in Fig.1, is considered. The model consists of a so urce (S), two p arallel half-duplex relays ( Re 1 , Re 2 ) and a destination (D). The corresponding in dex for the nodes are 0, 1, 2, 3, as shown in the Fig. 1. W e assum e that all the nodes are equipped with a singl e antenn a. Also, du e to the long distance or strong shadowing, no link is ass umed between the source and the destination , as well as between the relays. The background noise at each recei ver i s cons idered to be additive white Gaussian n oise (A WGN). The channel gain between node i and j is assumed to be cons tant and is represented by g ij as shown in the Fig. 1 . In addit ion, the channel st ate information knowledge at di f ferent n odes is as follows Source Relay # 1 Relay # 2 Destination g 13 g 23 X 1 X 2 g 02 g 01 Y 1 Y 2 X 0 Y 3 Fig. 1. The diamond relay chann el • Source kn ows all the channel gai ns ( i.e. , g 01 , g 02 , g 13 and g 23 ). • The relay i knows i ts i nward and outward channel gains ( g 0 i and g i 3 ). • The desti nation knows its inward channel gains ( g 13 and g 23 ). For this model, the successive relayi ng scheme i s performed in t wo stages 1 : 1) In t he first λ portion of the transm ission ti me 2 , S and Re 2 transmit to Re 1 and D, respectiv ely . 2) In the remaining (1 − λ ) porti on of the transm ission time, S and Re 1 transmit to Re 2 and D, respectiv ely . Note th at the relays decode and then re-encode what they ha ve receiv ed prior to their transmissio n. The other possible scheduling algorithm for the h alf-duplex diamond channel is called s imultaneous relaying [7] or transm ission p olicy I [1] and is as fol lows: 1) In the first λ po rtion of the transmi ssion time, S broadcasts its data to both relays. 2) In the remain ing (1 − λ ) portion of the transmis sion time, relays cooperativ ely transmit to D. It is possible to com bine bot h scheduli ng methods, h owe ver in this work, successive relaying is considered only . The receiv ed discrete-time complex baseband equiva lent signals at R e 1 1 Scheduling is assumed to be done in adv ance. 2 T otal transmission t ime is T time slots. Re 2 , and D are respectively giv en by Y 1 [ m ] = h 01 X 0 [ m ] + N 1 [ m ] Y 2 [ n ] = h 02 X 0 [ n ] + N 2 [ n ] Y 3 [ m ] = h 23 X 2 [ m ] + N 3 [ m ] Y 3 [ n ] = h 13 X 1 [ n ] + N 3 [ n ] , (1) where X i and Y j are the t ransmitted and recei ved si gnals from and to node i and j , respectiv ely (see Fig. 1). m ∈ { 1 , ..., λT } and n ∈ { λT + 1 , ..., T } denote the t ransmission time i ndex corresponding to the two stages 3 . h ij is the comp lex channel coefficient and is connected to g ij by g ij = | h ij | 2 . N j is the noise at node j and N j ∼ C N (0 , σ 2 j ) , for j = 1 , 2 , 3 . W e assume a verage power constraint for S, R e 1 and Re 2 and denote the constraints by P S , P Re 1 and P Re 2 , respectiv ely , i .e. for λT tuple and (1 − λ ) T tuple sub-codewords, we ha ve 1 λT λT X k =1 | x 01 [ k ] | 2 ≤ P S 1 (1 − λ ) T T X k = λT +1 | x 02 [ k ] | 2 ≤ P S 1 λT λT X k =1 | x 23 [ k ] | 2 ≤ P Re 2 1 (1 − λ ) T T X k = λT +1 | x 13 [ k ] | 2 ≤ P Re 1 , (2) where x ij [ k ] is the symbo l corresponding to the k th index, in the sub-code word transmitt ed by node i to no de j . Note that instead of allocating P S to both S → Re 1 and S → Re 2 links, we could allocate different powers t o th em i.e. , ρP S and µP S with the condition ρλ + µ (1 − λ ) = 1 for ρ ≥ 0 and µ ≥ 0 . Howev er du e t o th e s implicity of implementati on and also exposition, we choose ρ = µ = 1 . In addition, power budget at th e relay nodes assumed to be fixed over time. A rate R is said t o be achie vable for t his scheme, if for T → ∞ , D can decode the message wi th error probability ǫ → 0 . 3 It is assumed that λT is an i nteger . T ABLE I T R A N S M I S S I O N S TA T E S I N D I A M O N D R E L A Y C H A N N E L . State R 1 R 2 T i me S 1 Rx Rx t 1 S 2 Tx Rx t 2 S 3 Rx Tx t 3 S 4 Tx Tx t 4 In t he next three s ections, we first derive the cut-set upper bound and the achiev able rate using the successiv e relaying p rotocol, and then we state t he condition for opt imality of the schem e. I I I . C U T - S E T U P P E R B O U N D The upper b ound for half-duplex networks is given in [8] based on writing the well known cut-set bounds for different transm ission states shown in table I and t hen adding them up. Tx and Rx deno te transmi t and receive modes i n the tabl e, respectiv ely . Applying t he procedure t o th e d iamond relay channel, we have [1] R C 1 = t 1 I ( X 0 ; Y 1 , Y 2 ) + t 2 I ( X 0 ; Y 2 ) + t 3 I ( X 0 ; Y 1 ) + t 4 . 0 R C 2 = t 1 I ( X 0 ; Y 2 ) + t 2 ( I ( X 0 ; Y 2 ) + I ( X 1 ; Y 3 )) + t 3 . 0 + t 4 I ( X 1 ; Y 3 ) R C 3 = t 1 I ( X 0 ; Y 1 ) + t 2 . 0 + t 3 ( I ( X 0 ; Y 1 ) + I ( X 2 ; Y 3 )) + t 4 I ( X 2 ; Y 3 ) R C 4 = t 1 . 0 + t 2 I ( X 1 ; Y 3 ) + t 3 I ( X 2 ; Y 3 ) + t 4 I ( X 1 , X 2 ; Y 3 ) , where C j for j = 1 , ..., 4 represents the j th cut-set. The tim e-sharing coef ficients t i s satisfy P 4 i =1 t i = 1 . Using Gaussian codebooks and defining C ij = log 2 (1 + g ij P ij σ 2 j ) with P ij as the power allocated to th at link, we have the following optimization problem: max t i ≥ 0 R s.t. R ≤ t 1 C 012 + t 2 C 02 + t 3 C 01 + t 4 . 0 R ≤ t 1 C 02 + t 2 ( C 02 + C 13 ) + t 3 . 0 + t 4 C 13 R ≤ t 1 C 01 + t 2 . 0 + t 3 ( C 01 + C 23 ) + t 4 C 23 R ≤ t 1 . 0 + t 2 C 13 + t 3 C 23 + t 4 C 123 4 X i =1 t i = 1 , (3) where P 01 = P S , P 02 = P S , P 13 = P R 1 , P 23 = P R 2 , C 012 = log 2 [1 + ( g 01 σ 2 1 + g 02 σ 2 2 ) P S ] , and C 123 = log 2 [1 + 1 σ 2 3 ( p g 13 P Re 1 + p g 23 P Re 2 ) 2 ] . I V . A C H I E V A B L E R A T E U N D E R S U C C E S S I V E R E L A Y I N G It has been s hown in [1] (Theorem 4.1) that t he maximu m achiev able rate under th is transmissio n policy , is given by R S R = max { R 1 , R 2 } (4) R 1 = λ 1 C 01 + min { λ 1 C 23 , (1 − λ 1 ) C 02 } (5) R 2 = λ 2 C 23 + min { (1 − λ 2 ) C 13 , λ 2 C 01 } (6) with λ 1 = C 13 C 13 + C 01 λ 2 = C 02 C 23 + C 02 . (7) It has als o been proved that the d ecode and forward scheme is the best forwarding scheme under th e successiv e relaying protocol (Theorem 4 .2 in [1]). In [1], it is stated that for the case o f P S = P Re 1 = P Re 2 , t he scheme achiev es the cut-set up per bound if the channel gains are such t hat C 02 = C 13 and C 01 = C 23 (corollary 4.3). Here we generalize t he condition of op timality in th eorem 1. Before proving the theorem, it is note worthy to fi nd some special cases in the successive relaying scheme. Lemma 1 states su ch cases. In particular , W e are interested in the cases where R 1 = R 2 in (5) and (6). Lemma 1 Assu ming all li nks have non-zer o capacit y , the conditions for R 1 = R 2 ar e o ne of t he followings: C 01 C 02 = C 13 C 23 (8) C 01 = C 02 if C 01 C 02 ≤ C 13 C 23 (9) C 13 = C 23 if C 01 C 02 ≥ C 13 C 23 . (10) Pr oof: See [9]. The rate in (4) associated wi th each condit ion of equality (8-10) is: R = C 01 C 13 + C 01 ( C 13 + C 02 ) (11) R = C 02 (12) R = C 13 . (13) Now we want to check wheth er t he special cases s tated in (8)-(10) can m eet the cut-set bound and hence achiev e the capacity . Our num erical analysis show that the rates obtained for condition s (9) and (10) can not meet the bound. Theorem 1 states that the first case indeed meets the cut-set bound and hence give s the capacity of the di amond channel for the specified channel gains and power resources. Theor em 1 Assuming a diamond r elay channel wit h non-zer o capacity links, the successive r elaying scheme achieves the capacity of the diamond channel if C 01 C 02 = C 13 C 23 . Pr oof: The cut -set bounds can be writ ten as R C 1 = t 1 C 012 + t 2 C 02 + t 3 C 01 + t 4 . 0 R C 2 = t 1 C 02 + t 2 ( C 02 + C 13 ) + t 3 . 0 + t 4 C 13 R C 3 = t 1 C 01 + t 2 . 0 + t 3 ( C 01 + C 23 ) + t 4 C 23 R C 4 = t 1 . 0 + t 2 C 13 + t 3 C 23 + t 4 C 123 . (14) W e consider cuts 2 and 3 and tr y to maximize the mi nimum of these cuts by finding the best time-sharing vector t ∗ , ( t ∗ 1 , ..., t ∗ 4 ) . Lemma 2 The time-sharing vector t hat maximizes the minimu m of the cut s 2 and 3 , is obtained by choosing t ∗ = [0 , C 01 C 13 + C 01 , C 02 C 02 + C 23 , 0] , and is in s uch a way tha t the cuts give the same rate. Lemma 3 The cut-set rate of cut s 2 and 3 with the time-sha ring vector t ∗ is C 01 ( C 13 + C 02 ) C 13 + C 01 , the same a s the rate obtained in (11). Now assume that we incr eas e t ∗ 1 and t ∗ 4 fr om 0 to ǫ ≥ 0 and η ≥ 0 , r espectively . T o satisfy the const raint of (3), t ∗ 2 and t ∗ 3 have to be decr eased by γ and δ , r espectively . Note that one of the γ and δ can be ne g ative. T o hold t he conditio n (3), the fo llowing r elat ion should ex ists between the adjustments γ + δ = ǫ + η . (15) Now let us calculate the rate change of the cuts 2 and 3. B y considering C 01 C 02 = C 13 C 23 , we have ∆ R C 2 = ( C 02 ǫ − ( C 02 + C 13 ) γ + C 13 η ) (16) ∆ R C 3 = ( C 13 ǫ − ( C 02 + C 13 ) δ + η ) C 01 C 13 , (17) wher e ∆ R C i i = 1 , ..., 4 is th e rate differ ence between the rate obtained using the modified time-sharing vector and the rate acquir ed by the ti me-sharing vector t ∗ . Using (15) and substitut ing δ in (17), we have ∆ R C 3 = ( − C 02 ǫ + ( C 02 + C 13 ) γ − C 13 η ) C 01 C 13 . (18) Note that the signs of the rate changes in (16) and (18) ar e differ ent. Considering the fact tha t with the gi ven time-sharin g vector in l emma 2, i.e. t ∗ , we had R C 2 = R C 3 , but with the adjustment s actuall y we decr ease the minimum of the rates which conclud es th e pr oof. It is in ter esting to see that t he optimu m time-sha ring vector t ∗ makes all the cut-set bounds to b e equal. Ther efor e the su ccessive relaying scheme achieves the capacity . V . C O N C L U S I O N In this report, the condit ion for optimali ty of th e successiv e relayin g scheme in a diamond-shaped relay channel , has been generalized from C 01 = C 23 and C 02 = C 13 to a more general form of C 01 C 02 = C 13 C 23 . V I . A C K N O W L E D G E M E N T H. Bagheri would like t o thank Mr . V ahid Pourahmadi for helpful discussions. R E F E R E N C E S [1] F . Xue and S. Sandhu, “Coo peration i n a half-duplex gaussian diamond relay channel”, Info. Theory , vol. 53, no. 10, pp. 3806–38 14, Oct. 2007. [2] B. W ang, J. Z hang, and A. Host-Madsen, “On the capacity of mimo relay channels”, IE EE T rans. on Info. Theory , vol. 51, no. 1, pp. 29–43, Jan. 2005 . [3] B. Schein and R. Gallager , “The gau ssian parallel relay netwo rk”, in in Pro c. IEEE Int. Symp. Inf. Theory , 200 0. [4] B. Schein, Distributed coor dination in network information theory , Ph.d., MIT , Cambridge, MA, 2001. [5] B. Rankov and A. W ittneben, “Spectral efficient signaling for half-duplex relay channels”, in in Proc . Asi lomar Conf. Signals, syst., comput. , Pacific Grove, CA, Nov . 2005. [6] W . Chang, S. Chung, and Y . H. Lee, “Capacity bounds for alternating two-p ath relay chan nels”, in in Pro c. of the Allerton Confer ence on Communications, Contr ol and C omputing , Monticello, IL, Oct. 2007. [7] S . S. Changiz Rezaei, S. Oveis Gharan, and A. K. Khan dani, “Cooperativ e strateg ies for half-duplex parallel relay channel: Simultaneous relaying versu s successi ve relaying” , T ech. Rep. UW -ECE 2008-02, Uni versity of W aterloo, 2008. [8] M. A. Khojastepour , A. Sabharwal, and B. Aazhang, “Bounds on achiev able rat es for general multi-terminal networks with practical constraints”, in Inf. Pr ocess. Sens. Netw .: Second Int. W ork. , Palo Alto, CA, Apr . 2003. [9] H. Bagheri, A. S. Motahari, and A. K. Khandani, “Scheduling for dual-hop communication with half-duplex relays”, T ech. Rep. UW -ECE 2008-05, Uni versity of W aterloo, 2008. On the Capaci ty of the Diamond Half-Duple x Relay Channel Hossein Bagheri , A bolfazl S. Mot ahari, and Ami r K. Khandani Dept. of Electrical Engineer ing University of W aterloo W aterloo, ON, Canada N2L3G1 { hbag heri, ab olfazl, khandani } @cst.uwaterloo.ca Abstract W e consider a diamond-sh aped dual-ho p commu nication system con sisting a source, two pa r- allel ha lf-duplex relays an d a destination . In a sing le a ntenna con figuration, it has been p reviously shown that a two-phase node- scheduling algorithm, along with the d ecode and fo rward strate gy can achieve the capacity of the diamo nd cha nnel fo r a cer tain sy mmetric channel gains [1]. In this pa per , we o btain a mo re general condition for th e optimality o f th e scheme in terms of power resources and chann el gain s. I n particular, it is proved that if the prod uct of th e capacity of the simultaneou sly activ e link s are equal in both tran smission p hases, the scheme achieves the capacity of the chann el. I . I N T R O D U C T I O N Half-duplex relays i.e., relays t hat can not transmi t and recei ve at the same ti me, hav e recently attracted enormous attention due to their simplicity and cost ef ficiency . Some capacity results for the case of a single half-duplex relay are presented in [2]. T o realize multiple-antenna benefits wit hout increasing th e w eight and s ize of t he equipments, m ultiple relays come i nto play . A s imple model for in vestigati ng the potential benefits of the multiple relays is depicted in Fig. 1. The end-to-end capacity of th e relay channel h as been studied in [1], [3]–[7] and is referred to as diamond r elay channel in [1]. Schein in [3] and [4] Financial supp ort pro vided by Nortel and t he corresponding matching funds by the Natural Sciences and E ngineering Research Council of Canada (NSERC), and Ontario Centres of Excellence (OCE) are gratefully acknowledg ed. established upper and lo w er boun ds on the capacity of the full-dupl ex diamond channel. Half- duplex relays ha ve been con sidered in [1], [5]–[7]. Xue, and Sandhu in [1] proposed several communication schemes including mult ihop wit h spatial reuse, scale-forward, broadcast- multiaccess with common message, compress-forward, as well as hybrid ones. They assumed that there i s no interference between the transm itting and receiving relays. Howe ver [5]–[7] considered s uch int erference and used di f ferent n ames of two-way [5], [6] and s uccessive [7] relaying, respective ly for th e multihop with spatial r euse prot ocol. In t his work, we follow the set-up of [1] and assume t here i s no s uch int erference. In t his paper , we are interested in s ituations where t he s imple st rategy o f successive relaying achieves the capacity of the diamond channel. Surprisingly , we prove t hat when the product of the capacity of the sim ultaneously active links in both transmission phases are equal, the scheme achieves the capacity . Note that t his condition includes the result indicated in [1] as a special case. A. Notations Throughout this paper , boldface letters are used to denote vectors. A → B represents the l ink from node A to node B . The notation also means: appr oaches to when the right hand si de of → is a number . A circularly symmetric com plex Gaussian rando m variable Z with mean 0 and variance σ 2 , is represented by Z ∼ C N (0 , σ 2 ) . I I . T R A N S M I S S I O N M O D E L OV E R D I A M O N D R E L AY C H A N N E L In t his work, a dual -hop wireless system depicted in Fig.1, is considered. The model consists of a so urce (S), two p arallel half-duplex relays ( Re 1 , Re 2 ) and a destination (D). The corresponding in dex for the nodes are 0, 1, 2, 3, as shown in the Fig. 1. W e assum e that all the nodes are equipped with a singl e antenn a. Also, du e to the long distance or strong shadowing, no link is ass umed between the source and the destination , as well as between the relays. The background noise at each recei ver i s cons idered to be additive white Gaussian n oise (A WGN). The channel g ain between node i and j is assumed to be cons tant and is represented by g ij as shown in the Fig. 1 . In addit ion, the channel st ate information knowledge at di f ferent n odes is as follows Source Relay # 1 Relay # 2 Destination g 13 g 23 X 1 X 2 g 02 g 01 Y 1 Y 2 X 0 Y 3 Fig. 1. The diamond relay chann el • Source kn ows all the channel gai ns ( i.e. , g 01 , g 02 , g 13 and g 23 ). • The relay i knows i ts i nward and outward channel gains ( g 0 i and g i 3 ). • The desti nation knows its inward channel gains ( g 13 and g 23 ). For this model, the successive relayi ng scheme i s performed in t wo stages 1 : 1) In t he first λ portion of the transm ission ti me 2 , S and Re 2 transmit to Re 1 and D, respectiv ely . 2) In the remaining (1 − λ ) porti on of the transm ission time, S and Re 1 transmit to Re 2 and D, respectiv ely . Note th at the relays decode and then re-encode what they ha ve receiv ed prior to their transmissio n. The other possible scheduling algorithm for the h alf-duplex diamond channel is called s imultaneous relaying [7] or transm ission p olicy I [1] and is as fol lows: 1) In the first λ po rtion of the transmi ssion time, S broadcasts its data to both relays. 2) In the remain ing (1 − λ ) portion of the transmis sion time, relays cooperativ ely transmit to D. It is possible to com bine bot h scheduli ng methods, h owe ver in this work, successive relaying is considered only . The receiv ed discrete-time complex baseband equiva lent signals at R e 1 1 Scheduling is assumed to be done in adv ance. 2 T otal transmission t ime is T time slots. Re 2 , and D are respectively giv en by Y 1 [ m ] = h 01 X 0 [ m ] + N 1 [ m ] Y 2 [ n ] = h 02 X 0 [ n ] + N 2 [ n ] Y 3 [ m ] = h 23 X 2 [ m ] + N 3 [ m ] Y 3 [ n ] = h 13 X 1 [ n ] + N 3 [ n ] , (1) where X i and Y j are the t ransmitted and recei ved si gnals from and to node i and j , respectiv ely (see Fig. 1). m ∈ { 1 , ..., λT } and n ∈ { λT + 1 , ..., T } denote the t ransmission time i ndex corresponding to the two stages 3 . h ij is the comp lex channel coefficient and is connected to g ij by g ij = | h ij | 2 . N j is the noise at node j and N j ∼ C N (0 , σ 2 j ) , for j = 1 , 2 , 3 . W e assu me a verage power constraint for S, R e 1 and Re 2 and denote the constraints by P S , P Re 1 and P Re 2 , respectiv ely , i .e. for λT tuple and (1 − λ ) T tuple sub-codewords, we ha ve 1 λT λT X k =1 | x 01 [ k ] | 2 ≤ P S 1 (1 − λ ) T T X k = λT +1 | x 02 [ k ] | 2 ≤ P S 1 λT λT X k =1 | x 23 [ k ] | 2 ≤ P Re 2 1 (1 − λ ) T T X k = λT +1 | x 13 [ k ] | 2 ≤ P Re 1 , (2) where x ij [ k ] is the symbo l corresponding to the k th index, in the sub-code word transmitt ed by node i to no de j . Note that instead of allocating P S to both S → Re 1 and S → Re 2 links, we could allocate different powers t o th em i.e. , ρP S and µP S with the condition ρλ + µ (1 − λ ) = 1 for ρ ≥ 0 and µ ≥ 0 . Howev er du e t o th e s implicity of implementati on and also exposition, we choose ρ = µ = 1 . In addition, power budget at th e relay nodes assumed to be fixed over time. A rate R is said t o be achie vable for t his scheme, if for T → ∞ , D can decode the message wi th error probability ǫ → 0 . 3 It is assumed that λT is an i nteger . T ABLE I T R A N S M I S S I O N S TA T E S I N D I A M O N D R E L A Y C H A N N E L . State R 1 R 2 T i me S 1 Rx Rx t 1 S 2 Tx Rx t 2 S 3 Rx Tx t 3 S 4 Tx Tx t 4 In t he next three s ections, we first derive the cut-set upper bound and the achiev able rate using the successiv e relaying p rotocol, and then we state t he condition for opt imality of the schem e. I I I . C U T - S E T U P P E R B O U N D The upper b ound for half-duplex networks is given in [8] based on writing the well known cut-set bounds for different transm ission states shown in table I and t hen adding them up. Tx and Rx deno te transmi t and receive modes i n the tabl e, respectiv ely . Applying t he procedure t o th e d iamond relay channel, we have [1] R C 1 = t 1 I ( X 0 ; Y 1 , Y 2 ) + t 2 I ( X 0 ; Y 2 ) + t 3 I ( X 0 ; Y 1 ) + t 4 . 0 R C 2 = t 1 I ( X 0 ; Y 2 ) + t 2 ( I ( X 0 ; Y 2 ) + I ( X 1 ; Y 3 )) + t 3 . 0 + t 4 I ( X 1 ; Y 3 ) R C 3 = t 1 I ( X 0 ; Y 1 ) + t 2 . 0 + t 3 ( I ( X 0 ; Y 1 ) + I ( X 2 ; Y 3 )) + t 4 I ( X 2 ; Y 3 ) R C 4 = t 1 . 0 + t 2 I ( X 1 ; Y 3 ) + t 3 I ( X 2 ; Y 3 ) + t 4 I ( X 1 , X 2 ; Y 3 ) , where C j for j = 1 , ..., 4 represents the j th cut-set. The tim e-sharing coef ficients t i s satisfy P 4 i =1 t i = 1 . Using Gaussian codebooks and defining C ij = log 2 (1 + g ij P ij σ 2 j ) with P ij as the power allocated to th at link, we have the following optimization problem: max t i ≥ 0 R s.t. R ≤ t 1 C 012 + t 2 C 02 + t 3 C 01 + t 4 . 0 R ≤ t 1 C 02 + t 2 ( C 02 + C 13 ) + t 3 . 0 + t 4 C 13 R ≤ t 1 C 01 + t 2 . 0 + t 3 ( C 01 + C 23 ) + t 4 C 23 R ≤ t 1 . 0 + t 2 C 13 + t 3 C 23 + t 4 C 123 4 X i =1 t i = 1 , (3) where P 01 = P S , P 02 = P S , P 13 = P R 1 , P 23 = P R 2 , C 012 = log 2 [1 + ( g 01 σ 2 1 + g 02 σ 2 2 ) P S ] , and C 123 = log 2 [1 + 1 σ 2 3 ( p g 13 P Re 1 + p g 23 P Re 2 ) 2 ] . I V . A C H I E V A B L E R A T E U N D E R S U C C E S S I V E R E L A Y I N G It has been s hown in [1] (Theorem 4.1) that t he maximu m achiev able rate under th is transmissio n policy , is given by R S R = max { R 1 , R 2 } (4) R 1 = λ 1 C 01 + min { λ 1 C 23 , (1 − λ 1 ) C 02 } (5) R 2 = λ 2 C 23 + min { (1 − λ 2 ) C 13 , λ 2 C 01 } (6) with λ 1 = C 13 C 13 + C 01 λ 2 = C 02 C 23 + C 02 . (7) It has als o been proved that the d ecode and forward scheme is the best forwarding scheme under th e successiv e relaying protocol (Theorem 4 .2 in [1]). In [1], it is stated that for the case o f P S = P Re 1 = P Re 2 , t he scheme achiev es the cut-set up per bound if the channel gains are such t hat C 02 = C 13 and C 01 = C 23 (corollary 4.3). Here we generalize t he condition of op timality in th eorem 1. Before proving the theorem, it is note worthy to fi nd some special cases in the successive relaying scheme. Lemma 1 states su ch cases. In particular , W e are interested in the cases where R 1 = R 2 in (5) and (6). Lemma 1 Assu ming all li nks have non-zer o capacit y , the conditions for R 1 = R 2 ar e o ne of t he followings: C 01 C 02 = C 13 C 23 (8) C 01 = C 02 if C 01 C 02 ≤ C 13 C 23 (9) C 13 = C 23 if C 01 C 02 ≥ C 13 C 23 (10) Pr oof: See [9]. The rate in (4) associated wi th each condit ion of equality (8-10) is: R = C 01 C 13 + C 01 ( C 13 + C 02 ) (11) R = C 02 (12) R = C 13 (13) Now we want to check wheth er t he special cases s tated in (8)-(10) can m eet the cut-set bound and hence achiev e the capacity . Our num erical analysis show that the rates obtained for condition s (9) and (10) can not meet the bound. Theorem 1 states that the first case indeed meets the cut-set bound and hence give s the capacity of the di amond channel for the specified channel gains and power resources. Theor em 1 Assuming a diamond r elay channel wit h non-zer o capacity links, the successive r elaying scheme achieves the capacity of the diamond channel if C 01 C 02 = C 13 C 23 . Pr oof: The cut -set bounds can be writ ten as R C 1 = t 1 C 012 + t 2 C 02 + t 3 C 01 + t 4 . 0 R C 2 = t 1 C 02 + t 2 ( C 02 + C 13 ) + t 3 . 0 + t 4 C 13 R C 3 = t 1 C 01 + t 2 . 0 + t 3 ( C 01 + C 23 ) + t 4 C 23 R C 4 = t 1 . 0 + t 2 C 13 + t 3 C 23 + t 4 C 123 (14) W e consi der cuts 2 and 3 and try to maximize the minimum of these cuts by finding the best time-sharing vector t ∗ , ( t ∗ 1 , ..., t ∗ 4 ) . Lemma 2 The time-sharing vector t hat maximizes the minimu m of the cut s 2 and 3 , is obtained by choosing t ∗ = [0 , C 01 C 13 + C 01 , C 02 C 02 + C 23 , 0] , and is in s uch a way tha t the cuts give the same rate. Lemma 3 The cut-set rate of cut s 2 and 3 with the time-sha ring vector t ∗ is C 01 ( C 13 + C 02 ) C 13 + C 01 , the same a s the rate obtained in (11). Now assume that we incr eas e t ∗ 1 and t ∗ 4 fr om 0 to ǫ ≥ 0 and η ≥ 0 , r espectively . T o satisfy the const raint of (3), t ∗ 2 and t ∗ 3 have to be decr eased by γ and δ , r espectively . Note that one of the γ and δ can be ne g ative. T o hold the condition (3), t he following relation should ex ists between the adjustments γ + δ = ǫ + η (15) Now let us calculate the rate change of the cuts 2 and 3. B y considering C 01 C 02 = C 13 C 23 , we have ∆ R C 2 = ( C 02 ǫ − ( C 02 + C 13 ) γ + C 13 η ) (16) ∆ R C 3 = ( C 13 ǫ − ( C 02 + C 13 ) δ + η ) C 01 C 13 , (17) wher e ∆ R C i i = 1 , ..., 4 is th e rate differ ence between the rate obtained using the modified time-sharing vector and the rate acquir ed by the ti me-sharing vector t ∗ . Using (15) and substitut ing δ in (17), we have ∆ R C 3 = ( − C 02 ǫ + ( C 02 + C 13 ) γ − C 13 η ) C 01 C 13 . (18) Note that the signs of the rate changes in (16) and (18) ar e differ ent. Considering the fact tha t with the gi ven time-sharin g vector in l emma 2, i.e. t ∗ , we had R C 2 = R C 3 , but with the adjustment s actuall y we decr ease the minimum of the rates which conclud es th e pr oof. It is in ter esting to see that t he optimu m time-sha ring vector t ∗ makes all the cut-set bounds to b e equal. Ther efor e the su ccessive relaying scheme achieves the capacity . V . C O N C L U S I O N In this report, the condit ion for optimali ty of th e successiv e relayin g scheme in a diamond-shaped relay channel , has been generalized from C 01 = C 23 and C 02 = C 13 to a more general form of C 01 C 02 = C 13 C 23 . V I . A C K N O W L E D G E M E N T H. Bagheri would like t o thank Mr . V ahid Pourahmadi for helpful discussions. R E F E R E N C E S [1] F . Xue and S. Sandhu, “Coo peration i n a half-duplex gaussian diamond relay channel”, Info. Theory , vol. 53, no. 10, pp. 3806–38 14, Oct. 2007. [2] B. W ang, J. Z hang, and A. Host-Madsen, “On the capacity of mimo relay channels”, IE EE T rans. on Info. Theory , vol. 51, no. 1, pp. 29–43, Jan. 2005 . [3] B. Schein and R. Gallager , “The gau ssian parallel relay netwo rk”, in in Pro c. IEEE Int. Symp. Inf. Theory , 200 0. [4] B. Schein, Distributed coor dination in network information theory , Ph.d., MIT , Cambridge, MA, 2001. [5] B. Rankov and A. W ittneben, “Spectral efficient signaling for half-duplex relay channels”, in in Proc . Asi lomar Conf. Signals, syst., comput. , Pacific Grove, CA, Nov . 2005. [6] W . Chang, S. Chung, and Y . H. Lee, “Capacity bounds for alternating two-p ath relay chan nels”, in in Pro c. of the Allerton Confer ence on Communications, Contr ol and C omputing , Monticello, IL, Oct. 2007. [7] S . S. Changiz Rezaei, S. Oveis Gharan, and A. K. Khan dani, “Cooperativ e strateg ies for half-duplex parallel relay channel: Simultaneous relaying versu s successi ve relaying” , T ech. Rep. UW -ECE 2008-02, Uni versity of W aterloo, 2008. [8] M. A. Khojastepour , A. Sabharwal, and B. Aazhang, “Bounds on achiev able rat es for general multi-terminal networks with practical constraints”, in Inf. Pr ocess. Sens. Netw .: Second Int. W ork. , Palo Alto, CA, Apr . 2003. [9] H. Bagheri, A. S. Motahari, and A. K. Khandani, “Scheduling for dual-hop communication with half-duplex relays”, T ech. Rep. 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