Cooperative Strategies for the Half-Duplex Gaussian Parallel Relay Channel: Simultaneous Relaying versus Successive Relaying

Cooperati v e Strate gies for the Half-Duple x Gaussian P arallel Relay Channel: Simultaneous Relaying v ersus Successi ve Relaying Seyed Saeed Changiz Rezaei, Shahab Oveis Gharan , and Amir K. K handani Coding & Signal Tran smission Laboratory Department of Electrical & Computer Engineering Univ ersity of W aterloo W aterloo, ON, N2L 3G1 sschangi, shahab, khandan i@cst.uwaterloo.ca Abstract This study in vestigates the problem of communication for a ne twork composed of two half-duple x parallel relays with additi ve white Gaussian noise. T wo protocols, i.e., Simultaneous and Successive relaying, associated wi th two possible relay orderings are prop osed. The simultaneous relaying protocol i s based on Dynamic Decod e and F orwar d (DDF) scheme. For the successiv e relaying pro tocol: (i) a Non-Coope rative scheme based on the Dirty P aper Coding (DPC) , and (ii) a Cooper ative schem e based on the Block Markov Enco ding (BME) are considered. Furthermore, the composite scheme of emplo ying BME at one r elay and DPC at another always achiev es a better rate when comp ared to the Cooperative scheme. A “Simultaneous-Successive R elaying based on Dirty paper coding scheme” (SSRD) is also prop osed. The op timum ordering of the relays and hence the capacity of the half-duplex Gaussian parallel relay channel in the low and high signal-to-noise r atio (SNR) scenarios is deri ved. In the lo w SNR scenario, it is re vealed that under certain conditions for the chann el coef ﬁcients, the ratio of the achie vable r ate of the simultaneo us relaying based on DDF to the cut-set bound t ends to be 1. On the other hand, as S NR goes t o inﬁnity , it is prove d that successi ve relaying, based on the DPC, asymptotically achiev es the capacity of t he netwo rk. I . I N T R O D U C T I O N A. Motivation The co ntinuous gr owth in wireless co mmunicatio n has motiv ated in formatio n theoretists to extend shanno n’ s informa tion theoretic arguments for a single user chan nel to the scenarios that in volve commu nication among multiple users. Financi al suppo rts provided by Nortel, and the correspo nding match ing fund s by the Fede ral go ve rnment: Natural Sci ences and Engineering Researc h Council of Ca nada (NSE RC) and Province of Ontari o: Ontario Centres of Exce llence (OCE) are gratefully acknowl edged. 2 In this regard, coo perative wireless c ommunica tion h as bee n the focus of attention du ring recen t years. Due to rap id decrease of the tran smitted signal power with d istance, the ide a of multi- hopped commun ication has been pro posed. In mu lti-hopped commu nication, som e intermediate n odes as rela ys are exploited to facilitate d ata transmission f rom the source to the destination. Using this technique lead s to sa ving battery power as well as increasing the ph ysical coverage a rea. Moreover, relays by em ulating distributed tr ansmit antenn a, can form spatial div ersity and comb at the multi-pa th fading e ffect of the wir eless media. Motiv ated by practical co nstraints, half-d uplex relays which canno t transmit and receive at the same time and in the same f requen cy ban d are of great im portance . Her e, ou r goal is to stud y and an alyze the perfor mance lim its of a half- duplex parallel relay chann el. B. History Relay chan nel is a th ree termin al n etwork wh ich was introd uced for the ﬁrst time b y V an der Meulen in 1 971 [1]. T he most imp ortant cap acity results of the relay channel w ere r eported by Cover and El Gamal [2] . T wo relaying strategies are pro posed in [2] . In one strategy , the relay decodes the transmitted message and forwards the re-enco ded version to the destination, wh ile in an other one the relay does not decod e th e message, but send s the quantized received values to the de stination. Moreover , several work s on m ulti-relay chann els exist in the literature (See [3]– [11], [23], [29 ]–[36 ]). Schein in [3], [4 ] establishes uppe r and lower bo unds on th e capacity of a full-d uplex parallel relay ch annel in which the channel consists of a so urce, two relays and a d estination, where there is no direct link betwee n the source and the destination, an d also between the two relays. Generally , the best rate r eported for the full-du plex G aussian par allel relay chan nel is based on the Decode -Forward (DF) or Amp lify-Forward (AF) sch emes, with time sharing [3] , [4]. Xie and Kumar generalize the block Markov encoding scheme in [2] for a network of multip le relays [5]. Gastpar, Kramer, and Gupta extend compress an d fo rward scheme to a multiple relay ch annel by introdu cing th e c oncept of anten na po lling in [6]–[ 8]. I n [9], Amichai, Sham ai, Stein berg and Kramer consider a p arallel relay setup, in which a no madic so urce sends its inf ormation to a remo te d estination via some r elays with lossless link s to the destination. They investigate the case that th ese relays d o not have any decod ing capability , so signals received at the relays must b e compr essed. The a uthors also fu lly characterize the cap acity of this c ase for the Gaussian chann el. In [10] , Maric and Y ates investigate DF an d AF schemes in a parallel-r elay network. Motiv ated by ap plications in sensor networks, th ey a ssume la rge ba ndwidth reso urces allowing o rthogo nal tran smissions at different no des. They characterize o ptimum resource allo cation f or AF and DF and show that the wide-band regime min imizes th e en ergy cost per info rmation bit in DF , while AF sho uld work in the band -limited regime to achieve the best rate. Razaghi and Y u in [11] pr opose a parity-f orwarding scheme for f ull-duplex m ultiple relay . They sho w th at p arity-for warding can achieve the cap acity in a n ew fo rm of degraded r elay networks. Radios that can rec ei ve and transm it simultaneo usly in th e same fr equency band r equire co mplex and expensive compon ents [18] . Hence, Khojastepou r and Aazhang in [1 3], [14] call the half- duplex relay as “ Cheap Re lay ”. Recently , h alf-dup lex relay ing has d rawn a great deal of atten tion (See [1 3]–[1 9], [23], [2 9]–[3 6]). Zahedi an d DRAFT 3 El Gam al consider two d ifferent cases of fr equency division Gau ssian relay ch annel, der i ving lower and upper bound s o n the cap acity [1 5]. T hey also derive sing le letter char acterization of the capacity of frequen cy division additive white Gaussian noise (A WGN) relay channe l with simp le linear r elaying scheme [1 6], [17 ]. Th e p roblem of time di vision relaying is also consid ered b y Host-Mad sen and Zhan g [18]. By considering fadin g scenarios, and assuming channel state infor mation (CSI), they study u pper and lower bou nds o n the o utage capacity and the Ergodic capacity . In [1 9], Lian g and V eeralli pr esent a Gaussian o rthogo nal relay m odel, in which the re lay-to- destination channel is orthog onal to the source-to-re lay and source-to- destination chann el. They sho w th at wh en the source-to- relay chan nel is better than th e source- to-destination chan nel an d the sign al-to-noise ratio (SNR) of the relay-to- destination is less than a given threshold , optim izing re source allo cation ca uses the lower an d the u pper bound s to coincide with e ach other . C. Contributions and Relation to Pr evious W orks In this paper, we study tr ansmission strategies f or a network with a source, a destination , and two half-d uplex relays with additive white G aussian noise which coop erate with each o ther to facilitate d ata transmission from the source to the d estination. Furtherm ore, it is assumed that n o direct link exists between the sou rce and th e destination. Half-dup lex relaying , in mu ltiple relay networks, is studied in [23], [ 29]–[3 6]. Gastpar in [23] shows that in a Gaussian parallel relay chann el with inﬁnite nu mber of r elays, the optimum coding scheme is AF . Rankov and W ittneben in [29], [30] further study the pr oblem of half-dup lex relay ing in a two-hop c ommun ication scenario. In their study , they also consider a parallel relay setup with two re lays wh ere there is no d irect link b etween the source and the destination , while there exists a link between the relays. Th eir relaying pro tocols ar e based o n either AF or DF , in which the relays successiv ely for ward their messages fro m the source to the destination. W e call this protoco l “ Successive Relaying ” in the seq uel. Xue and San dhu in [31] fur ther stu dy d ifferent half -duplex relay ing protoco ls for th e Gaussian parallel relay channel. Since they assume th at there is no link between th e relay s, th ey refer to the ir parallel ch annel as a Dia mond Relay Chan nel . In th is work, ou r prim ary objec ti ve is to ﬁn d the b est o rdering of th e relay s in th e inten ded set-up. W e consider two relaying p rotocols, i.e., simultaneo us r elaying versus successive relayin g, associated with two p ossible relay orderin gs. For simultaneo us relay ing, each rela y exploits “Dynamic DF (DDF)” . It shou ld be no ted tha t the DD F scheme conside red h ere is slightly different from th e DDF introdu ced in [3 4] and [3 5]. In those works, the DDF scheme is app lied to the set-up of the multiple relay network in which th e no des only have the CSI of their r eceiving channel. In the DDF sch eme described in [34], the sourc e is broadca sting the message to all the network nodes during whole per iod of transmission and ea ch relay , listens to the tr ansmitted signal of the so urce a nd other relay s until it ca n decod e the transmitted message. Conseq uently , it transmits its signal co herently with th e source and other active relays in the r emaining time. Howe ver , in our set-up, all the nod es are assumed to ha ve all the channel coefﬁcients. Therefo re, in a ﬁxed pr e-assigned portion of the time, the relays r eceiv e the signal transmitted f rom the source, an d in the r emaining time slot they transmit the r e-encod ed version o f the de coded message together . In other word s, th e relays o perate in a synchro nous manner . DRAFT 4 For succ essi ve re laying, we study a Non- Cooperative scheme ba sed on “Dirty Paper Cod ing (DPC)” and also a Cooperative scheme based on “Block Mar kov Enc oding (BME)”. It is worth noting th at th e a uthors in [36] also propo se successi ve relaying protocol for the set up with two parallel relays and direct link s between the r elays and between the source and th e destination. They pr opose a simple repetition cod ing at th e r elays, and show that th eir scheme can re cover the loss in the multiplexing g ain, wh ile achieving d iv ersity g ain of 2. W e derive the optimum relay or dering in low an d high SNR scenario s. In low SNR scenarios and un der certain channel cond itions, we show that the ratio o f the achiev able r ate o f DDF for simultan eous relay ing to the c ut-set bound tends to one . On the other han d, in high SNR scenarios, we prove tha t the prop osed DPC for successiv e relaying asympto tically ach iev es th e capacity . After this work was co mpleted, we becam e aware o f [32] which h as in depend ently proposed an ac hiev able rate based o n the comb ination of su perposition cod ing, BME an d DPC. In their scheme, the intend ed message “ w ” is split into a m essage which is transmitted to the destination by explo iting cooperatio n between the relays “ w r ” an d a message wh ich is transmitted to the d estination withou t using any c ooperatio n between the r elays “ w d ” . Hence, the signal assoc iated with “ w d ” , tr ansmitted b y o ne re lay , can be con sidered as interfer ence on the other relay . “ w r ” is transmitted by using BME and “ w d ” is transmitted by employing DPC. Theref ore, in their g eneral scheme, the associated signals with th ese two messages ar e super imposed and tr ansmitted. As th e chan nel between the two relays b ecome stron g, their prop osed scheme is converted to BME. On the oth er hand, as th e channel be comes weak, their p roposed scheme b ecomes DPC. Unlike [32 ], in which the au thors o nly consid er successiv e rela ying and pro pose a combine d BME and DPC, as the main result of this paper, simultan eous and succ essi ve relay ing pro tocols are combine d and a “Simultaneous- Successiv e Relaying based on Dirty paper c oding” (SSRD) scheme with a new ach ie vable rate is pr oposed. I t is shown that in the low SNR scenario and under certain channel conditions, SSRD scheme is con verted to simultaneous relaying b ased on DDF , while in the h igh SNR scenarios, when the r atio of the relay powers to the sourc e power remain con stant, it becomes succ essi ve relay ing based on DPC (to ach iev e the capacity ). Besides this ma in result, som e other results obtained in this paper a re as follows: • T wo d ifferent types of deco ding, i.e., successive and b ackwar d de coding, at the destina tion for the BME scheme are prop osed. W e prove that the achievable ra te of BME with back ward decoding is greater th an that of BME with successive deco ding, i.e., C low B M E back ≥ C low B M E succ . • It is proved that BME with backward decoding lead s to a simp le stra tegy in which at mo st, one of the relays is required to co operate with the oth er relay in sending the b in ind ex of th e o ther relay’ s m essage. Acco rdingly , in the Gaussian case, the co mbination of BME at on e relay and DPC at the other relay always ac hiev es a better rate than the simp le BME. • In th e d egraded case, wher e the destina tion receives a degra ded version of th e r eceiv ed signals at the relay s, BME with back ward deco ding achieves the successive cut-set bo und. The re st of th e pape r is organized as f ollows: In sectio n II, the system model is introdu ced. I n section I II, the achiev able rates an d cod ing schemes for a half-d uplex relay network are de riv ed. O ptimality r esults are discussed DRAFT 5 in section I V . Simulatio n results are presen ted in section V . Finally , section VI con cludes the pap er . D. No tation Throu ghout the paper, the supe rscript H stands fo r matrix op eration of co njugate tr ansposition. Lowercase bold letters and regular letters represent vector s and scalars, respec ti vely . For any two fun ctions f ( n ) an d g ( n ) , f ( n ) = O ( g ( n )) is eq uiv alent to lim n →∞    f ( n ) g ( n )    < ∞ , and f ( n ) = Θ( g ( n )) is equ iv alent to lim n →∞ f ( n ) g ( n ) = c , where 0 < c < ∞ . And C ( x ) , 1 2 log 2 (1 + x ) . Fur thermore, for the sake of brevity , A ( n ) ǫ denotes the set of weak ly jointly typical sequen ces f or any in tended set of rand om variables. I I . S Y S T E M M O D E L W e consider a Gaussian network which con sists of a source , two half-dup lex r elays, and a d estination, and there is no direct link between the source a nd th e destinatio n. Her e we d eﬁne fo ur time slots accor ding to the tr ansmitting and receiving mode of each relay (See Fig. ?? ), where t b denotes the dur ation of time slot b ( P 4 b =1 t b = 1 ) . Nodes 0, 1, 2, and 3 rep resent the sourc e, relay 1, relay 2, and the destination, respectively . Mo reover , the transmitting and receiving signals at no de a during time slot b are repre sented by x ( b ) a and y ( b ) a , respectively . Hence , at each node c ∈ { 1 , 2 , 3 } , we have y ( b ) c = X a ∈{ 0 , 1 , 2 } h ac x ( b ) a + z ( b ) c . (1) where h ac , s de note ch annel coefﬁcients fro m node a to node c , an d z ( b ) c is the A WGN term with zero mean and variance of “ 1 ” per dim ension. Noting the tra nsmission strategies in Fig. ?? , we h av e y (1) 1 = h 01 x (1) 0 + h 21 x (1) 2 + z (1) 1 , (2) y (1) 3 = h 23 x (1) 2 + z (1) 3 , (3) y (2) 2 = h 02 x (2) 0 + h 12 x (2) 1 + z (2) 2 , (4) y (2) 3 = h 13 x (2) 1 + z (2) 3 , (5) y (3) k = h 0 k x (3) 0 + z (3) k , k ∈ { 1 , 2 } , (6) y (4) 3 = 2 X k =1 h k 3 x (4) k + z (4) 3 . (7) Throu ghout th e paper, we assume that h 01 ≥ h 02 unless speciﬁed otherwise, and fro m reciproc ity assump tion, we have h 12 = h 21 . Further more, th e power constraints P 0 , P 1 , and P 2 should be satisﬁed f or the source, the ﬁrst re lay , and the seco nd r elay , resp ecti vely . Hen ce, den oting the p ower con sumption of node a at time slot b by P ( b ) a = E h x ( b ) H a x ( b ) a i , we have P (1) 0 + P (2) 0 + P (3) 0 = P 0 , (8) P (2) 1 + P (4) 1 = P 1 , P (1) 2 + P (4) 2 = P 2 . DRAFT 6 Source Destination Rela y 1 Rela y 2 h 01 h 12 h 23 the v ectors x (1) 0 and x (1) 2 . The ﬁrst rela y and the destination rece iv e y (1) 1 and y (1) 3 , respectiv ely . The source and the second rela y transmit a) Time slot 1 with duration t 1 : The source and the ﬁrst rela y transmit the v ectors x (2) 0 and x (2) 1 . The sec ond rela y and the destination receive y (2) 2 and y (2) 3 , resp ectiv ely . Source Destination Rela y 1 Rela y 2 h 12 h 13 h 02 b) Time slot 2 with duration t 2 : Source Destination Rela y 1 Rela y 2 h 01 h 02 The source transmits the vector x (3) 0 . The ﬁrst and the second rela y rec eiv e y (3) 1 and y (3) 2 , respectiv ely . c) Time slot 3 with duration t 3 : Source Destination Rela y 1 Rela y 2 h 13 h 23 The destination receiv es y (4) 3 . The relays transm it the vectors x (4) 1 and x (4) 2 . d) Time slot 4 with duration t 4 : Fig. 1. System Model. I I I . A C H I E V A B L E R AT E S A N D C O D I N G S C H E M E S In th is section , we prop ose two coop erative pr otocols, i.e. Succe ssive and Simu ltaneous relayin g proto cols, for a half-du plex Gaussian parallel relay channe l. A. Successive Relayin g Pr otocol In Suc cessive relaying protoco l, the relays are no t allowed to receive and transmit simu ltaneously , i.e. t 3 = t 4 = 0 , and the relations between the transmitted and the received signals at the r elays and at the de stination f ollow fr om (2)-(5). For the successive relaying proto col, we p ropose a Non-Coo perative and a Cooperative Coding scheme in the sequel. In the p roposed sch emes, the tim e is divided into odd and even time slots with the duration t 1 and t 2 , respectively . Accor dingly , at each odd and even time slots, the sou rce transmits a n ew message to one of the relays, and the de stination receives a n ew message from the other r elay , successively (See Fig . 2). DRAFT 7 R (2) R (1) R (1) R (2) R (2) R (1) Fig. 2. Information ﬂow transfer for successi v e relaying protocol for two relays. 1) Non-Coop erative Coding: In th e Non -Cooperative Codin g schem e, each relay considers the other ’ s sig nal as interferen ce. Since th e source k nows each relay ’ s message, it can apply the Gelfand-Pinsker’ s co ding scheme to transmit its message to th e other relay . Th e f ollowing Theo rem g i ves th e achievable rate of this sch eme. Source Destination Time Slot 2 with duration t 2 R (2) R (1) Source Destination Time Slot 1 with duration t 1 R (1) R (2) Fig. 3. Successiv e relaying protocol based on Non-Cooperati ve Coding. Theorem 1 F or the ha lf-duplex parallel r elay chan nel, assuming successive relaying, the following rate C low DP C is achievable: C low DP C = max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 R (1) + R (2) , (9) subject to: R (1) ≤ min  t 1 ( I ( U (1) 0 ; Y (1) 1 ) − I ( U (1) 0 ; X (1) 2 )) , t 2 I ( X (2) 1 ; Y (2) 3 )  , (10) R (2) ≤ min  t 2 ( I ( U (2) 0 ; Y (2) 2 ) − I ( U (2) 0 ; X (2) 1 )) , t 1 I ( X (1) 2 ; Y (1) 3 )  . (11) with pr obabilities: p ( x (1) 2 , u (1) 0 , x (1) 0 ) = p ( x (1) 2 ) p ( u (1) 0 | x (1) 2 ) p ( x (1) 0 | u (1) 0 , x (1) 2 ) , p ( x (2) 1 , u (2) 0 , x (2) 0 ) = p ( x (2) 1 ) p ( u (2) 0 | x (2) 1 ) p ( x (2) 0 | u (2) 0 , x (2) 1 ) . DRAFT 8 Pr o of: See Appe ndix A. From Theorem 1, the achievable rate o f the pro posed scheme for the Gaussian case can be ob tained a s fo llows. corollary 1 F or the half-d uplex Gaussian p arallel r elay chann el, assuming successive relaying pr otocol wit h p ower constraint at the so ur ce a nd at each relay , DPC a chieves the following rate: C low DP C = max  R (1) + R (2)  , (12) subject to: R (1) ≤ min t 1 C h 2 01 P (1) 0 t 1 ! , t 2 C  h 2 13 P 1 t 2  ! , R (2) ≤ min t 2 C h 2 02 P (2) 0 t 2 ! , t 1 C  h 2 23 P 2 t 1  ! , P (1) 0 + P (2) 0 = P 0 , t 1 + t 2 = 1 , 0 ≤ t 1 , t 2 , P (1) 0 , P (2) 0 . Pr o of: From Costa’ s Dirty Paper Coding [28] , b y having U (1) 0 = X (1) 0 + h 01 h 12 P (1) 0 h 2 01 P (1) 0 + t 1 X (1) 2 , (13) U (2) 0 = X (2) 0 + h 02 h 12 P (2) 0 h 2 02 P (2) 0 + t 2 X (2) 1 . (14) where X (1) 0 ∼ N (0 , P (1) 0 ) , X (2) 0 ∼ N (0 , P (2) 0 ) , X (1) 2 ∼ N (0 , P 2 ) , and X (2) 1 ∼ N (0 , P 1 ) , and ap plying th em to Theorem 1, we obtain corollary 1. ( ˆ s ( b  2) 2 ; ˆ w ( b  2) ) x (2) 1 ( w ( b  1) | s ( b  2) 2 ) ; u (2) 1 ( s ( b  2) 2 ) ( ˆ w ( b  1) ; ˆ w ( b ) ) x (2) 0 ( w ( b ) | w ( b  1) ; s ( b  2) 2 ) ( ˆ w ( b  1) ; ˆ w ( b ) ) x (1) 2 ( w ( b  1) | s ( b  2) 1 ) ; u (1) 2 ( s ( b  2) 1 ) ( ˆ s ( b  2) 1 ; ˆ w ( b  2) ) x (1) 0 ( w ( b ) | w ( b  1) ; s ( b  2) 1 ) Fig. 4. Successiv e relaying protocol based on Cooperati ve Coding. 2) Cooperative Coding : In this type of coding scheme, we assume that, at each time slot, the receiving rela y decodes not only the new transmitted message from the source, but also the previous message transmitted fr om the transmitting relay (See Figs. 2 and 4) . Our propo sed cod ing scheme is based on binnin g, superposition coding, and Block Markov En coding . The source sends B me ssages w (1) , w (2) , · · · , w ( B ) in B + 2 time slots. Generally , th is scheme can be described as follo ws ( See Figs. 4 an d 5). In time slot b , the r elay ( b + 1) mod 2 +1 decodes the transm itted m essages w ( b ) and w ( b − 1) from the sour ce a nd the oth er relay , r espectiv ely . In time slot DRAFT 9 x (1) 0 ( w (3) | w (2) ; s (1) 1 ) x (1) 0 ( w (1) | 1 ; 1) x (1) 2 (1 | 1) ; u (1) 2 (1) x (2) 0 ( w (2) | w (1) ; 1) x (1) 2 ( w (2) | s (1) 1 ) ; u (1) 2 ( s (1) 1 ) x (2) 1 ( w (3) | s (2) 2 ) ; u (2) 1 ( s (2) 2 ) x (2) 0 ( w (4) | w (3) ; s (2) 2 ) x (2) 1 ( w (1) | 1) ; u (2) 1 (1) Rela y 1 Source Rela y 2 Blo c k 2 Blo c k 1 Blo c k 3 Blo c k 4 Fig. 5. Decode-and-forw ard for successi ve relaying protocol. b + 1 , it b roadcasts w ( b ) and the bin ind ex of w ( b − 1) , s ( b − 1) ( b +2) mod 2+1 , to the destination using the b inning fu nction deﬁned next. Deﬁnition (The Binning Function) : The binning function f (( b +1) mod 2+1) B in ( w ( b − 2) ) : W = { 1 , 2 , · · · , 2 nR (( b +1) mod 2+1) } − → { 1 , 2 , . . . , 2 nr (( b +1) mod 2+1) Bin } is deﬁned by f (( b +1) mod 2+1) B in ( w ( b − 2) ) = s ( b − 2) ( b +1) mod 2+1 , wher e f (( b +1) mod 2+1) B in ( . ) assigns a random ly un iform distributed integer b etween 1 an d 2 nr (( b +1) mod 2+1) Bin indepen dently to each memb er of W . As ind icated in Fig. 5, in the ﬁrst time slot, the sourc e tr ansmits the codeword x (1) 0 ( w (1) | 1 , 1) to the ﬁrst relay , while the seco nd r elay tran smits a doub ly indexed co dew ord x (1) 2 (1 | 1) and the codeword u (1) 2 (1) to the ﬁrst relay and to the destination . In the second time slot, the source tran smits th e codeword x (2) 0 ( w (2) | w (1) , 1) to the second relay , a nd having decod ed the me ssage w (1) , the ﬁrst relay broadcasts the cod ew ords x (2) 1 ( w (1) | 1) and u (2) 1 (1) to the secon d rela y and to the d estination. It should be noted th at the destinatio n cannot deco de the message w (1) at the end of this time slot; howev er , th e second relay de codes w (1) and w (2) messages. Using the b inning function, it ﬁnds the bin index o f w (1) accordin g to s (1) 1 = f (1) B in ( w (1) ) . In the third time slot, the sourc e transm its the codeword x (1) 0 ( w (3) | w (2) , s (1) 1 ) to the ﬁrst relay , and the second relay broadcasts the codewords x (1) 2 ( w (2) | s (1) 1 ) and u (1) 2 ( s (1) 1 ) to the ﬁrst relay and to the destinatio n. T wo typ es of decodin g can be used at the d estination: su ccessi ve decod ing and b ackward decod ing. Successiv e decodin g at the destination can be d escribed as f ollows. At the end of the b th time slot, the destinatio n cannot decod e the message w ( b − 1) ; howe ver , h aving dec oded the b in index s ( b − 2) ( b +1) mod 2+1 from the received vector of the b th time slot, it ca n deco de the message w ( b − 2) from s ( b − 2) ( b +1) mod 2+1 and th e received vecto r of the ( b − 1) th time slot. On the other hand, backward deco ding can b e explained as fo llows . Ha ving received the sequen ce o f the B + 2 ’th time slot, the ﬁnal destinatio n starts deco ding the inten ded messages. In the tim e slot B + 2 , one of the relays transmits the dumm y message “1” along with the bin index of the message w ( B ) to the destination. Having recei ved th is bin index, th e d estination decodes it, an d then back wardly decod es m essages w ( b ) , b = B , B − 1 , · · · , 1 an d their bin indices. The f ollowing Theor em gives the ach iev able rate of the p roposed scheme. Theorem 2 F or the ha lf-duplex parallel r elay chann el, a ssuming successive relaying, th e BME scheme achieves DRAFT 10 the rates C low B M E succ and C low B M E back using successive and ba ckw ar d d ecoding, r espectively: C low B M E succ = R (1) + R (2) ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min ( min  t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  , t 2 I  X (2) 1 ; Y (2) 3 | U (2) 1  + t 1 I  U (1) 2 ; Y (1) 3  + min  t 1 I  X (1) 2 ; Y (1) 3 | U (1) 2  + t 2 I  U (2) 1 ; Y (2) 3  , t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  , t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1 | U (1) 2  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2 | U (2) 1  . (15) with pr obabilities p ( x (1) 0 , x (1) 2 , u (1) 2 ) = p ( u (1) 2 ) p ( x (1) 2 | u (1) 2 ) p ( x (1) 0 | x (1) 2 , u (1) 2 ) , p ( x (2) 0 , x (2) 1 , u (2) 1 ) = p ( u (2) 1 ) p ( x (2) 1 | u (2) 1 ) p ( x (2) 0 | x (2) 1 , u (2) 1 ) , p ( x (1) 2 , u (1) 2 ) = p ( u (1) 2 ) p ( x (1) 2 | u (1) 2 ) , p ( x (2) 1 , u (2) 1 ) = p ( u (2) 1 ) p ( x (2) 1 | u (2) 1 ) . C low B M E back = R (1) + R (2) ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min  t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2  , t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  , t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  . (16) with pr obabilities p ( x (1) 0 , x (1) 2 ) = p ( x (1) 2 ) p ( x (1) 0 | x (1) 2 ) , p ( x (2) 0 , x (2) 1 ) = p ( x (2) 1 ) p ( x (2) 0 | x (2) 1 ) . Pr o of: See Appe ndix B. Now , the following set of propositions and corollaries inv estigate the Non-Coop erative and Cooper ati ve sch emes and comp are th em with each other . Proposition 1 The BME with backwar d decoding achieves a be tter rate than the one with successive d ecoding, i.e., C low B M E back ≥ C low B M E succ . Pr o of: For th e ﬁrst term of m inimization ( 15), we have min  t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  , t 2 I  X (2) 1 ; Y (2) 3 | U (2) 1  + t 1 I  U (1) 2 ; Y (1) 3  + min  t 1 I  X (1) 2 ; Y (1) 3 | U (1) 2  + t 2 I  U (2) 1 ; Y (2) 3  , t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  ≤ min  t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  , t 1 I  X (1) 2 , U (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 , U (2) 1 ; Y (2) 3  . (17) DRAFT 11 Let us focus on t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  : t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  ( a ) = t 1 H  Y (1) 1 | X (1) 2 , U (1) 2  − t 1 H  Y (1) 1 | X (1) 0 , X (1) 2  + t 2 H  Y (2) 2 | X (2) 1 , U (2) 1  − t 2 H  Y (2) 2 | X (2) 0 , X (2) 1  ( b ) ≤ t 1 H  Y (1) 1 | X (1) 2  − t 1 H  Y (1) 1 | X (1) 0 , X (1) 2  + t 2 H  Y (2) 2 | X (2) 1  − t 2 H  Y (2) 2 | X (2) 0 , X (2) 1  ( c ) = t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  . (18) ( a ) an d ( c ) fo llow from the deﬁnition of m utual infor mation, the fact th at U (1) 2 − →  X (1) 0 , X (1) 2  − → Y (1) 1 and U (2) 1 − →  X (2) 0 , X (2) 1  − → Y (2) 2 form Markov chain, and ( b ) follows from the fact th at cond itioning red uces entropy . Inequ ality ( b ) become s eq uality if p ( x (1) 0 , x (1) 2 , u (1) 2 ) = p ( u (1) 2 ) p ( x (1) 2 ) p ( x (1) 0 | x (1) 2 ) and p ( x (2) 0 , x (2) 1 , u (2) 1 ) = p ( u (2) 1 ) p ( x (2) 1 ) p ( x (2) 0 | x (2) 1 ) . Using the similar argumen t for t 1 I  X (1) 2 , U (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 , U (2) 1 ; Y (2) 3  , t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1 | U (1) 2  , and t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2 | U (2) 1  in (15) and (1 7), and the fact U (1) 2 − → X (1) 2 − → Y (1) 3 , U (2) 1 − → X (2) 1 − → Y (2) 3 , U (1) 2 − →  X (1) 0 , X (1) 2  − → Y (1) 1 , U (2) 1 − →  X (2) 0 , X (2) 1  − → Y (2) 2 form Mar kov chain, and Appendix B, along with comparing C low B M succ and C low B M back in Theorem 2, we have C low B M back ≥ C low B M succ . From Theo rem 2, we h a ve the following co rollary for the G aussian case. corollary 2 F or the half-d uplex Gaussian p arallel r elay chann el, assuming successive relaying pr otocol wit h p ower constraint at the so ur ce a nd each r elay , BME achieves the following rates C low B M E succ = max min  C low B M E 1 + C low B M E 2 , t 1 C   h 2 01 P (1) 0 + h 2 12 θ 2 P 2 + 2 h 01 h 12 q ¯ α 1 θ 2 P (1) 0 P 2 t 1   , t 2 C   h 2 02 P (2) 0 + h 2 12 θ 1 P 1 + 2 h 02 h 12 q ¯ α 2 θ 1 P (2) 0 P 1 t 2     . (19) C low B M E back = max min   t 1 C   h 2 01 P (1) 0 + h 2 12 P 2 + 2 h 01 h 12 q ¯ β 1 P (1) 0 P 2 t 1   , t 2 C   h 2 02 P (2) 0 + h 2 12 P 1 + 2 h 02 h 12 q ¯ β 2 P (2) 0 P 1 t 2   , t 1 C h 2 01 β 1 P (1) 0 t 1 ! + t 2 C h 2 02 β 2 P (2) 0 t 2 ! , t 1 C  h 2 23 P 2 t 1  + t 2 C  h 2 13 P 1 t 2  . (20) DRAFT 12 subject to: C low B M E 1 = min t 1 C h 2 01 α 1 P (1) 0 t 1 ! , t 1 C  h 2 23 ¯ θ 2 P 2 h 2 23 θ 2 P 2 + t 1  + t 2 C  h 2 13 θ 1 P 1 t 2  ! , (21) C low B M E 2 = min t 2 C h 2 02 α 2 P (2) 0 t 2 ! , t 2 C  h 2 13 ¯ θ 1 P 1 h 2 13 θ 1 P 1 + t 2  + t 1 C  h 2 23 θ 2 P 2 t 1  ! , (22) P (1) 0 + P (2) 0 = P 0 , t 1 + t 2 = 1 , 0 ≤ α 1 , α 2 ≤ 1 , 0 ≤ β 1 , β 2 ≤ 1 , 0 ≤ θ 1 , θ 2 ≤ 1 . wher e ¯ θ i = 1 − θ i , ¯ α i = 1 − α i , an d ¯ β i = 1 − β i for i = 1 , 2 . Pr o of: Let V (1) 0 ∼ N (0 , α 1 P (1) 0 ) , V (2) 0 ∼ N (0 , α 2 P (2) 0 ) , V (1) 2 ∼ N (0 , θ 2 P 2 ) , V (2) 1 ∼ N (0 , θ 1 P 1 ) , U (1) 2 ∼ N (0 , ¯ θ 2 P 2 ) and U (2) 1 ∼ N (0 , ¯ θ 1 P 1 ) , wh ich are inde pendent o f each othe r . Letting X (1) 0 = V (1) 0 + r ¯ α 1 P (1) 0 θ 2 P 2 V (1) 2 , X (2) 0 = V (2) 0 + r ¯ α 2 P (2) 0 θ 1 P 1 V (2) 1 , X (1) 2 = V (1) 2 + U (1) 2 , X (2) 1 = V (2) 1 + U (2) 1 and using the result in the expre ssion for the achievable rate obtained in Th eorem 1, w e obtain C low B M E succ for the Gaussian case, as given in [3 2] and (19), (21), and (22), respectively . For backward decodin g, let V (1) 0 ∼ N (0 , β 1 P (1) 0 ) , V (2) 0 ∼ N (0 , β 2 P (2) 0 ) , X (1) 2 ∼ N (0 , P 2 ) , and X (2) 1 ∼ N (0 , P 1 ) , which are independent of each oth er . By setting X (1) 0 = V (1) 0 + r ¯ β 1 P (1) 0 P 2 X (1) 2 , X (2) 0 = V (2) 0 + r ¯ β 2 P (2) 0 P 1 X (2) 1 and u sing the result in the expr ession for the achievable r ate obtain ed in Theorem 1, we ob tain C low B M E back for the Gau ssian ca se, as given in (20). Proposition 2 In symmetric scen arios, wher e h 01 = h 02 , h 13 = h 23 , an d P 1 = P 2 , Non- Cooperative DPC scheme outperforms Coop erative BME scheme, i.e. C low B M E back ≤ C low DP C . Pr o of: Due to th e symmetric assump tion, we have t 1 = t 2 = 1 2 , P (1) 0 = P (2) 0 = P 0 2 , and β 1 = β 2 = 1 2 . Hence, from (20), we have C low B M E back ≤ min  C  h 2 01 P 0 2  , C  2 h 2 13 P 1   . (23) And also C low DP C in (1 2) b ecomes C low DP C = min  C  h 2 01 P 0  , 1 2 C  h 2 01 P 0  + 1 2 C  2 h 2 13 P 1  , C  2 h 2 13 P 1   . (24) Comparing (2 3) an d (24), we have C low B M E back ≤ C low DP C . According to the discussion in Ap pendix B, r (1) B in = 0 or r (2) B in = 0 . In othe r word s, in the Cooperative BME scheme based on b ackward decoding , at most on e relay is necessary to use b inning func tion for the m essage it DRAFT 13 receives fro m ano ther , and the other relay is no t necessary to cooperate with this relay . Therefo re, we pro pose a composite BME-DPC sch eme. In this scheme, one of the relays d ecodes th e o ther r elay’ s message. Having d ecoded that, it then u ses the b inning functio n to co operate with th e other relay . On the other hand, using the Gelfand- Pinsker’ s result the source canc els the in terference due to one relay on the other . Henc e, we have the fo llowing Theorem . Theorem 3 The composite BME- DPC scheme, achieves th e following rate: C low B M E − D P C = max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min  t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  , t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + t 2  I  U (2) 0 ; Y (2) 2  − I  U (2) 0 ; X (2) 1  , t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  , t 2  I  U (2) 0 ; Y (2) 2  − I  U (2) 0 ; X (2) 1  + t 2 I  X (2) 1 ; Y (2) 3  . (25) Pr o of: Assuming r (1) B in = 0 , and u sing Theorem 1 and Theorem 2 along with a similar argumen t as in Appen dix B, Theo rem 3 is imm ediate. corollary 3 F or the Gaussian case, th e comp osite BME-DP C scheme achieves th e following rate C low B M E − D P C . Furthermore , C low B M E − D P C ≥ C low B M E back . In oth er words, the composite BME-DPC scheme a lways achieves a better rate than the BME scheme for th e Gaussian scena rio. C low B M E − D P C = R (1) + R (2) ≤ max min   t 1 C   h 2 01 P (1) 0 + h 2 12 P 2 + 2 h 01 h 12 q ¯ αP (1) 0 P 2 t 1   , t 1 C h 2 01 αP (1) 0 t 1 ! + t 2 C h 2 02 P (2) 0 t 2 ! , t 1 C  h 2 23 P 2 t 1  + t 2 C  h 2 13 P 1 t 2  , t 2 C h 2 02 P (2) 0 t 2 ! + t 2 C  h 2 13 P 1 t 2  ! . (26) subject to : P (1) 0 + P (2) 0 = P 0 , t 1 + t 2 = 1 , 0 ≤ t 1 , t 2 , P (1) 0 , P (2) 0 , 0 ≤ α ≤ 1 . wher e ¯ α = 1 − α . Pr o of: As in Th eorem 3, we assume that r (1) B in = 0 . Now , we show that every rate pairs  R (1) , R (2)  satisfying ( 101)-(10 7 ) satisfy (2 6). After specia lizing (1 01)-(107) f or the Gau ssian c ase and compa ring with (26), one observes that th e second term in minimization (10 1) does not e xist. Substituting r (1) B in = 0 in (102)-(10 7), DRAFT 14 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000 111111 00000 11111 Rela y 2 h 01 h 23 Destination h 13 h 02 t 3 t 4 Source Rela y 1 Fig. 6. Simultaneous relaying protocol for two relays. one can ob tain the other three cor respondin g term s. Com paring those term s with (26), it can b e readily seen that C low B M E − D P C ≥ C low B M E back . Remark 1 Assuming r (1) B in = 0 , as in Theor em 3 and cor ollary 3, t he destination jointly decodes the curr ent message and the bin in dex of the next message at the end of even time slots an d then it can decode the next message at the end of odd time slots. Ther efor e, using ba ckw ar d d ecoding is no t necessary in th e BME-DPC scheme. B. Simultaneo us Relaying P r oto col Figure 6 sh ows simultaneou s re laying protoco l. In simu ltaneous relaying, in on e time slot of d uration t 3 the source transmits its signal sim ultaneously to the two relays. In the next time slot of duration t 4 , two re lays transmit their signal coh erently to th e d estination. Hence, in th is p rotocol, t 1 = t 2 = 0 an d ou r system mod el follows f rom (6) and (7). 1) Dynamic Decod e-and -F orwar d ( DDF): I n DDF schem e each relay d ecodes the transmitted m essage from the sou rce in time slot t 3 (Broadcast (BC) State), and forwards its re-enco ded version in time slot t 4 (Multiple Access (MA C) State). The following Theo rem g i ves the achiev able ra te of the DDF scheme f or the gen eral discrete memory less chann els. Theorem 4 F or the h alf-duplex parallel r elay chann el, assuming simulta neous relaying and the fact th at what th e second r e lay r e ceives is a d e graded ve rsion o f what the ﬁrst r elay r eceive s, the following rate C low DD F is achievable: C low DD F = max 0 ≤ t 3 ,t 4 ,t 3 + t 4 =1 R p + R c , (27) subject to: R p ≤ min  t 3 I ( X (3) 0 ; Y (3) 1 | U (3) 0 ) , t 4 I ( X (4) 1 ; Y (4) 3 | X (4) 2 )  , (28) R c ≤ t 3 I ( U (3) 0 ; Y (3) 2 ) , (29) R p + R c ≤ t 4 I ( X (4) 1 , X (4) 2 ; Y (4) 3 ) . (30) DRAFT 15 with pr obabilities: p ( u (3) 0 , x (3) 0 ) = p ( u (3) 0 ) p ( x (3) 0 | u (3) 0 ) , p ( x (4) 1 , x (4) 2 ) = p ( x (4) 1 ) p ( x (4) 2 | x (4) 1 ) . Pr o of: The achievable r ate of DDF is equal to C low DD F = R p + R c , wher e ( R p , R c ) should be both in the capacity region of BC (cor respondin g to the BC state) and MAC (co rrespond ing to the MAC state). Ap plying the superpo sition coding of the degrad ed BC [ 12] the fo llowing rates are a chiev able fo r the ﬁrst hop: R p ≤ t 3 I ( X (3) 0 ; Y (3) 1 | U (3) 0 ) , R c ≤ t 3 I ( U (3) 0 ; Y (3) 2 ) . (31) with pr obability p ( u (3) 0 , x (3) 0 ) = p ( u (3) 0 ) p ( x (3) 0 | u (3) 0 ) . And using the sup erposition codin g of the extended MA C (See [25], [2 6]) the fo llowing rates are ach ie vable for the secon d ho p: R p ≤ t 4 I ( X (4) 1 ; Y (4) 3 | X (4) 2 ) , R p + R c ≤ t 4 I ( X (4) 1 , X (4) 2 ; Y (4) 3 ) . (32) with pr obability p ( x (4) 1 , x (4) 2 ) = p ( x (4) 1 ) p ( x (4) 2 | x (4) 1 ) . In the Gaussian case (assuming h 01 ≥ h 02 ), the source splits its total available power P 0 to P (3) 0 ,p and P (3) 0 ,c associated with the “Private” an d the “Common ” messages, respectively . Letting X (3) 0 ∼ N (0 , P 0 ) , U (3) 0 ∼ N  0 , P (3) 0 ,c  , a nd X (4) 1 ∼ N ( 0 , P 1 ) , assuming that r elay 1 a nd re lay 2 transmit their codewords associa ted with the com mon me ssage with N  0 , P (4) 1 ,c  and N (0 , P 2 ) , and u sing (31) and (32) we have the fo llowing corollary . DRAFT 16 corollary 4 F or the h alf-duplex Gau ssian parallel relay channel, assuming simultaneo us relaying p r o tocol with power con straints at the source and a t ea ch relay , DDF achieves the following rate C low DD F = R p + R c , (33) subject to: R p ≤ min t 3 C h 2 01 P (3) 0 ,p t 3 ! , t 4 C h 2 13 P (4) 1 ,p t 4 !! , R c ≤ t 3 C h 2 02 P (3) 0 ,c t 3 + h 2 02 P (3) 0 ,p ! , R p + R c ≤ t 4 C      h 2 13 P (4) 1 ,p +  h 13 q P (4) 1 ,c + h 23 √ P 2  2 t 4      , P (3) 0 ,p + P (3) 0 ,c = P 0 , P (4) 1 ,p + P (4) 1 ,c = P 1 , t 3 + t 4 = 1 , 0 ≤ t 3 , t 4 , P (3) 0 ,p , P (3) 0 ,c , P (4) 1 ,p , P (4) 1 ,c . Interestingly , successive decoding at th e destinatio n does no t degrad e the perform ance o f DDF schem e in the Gaussian scenario as shown in the fo llowing Proposition . Proposition 3 The rate of DDF scheme is achievable b y successive decoding o f the common an d private me ssages at the d estination. Pr o of: Con sider the sum rate for b oth th e comm on m essage and th e p riv ate message for the extended m ultiple access chan nel fro m relays to the destination , R p + R c ≤ t 4 C   h 2 13 P (4) 1 ,p + ( h 13 q P (4) 1 ,c + h 23 √ P 2 ) 2 t 4   . (34) It can be readily veriﬁed that su bject to the constraint P (4) 1 ,p + P (4) 1 ,c = P 1 , the right-han d side of (34) is a decreasing function of P (4) 1 ,p or equiv alently an increasing func tion of P (4) 1 ,c . Now , let us equ ate R p in (34) with the private rate ´ R p of another MA C which is ach ie ved by succe ssi ve deco ding of com mon and pr i vate m essages. Therefore , we have R p = ´ R p = t 4 C h 2 13 ´ P (4) 1 ,p t 4 ! ≤ t 4 C h 2 13 P (4) 1 ,p t 4 ! . (35) According to ( 35), we have (See Fig. 7) ´ P (4) 1 ,p ≤ P (4) 1 ,p = ⇒ R p + R c ≤ ´ R p + ´ R c , R c ≤ ´ R c . Hence, ( R p , R c ) lies in th e corner poin t of the extended MAC with parameter s ( ´ P (4) 1 ,p , ´ P (4) 1 ,c ) , i.e. su ccessi ve deco ding of comm on an d priv ate me ssages achieves the DF r ate. DRAFT 17 Common R ate R c ´ R c ´ R p = R p Priv ate Rate Fig. 7. The order of decoding “C ommon” and “Private” messages. C. Simultan eous-Succ essive Relaying Pr otocol ba sed on Dirty p aper codin g (S SRD) Source Destination Rela y 1 Rela y 2 R 1 R 2 a) Time slot 1 with duration t 1 Source Destination Rela y 1 Rela y 2 R 4 b) Time slot 2 with duration t 2 R 3 Source Destination Rela y 1 Rela y 2 d) Time slot 4 with duration t 4 ( R 7 ; R 9 ) ( R 8 ; R 9 ) Source Destination Rela y 1 Rela y 2 R 6 ( R 5 ; R 6 ) c) Time slot 3 with duration t 3 Fig. 8. SSRD Scheme for the Half-Duplex Parallel Relay Channel. In this section, we p ropose an achiev able rate for the half-d uplex parallel re lay channe l. Our achievable scheme is ba sed on the combin ation o f the successiv e rela ying pro tocol based o n DPC scheme and simultaneou s r elaying protoco l based o n DDF (SSRD sch eme). Hence, we have the f ollowing Theor em. Theorem 5 Considering F ig. 8, for the h alf-duplex parallel r elay chann el, SS RD scheme a chieves the following DRAFT 18 rate C low S S RD : C low S S RD =min ( R 1 + R 4 + R 5 + R 6 , R 2 + R 3 + R 7 + R 8 + R 9 ) , (36) subject to: R 9 ≤ R 6 , R 1 + R 5 ≤ R 3 + R 7 , R 4 ≤ R 2 + R 8 . (37 ) Pr o of: SSRD scheme is illustrated in Fig. 8. As indicate d in the ﬁg ure, transmission is perfo rmed in 4 time slots. Relay 1 transmits its priv ate message wh ich was received in time slots t 1 and t 3 (correspo nding to rates R 1 and R 5 ) in time slots t 2 and t 4 (correspo nding to rates R 3 and R 7 ). On the other hand, relay 2 transmits its pri vate message wh ich ha s be en received in time slot t 2 (correspo nding to rate R 4 ) in time slots t 1 and t 4 (correspo nding to rates R 2 and R 8 ). Furthermore, th e two re lays sen d the co mmon message they hav e already received in time slot t 3 (correspo nding to rate R 6 ) coh erently in time slot t 4 (correspo nding to rate R 9 ). As observed, here w e consid er the pri vate rate for b oth r elays in the M A C state, i.e. time slot t 4 . This is due to the reason that relay 2 also r eceiv es the pr i vate message in time slot t 2 . Hence , f rom the above descr iption and Fig. 8, we h av e C low S S RD =min ( R 1 + R 4 + R 5 + R 6 , R 2 + R 3 + R 7 + R 8 + R 9 ) , (38) subject to: R 9 ≤ R 6 , R 1 + R 5 ≤ R 3 + R 7 , R 4 ≤ R 2 + R 8 . (39) Using corollar ies 1, 4, a nd Pro position 3, f or the Gaussian case we h av e DRAFT 19 C low S S RD =min t 1 C h 2 01 P (1) 0 t 1 ! + t 2 C h 2 02 P (2) 0 t 2 ! + t 3 C h 2 01 P (3) 0 ,p t 3 ! + t 3 C h 2 02 P (3) 0 ,c t 3 + h 2 02 P (3) 0 ,p ! , t 1 C h 2 23 P (1) 2 t 1 ! + t 2 C h 2 13 P (2) 1 t 2 ! + t 4 C h 2 13 P (4) 1 ,p + h 2 23 P (4) 2 ,p t 4 ! + t 4 C       h 13 q P (4) 1 ,c + h 23 q P (4) 2 ,c  2 t 4 + h 2 13 P (4) 1 ,p + h 2 23 P (4) 2 ,p           , (40) subject to: t 4 C       h 13 q P (4) 1 ,c + h 23 q P (4) 2 ,c  2 t 4 + h 2 13 P (4) 1 ,p + h 2 23 P (4) 2 ,p      ≤ t 3 C h 2 02 P (3) 0 ,c t 3 + h 2 02 P (3) 0 ,p ! , t 1 C h 2 01 P (1) 0 t 1 ! + t 3 C h 2 01 P (3) 0 ,p t 3 ! ≤ t 2 C h 2 13 P (2) 1 t 2 ! + t 4 C h 2 13 P (4) 1 ,p t 4 ! , t 2 C h 2 02 P (2) 0 t 2 ! ≤ t 1 C h 2 23 P (1) 2 t 1 ! + t 4 C h 2 23 P (4) 2 ,p t 4 ! , P (1) 0 + P (2) 0 + P (3) 0 ,p + P (3) 0 ,c = P 0 , P (2) 1 + P (4) 1 ,p + P (4) 1 ,c = P 1 , P (1) 2 + P (4) 2 ,p + P (4) 2 ,c = P 2 , t 1 + t 2 + t 3 + t 4 = 1 , 0 ≤ t 1 , t 2 , t 3 , t 4 , P (1) 0 , P (2) 0 , P (3) 0 ,p , P (3) 0 ,c , P (2) 1 , P (4) 1 ,p , P (4) 1 ,c , P (1) 2 , P (4) 2 ,p , P (4) 2 ,c . According to corollary 3, anoth er comb ined simultaneo us-successiv e re laying p rotocol based on BME is not necessary . Howe ver , a “Simu ltaneous-Succe ssi ve Relaying pr otocol based on BME- DPC”, can be easily d erived. Assuming the ﬁrst relay d ecodes th e second on e’ s m essage, the achievable ra te o f this n ew scheme would be th e same a s C low S S RD . Ho wev er, since the me ssages for the second relay are common, R 8 in the expression of the achiev able ra te is zero . Furtherm ore, the fo llowing constraints instead of (39) sho uld be satisﬁed: R 9 ≤ R 4 + R 6 , R 1 + R 5 ≤ R 3 + R 7 , R 1 + R 4 ≤ t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  . (41) I V . O P T I M A L I T Y R E S U L T S In this section, an up per bound for th e half-duplex para llel rela y channel is d eriv ed an d in vestigated. Th e authors in [27] p roposed some u pper b ounds on the achievable rate for gener al half-du plex multi-terminal network s. Here, we explain their results brieﬂy an d apply them to our h alf-dup lex parallel relay network . Authors in [2 7] deﬁne the con cept o f state for a ha lf-duplex network with N nodes. Th e state o f the n etwork is a valid partitioning of its no des into two sets of the “sender node s” and th e “r eceiver nodes” such that ther e is no DRAFT 20 active link th at arrives a t a sende r node , and ˆ t m is the p ortion of the time that network is used in state m wher e m ∈ { 1 , 2 , . . . , M } . The following Theo rem for the u pper boun d of the inf ormation ﬂow fr om th e su bset S 1 to the subset S 2 of the nodes, where S 1 and S 2 are disjoin t is proved in [27]. Theorem 6 F or a gen eral half-d uplex network with N nodes an d a ﬁnite n umber of states, M , the maximum achievable informa tion rates { R ij } fr om a node set S 1 to a disjoint no de set S 2 , S 1 , S 2 ⊂ { 0 , 1 , . . . , N − 1 } , is bound ed by X i ∈ S 1 ,j ∈ S 2 R ij ≤ sup p ( x ( m ) 0 ,x ( m ) 2 ,...,x ( m ) N − 1 ) , ˆ t m min S M X m =1 ˆ t m I  X ( m ) S ; Y ( m ) S | X ( m ) S c  . (42) for some joint pr obability distribution p ( x ( m ) 0 , x ( m ) 2 , . . . , x ( m ) N − 1 ) when the minimization is over all the sets S ⊂ { 0 , 1 , . . . , N − 1 } subject to S T S 1 = S 1 , S T S 2 = ∅ a nd the supr emum is over all th e no n-negative ˆ t m subject to P M i =1 ˆ t m = 1 . Here , x ( m ) S , y ( m ) S , and x ( m ) S c denote the signals transmitted a nd received b y n odes in set S , an d transmitted b y nodes in set S c , du ring state m , respectively . From Theo rem 6, the m aximum ac hiev able rate C low is uppe r bound ed as C low ≤ C up , min  ˆ t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + ˆ t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  + ˆ t 3 I  X (3) 0 ; Y (3) 1 , Y (3) 2  , ˆ t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2 , Y (2) 3  + ˆ t 3 I  X (3) 0 ; Y (3) 2  + ˆ t 4 I  X (4) 1 ; Y (4) 3 | X (4) 2  , ˆ t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1 , Y (1) 3  + ˆ t 3 I  X (3) 0 ; Y (3) 1  + ˆ t 4 I  X (4) 2 ; Y (4) 3 | X (4) 1  , ˆ t 1 I  X (1) 2 ; Y (1) 3  + ˆ t 2 I  X (2) 1 ; Y (2) 3  + ˆ t 4 I  X (4) 1 , X (4) 2 ; Y (4) 3  , (43) subject to ˆ t 1 + ˆ t 2 + ˆ t 3 + ˆ t 4 = 1 . By setting ˆ t 3 = ˆ t 4 = 0 in (43), we ob tain an upper boun d on the successive r elaying proto col which we c all it successive cut-set b ound in th e sequel. Theorem 7 In a degraded h alf-duplex parallel r ela y chann el whe r e the d estination r eceives a degr aded version of the r eceived signals at r elays, i.e. X (1) 2 − → Y (1) 1 − → Y (1) 3 and X (2) 1 − → Y (2) 2 − → Y (2) 3 , BME based on backwar d decodin g achieves the su ccessive cut-set boun d. Pr o of: Setting ˆ t 3 = ˆ t 4 = 0 in (43) an d compar ing the result with ( 16) the Th eorem is proved. In high SNR scenarios, we have the fo llowing Theorem . Theorem 8 In high SNR scenarios, a ssuming no n-zer o sou r ce-r elay a nd r ela y-destination links, when power avail- able for the sou r ce and each r elay tends to inﬁn ity , time slots ˆ t 3 and ˆ t 4 in (43) tend to zer o as O  1 log P 0  . Furthermore , th e upper bo und on the ca pacity of the half-d uplex parallel relay chan nel in h igh SNR scenario s is C up = C low DP C + O  1 log P 0  . DRAFT 21 In o ther wor ds, DPC a chieves the capacity o f a half-d uplex Gaussian pa rallel r elay c hann el as S NR g oes to inﬁnity . Pr o of: Thr ougho ut the proo f, we assum e the power of the r elays goes to inﬁnity as P 1 = γ 1 P 0 , P 2 = γ 2 P 0 where γ 1 , γ 2 are co nstants inde pendent o f the SNR. Substituting X (1) 0 ∼ N (0 , ˆ P (1) 0 ) , X (2) 0 ∼ N (0 , ˆ P (2) 0 ) , X (3) 0 ∼ N (0 , ˆ P (3) 0 ) , X (2) 1 ∼ N (0 , ˆ P (2) 1 ) , X (4) 1 ∼ N (0 , ˆ P (4) 1 ) , X (1) 2 ∼ N (0 , ˆ P (1) 2 ) , and X (4) 2 ∼ N (0 , ˆ P (4) 2 ) in ( 43), an d assuming com plete co operation between the tr ansmitting and receiving no des for each cut in (43), we have C up ≤ min ˆ t 1 C h 2 01 ˆ P (1) 0 ˆ t 1 ! + ˆ t 2 C h 2 02 ˆ P (2) 0 ˆ t 2 ! + ˆ t 3 C ( h 2 01 + h 2 02 ) ˆ P (3) 0 ˆ t 3 ! , ˆ t 2 C   h 2 02 ˆ P (2) 0 ˆ t 2 + ( h 2 12 + h 2 13 ) ˆ P (2) 1 ˆ t 2 + 2 h 02 h 12 q ˆ P (2) 0 ˆ P (2) 1 ˆ t 2 + h 2 02 h 2 13 ˆ P (2) 0 ˆ P (2) 1 ˆ t 2 2   + ˆ t 3 C h 2 02 ˆ P (3) 0 ˆ t 3 ! + ˆ t 4 C h 2 13 ˆ P (4) 1 ˆ t 4 ! , ˆ t 1 C   h 2 01 ˆ P (1) 0 ˆ t 1 + ( h 2 12 + h 2 23 ) ˆ P (1) 2 ˆ t 1 + 2 h 01 h 12 q ˆ P (1) 0 ˆ P (1) 2 ˆ t 1 + h 2 01 h 2 23 ˆ P (1) 0 ˆ P (1) 2 ˆ t 2 1   + ˆ t 3 C h 2 01 ˆ P (3) 0 ˆ t 3 ! + ˆ t 4 C h 2 23 ˆ P (4) 2 ˆ t 4 ! , ˆ t 1 C h 2 23 ˆ P (1) 2 ˆ t 1 ! + ˆ t 2 C h 2 13 ˆ P (2) 1 ˆ t 2 ! + ˆ t 4 C   h 2 13 ˆ P (4) 1 + h 2 23 ˆ P (4) 2 + 2 h 13 h 23 q ˆ P (4) 1 ˆ P (4) 2 ˆ t 4     . (44) subject to : ˆ P (1) 0 + ˆ P (2) 0 + ˆ P (3) 0 = P 0 , ˆ P (2) 1 + ˆ P (4) 1 = P 1 , ˆ P (1) 2 + ˆ P (4) 2 = P 2 , ˆ t 1 + ˆ t 2 + ˆ t 3 + ˆ t 4 = 1 , 0 ≤ ˆ t 1 , ˆ t 2 , ˆ t 3 , ˆ t 4 , ˆ P (1) 0 , ˆ P (2) 0 , ˆ P (3) 0 , ˆ P (2) 1 , ˆ P (4) 1 , ˆ P (1) 2 , ˆ P (4) 2 . Furthermo re, fro m cor ollary 1, the achiev able ra te o f the DPC scheme c an be expressed as C low DP C = min t 1 C h 2 01 P (1) 0 t 1 ! + t 2 C h 2 02 P (2) 0 t 2 ! , t 2 C h 2 02 P (2) 0 t 2 ! + t 2 C  h 2 13 P 1 t 2  , t 1 C h 2 01 P (1) 0 t 1 ! + t 1 C  h 2 23 P 2 t 1  , t 1 C  h 2 23 P 2 t 1  + t 2 C  h 2 13 P 1 t 2  . (45) DRAFT 22 By setting P (1) 0 = P (2) 0 = P 0 2 and t 1 = t 2 = 0 . 5 in (45), expression (45) can be simp liﬁed as C low DP C ≥ 1 2 ln P 0 + c. (46) where c is some constant which depend s on c hannel co efﬁcients. Knowing tha t th e term corresp onding to each cut-set in (44) for th e o ptimum values of ˆ t 1 , · · · , ˆ t 4 is ind eed an upper-bou nd f or C low DP C , and by setting ˆ P (1) 0 = ˆ P (2) 0 = ˆ P (3) 0 = P 0 in (4 4), we have the fo llowing inequa lity between (46) and th e ﬁrst cu t o f (44). 1 2 ln P 0 + c ≤ ˆ t 1 2 ln  h 2 01 P 0 ˆ t 1  + ˆ t 2 2 ln  h 2 02 P 0 ˆ t 2  + ˆ t 3 2 ln  ( h 2 01 + h 2 02 ) P 0 ˆ t 3  + ˆ t 2 1 2 h 2 01 P 0 + ˆ t 2 2 2 h 2 02 P 0 + ˆ t 2 3 2( h 2 01 + h 2 02 ) P 0 =  1 − ˆ t 4  2 ln P 0 + ˆ t 1 2 ln h 2 01 + ˆ t 2 2 ln h 2 02 + ˆ t 3 2 ln  h 2 01 + h 2 02  − ˆ t 1 2 ln ˆ t 1 − ˆ t 2 2 ln ˆ t 2 − ˆ t 3 2 ln ˆ t 3 + ˆ t 2 1 2 h 2 01 P 0 + ˆ t 2 2 2 h 2 02 P 0 + ˆ t 2 3 2 ( h 2 01 + h 2 02 ) P 0 . (47) Note that in d eriving (46) and (47), the f ollowing inequ ality is applied to lower/upper-bound the co rrespond ing terms: ln( x ) ≤ ln(1 + x ) ≤ ln( x ) + 1 x , ∀ x > 0 . (48) Consequently , we have ˆ t 4 ≤ 1 ln P 0  2 c + ˆ t 1 ln h 2 01 + ˆ t 2 ln h 2 02 + ˆ t 3 ln  h 2 01 + h 2 02  − ˆ t 1 ln ˆ t 1 − ˆ t 2 ln ˆ t 2 − ˆ t 3 ln ˆ t 3  + 1 ln P 0  ˆ t 2 1 h 2 01 P 0 + ˆ t 2 2 h 2 02 P 0 + ˆ t 2 3 ( h 2 01 + h 2 02 ) P 0  . Hence, we can bou nd th e optimum value of ˆ t 4 in (4 4) as 0 ≤ ˆ t 4 ≤ O  1 log P 0  . (49) Similarly , b y co nsidering the fou rth cut in (44), we ca n der i ve another b ound on the op timum value of ˆ t 3 as follows: 0 ≤ ˆ t 3 ≤ O  1 log P 0  . (50) Applying the in equality between (4 6) and the term correspon ding to the second cu t in (44), knowing (from (49) and (50)) th e fact that ˆ t 3 ≤ c 3 ln P 0 , and ˆ t 4 ≤ c 4 ln P 0 (where c 3 and c 4 are co nstants), an d using ineq ualities (48), and ln(1 + x ) ≤ x, ∀ x ≥ 0 , (51) DRAFT 23 we obta in 1 2 ln P 0 + c ≤ ˆ t 2 2 ln h 2 02 h 2 13 γ 1 P 2 0 ˆ t 2 2 1 + ˆ t 2 γ 1 h 2 13 P 0 + ˆ t 2  h 2 12 + h 2 13  h 2 02 h 2 13 P 0 + ˆ t 2 h 12 h 2 13 h 02 √ γ 1 P 0 !! + ˆ t 3 2 ln  h 2 02 P 0 ˆ t 3  + ˆ t 4 2 ln  h 2 13 γ 1 P 0 ˆ t 4  + ˆ t 3 2 2  ˆ t 2 h 2 02 P 0 + ˆ t 2 γ 1 ( h 2 12 + h 2 13 ) P 0 + 2 ˆ t 2 h 02 h 12 √ γ 1 P 0 + h 2 02 h 2 13 γ 1 P 2 0  + ˆ t 2 3 2 h 2 02 P 0 + ˆ t 2 4 2 γ 1 h 2 13 P 0 ≤ ˆ t 2 ln P 0 + ˆ t 2 2 ln  h 2 02 h 2 13 γ 1 ˆ t 2 2  + ˆ t 2 2 2 γ 1 h 2 13 P 0 + ˆ t 2 2  h 2 12 + h 2 13  2 h 2 02 h 2 13 P 0 + ˆ t 2 2 h 12 2 h 2 13 h 02 √ γ 1 P 0 + c 3 2 ln P 0 ln h 2 02 − c 3 2 ln P 0 ln ˆ t 3 + c 3 2 + c 4 2 ln P 0 ln γ 1 h 2 13 − c 4 2 ln P 0 ln ˆ t 4 + c 4 2 + ˆ t 3 2 2  ˆ t 2 h 2 02 P 0 + ˆ t 2 γ 1 ( h 2 12 + h 2 13 ) P 0 + 2 ˆ t 2 h 02 h 12 √ γ 1 P 0 + h 2 02 h 2 13 γ 1 P 2 0  + ˆ t 2 3 2 h 2 02 P 0 + ˆ t 2 4 2 γ 1 h 2 13 P 0 Therefo re, we h av e 1 2 ln P 0 + c ≤ ˆ t 2 ln P 0 + ´ c + O  1 ln P 0  + O  1 P 0  . Hence, 1 2 − c 2 log P 0 ≤ ˆ t 2 . (52) Similarly , f rom the third cu t of (4 4), f or ˆ t 1 we have 1 2 − c 1 log P 0 ≤ ˆ t 1 . (53) From (52) an d (53), and also the fact that ˆ t 1 + ˆ t 2 + ˆ t 3 + ˆ t 4 = 1 , we obtain 1 2 − c 2 log P 0 ≤ ˆ t 2 ≤ 1 2 + c 1 log P 0 , (54) 1 2 − c 1 log P 0 ≤ ˆ t 1 ≤ 1 2 + c 2 log P 0 . (55) Hence, from (49), (50), (54), an d (5 5) as P 0 → ∞ , ˆ t 3 , ˆ t 4 → 0 and ˆ t 1 , ˆ t 2 → 0 . 5 . This proves th e ﬁr st par t o f the Theorem . Moreover , knowing th at each term cor respond ing to th e four cuts in ( 44) is greater than 0 . 5 ln( P 0 ) + c an d as ˆ t 1 , ˆ t 2 are strictly above zero (ap proach ing 0 . 5 ), we can easily conclud e that ˆ P (1) 0 , ˆ P (2) 0 , ˆ P (2) 1 , ˆ P (1) 2 ∼ Θ ( P 0 ) . ( 56) DRAFT 24 Now , we prove th at the DPC scheme with the para meters t 1 = ˆ t 1 + ˆ t 3 + ˆ t 4 2 , t 2 = ˆ t 2 + ˆ t 3 + ˆ t 4 2 , P (1) 0 = ˆ P (1) 0 and P (2) 0 = ˆ P (2) 0 , wher e ˆ t 1 , · · · , ˆ t 4 , ˆ P (1) 0 , ˆ P (2) 0 are the parameter s corre sponding to the m aximum value o f (44), ach iev es the cap acity with a gap no mor e than O  1 log P 0  . T o prove this, we show that each of the fo ur terms in ( 45) is no more than O  1 log P 0  below the corre sponding term (fr om the same cut) in (44). T o show this, for th e ﬁrst cu t we have ˆ t 1 C h 2 01 ˆ P (1) 0 ˆ t 1 ! + ˆ t 2 C h 2 02 ˆ P (2) 0 ˆ t 2 ! + ˆ t 3 C ( h 2 01 + h 2 02 ) ˆ P (3) 0 ˆ t 3 ! − t 1 C h 2 01 P (1) 0 t 1 ! − t 2 C h 2 02 P (2) 0 t 2 ! ( a ) ≤ ˆ t 1 2 ln h 2 01 ˆ P (1) 0 ˆ t 1 ! + ˆ t 2 2 ln h 2 02 ˆ P (2) 0 ˆ t 2 ! + ˆ t 3 C ( h 2 01 + h 2 02 ) ˆ P (3) 0 ˆ t 3 ! −  ˆ t 1 2 + ˆ t 3 + ˆ t 4 4  ln h 2 01 ˆ P (1) 0 t 1 ! −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln h 2 02 ˆ P (2) 0 t 2 ! + ˆ t 2 1 2 h 2 01 ˆ P (1) 0 + ˆ t 2 2 2 h 2 02 ˆ P (2) 0 ( b ) . ˆ t 1 2 ln h 2 01 ˆ P (1) 0 ˆ t 1 ! + ˆ t 2 2 ln h 2 02 ˆ P (2) 0 ˆ t 2 ! + ˆ t 3 2 ln  ( h 2 01 + h 2 02 ) P 0 ˆ t 3 + ˆ t 1  −  ˆ t 1 2 + ˆ t 3 + ˆ t 4 4  ln h 2 01 ˆ P (1) 0 t 1 ! −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln h 2 02 ˆ P (2) 0 t 2 ! + O  1 log P 0  ( c ) . ˆ t 3 2 ln   P 0 q ˆ P (1) 0 ˆ P (2) 0   − ˆ t 4 4 ln  ˆ P (1) 0 ˆ P (2) 0  + O  1 log P 0  ( d ) . O  1 log P 0  . (57) Here, ( a ) fo llows from (48), noting the f unction ˆ t 1 ln( P 0 − x − y ) + ˆ t 2 ln( y ) + ˆ t 3 ln  ˆ t 3 +  h 2 01 + h 2 02  x  takes its maximum value at x ≤ ˆ t 3 ˆ t 3 + ˆ t 1 P 0 and hen ce substituting ˆ P (3) 0 = ˆ t 3 ˆ t 3 + ˆ t 1 P 0 and ﬁnally noting ˆ P (1) 0 , ˆ P (2) 0 ∼ Θ( P 0 ) result in ( b ) , ( c ) follows from ˆ t 3 , ˆ t 4 ∼ O  1 log P 0  and ln  t 1 ˆ t 1  ∼ O  1 log P 0  , and ﬁn ally ( d ) follows from ˆ P (1) 0 , ˆ P (2) 0 ∼ Θ( P 0 ) . DRAFT 25 Next, we bou nd the difference between the term s in the fo urth cut of (44) and th e fou rth ter m in C low DP C ˆ t 1 C h 2 23 ˆ P (1) 2 ˆ t 1 ! + ˆ t 2 C h 2 13 ˆ P (2) 1 ˆ t 2 ! + ˆ t 4 C   h 2 13 ˆ P (4) 1 + h 2 23 ˆ P (4) 2 + 2 h 13 h 23 q ˆ P (4) 1 ˆ P (4) 2 ˆ t 4   − t 1 C  h 2 23 P 2 t 1  − t 2 C  h 2 13 P 1 t 2  ( a ) . ˆ t 1 2 ln h 2 23 ˆ P (1) 2 ˆ t 1 ! + ˆ t 2 2 ln h 2 13 ˆ P (2) 1 ˆ t 2 ! + ˆ t 4 C   h 2 13 ˆ P (4) 1 + h 2 23 ˆ P (4) 2 + 2 h 13 h 23 q ˆ P (4) 1 ˆ P (4) 2 ˆ t 4   −  ˆ t 1 2 + ˆ t 3 + ˆ t 4 4  ln  h 2 23 P 2 t 1  −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln  h 2 13 P 1 t 2  + O  1 P 0  ( b ) . ˆ t 1 2 ln  h 2 23 P 2 ˆ t 1  + ˆ t 2 2 ln  h 2 13 P 1 ˆ t 2  + ˆ t 4 ln h 13 s P 1 ˆ t 2 + ˆ t 4 + h 23 s P 2 ˆ t 1 + ˆ t 4 ! −  ˆ t 1 2 + ˆ t 3 + ˆ t 4 4  ln  h 2 23 P 2 t 1  −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln  h 2 13 P 1 t 2  + O  1 P 0  ( c ) . ˆ t 4 2 ln   2 q ( ˆ t 1 + ˆ t 4 )( ˆ t 2 + ˆ t 4 ) + h 13 h 23 ( ˆ t 2 + ˆ t 4 ) r P 1 P 2 + h 23 ( ˆ t 1 + ˆ t 4 ) h 13 r P 2 P 1   − ˆ t 3 4 ln ( P 1 P 2 ) + O  1 log P 0  ( d ) . O  1 log P 0  . (58) Here, ( a ) fo llows f rom ( 48) and no ting ˆ P (2) 1 , ˆ P (1) 2 ∼ Θ( P 0 ) , notin g the fu nction ˆ t 1 ln( P 2 − y ) + ˆ t 2 ln( P 1 − x ) + ˆ t 4 ln  ˆ t 4 +  h 13 √ x + h 23 √ y  2  takes its maximu m value at x ≤ ˆ t 4 ˆ t 4 + ˆ t 2 P 1 , y ≤ ˆ t 4 ˆ t 4 + ˆ t 1 P 2 and hence substitutin g ˆ P (4) 1 = ˆ t 4 ˆ t 4 + ˆ t 2 P 1 and ˆ P (4) 2 = ˆ t 4 ˆ t 4 + ˆ t 1 P 2 result in ( b ) , ( c ) follo ws f rom ˆ t 3 , ˆ t 4 ∼ O  1 log P 0  and ˆ t 1 , ˆ t 2 ∼ 0 . 5 + O  1 log P 0  , and ﬁnally ( d ) follows fro m the facts that P 1 P 2 ∼ Θ(1) , ˆ t 1 + ˆ t 4 , ˆ t 2 + ˆ t 4 ∼ Θ(1 ) , and ˆ t 4 ∼ O ( 1 log P 0 ) . Next, we bou nd the difference between the term s in the secon d cut of (44) and the second ter m in C low DP C ˆ t 2 C   h 2 02 ˆ P (2) 0 ˆ t 2 + ( h 2 12 + h 2 13 ) ˆ P (2) 1 ˆ t 2 + 2 h 02 h 12 q ˆ P (2) 0 ˆ P (2) 1 ˆ t 2 + h 2 02 h 2 13 ˆ P (2) 0 ˆ P (2) 1 ˆ t 2 2   + ˆ t 3 C h 2 02 ˆ P (3) 0 ˆ t 3 ! + ˆ t 4 C h 2 13 ˆ P (4) 1 ˆ t 4 ! − t 2 C h 2 02 P (2) 0 t 2 ! − t 2 C  h 2 13 P 1 t 2  ( a ) . ˆ t 2 2 ln h 2 02 h 2 13 ˆ P (2) 0 ˆ P (2) 1 ˆ t 2 2 ! + ˆ t 3 C h 2 02 ˆ P (3) 0 ˆ t 3 ! + ˆ t 4 C h 2 13 ˆ P (4) 1 ˆ t 4 ! −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln h 2 02 h 2 13 ˆ P (2) 0 P 1 t 2 2 ! + O  1 P 0  ( b ) . ˆ t 2 2 ln h 2 02 h 2 13 ˆ P (2) 0 P 1 ˆ t 2 2 ! + ˆ t 3 2 ln  h 2 02 P 0 ˆ t 3 + ˆ t 2  + ˆ t 4 2 ln  h 2 13 P 1 ˆ t 4 + ˆ t 2  −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln h 2 02 ˆ P (2) 0 t 2 ! −  ˆ t 2 2 + ˆ t 3 + ˆ t 4 4  ln  h 2 13 P 1 t 2  + O  1 P 0  ( c ) . ˆ t 3 4 ln P 2 0 ˆ P (2) 0 P 1 ! + ˆ t 4 4 ln P 1 ˆ P (2) 0 ! + O  1 log P 0  ( d ) . O  1 log P 0  . (59) Here, ( a ) follows f rom (48), the fact that P (2) 0 = ˆ P (2) 0 ∼ Θ ( P 0 ) and upp er-bounding ˆ P (3) 0 ≤ P 0 , ˆ P (4) 1 ≤ P 1 , noting the facts that ˆ P (2) 0 + ˆ P (3) 0 ≤ P 0 and ˆ P (2) 1 + ˆ P (4) 1 = P 1 , the functio ns ˆ t 2 ln( P 0 − x ) + ˆ t 3 ln  ˆ t 3 + h 2 02 x  and DRAFT 26 ˆ t 2 ln( P 1 − y )+ ˆ t 4 ln  ˆ t 4 + h 2 13 y  are maximized at x ≤ ˆ t 3 ˆ t 2 + ˆ t 3 P 0 and y ≤ ˆ t 4 ˆ t 2 + ˆ t 4 P 1 , hence, substituting ˆ P (3) 0 = ˆ t 3 ˆ t 2 + ˆ t 3 P 0 and ˆ P (4) 1 = ˆ t 4 ˆ t 2 + ˆ t 4 P 1 upper-boun ds the e xpression which results in ( b ) , ( c ) follows from ˆ t 3 , ˆ t 4 ∼ O  1 log P 0  , ˆ t 1 , ˆ t 2 ∼ 0 . 5 + O  1 log P 0  , and ﬁnally ( d ) follows fro m the fact that ˆ P (2) 0 , P 1 ∼ Θ ( P 0 ) an d also ˆ t 3 , ˆ t 4 ∼ O  1 log P 0  . Noting that the second and the third cu ts are the same, and using the sam e argumen t as in (59), we can bo und the difference b etween the term s in the third cut o f (44) an d the third ter m in C low DP C as ˆ t 1 C   h 2 01 ˆ P (1) 0 ˆ t 1 + ( h 2 12 + h 2 23 ) ˆ P (1) 2 ˆ t 1 + 2 h 01 h 12 q ˆ P (1) 0 ˆ P (1) 2 ˆ t 1 + h 2 01 h 2 23 ˆ P (1) 0 ˆ P (1) 2 ˆ t 2 1   + ˆ t 3 C h 2 01 ˆ P (3) 0 ˆ t 3 ! + ˆ t 4 C h 2 23 ˆ P (4) 2 ˆ t 4 ! − t 1 C h 2 01 P (1) 0 t 1 ! − t 1 C  h 2 23 P 2 t 1  ≤ O  1 log P 0  . (60) Observing (57), ( 58), (59) an d (6 0), com pletes the proo f of the Theor em. Theorem 9 In low SNR scena rios, assumin g P 1 = γ 1 P 0 , P 2 = γ 2 P 0 with γ 1 , γ 2 constants ind ependen t of the SNR, when the power available fo r the so ur ce and each relay tends to zer o and  h 13 √ γ 1 + h 23 √ γ 2  2 ≤ min  h 2 01 , h 2 02  , the ratio of the achievable rate of the simultaneou s relaying pr otocol ba sed on DDF to cut-set upp er b ound g oes to 1. I n this scena rio t 3 = t 4 = 1 2 , an d no private messages shou ld be transmitted. Pr o of: By the same argu ment as in Theorem 8 and consider ing only the fo urth cut, we obtain anothe r up per bound on the capacity . By th e following inequality ln(1 + x ) ≤ x. (61) we can b ound the up per b ound on th e capacity as C up ≤  h 13 √ γ 1 + h 23 √ γ 2  2 P 0 2 ln 2 . (62) Now , assuming t 1 = t 2 = 0 , t 3 = t 4 = 1 2 , an d transm itting just th e com mon message, we can ac hiev e the following rate C low DD F : C low DD F = min  1 2 C  2 h 2 02 P 0  , 1 2 C  2 ( h 13 √ γ 1 + h 23 √ γ 2 ) 2 P 0   . (63) According to th e T aylor expansion of ln(1 + x ) at x = 0 , we have x − x 2 2 ≤ ln (1 + x ) , (64) Hence, 1 ln 2 min h 2 02 P 0 2 − h 4 02 P 2 0 2 ,  h 13 √ γ 1 + h 23 √ γ 2  2 P 0 2 −  h 13 √ γ 1 + h 23 √ γ 2  4 P 2 0 2 ! ≤ C low DD F . (65) By (62), ( 65), and  h 13 √ γ 1 + h 23 √ γ 2  2 ≤ min  h 2 01 , h 2 02  , we have lim P 0 → 0 C low DD F C up → 1 . (66) DRAFT 27 V . S I M U L AT I O N R E S U L T In this section, the achiev able rate of different p roposed schemes, i.e., SSRD, DPC, BME, and BME- DPC are compare d with each other an d with the u pper boun d in d ifferent chann el con ditions. Figure 9 compares the achiev able rate of the SSRD scheme with that of the DPC scheme for successi ve r elaying and the DDF scheme for simultaneo us relay ing pro tocols. H ere the symmetric scenario in which P 1 = P 2 and h 01 = h 02 = h 12 = h 13 = h 23 = 1 is considered . Th e u pper boun d is also inclu ded in the ﬁgu re. In orde r to satisfy the co ndition in Theo rem 9, i.e.,  h 13 √ γ 1 + h 23 √ γ 2  2 ≤ min  h 2 01 , h 2 02  , in Figs. 9 a and b, we also assume P 0 = P 1 + 1 0 ( dB ) = P 2 + 1 0 ( dB ) and P 0 = P 1 + 5 ( dB ) = P 2 + 5 ( dB ) , respecti vely . As the Figs. 9a and b sho w , SSRD ach ie vable r ate almost coincides with the upper bound ov er all range s of SNR. As proved in the previous section, in high SNR scenario , SSRD scheme coincid es with DPC and the successiv e relaying protoco l becomes optimu m, while in low SNR scenar io it coin cides with DDF and the simultan eous relaying pro tocol is optimum . On th e other ha nd, in Figs. 9 c and d we a ssume that P 0 = P 1 = P 2 and P 0 = P 1 − 5( dB ) = P 2 − 5( dB ) . In this situatio n, the con dition in Theor em 9 is no lon ger satisﬁed. Therefo re, as th ese ﬁg ures show , the ra tio o f the achiev able rate of the SSRD schem e to the cut-set bo und, i.e. , C low S S RD C up does not tend to one. Further more, the achiev able ra tes of the SSRD, DPC, an d DDF schem es coincid e with each other . Figure 1 0 co mpares the achiev able rate of different successive schem es with each other and th e suc cessi ve cu t-set bound . It shows as the in ter r elay ch annel beco mes stron ger , BME sch eme ca n ach iev e the successi ve cut-set b ound, while the achievable rate of the DPC is indepe ndent of th at channel. Furth ermore, this ﬁg ure in dicates BME-DPC giv es a better achiev able r ate with respec t to BME with successive decod ing which was p roposed in [3 2]. V I . C O N C L U S I O N In this paper, we investigated the problem of cooper ati ve strategies for a half-d uplex parallel relay chann el with two relays. W e d eriv ed the optimu m relay or dering and h ence the asympto tic capacity of the ha lf-duplex Gaussian parallel relay ch annel in low a nd high SNR scenar ios. Simultaneo us and Successive relayin g protoco ls, associated with two possible r elay ord erings were p roposed . For simultan eous relaying, each relay em ploys DDF . On the oth er hand, for successi ve relaying, we prop osed a Non-Coope rative Coding scheme based on DPC and a Cooperative Codin g scheme based on BME. Mo reover , a coding sch eme based on the comb ination of DPC an d BME, in which one of the r elays uses DPC while the other one employs BME was pro posed. W e showed that this composite sche me ach ie ves a better r ate with respect to cooper ati ve coding based o n BME with backward or successive deco ding in the Ga ussian case. W e also prop osed the SSRD scheme as a co mbination o f the simultan eous an d successiv e protocols based on DPC. In high SNR scenar ios, we proved that our Non-Coo perative Codin g scheme based on DPC asymptotically achieves the capacity . Hence, in the high SNR scenario, the optimu m relay ordering is Successive . On the o ther hand, in low SNR wh ere ( h 13 γ 1 + h 23 γ 2 ) 2 ≤ min  h 2 01 , h 2 02  , DDF ach iev es the capacity . Hence, in lo w SNR scenario and und er the con dition speciﬁed ab ove for the chann el coefﬁcients, th e optimu m re lay orderin g is Simultaneo us . DRAFT 28 −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 Relay Power (dB) Rate (bit per dimension) a) P 0 =P 1 +10 (dB)=P 2 +10 (dB) −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 b) P 0 =P 1 +5 (dB)=P 2 +5(dB) Relay Power (dB) Rate (bit per dimension) −20 −18 −16 −14 −12 −10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Relay Power (dB) Rate (bit per dimension) c) P 0 =P 1 =P 2 −20 −18 −16 −14 −12 −10 0 0.01 0.02 0.03 0.04 0.05 Relay Power (dB) Rate (bit per dimension) d) P 0 =P 1 −5 (dB)=P 2 −5 (dB) Cut−set Bound SSRD DPC DDF Cut−set Bound SSRD DPC DDF Cut−set Bound SSRD DPC DDF Cut−set Bound SSRD DPC DDF Fig. 9. Rate versus relay po wer . A P P E N D I X A Pr o of of Theo r em 1 Codeboo k Construction : Let us divide time slot num ber b , b = 1 , 2 , · · · , B + 1 into odd and even numb ers. At odd and even tim e slots, source gener ates 2 nr (1) AU X and 2 nr (2) AU X sequences u (1) 0 ( q 1 ) and u (2) 0 ( q 2 ) according to Q t 1 n i =1 p ( u (1) 0 ,i ) and Q t 2 n i =1 p ( u (2) 0 ,i ) , respectively . Then, sou rce thr ows u (1) 0 and u (2) 0 sequences unifor mly into 2 nR (1) and 2 nR (2) bins, respectively . Let us d enote B 1 ( w ( b ) ) an d B 2 ( w ( b ) ) as the set of sequences at the od d or ev en tim e slot that belong to the w ( b ) ’th bin, respectively (fo r od d time slots, w ( b ) ≤ 2 nR (1) , and for the even time slots, w ( b ) ≤ 2 nR (2) ). Relay 1 and relay 2 generate 2 nR (1) and 2 nR (2) i.i.d x (2) 1 and x (1) 2 sequences according to prob abilities Q t 2 n i =1 p  x (2) 1 ,i  and Q t 1 n i =1 p  x (1) 2 ,i  . Furthermore, for all q 1 and q 2 , the source generates doub le indexed co debook s x (1) 0  w ( b ) | w ( b − 1) , q 1  and x (2) 0  w ( b ) | w ( b − 1) , q 2  accordin g to Q t 1 n i =1 p ( x (1) 0 ,i | x (1) 2 ,i , u (1) 0 ,i ) and Q t 2 n i =1 p ( x (2) 0 ,i | x (2) 1 ,i , u (2) 0 ,i ) , respectively . Encodin g: Encodin g at th e so ur ce: DRAFT 29 −30 −20 −10 0 10 20 30 40 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 h 12 2 (dB) Rate(bit per dimension) P 0 = P 1 = P 2 = 10(dB), h 01 2 = h 23 2 = 1(dB), h 02 2 = h 13 2 = 10(dB). Successive Cut−Set Bound BME with successive decoding DPC BME−DPC Fig. 10. Rate versus inter relay gain. At th e o dd tim e slot b , the sou rce inten ds to send the message w ( b ) to the ﬁrst relay . In ord er to do that, since source knows what it h as tra nsmitted d uring th e last time slot to the second relay , it ch ooses a cod ew ord u (1) 0 ( q 1 ) such that u (1) 0 ( q 1 ) ∈ B 1 ( w ( b ) ) and  u (1) 0 ( q 1 ) , x (1) 2  w ( b − 1)   ∈ A ( n ) ǫ . Such a task can be done almost surely , if r (1) AU X − R (1) ≥ t 1 I  U (1) 0 ; X (1) 2  (See [1 2]). Follo wing that it sends x (1) 0 ( u (1) 0 , x (1) 2 ) . At the even tim e slot b , the source sends th e m essage w ( b ) to the second relay in the similar m anner . Such a task can b e don e almo st sure ly if r (2) AU X − R (2) ≥ t 2 I  U (2) 0 ; X (2) 1  . Encodin g at relay 1: At the ev en time slot b , relay 1 enco des w ( b − 1) ∈ { 1 , · · · , 2 nR (1) } to x (2) 1  w ( b − 1)  . Encodin g at relay 2: At the odd time slot b , relay 2 en codes w ( b − 1) ∈ { 1 , · · · , 2 nR (2) } to x (1) 2  w ( b − 1)  . Decoding : Decoding at r elay 1: At the odd tim e slot b , re lay 1 d eclares ˆ w ( b ) = w ( b ) iff all the sequ ences u (1) 0 ( q 1 ) wh ich are jointly typ ical with DRAFT 30 y (1) 1 belong to a uniq ue bin B 1 ( ˆ w ( b ) ) . Therefo re, in o rder to make the probab ility o f error zero, from [12], we have r (1) AU X ≤ t 1 I  U (1) 0 ; Y (1) 1  . (67) According to ( 67) and the encod ing cond ition at source, w e h av e R (1) ≤ t 1  I ( U (1) 0 ; Y (1) 1 ) − I ( U (1) 0 ; X (1) 2 )  . (68) Decoding at r elay 2: At the e ven time slot b , relay 2 declares ˆ w ( b ) = w ( b ) iff all the sequences u (2) 0 ( q 2 ) which are jointly typical with y (2) 2 belong to a uniq ue bin B 2 ( ˆ w ( b ) ) . Therefo re, in o rder to make the probab ility o f error zero, from [12], we have r (2) AU X ≤ t 2 I  U (2) 0 ; Y (2) 2  . (69) According to ( 69) and the encod ing cond ition at source, w e h av e R (2) ≤ t 2  I ( U (2) 0 ; Y (2) 2 ) − I ( U (2) 0 ; X (2) 1 )  . (70) Decoding at the ﬁna l d estination: At the od d time slot b , d estination declares ˆ w ( b − 1) = w ( b − 1) iff  x (1) 2  ˆ w ( b − 1)  , y (1) 3  ∈ A ( n ) ǫ . Hence, in or der to make the p robability of er ror zero, fro m [1 2], we have R (1) ≤ t 1 I ( X (1) 2 ; Y (1) 3 ) . (71) Similarly , at the even time slot b , we have R (2) ≤ t 2 I ( X (2) 1 ; Y (2) 3 ) . (72) From the enc oding at the so urce and ( 67)-(72), we ob tain (9)-(11). A P P E N D I X B Pr o of of Theo r em 2 Codeboo k Construction : Let us d i vide the time slots b , b = 1 , 2 , · · · , B + 2 into o dd and even time slots. The sourc e g enerates two codebo oks x (1) 0  w ( b ) | w ( b − 1) , s ( b − 2) 1  and x (2) 0  w ( b ) | w ( b − 1) , s ( b − 2) 2  of size 2 nR (1) and 2 nR (2) correspo nding to ev en and odd time slots, respecti vely . The ﬁrst codebook is generated according to the probability p ( x (1) 0 , x (1) 2 , u (1) 2 ) = Q t 1 n i =1 p ( u (1) 2 ,i ) p ( x (1) 2 ,i | u (1) 2 ,i ) p ( x (1) 0 ,i | x (1) 2 ,i , u (1) 2 ,i ) , and the seco nd codebo ok is generate d accord ing to th e prob ability p ( x (2) 0 , x (2) 1 , u (2) 1 ) = Q t 2 n i =1 p ( u (2) 1 ,i ) p ( x (2) 1 ,i | u (2) 1 ,i ) p ( x (2) 0 ,i | x (2) 1 ,i , u (2) 1 ,i ) . On the other ha nd, relay 2 gener ates 2 nr (1) Bin i.i.d code words u (1) 2 and 2 nR (2) i.i.d code words x (1) 2 accordin g to the pro babilities p ( u (1) 2 ) = Q t 1 n i =1 p ( u (1) 2 ,i ) and p ( x (1) 2 | u (1) 2 ) = Q t 1 n i =1 p ( x (1) 2 ,i | u (1) 2 ,i ) at each od d time slot and relay 1 generates 2 nr (2) Bin i.i.d c odew ords u (2) 1 and 2 nR (1) i.i.d c odew ords x (2) 1 accordin g to the probab ilities p ( u (2) 1 ) = Q t 2 n i =1 p ( u (2) 1 ,i ) and p ( x (2) 1 | u (2) 1 ) = Q t 2 n i =1 p ( x (2) 1 ,i | u (2) 1 ,i ) at each even time slo t, r espectiv ely . DRAFT 31 Encodin g: Encodin g at th e so ur ce: At the odd time slot b , so urce encodes w ( b ) ∈ { 1 , · · · , 2 nR (1) } to x (1) 0  w ( b ) | w ( b − 1) , s ( b − 2) 1  and at th e e ven time slot b , it encod es w ( b ) ∈ { 1 , · · · , 2 nR (2) } to x (2) 0  w ( b ) | w ( b − 1) , s ( b − 2) 2  and sends the m in odd a nd even time slots, respectively . Encodin g at relay 1: At the even time slot b , relay 1 encod es the bin ind ex s ( b − 2) 2 of the message w ( b − 2) it has received from relay 2 in the previous time slot to u (2) 1  s ( b − 2) 2  . Follo wing that, it enco des w ( b − 1) which was re ceiv ed fr om th e so urce in time slot b − 1 to x (2) 1  w ( b − 1) | s ( b − 2) 2  and sends it. Encodin g at relay 2: At th e odd time slot b , relay 2 en codes the bin index s ( b − 2) 1 of the message w ( b − 2) it has received from relay 1 in the previous time slot to u (1) 2  s ( b − 2) 1  . Follo wing that, it enco des w ( b − 1) which was re ceiv ed fr om th e so urce in time slot b − 1 to x (1) 2  w ( b − 1) | s ( b − 2) 1  and sends it. Decoding : Decoding at r elay 1: Knowing w ( b − 2) and consequen tly s ( b − 2) 1 , at tim e slot b , relay 1 declares ( ˆ w ( b − 1) , ˆ w ( b ) ) = ( w ( b − 1) , w ( b ) ) iff there exits a u nique ( ˆ w ( b − 1) , ˆ w ( b ) ) such that  x (1) 0  ˆ w ( b ) | ˆ w ( b − 1) , s ( b − 2) 1  , x (1) 2  ˆ w ( b − 1) | s ( b − 2) 1  , u (1) 2 ( s ( b − 2) 1 ) , y (1) 1  ∈ A ( n ) ǫ . Hence, in or der to m ake prob ability of err or zer o, fr om the E xtended M A C c apacity r egion (See [1 2], [24 ], [2 5], and [26 ]), we have R (1) ≤ t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  , (73) R (1) + R (2) ≤ t 1 I ( X (1) 0 , X (1) 2 ; Y (1) 1 | U (1) 2 ) . (74) Decoding at r elay 2: Knowing w ( b − 2) and consequen tly s ( b − 2) 2 , at tim e slot b , relay 2 declares ( ˆ w ( b − 1) , ˆ w ( b ) ) = ( w ( b − 1) , w ( b ) ) iff there exits a u nique ( ˆ w ( b − 1) , ˆ w ( b ) ) such that  x (2) 0  ˆ w ( b ) | ˆ w ( b − 1) , s ( b − 2) 2  , x (2) 1  ˆ w ( b − 1) | s ( b − 2) 2  , u (2) 1 ( s ( b − 2) 2 ) , y (2) 2  ∈ A ( n ) ǫ . Hence, in or der to m ake the pro bability of er ror ze ro, f rom E xtended M A C c apacity r egion (See [1 2], [24 ], [2 5], and [26 ]), we have R (2) ≤ t 2 I ( X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1 ) , (75) R (1) + R (2) ≤ t 2 I ( X (2) 0 , X (2) 1 ; Y (2) 2 | U (2) 1 ) . (76) Decoding at the ﬁna l d estination: Decoding at th e ﬁnal destin ation can be d one either S uccessively or Backwar dly as follows. DRAFT 32 1) Succe ssive Deco ding: At the end of odd time slot b , d estination ﬁrst d eclares the bin in dex ˆ s ( b − 2) 1 = s ( b − 2) 1 of the message w ( b − 2) iff there exists a un ique ˆ s ( b − 2) 1 such that  u (1) 2 ( ˆ s ( b − 2) 1 ) , y (1) 3  ∈ A ( n ) ǫ . Henc e, in orde r to make the p robab ility o f error zero, fro m [1 2] we have r (1) B in ≤ t 1 I ( U (1) 2 ; Y (1) 3 ) . (77) Having decod ed the bin index s ( b − 2) 1 of the message w ( b − 2) , destination can resolve its uncertainty about the message w ( b − 2) and declares ˆ w ( b − 2) = w ( b − 2) iff there exists a unique ˆ w ( b − 2) such that  x (2) 1 ( ˆ w ( b − 2) | s ( b − 3) 2 ) , u (2) 1 ( s ( b − 3) 2 ) , y (2) 3  ∈ A ( n ) ǫ . Hence , in or der to make the pr obability of err or zer o, from [12 ] we have R (1) − r (1) B in ≤ t 2 I ( X (2) 1 ; Y (2) 3 | U (2) 1 ) . (78) Using the sam e argumen t for the even tim e slot b , we have r (2) B in ≤ t 2 I ( U (2) 1 ; Y (2) 3 ) , (79) R (2) − r (2) B in ≤ t 1 I ( X (1) 2 ; Y (1) 3 | U (1) 2 ) . (80) From (77), ( 78), (79), and (80), R (1) and R (2) are bo unded as follows R (1) ≤ t 2 I  X (2) 1 ; Y (2) 3 | U (2) 1  + t 1 I  U (1) 2 ; Y (1) 3  , (81) R (2) ≤ t 1 I ( X (1) 2 ; Y (1) 3 | U (1) 2 ) + t 2 I ( U (2) 1 ; Y (2) 3 ) . (82) From (73)-( 76), (8 1), and (82), th e achievable rate of BME scheme based o n succe ssi ve dec oding is equ al to C low B M succ = R (1) + R (2) ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min ( (83) min  t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  , t 2 I  X (2) 1 ; Y (2) 3 | U (2) 1  + t 1 I  U (1) 2 ; Y (1) 3  + min  t 1 I  X (1) 2 ; Y (1) 3 | U (1) 2  + t 2 I  U (2) 1 ; Y (2) 3  , t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  , t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1 | U (1) 2  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2 | U (2) 1  . 2) Backwar d Dec oding: Follo wing receiving the sequenc e correspond ing to the B + 2 ’th time slot, d estination starts dec oding the messages in a backward manner, i.e. from w ( B ) back to w (1) . At the end of o dd time slot b , knowing the value s ( b − 1) 2 from the received signal in time slot b + 1 , d estination declar es  ˆ w ( b − 1) , ˆ s ( b − 2) 1  =  w ( b − 1) , s ( b − 2) 1  iff there exists a uniqu e pair  ˆ w ( b − 1) , ˆ s ( b − 2) 1  such tha t f (2) B in  ˆ w ( b − 1)  = s ( b − 1) 2 and  x (1) 2  ˆ w ( b − 1) , ˆ s ( b − 2) 1  , u (1) 2  ˆ s ( b − 2) 1  , y (1) 3  ∈ A ( n ) ǫ . Similarly , at the end of ev en time slot b , k nowing the value s ( b − 1) 1 for th e recei ved signal in time slo t b + 1 , destination de clares  ˆ w ( b − 1) , ˆ s ( b − 2) 2  =  w ( b − 1) , s ( b − 2) 2  iff there exists a un ique p air  ˆ w ( b − 1) , ˆ s ( b − 2) 2  such that f (1) B in  ˆ w ( b − 1)  = s ( b − 1) 1 and  x (2) 1  ˆ w ( b − 1) , ˆ s ( b − 2) 1  , u (2) 1  ˆ s ( b − 2) 2  , y (2) 3  ∈ A ( n ) ǫ . Hence, in order to make the DRAFT 33 probab ility of e rror zero, fr om [ 12] we have r (1) B in ≤ R (1) , (84) r (2) B in ≤ R (2) , (85) R (2) − r (2) B in ≤ t 1 I  X (1) 2 ; Y (1) 3 | U (1) 2  , (86) R (2) − r (2) B in + r (1) B in ≤ t 1 I  X (1) 2 , U (1) 2 ; Y (1) 3  , (87) R (1) − r (1) B in ≤ t 2 I  X (2) 1 ; Y (2) 3 | U (2) 1  , (88) R (1) − r (1) B in + r (2) B in ≤ t 2 I  X (2) 1 , U (2) 1 ; Y (2) 3  . (89) Hence, by employing BME and Backward decoding , the fo llowing rate is achievable sub ject to (73)-(76) and (84)-(89) constrain ts. C low B M E back = R (1) + R (2) . (90) Optimum inpu t d istrib utions Now , we prove there exists in put pro bability distributions ( p ( x (1) 0 , x (1) 2 , u (1) 2 ) an d p ( x (2) 0 , x (2) 1 , u (2) 1 ) ) which max- imize (90) an d h av e the fo llowing property : u (1) 2 is in depende nt f rom ( x (1) 0 , x (1) 2 ) and u (2) 1 is indepen dent from ( x (2) 0 , x (2) 1 ) . T o prove this, co nsider p ( x (1) 0 , x (1) 2 , u (1) 2 ) and p ( x (2) 0 , x (2) 1 , u (2) 1 ) along with t 1 , t 2 which maximize (90) subject to th e required con straints. Let us d eﬁne ˆ p ( x (1) 0 , x (1) 2 , u (1) 2 ) and ˆ p ( x (2) 0 , x (2) 1 , u (2) 1 ) as ˆ p ( x (1) 0 , x (1) 2 , u (1) 2 ) = p ( u (1) 2 ) p ( x (1) 0 , x (1) 2 ) , (91) ˆ p ( x (2) 0 , x (2) 1 , u (2) 1 ) = p ( u (2) 1 ) p ( x (2) 0 , x (2) 1 ) , (92) Now , we show that ˆ p ( x (1) 0 , x (1) 2 , u (1) 2 ) an d ˆ p ( x (2) 0 , x (2) 1 , u (2) 1 ) along with t 1 , t 2 achieve at least the same rate as th e optimum o ne. Le t us denote the values of mutual in formation and entropy with respect to the inpu t d istributions p, ˆ p by I p , H p and I ˆ p , H ˆ p , respec ti vely . The r ight-han d sides of (86)-(89) with r espect to p can be up per-bounde d by the ones corr espondin g to ˆ p as follows t 1 I p  X (1) 2 ; Y (1) 3 | U (1) 2  ( a ) ≤ t 1 I p  X (1) 2 ; Y (1) 3  = t 1 I ˆ p  X (1) 2 ; Y (1) 3  , (93) t 1 I p  X (1) 2 , U (1) 2 ; Y (1) 3  ( a ) = t 1 I p  X (1) 2 ; Y (1) 3  = t 1 I ˆ p  X (1) 2 ; Y (1) 3  , (94) t 2 I p  X (2) 1 ; Y (2) 3 | U (2) 1  ( b ) ≤ t 2 I p  X (2) 1 ; Y (2) 3  = t 2 I ˆ p  X (2) 1 ; Y (2) 3  , (95) t 2 I p  X (2) 1 , U (2) 1 ; Y (2) 3  ( b ) = t 2 I p  X (2) 1 ; Y (2) 3  = t 2 I ˆ p  X (2) 1 ; Y (2) 3  . (96) where ( a ) f ollows from the fact that U (1) 2 − → X (1) 2 − → Y (1) 3 forms a Mar kov c hain an d ( b ) follows from the fact that U (2) 1 − → X (2) 1 − → Y (2) 3 forms a Markov chain. Mo reover as in distribution ˆ p , u (1) 2 and u (2) 1 are ind ependen t from ( x (1) 0 , x (1) 2 ) and ( x (2) 0 , x (2) 1 ) , it can b e easily veriﬁed that the rig ht-hand sides of (9 3)-(96) are equa l to the right-ha nd sides o f (86)-(89) with the input d istribution ˆ p , respecti vely . Hen ce, b y utilizing ˆ p instead of p , the region DRAFT 34 that satisﬁes (86)-(89) is enla rged. Now , let us consider the right-ha nd sides o f (73)-(76). t 1 I p  X (1) 0 ; Y (1) 1 | X (1) 2 , U (1) 2  ( a ) ≤ t 1 I p  X (1) 0 ; Y (1) 1 | X (1) 2  = t 1 I ˆ p  X (1) 0 ; Y (1) 1 | X (1) 2  (97) t 1 I p  X (1) 0 , X (1) 2 ; Y (1) 1 | U (1) 2  ( a ) ≤ t 1 I p  X (1) 0 , X (1) 2 ; Y (1) 1  = t 1 I ˆ p  X (1) 0 , X (1) 2 ; Y (1) 1  (98) t 2 I p  X (2) 0 ; Y (2) 2 | X (2) 1 , U (2) 1  ( b ) ≤ t 2 I p  X (2) 0 ; Y (2) 2 | X (2) 1  = t 2 I ˆ p  X (2) 0 ; Y (2) 2 | X (2) 1  (99) t 2 I p  X (2) 0 , X (2) 1 ; Y (2) 2 | U (2) 1  ( b ) ≤ t 2 I p  X (2) 0 , X (2) 1 ; Y (2) 2  = t 2 I ˆ p  X (2) 0 , X (2) 1 ; Y (2) 2  (100) where ( a ) follows from the fact tha t U (1) 2 − → ( X (1) 2 , X (1) 0 ) − → Y (1) 1 form a Markov chain and ( b ) follows fro m the fact that U (2) 1 − → ( X (2) 1 , X (2) 0 ) − → Y (2) 2 form a Ma rkov ch ain. Similarly , we observe that the right-han d sides of (97)-(100) r epresent the r ight-hand sides of ineq ualities ( 73)-(76) with the inpu t distribution ˆ p . Hence, the r egion of ( R (1) , R (2) ) that satisﬁes (73)-(7 6) and (84)-(89) is enlarged by utilizing the in put d istribution ˆ p instead of p . This pr oves the indepen dency of input d istributions with u (1) and u (2) in the o ptimum distribution. Simplifying the achievable rate As we can assum e that the inpu t d istributions are of the form (9 1) and ( 92), the ac hiev able rate can be simpliﬁed as follows. C low B M E back = R (1) + R (2) ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min  t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2  , (101) subject to r (1) B in ≤ R (1) , (102) r (2) B in ≤ R (2) , (103) R (1) ≤ t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  , (104) R (2) ≤ t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  , (105) R (2) − r (2) B in + r (1) B in ≤ t 1 I  X (1) 2 ; Y (1) 3  , ( 106) R (1) − r (1) B in + r (2) B in ≤ t 2 I  X (2) 1 ; Y (2) 3  . ( 107) with inp ut distributions p ( x (1) 0 , x (1) 2 ) = p ( x (1) 2 ) p ( x (1) 0 | x (1) 2 ) , p ( x (2) 0 , x (2) 1 ) = p ( x (2) 1 ) p ( x (2) 0 | x (2) 1 ) . DRAFT 35 Now , we show tha t (1 01)-(10 7 ) is eq uiv alent to C low B M E back ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min  t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2  , t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  , t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  . ( 108) First, it is easy to verify tha t (101)-(107) imply (108). Now , in o rder to pr ove th at th e conv erse is also true, we show that f or every possible rate r satisfying (10 8), th ere exists a qu ad-tupp le  R (1) , R (2) , r (1) B in , r (2) B in  such that R (1) + R (2) = r ,  R (1) , R (2) , r (1) B in , r (2) B in  satisﬁes ( 101)-(107), and moreover at least one o f bin rates is eq ual to zero, i.e. r (1) B in = 0 or r (2) B in = 0 . Let u s deﬁne R (1) , min  r , t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  , R (2) , r − R (1) . As r satisﬁ es (108), we co nclude that ( R (1) , R (2) ) satisﬁes (10 1), (1 04), and (105). Further more, as R (1) + R (2) = r ≤ t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  , we co nclude that either R (1) ≤ t 2 I  X (2) 1 ; Y (2) 3  or R (2) ≤ t 1 I  X (1) 2 ; Y (1) 3  . For the sak e of symmetry , let u s assume that th e ﬁrst case has occurred, i.e. R (1) ≤ t 2 I  X (2) 1 ; Y (2) 3  . No w , we deﬁne r (1) B in , 0 and r (2) B in , max  0 , R (2) − t 1 I  X (1) 2 ; Y (1) 3  . Obviously , (102), (103), and (106) are valid. Conside ring ( 107), we have R (1) − r (1) B in + r (2) B in = R (1) + max  0 , r − R (1) − t 1 I  X (1) 2 ; Y (1) 3  ( a ) ≤ t 2 I  X (2) 1 ; Y (2) 3  (109) where ( a ) follows fro m the facts that r ≤ t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  and R (1) ≤ t 2 I  X (2) 1 ; Y (2) 3  . Hence, ( 107) is also valid. The second case in which R (2) ≤ t 1 I  X (1) 2 ; Y (1) 3  can be dealt with in a similar manner . Hence, from the above ar gumen t, the ach iev able r ate of BME scheme with backward decoding can be simpliﬁed as follows: C low B M E back ≤ max 0 ≤ t 1 ,t 2 ,t 1 + t 2 =1 min  t 1 I  X (1) 0 , X (1) 2 ; Y (1) 1  , t 2 I  X (2) 0 , X (2) 1 ; Y (2) 2  , t 1 I  X (1) 0 ; Y (1) 1 | X (1) 2  + t 2 I  X (2) 0 ; Y (2) 2 | X (2) 1  , t 1 I  X (1) 2 ; Y (1) 3  + t 2 I  X (2) 1 ; Y (2) 3  , ( 110) with pr obabilities p ( x (1) 0 , x (1) 2 ) = p ( x (1) 2 ) p ( x (1) 0 | x (1) 2 ) , p ( x (2) 0 , x (2) 1 ) = p ( x (2) 1 ) p ( x (2) 0 | x (2) 1 ) . R E F E R E N C E S [1] E. C. v an-de r Meule n, “Three-t erminal communic ation channe ls, ” Adv . Appl. Pr ob ., vol. 3, pp. 120-15 4, 1971. [2] T . M. Cov er and A. E l Gamal , “Capaci ty Theorems for th e Relay Channel , ” IEEE T ran sactions on Informat ion Theory , V ol. 25, No. 5, pp. 572-584, Sept ember 197 9. [3] B. Schei n and R. G. Gallager , “The Gaussian para llel relay net work, ” in Pr oc IEEE Int. Symp. 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Cooperative Strategies for the Half-Duplex Gaussian Parallel Relay Channel: Simultaneous Relaying versus Successive Relaying

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