Coding Strategies for Noise-Free Relay Cascades with Half-Duplex Constraint

Coding Strate gies for Noise-Free Relay Cascades with Half-Duple x Constraint T obias Lutz, Christoph Haus l, an d Ralf K ¨ otter Institute for Communication s Engineer ing, TU M ¨ unchen , Munich, Germany Email: { tobi.lutz, christoph.ha usl, ralf.koetter } @tum.de Abstract —T wo types of noise-free relay cascades are inv esti- gated. Networks where a source communicates with a distant recei ver via a cascade of half-duplex constrained rela ys, and networks where not only the source but also a sin gle relay node intends to transmit information to the same d estination. W e introduce two relay channel models, captu ring the half-duplex constraint, and within the framework of th ese models capacity is determined for th e first network type. It t urns out that capacity is sign ificantly higher than the rates wh ich are achievable with a straightforward time-sharing approach. A capacity achieving coding strategy is p resented based on allocating the transmit and rece ive time slots of a node in d ependence of th e node’ s prev iously receiv ed data. For the n etworks of t he second type, an upper bound to the rate region is d eriv ed from th e cut-set bound. Fu rther , achiev abili ty of the cut-set boun d in the single relay case is shown given that th e sour ce rate exceeds a certain minimum value. I . I N T RO D U C T I O N The focus of this paper is on half- duplex constrained relay line ne tworks, i. e. o n multi-ho pping networks wh ere the interme diate relay nod es are a rranged in a cascade and , further, are not able to tran smit and receiv e simultaneo usly . W e consider networks with a s ingle source-destinatio n pair and networks where in addition to the source a single relay node intends to transmit own inform ation. Sin ce the main inter est is to gain a better under standing of half-dup lex c onstrained transmission, we assume noiseless network links in ord er to av oid detra ction f rom the actual topic. The classical relay channel goes back to v a n der Meulen [1]. Further significan t results concern ing capacity and coding schemes were obtained in [ 2]. More r ecently , the fo cus of attention shifted towards relay network s and an ach iev ab le rate fo rmula for relay line networks with a single sour ce- destination pair togethe r with a random coding scheme ap- peared in [3]. A com prehensive literature su rvey as well as a classification of rando m codin g strategies is given in [4] . There ha s also been work on d etermining the capacity or rate region of various half- duplex constrain ed relay chan nels [5], [6] and networks [7 ], howe ver, under th e assumption that the time-division sched ule is d etermined a priori. An obvious approach in order to han dle the half-dup lex constraint in a line network is to use a tr ansmission schem e in wh ich even numbered relay s send in say even numb ered time slots and receive d uring od d numb ered time slots while This work was supported by the European Commission in the framewo rk of the FP7 (contra ct n. 215252) and by D ARP A under the ITMANET program. odd nu mbered relays b ehave v ice versa. If the sour ce uses a binary alph abet, the rate becomes 0 . 5 b its per use while a ternary alphabet yields a rate of 0 . 5 log 2 3 bits per use. By allowing randomly allo cated tr ansmit an d receive time slots, higher rates are possible as was poin ted out in [ 8]. In [9] , the same author uses an entirely bina ry , noiseless mo del for the sin gle relay c hannel su ch th at the half -duplex constrain t is included. It is shown there that capa city is equal to 0 . 7729 bits per use what d emonstrates tha t time-sh aring falls c onsiderably short of the theoretical achiev ability . By the way , the sam e channel model was used in [10] in a different context. T wo coding schemes for this particular model were outlined therein, which, in hindsigh t, can be interpreted as half-duplex schemes. W e will introduce two further ch annel m odels fo r h alf- duplex constrained relays. W ithin the framew ork of these mod- els, it is sh own that the capacity of a half-du plex constrained single relay channel is eq ual to 0 .8295 b its per use if the relay is able to distinguish binary symbols and 1.138 9 bits per use if, in ad dition, the relay is capable of detecting time slots with out transmission. Furthermore, it is shown that the capacity of each relay cascade with finite leng th is greater than on e bit per use assumed the latter r elay m odel is utilized. Th e key id ea of the achiev able scheme is to dete rmine the slot allocation of each relay node in depend ence of the d ata rec eiv ed by the relay before. W ith regard to half-duplex constrained line networks, where not o nly the sourc e but also a single relay node intends to transmit own info rmation to the same destination, an upper bound to the r ate region is d erived. W e finally show th at in th e special case of a single relay ch annel (with source and relay source), a slightly different version of the introdu ced co ding scheme is ab le to achieve a se gment on th e cut-set b ound, provided that the source rate exceeds a certain minimum v alu e. Notation: | S | d enotes th e card inality of set S an d P ( S ) the power set of S . Further, S ¯ i := S \{ i } while { f ( i ) : 1 ≤ i ≤ m } me ans { f (1) , f (2) , . . . , f ( m ) } . Th e co nditional pmf p Y | X ( y , x ) is indicated as p ( y | x ) whenever the ran dom variables can b e figured out fro m the arguments. Further, the vector x [0: m ] := ( x 0 , x 1 , . . . , x m ) sum marizes realizations o f the random variables X 0 , X 1 , . . . , X m . The entropy e x pression H ( Y i | X ( k : k> 1) ) equ als H ( Y i | X k ) in case k > 1 an d H ( Y i ) in case k ≤ 1 . W e will abbreviate p X i X i +1 ( a, b ) as p i ab . I I . N E T W O R K M O D E L W e consider a d iscrete, memoryless lin e ne twork co mposed of m + 2 nodes whereas each node is ch aracterized by a uniq ue P S f r a g r e p l a c e m e n t s X 0 X i X i X i − 1 X i − 1 Y i Y i Y i − 1 Y i − 1 Y m +1 Relay Relay i i − 1 1 2 Fig. 1. The considered m ultipl e rela y c ascade (top) and an exce rpt. If relay i is transmitting, the switc h is in posit ion 1 otherwi se in position 2 . number fr om the integer set { 0 , . . . , m + 1 } . The integers 0 and m + 1 are allocated to source and d estination, r espectiv ely . The remainin g nod es 1 to m represent h alf-dup lex constrain ed relays (abbr eviated as HD relays). A graphical represen tation is giv en in Fig. 1. Th e output of the i th n ode, which is the input to the channel b etween nod e i and i + 1 , is deno ted as X i , i ∈ { 0 , . . . , m } , and takes values on the alphabet X i = { 0 , 1 , N } , where N is meant to signify a ch annel use in which node i is n ot tran smitting. Corr espondin gly , the input of th e i th node, wh ich is the ou tput of the ch annel b etween node i − 1 and node i , is Y i , i ∈ { 1 , . . . , m + 1 } , with values from the alphab et Y i . Each message w 0 , sent via multiple ho ps from source n ode 0 to sink node m + 1 , is un iformly drawn from the index set W 0 = { 1 , . . . , 2 nR 0 } , where n is the block length of the en coding scheme and R 0 the tran smission rate. Apart from th e source nod e, th ere is p ossibly a single relay node r ∈ { 1 , · · · , m } , which intend s to tran smit independ ent indices taken fro m W r = { 1 , . . . , 2 nR r } to the d estination. Again, the tran smission scheme is m ulti-hop ping since the informa tion flow associated with message w r has to pass all nodes with indic es greater than r . W e assume n oiseless links what re sults in a d eterministic network, i. e. the entries in p ( y [1: m +1] | x [0: m ] ) are either 0 or 1 . In order to intro duce the half-du plex con straint, we impose f ollowing channel mo del onto each relay node i ∈ { 1 , . . . , m } Y i =  X i − 1 , if X i = N X i , if X i ∈ { 0 , 1 } , (1) where Y m +1 = X m . Relay model (1) is deno ted a s ternary since the reception alphabet of ea ch relay node is Y i = { 0 , 1 , N } . It can easily be verified that Y i = { 0 , 1 } when ( N,N ) is excluded from the Cartesian product X i − 1 × X i , and in this case the model is referred to as bina ry . The inter pretation of both mode ls is as follows: in case relay i sends binary data, i. e. x i ∈ { 0 , 1 } , it o nly hear s itself and, thu s, can not listen to relay i − 1 or , equiv alen tly , relay i and relay i − 1 are disconnected . Conversely , if relay i is quiet, i. e. x i = N, it is sen siti ve for the channel input o f relay i − 1 . T he fe edback interpretatio n of the relay nodes as sho wn in Fig. 1 results from these consider ations. As a conseq uence of the un derlying model, the conditiona l chan nel p mf can be factore d as p  y [1: m +1] | x [0: m ]  = p  y 1 | x [0:1]  · · · p  y m | x [ m − 1: m ]  p ( y m +1 | x m ) . (2) Moreover , we will assume that the channel in puts X 0 , X 1 , . . . , X m form a Markov chain what seems to be unm otiv ated at first glance but tu rns out to be without loss of optimality as explained in Remark 3. I I I . C O D I N G T H E O R E M S Theor em 1: The zero-erro r cap acity of the relay network defined ab ove, wher e on ly th e sour ce but no relay tr ansmits own info rmation, is g i ven by C = max p ( x [0: m ] ) min { H ( Y 1 | X 1 ) , . . . , H ( Y m | X m ) , H ( Y m +1 ) } . (3) Pr oof: The proo f is given in the Appen dix. Ach iev ability is shown in the next section. Example 1 (Single HD Relay Cha nnel, m = 1 ): The con- sidered chan nel with a ternary relay falls into the class of degraded relay channe ls [2]. At each time instance, the relay is either listening or transmitting. When the relay tran smits, i. e. x 1 ∈ { 0 , 1 } , the sour ce inpu t ca nnot b e detected by the relay and, consequen tly , the source sho uld not transmit. Thus, it can be assum ed w .l.o.g. that p 0 00 = p 0 01 = p 0 10 = p 0 11 = 0 . Hence, the source input is not ran dom when x 1 ∈ { 0 , 1 } and together with (1), equation (3) reduces to C = max p ( x [0:1] ) min { H ( X 0 | X 1 = N ) p X 1 ( N ) , H ( X 1 ) } . (4) Howe ver, w hen the relay is listening , i. e. x 1 = N, the source should make optimum use of the ch annel by enco ding with unifor mly distributed input symbo ls, i. e. p 0 0 N = p 0 1 N = p 0 NN . Furthermo re, in order to achieve the m aximum infor mation flow H ( X 1 ) fro m the relay to the sink or, like wise, from a sym metry argumen t, we can cho ose p 0 N 0 = p 0 N 1 . Th ese consideratio ns yield a single degree of f reedom in (4). Since the maximum d oes n ot occur in the maximum of one of the two co ncave functions, (4) is solved by H ( X 0 | X 1 ) = H ( X 1 ) . The resulting assignment is p 0 0 N = p 0 1 N = p 0 NN = 0 . 2395 and p 0 N 0 = p 0 N 1 = 0 . 1407 , which yields C = 1 . 1389 bits per channel use. Remark 1: Evaluation of ca pacity for the b inary HD m odel is almost alon g the same lines as in Examp le 1. Howe ver, the channel in put x 0 x 1 = NN is not allo wed in the binary mod el and, thus, we a priori h av e p 0 NN = 0 , which yields C = 0 . 8295 bits per channel use. Example 2 (Infinite HD Relay Channel, m → ∞ ) : All re- lays in the c ascade behav e accordin g to the ternary model. Due to the Markov proper ty o f the channel inputs, th e joint pm f p ( x [0: m ] ) is co mpletely characterized by p ( x [0:1] ) , p ( x [1:2] ) , . . . , p ( x [ m − 1: m ] ) . Furth er , H ( Y i | X i ) = H ( X i − 1 | X i ) , which follows fr om (1). The id ea is now to find a probab ility assignment such that the p ( x [ i − 1: i ] ) are eq ual f or all i ∈ { 1 , 2 , . . . , m } without violatin g any o ptimality req uiremen ts. If we can find such a pr obability assignment, ca pacity simply follows by maximizing a single H ( X i − 1 | X i ) for th at p artic- ular assignme nt. W e now pick an arb itrary positive integer i and try to make p i − 1 kl and p i kl equal fo r all combinations k , l ∈ { 0 , 1 , N } . By the same argume nts as in Example 1, we c an choose w .l.o.g. p i − 1 00 = p i − 1 01 = p i − 1 10 = p i − 1 11 = 0 , and the same is valid for p ( x [ i : i +1] ) . As a simple con sequence, p i − 1 N 0 = p i 0 N and p i − 1 N 1 = p i 1 N and, from a symmetr y argument, p i − 1 N 0 = p i − 1 N 1 and p i 0 N = p i 1 N . Fu rther regardin g our ob jectiv e, we have to require that p i − 1 kN = p i kN for k ∈ { 0 , 1 } . Since index i has been p icked arbitrar ily at the beginnin g, the procedur e is valid for each p ( x [ i − 1: i ] ) and p ( x [ i : i +1] ) , 1 ≤ i ≤ m − 1 , wh at is sufficient in order to achiev e equal p mfs with a co mmon, single degree of freedom ( e. g. p i 0 N ) . Hence, H ( X i − 1 | X i ) , 1 ≤ i ≤ m , is easy to optimize yielding H ( X i − 1 | X i ) = 1 bit achieved at p i 0 N = 1 6 . The cap acity C is, therefore, equ al to 1 bit per channel use. Remark 2: Application of the binary HD relay model yields C = 0 . 5 bits per chann el use for all r elay cascades co mposed of two or more bin ary HD relays. Therefo re, th e op timum transmission strategy is just a straightforward time-sh aring approa ch. The reason lies simply in th e fact that the relays cannot en code parts of their information by means of the slot allocation since the subsequen t relay is not able to r ecognize when nothing (i. e. symbol N) was sent. Theor em 2: The ra te region of the re lay n etwork de fined above with tw o so urces, na mely source node 0 an d relay node r , is char acterized by R 0 ≤ max p ( x [0: m ] ) min { H ( Y i | X i ) : 1 ≤ i ≤ m } (5) R r ≤ max p ( x [0: m ] ) min  H  Y i | X r − 1 , X ( i : i ≤ m )  : r + 1 ≤ i ≤ m + 1  (6) and ( 7 ) shown at th e bottom of the pag e. The maximization of the equations is performed jo intly regarding p ( x [0: m ] ) . Pr oof: The proo f is g iv en in the App endix. Example 3 (HD Single Relay Network with T wo Sources): The ternary relay network co nsidered her e is characteriz ed b y m = 1 and r = 1 and together with (1), Theorem 2 becomes R 0 ≤ H ( X 0 | X 1 ) (8) R 1 ≤ H ( X 1 | X 0 ) (9) R 0 + R 1 ≤ H ( X 1 ) . (10) An outer bou nd on the rate region of the con sidered line network is obviously g i ven by R 0 + R 1 ≤ log 2 3 bits (Fig. 2, graph (a)) since th e sum-rate can n ev er b e larger than the maximum o f H ( X 1 ) . W e first try to de termine wh ether po ints on this outer bound , besides ( R 0 , R 1 ) = (0 , log 2 3) bits, are delivered by e quations (8) to (1 0) what ine vitably requires a uniform p X 1 ( x 1 ) . Since H ( X 0 | X 1 ) has to b e smaller or equal to H ( X 1 ) , we are allowed to assume equ ality in ( 8) what f ollows from Theor em 1. By mak ing the sam e op timality assumptions regarding p ( x [0:1] ) as in Example 1, we get R 0 = 1 3 log 2 3 b its and, consequently , R 1 ≤ 2 3 log 2 3 bits. Note that this value for R 1 does not contrad ict w ith (9), i. e. it is smaller than H ( X 1 | X 0 ) concernin g the assumed input distribution. The obtained p oint lies on the outer bound and it f ollows from a tim e-sharing argumen t that all points on th e line b etween (0 , log 2 3) b its an d ( 1 3 log 2 3 , 2 3 log 2 3) b its are part of the rate region bo und characterized b y (8) to (10). In th e sequel, we maintain the optimality assump tions r e- garding p ( x [0:1] ) an d focu s on the remaining interval 1 3 log 2 3 < R 0 ≤ 1 . 1389 bits, where 1 . 1389 bits is the capacity of a single tern ary HD r elay ch annel (E xample 1). Again, R 0 = H ( X 0 | X 1 ) but now p X 1 ( x 1 ) is not u niform anymore (due to R 0 > 1 3 log 2 3 ) yielding a su m-rate strictly smaller than log 2 3 bits. An upper bound on R 1 is gi ven by H ( X 1 ) − R 0 . It remains to ch eck wheth er this expression is smaller or eq ual to the right hand side of (9 ) in the con sidered inter val for R 0 for the assumed input d istribution. Howe ver, th is is satisfied and, therefore, the com plete u pper b ound o n the r ate region accordin g to (8)- (10) is char acterized by R 1 ≤ ( log 2 3 − R 0 , 0 ≤ R 0 ≤ 1 3 log 2 3 H b  R 0 log 2 3  +  1 − R 0 log 2 3  − R 0 , 1 3 log 2 3 < R 0 ≤ C, where H b ( · ) denotes the binary entropy function and C = 1.138 9 bits per channel use. A graphical r epresentation is gi ven in Fig. 2, graph ( b). I V . C O D I N G S T R A T E G I E S A. Achievability of C in Theorem 1 A coding strategy is presented capable of achieving C in Theorem 1 . As it is stand ard in achiev ability pro ofs, blocks of transmissions are used such that in B b locks a sequence of B − m ind ices w 0 ∈ W 0 is sen t from the source to the destination. As B → ∞ , the rate R 0 ( B − m ) B → R 0 . The ide a beh ind the coding strategy is th e following. Based on the feedf orward proper ty o f the considered line network and due to the fact that each n ode is aware of the enco ding strategy used by node s with larger ind ices, n ode i , 0 ≤ i ≤ m , knows at each time instance the cod ew ord, which will be sen t b y nodes l > i in the up coming transmission blo ck. Thus, eac h nod e is able to adapt its transmission to the codew ord chosen by the next node what can be exploited in order to prevent that concurr ently sent codewords of adjacen t nodes occupy the same time slots with binary symbols { 0 , 1 } . Different te chniques for encodin g ar e used by the source and the ternary r elays. While th e source utilizes a ternary alphab et { 0 , 1 , N } for enc oding, the r elays represen t their messages by a combinatio n of binary symbols { 0 , 1 } and the allocation of binary symb ols to the slots of a transmission b lock. Let n i denote the nu mber of binary sym bols used by r elay i du ring a single transmission block. Th en, at most 2 n m  n n m  indices can be encoded by relay m where 2 n m denotes the number of distinctive in dices when th e binar y symbols are located at fixed slots while  n n m  denotes the n umber of possible slot allocations. Due to the half- duplex constra int, the effecti ve codeword length of relay m − 1 reduc es to n − n m . Th is results from th e fact that relay m cannot pay atten tion to r elay m − 1 when relay m sen ds binary symbols and, therefore, the number of indice s, encodable by relay m − 1 , is at most 2 n m − 1  n − n m n m − 1  . R 0 + R r ≤ max p ( x [0: m ] ) min  min { H ( Y i | X i ) : 1 ≤ i ≤ r − 1 } + min { H ( Y k | X ( r − 1: r − 1 ≥ 1) , X ( k : k ≤ m ) ) : r + 1 ≤ k ≤ m + 1 } , H ( Y m +1 )  (7) The same a rgumentation holds for each r elay in th e chain , i. e. relay i , 1 ≤ i ≤ m , is able to encode at most 2 n i  n − n i +1 n i  indices p er tr ansmission block wher e n m +1 = 0 sinc e the sink node listens all the time. Finally , the e ffecti ve length of the source codeword is n − n 1 what enables th e source to enc ode a maximum of 3 n − n 1 indices. The rate R = n − 1 log 2 |W 0 | is R 0 = min  n − n 1 n log 2 3 , n i n + 1 n log 2  n − n i +1 n i  : ∀ i  , (11) where 1 ≤ i ≤ m . Codeboo k Con struction: The source and all relays gene rate codewords according to the scheme d escribed in the previous paragra ph. Let w i ∈ W 0 indicate a m essage index forwarded by re lay i , and let s i ∈ S i denote a p articular slot allocatio n used by r elay i for encoding indices w i . Note that each s i consists of n − n i +1 slots, which can be embedded in at most  n n i +1  ways into a block of length n whereas the emb edding is a function of the concurre ntly used s i +1 , . . . , s m . The resulting slot allocations of length n , e mployed by relay i , are deno ted as z i ∈ Z i and dep end on s i , . . . , s m . Th e pr ocedure works as follows. Fix |W 0 | relay m codewords x n m ( w m ) . For each slot allocatio n z m used in rela y m co dew ords, construct |W 0 | relay m − 1 codewords x n m − 1 ( w m − 1 , z m ) . This ensures that relay m − 1 can encode eac h message w m − 1 indepen dently of the slot allocatio n used by relay m . The pro cedure repeats and, finally , for each slot allocation z 1 used in relay 1 codewords, construct |W 0 | source codewords x n 0 ( w 0 , z 1 ) . Encodin g (at the en d of block b − 1 ): Let w ( b ) 0 ∈ W 0 denote the new m essage chosen by the sour ce to be sent in block b , and let ˆ w ( b ) i ∈ W 0 denote the estimate of w ( b − i ) 0 made b y relay i at the end of block b − 1 . Further, ˆ s ( b ) i , which is a function of ˆ w ( b ) i , cor respond s to the slot allocatio n used by relay i in tran smission block b fo r encoding ˆ w ( b ) i whereas ˆ z ( b ) i is determined by ˆ s ( b ) i , . . . , ˆ s ( b ) m . Relay n ode m sends x n m ( ˆ w ( b ) m ) in b lock b . Since r elay nod e i , 1 ≤ i ≤ m − 1 , knows all previously sent ind ices ( ˆ w ( b − 1) i , ˆ w ( b − 2) i . . . ) , which equal ( ˆ w ( b ) i +1 , ˆ w ( b ) i +2 , . . . ) , it knows ˆ z ( b ) i +1 and en codes its latest index ˆ w ( b ) i with x n i ( ˆ w ( b ) i , ˆ z ( b ) i +1 ) . Similarly , the sou rce ch ooses x n 0 ( w ( b ) 0 , ˆ z ( b ) 1 ) for transmission in block b . Decoding (at the end of block b − 1 ): At the end of block b − 2 , relay i h as estimates ( ˆ w ( b − 1) i , ˆ w ( b − 2) i , . . . ) and, therefore, estimates of ( ˆ s ( b − 1) i , ˆ s ( b − 1) i +1 , . . . ) and of ˆ z b − 1 i . Th en, based on the received sequen ce x n i − 1 ( ˆ w ( b − 1) i − 1 , ˆ z ( b − 1) i ) du ring block b − 1 and due to the knowledge of the codebook used b y relay i − 1 , r elay i is able to determin e the u nknown index ˆ w ( b − 1) i − 1 . The destination knows the co deboo k used by r elay m and upon receiving x n m ( ˆ w ( b − 1) m ) , it can determ ine ˆ w ( b − 1) m . Both the co deboo k co nstruction and the n oise freedo m of the relay cascade guaran tee, that the decod ing steps can be perfo rmed with zero-err or pro bability . Achievability: Using the relation n − 1 log  n m  = H b  m n  [11, Th. 1.4.5] as n → ∞ , optimality assumptions regarding p ( x [ i : i +1] ) (sym metry , zer o pro babilities - see Examp le 2), the resultant identities n i n = p i 0 N + p i 1 N and n − n i − n i +1 n = p i NN , we 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 P S f r a g r e p l a c e m e n t s (a) (b) (c) R 0 (bits per use) R 1 (bits per use) Fig. 2. A single ternary HD relay channel with two sources is considered. (a) Bound due to s ingle source capaciti es. (b) Upper bound due to T heorem 2. (c) Region due to the coding strat egy with bloc k len gth n = 640 . obtain n i n + 1 n log 2  n − n i +1 n i  − → H  X i | X ( i +1: i +1 ≤ m )  , where 1 ≤ i ≤ m . Accord ing to th e model in (1 ), H ( X i | X i +1 ) = H ( Y i +1 | X i +1 ) what shows th at each entry in (1 1), except for the first, conv erges to the correspon ding entry in (3). T he first entry in (11) c orrespon ds to a sourc e, which uses unif ormly distributed in put sy mbols w hen relay 1 is listening. E valuation of H ( Y 1 | X 1 ) regarding a unifo rm p X 0 | X 1 ( x 0 , N ) yields p X 1 ( N ) log 2 3 . Hence, the first entry in (3) equals the first entry in ( 11). Remark 3: At this point, we are able to justify wh y it has been without loss of op timality to impose the Ma rkov property on the chan nel inputs. Assume that each pair o f ch annel inputs is statistically depend ent gi ven all remaining inputs. Th en the proced ure regardin g Theo rem 1 , as shown in the Ap pendix, yields max min { H ( Y i | X [ i : m ] ) , H ( Y m +1 ) : 1 ≤ i ≤ m } as simplified cut-set bou nd what is smaller or at mo st equal to the achiev able ra te. But since the cut-set bo und is an outer bound, only equality is valid, achiev ed e. g. by X 0 → · · · → X m . For non-Markovian inpu ts, the rate region bound as stated in Theorem 2 is still an upp er b ound (but eventually looser) . The Markov prop erty merely cancels conditional r andom variables from the entropies what does not r educe the region. B. Coding Strate gy for a HD Re lay Cascade with T wo S our ces A coding schem e based on similar ideas can be derived for a line network whe re a secon d relay n ode r intends to tran smit own info rmation. T wo main poin ts h av e to be co nsidered: • Relay sou rce r and all subsequen t relay nodes must be able to e ncode |W 0 | · |W r | different indice s since W 0 and W r are independe nt. • The slot allocations z r ∈ Z r , a pplied by re lay sour ce r , are completely determined by the sou rce indices w 0 . Theor em 3: Consider a terna ry single HD rela y chan nel where both source and relay send own infor mation. T he bound, described by equations (8 ) to (1 0), is achievable pr ovided that the source rate exceeds a thresho ld. Pr oof: Let t n 1 and (1 − t ) n 1 denote the numb er of binary symb ols used by the relay for enco ding e ach w 0 and w 1 , resp ectiv ely , where 0 ≤ t ≤ 1 . Further, all possible slot allocations of the relay represent in dices w 0 . If the number of source in dices m atches the num ber of relay codew ords fo r representin g source indices, or expressed in R 0 n − n 1 n log 2 3 = t n 1 n + 1 n log 2  n n 1  , 0 ≤ t ≤ 1 , (12) the cut-set bo und is achievable. Note that the lhs of (12) equals p X 1 ( N ) log 2 3 what in turn equ als H ( X 0 | X 1 ) , assumed the same p ( x [0:1] ) is used than in E xample 1. Furth er , R 1 = (1 − t ) n 1 n − 1 . As n → ∞ , R 0 + R 1 → H ( X 1 ) wha t results from [11 , Th. 1.4.5 ] u nder consider ation of the rhs of ( 12). The m inimum R 0 (threshold ) follows from (12) f or t = 0 . V . A P P E N D I X Pr oof of Theor em 1: An upp er b ound o n the capacity of each single sour ce-destination network with sou rce 0 and sink node m + 1 is given by [12, Th. 14.10 .1] C ≤ max p ( x [0: m ] ) min S ∈M I ( X 0 , X S c ; Y S , Y m +1 | X S ) , (13) where M = P ( { 1 , . . . , m } ) and S c is the comp lement of S in { 1 , . . . , m } . In case of a noise-free n etwork, (13) becom es C ≤ max p ( x [0: m ] ) min S ∈M H ( Y S , Y m +1 | X S ) . (14) Let S be no nempty and let l ∈ { 1 , . . . , m } denote the smallest integer in S . Then H ( Y S , Y m +1 | X S ) ( a ) ≥ H ( Y l | X S ¯ l , X l ) + H ( Y S ¯ l | X S ¯ l , X l , Y l ) ( b ) = H ( Y l | X l ) + H ( Y S ¯ l | X S ¯ l , X l , Y l ) ( c ) ≥ H ( Y l | X l ) , (15) where ( a ) follows fro m th e ch ain rule an d ( b ) f rom X S ¯ l → X l → Y l . Equality in ( a ) and ( c ) is achieved by the ascending index sets S = { l , l + 1 , . . . , m } , 1 ≤ l ≤ m , which compose the entries of a set say M a . Hence, for each S ′ ∈ M\{∅ } there exists an S ∈ M a such that H ( Y S , Y m +1 | X S ) ≤ H ( Y S ′ , Y m +1 | X S ′ ) . T ake e. g. S ′ = { l , l + v } , where 0 ≤ v ≤ m − l , a nd extend it to an ascending index set S = { l, l + 1 , . . . , m } . The claim , stated in th e sentence bef ore the last, holds. In summary , (15) yields the first m en tries in (3) whereas the remain ing entry , H ( Y m +1 ) , follows when S in (14) is replaced by th e empty set. Pr oof o f Theor em 2: The deriv ation of the individual rate boun ds is alm ost along the same lines as in the pro of of Theorem 1. Hence, we concentrate o n the sum-rate bound . An u pper boun d on the sum-rate of ea ch network with tw o sources 0 and r and a sink m + 1 is [1 2, Th. 14.10.1] R 0 + R r ≤ max p ( x [0: m ] ) min S ∈M I ( X 0 , X r , X S c ; Y S , Y m +1 | X S ) , (16) where M is the p ower set of M d ∪ M u := { 1 , . . . , r − 1 } ∪ { r + 1 , . . . , m } . Note th at the rhs of (1 6) simplifies to th e rhs of (14) d ue to the a ssumed no ise freedom. Let S d ∈ P ( M d ) and S u ∈ P ( M u ) where S = S d ∪ S u . First let S d and S u be n onempty , i. e. M ′ := P ( S ) ⊂ M . Further, let i and j be the minimum and m aximum values in S d whereas k d enotes the minimum v alue in S u . Then H ( Y S , Y m +1 | X S ) ( a ) ≥ H ( Y i , Y k | X S ) + H ( Y S d ¯ i , Y S u ¯ k | X S , Y i , Y k ) ( b ) = H ( Y i , Y k | X i , X j , X k ) + H ( Y S d ¯ i , Y S u ¯ k | X S , Y i , Y k ) ( c ) ≥ H ( Y i | X i ) + H ( Y k | X r − 1 , X k ) , (17) where ( a ) follows from the c hain r ule, ( b ) from ( X S d ¯ i , X S u ) → X i → Y i and ( X S d ¯ j , X S u ¯ k ) → ( X j , X k ) → Y k , and ( c ) from applying chain rule to the first term in ( b ) under consideratio n of ( X j , X k ) → ( X r − 1 , X k ) → Y k together with the d escribed Markov relation s. Equa lity in ( a ) and ( c ) is ach ie ved by the ascend ing sets S d = { i, i + 1 , . . . , r − 1 } , 1 ≤ i ≤ r − 1 , and S u = { k , k + 1 , . . . , m } , r + 1 ≤ k ≤ m , which compo se the entr ies S = S d ∪ S u of a set say M a . Then for each S ′ ∈ M ′ there exists a S ∈ M a such that H ( Y S , Y m +1 | X S ) ≤ H ( Y S ′ , Y m +1 | X S ′ ) . T ake e. g. S ′ = { i, i + v } ∪ { k , k + w } , where 0 ≤ v ≤ r − 1 − i and 0 ≤ w ≤ m − k , and extend S ′ to an ascending ind ex set S = { i, i + 1 , . . . , r − 1 } ∪ { k , k + 1 , . . . , m } . The inequality relation holds. In summary , th e procedure yields min { H ( Y i | X i ) : 1 ≤ i ≤ r − 1 } + min { H ( Y k | X r − 1 , X k ) : r + 1 ≤ k ≤ m } in (7), what follows from (17) taking into acc ount all com- binations o f i and k . The last en try in ( 7) an d th e modified version o f above equa tion in ( 7) result when, in addition, the sets S ∈ M\M ′ are considered ( S d , S u empty or both). R E F E R E N C E S [1] E. 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