Incomplete decode-and-forward protocol using distributed space-time block codes

1 Incomplete decode-and-forw ard protocol us ing distrib uted space-time block codes Charlotte Hucher 1 , Ghaya Rekaya -Ben Othman 1 and Ahmed Saadani 2 1 Ecole Nationale Superieure des T elecommunications, Paris 2 France T elecom Research&Dev eloppement Email: { hucher ,rekaya } @enst.fr , ahmed.saadani@orange-ftgroup.com Abstract In this work, we explor e the introduction of distributed space-time codes in decod e-and-f orward (DF) protoco ls. A first proto col named the Asymmetr ic DF is presented . It is based on two p hases of different lengths, define d so that sign als can be fully decoded at relays. This strategy brin gs full diversity but the symbo l rate is not optim al. T o solve this problem a second pro tocol named the Incomp lete DF is defined. It is based o n a n incom plete dec oding at th e r elays redu cing the length of the first p hase. Th is last strategy brin gs bo th fu ll diversity and full symbol rate. Th e outage pr obability an d the simula tion results show that the Incom plete DF has better p erform ance than any existing DF pro tocol and than the n on-or thogon al amplify-and-f orward (N AF) strategy using the same space-time codes. M oreover the div ersity-m ultiplexing g ain tradeoff (DMT) of this new DF protocol is proven to be the same as the one of th e NAF . Index T erms cooper ati ve diversity , relay chan nel, decode- and-fo rward (DF) , space-time blo ck codes (STBC) I . I N T RO D U C T I O N Div ersity techni ques h a ve been dev eloped in order to combat fading on wireless channels. Recently , a ne w di versity technique has been proposed with cooperative systems [1], [2]. Dif ferent nodes in the network cooperate in order to form a MIMO system array and exploit space- time di versity . Cooperation proto cols ha ve been classified i n three main families: amplify-and- forward (AF), decode-and-forward (DF), and com press-and-forward (CF). October 28, 2018 DRAFT 2 DF protocols requi re more processing than AF ones, as the signals have t o be decoded at relay b efore being forwa rded. Ho we ver , if signals are correctly decoded a t relays, performance are better t han thos e of AF prot ocols, as nois e is deleted. Moreover , in thi s paper , our work is mot iv ated by the potential advantages of DF protocols over AF protocols in some scenarios. For example, it has been proven in [3] that in a multih op context i t is necessary t o use a DF p rotocol at some relays t o re g enerate the signals. I ndeed a full AF st rategy would add more noise at each hop, which m akes signals no longer decodable. There are few propo sed DF protocols in l iterature. They us ually do not succeed to bring both full d iv ersity and full symbol rate. The L TW DF (named by its authors Laneman, Tse and W ornell [4]) has a full diversity order but a rate of 1 2 symbol per channel use (sym b . pcu). The NBK DF (named by its authors Nabar , Bolcskei and Kneubuhler [5]) has a rate of 1 symb. pcu but no diversity . Indeed, as signals hav e to be ful ly decoded at relays, t he first phase of the transmission needs 1 c hannel use for each information sy mbol. T w o dif ferent cases c an be implemented in the second ph ase: either th e source sends new sym bols to have a rate of 1 symb . pcu, b ut di versity is los t (these ne w symbols not being relayed); or the so urce sends the same symbols or a combinatio n of them to hav e div ersity , but the rate drops. The only propos ed solution to this problem is the Dynamic DF (DDF) protocol [6] which succeeds to bring both full diversity and a rate of 1 symb . pcu. Ho wev er its implementati on is quite complex and an usable DDF w as not proposed. T o define a DF protocol wi th both full rate and full diversity , we suggest to introduce distributed space-time block codes (STBC) in the same way they hav e been successfully used in AF strategies, and in particular with the non-orthogonal AF (N AF) [6], [7], as well as in t he Alamouti DF prot ocol [8]. W e first present a DF protocol with asym metric send ing and relayi ng phases. It brings full di versity , but the rate is only 2 3 symb . pcu. T o sol ve this problem of low symbol rate, we define an Incomplete DF based on an incom plete decodi ng at the relays . This protocol brings both full rate and full d iv ersity . Outage probabilities calculations and simulation results ha ve been conducted to v alidate these approac hes and to prove that Incomplete DF has better performance than any existing DF prot ocol and than the N AF using the same STBC. Moreover a theoretical s tudy sho ws that the Incomp lete DF has the same div ersity-mult iplexing gain tradeoff (DMT) than the NAF . October 28, 2018 DRAFT 3 I I . S Y S T E M M O D E L A N D N OT A T I O N S W e consider a wireless network with N + 1 sources and one destination . As the channel is shared in a TDMA manner , each user is allocated a different time slot , and the system c an be reduced to a relay channel with one source, N relays and one destination. The N + 1 sources will play th e role of the source in successio n, while the ot hers will be used as relays. The channel links are assumed to be Rayleigh di stributed and slow fading, so their coef ficients can be considered as constant du ring t he transmissio n of at least one frame. Besides, we suppose a symmetric scenario, i.e. all the channel links are subject to the same a verage signal-to-noise ratio (SNR). As this work focuses on the prot ocol, for sim plicity , a uniform energy distribution is assumed. Considered terminals are half-dupl ex; t hey cannot recei ve and transmit at the same tim e. They are equip ped with only one antenn a; the MIM O case i s not consid ered in thi s work. In the next sections, notation giv en on figure 1 will be u sed. The channel coeffi cient of the link between source S and desti nation D is g 0 , the one between s ource S and relay RS n , n ∈ { 1 , . . . , N } , is h n and the one between relay RS n and desti nation D is g n . There is no channel state information (CSI) at the source, t he destinati on is s upposed t o know all the channel coefficients g n , which is necessary for t he decoding of the in formation, and each relay RS n is assumed to kno w its corresponding source-relay channel coef ficient h n . In the paper , foll owing notati on are used. Boldace lo wer case letters v denote vectors. Boldface capital lett ers M denote m atrices. M † denote the transpose con jugate of m atrix M . P r stands for a probability . R , C , Q and Z st ands for the real, complex, ration al and integer field respectively . For each algebraic num ber field K , the ring of integers is denoted O K . I I I . T H E A S Y M M E T R I C D F P ROT O C O L The Asymmetric D F i s a first approach to the introduction of dis tributed space-time codes in DF p rotocols. It is com posed of 2 ph ases of dif ferent l engths. During the first phase, the source sends all the information sym bols in a non-coded manner , in order for t he r elays to be able to easily decode them . The space-time codew ord is then reconstructed in t he second phase. This protocol is t o be associated with a 2 N × 2 N algebraic ST code. October 28, 2018 DRAFT 4 A. T ransmission s cheme Let’ s consi der the 2 N × 2 N algebraic S T code C wh ich can be either a Threaded Algebraic Space-T ime (T AST) code [9], a perfect code [10 ] or quasi-perfect codes [11]. Th is f amilies of codes have a code word wh ich can be written in the following form X =           x 1 x 2 . . . x 2 N − 1 x 2 N γ σ ( x 2 N ) σ ( x 1 ) . . . σ ( x 2 N − 2 ) σ ( x 2 N − 1 ) . . . . . . . . . . . . . . . γ σ 2 N − 2 ( x 3 ) γ σ 2 N − 2 ( x 4 ) . . . σ 2 N − 2 ( x 1 ) σ 2 N − 2 ( x 2 ) γ σ 2 N − 1 ( x 2 ) γ σ 2 N − 1 ( x 3 ) . . . γ σ 2 N − 1 ( x 2 N ) σ 2 N − 1 ( x 1 )           =           l 1 l 2 . . . l 2N − 1 l 2N           (1) where the x k , k ∈ { 1 , . . . , 2 N } , are elements of the ring of integers O K of K , a c yclic extension field of Q ( i ) of dim ension 2 2 N (the X k are linear combinations of 2 N informatio n symbol s), σ is the generator of the Gallois group K / Q ( i ) and γ is an element of either K or Z ( i ) used to separate the layers of t he codew ord. Overall 4 N 2 information symbol s are s end in the codeword. Let’ s call l k , k ∈ { 1 , . . . , 2 N } , the l ines of t he codew ord matrix . The transmissio n frame for a N -relay channel is described in figure 2. The t ransmission of one frame lasts 2 N × 2 N + 2 N × N = 6 N 2 channel uses. There are two main phases: during the first one, which lasts 2 N × 2 N = 4 N 2 channel uses, the source sends the 4 N 2 symbols and the N relays li sten. During the second phase, which lasts 2 N × N = 2 N 2 channel uses, the source sends the N last li nes of th e codeword, while the N relays send the reconstructed version of the N first lines. Relay RS n , n ∈ { 1 , . . . , N } , sends the recoded version of the n th line e l n of the code word while source s ends the ( N + n ) th line l N + n . The dest ination keeps list ening du ring the whol e transmission . The symbol rate is t hen 4 N 2 6 N 2 = 2 3 symb . pcu. B. Selection between the Asymmetric DF protocols and the no n-cooperative case DF p rotocols assume that signals are correctly decoded at relays during the first phase of the transmissio n, which is obviously not always the case. That is why we ha ve to guarantee the first phase of the transmis sion to b e able to use DF protocols . In literature, a selection based on the source-relay links quality was m ade [12]. The us ed criterion is the outage probabil ity . Indeed, according to Shannon theorem, if the link between source and relay RS n , n ∈ { 1 , . . . , N } is in outage, no detection is possible at this relay without error . In the other case, the October 28, 2018 DRAFT 5 source-relay R S n link i s not in outage, detectio n is possi ble and we use a DF p rotocol assu ming that no e rror occurs at relay RS n . In our case, the outage ev ent of a source-re lay RS n link is defined by O =  log  1 + ρ | h n | 2  < 3 2 R  where ρ defined such that SNR ( dB ) = 10 log 10 ( ρ ) is the signal to noise ratio, R is the global spectral efficienc y , a nd so 3 2 R i s the s pectral efficienc y of the source-relay RS i link. Only relays that can decode correctly the sig nals (whose source-relay link is not in outage) are selected. If there are N u ≥ 1 of them selected, a DF protocol with N u relays is used, and if none of th em is g ood, we us e a n on-cooperativ e strategy . In practice, each relay can determine whether its source-relay link is in outage or not and send this i nformation to the desti nation. The destinati on then knows how many relays can b e used and so the scheme to be applied. The destinatio n broadcasts this i nformation to the other nodes of the network. This impl ementation aspect (channel estim ation and feedback) will not be detailed any more in this paper as we focus only on the protocol. C. P erf ormance of the Asymmetr ic DF protocol from si mulation r esults Simulation hav e been made t o compare t he performance of t he N AF and Asy mmetric DF schemes in the o ne-relay case. Both protocol s have been implement ed with a di stributed Golden code [13] and decoded with a sphere decoder . A more detailed presentation of the Gold en code is made i n subsection V I-A. The N AF is proposed i n [6]. The p rotocol is non-orth ogonal: the source and the relay transmit in the s ame time. I mplement ed with the distributed Golden code, the scheme is the follo wing: the source first sends coded signals αx 1 and αx 2 (defined i n equation (8)) while the relay listens. The source then sends the coded signal s iσ ( α ) σ ( x 2 ) and σ ( α ) σ ( x 1 ) while the relay forwards the receive d signals . The Asymmetric DF is implemented in t he way described in figure 2. The source first sends the information symbols s 1 , s 2 , s 3 and s 4 while the relay listens and decodes them. The source then sends t he coded signals iσ ( α ) σ ( x 2 ) and σ ( α ) σ ( x 1 ) while the relay send s the coded si gnals α e x 1 = α ( e s 1 + θ e s 2 ) and α e x 2 = α ( e s 3 + θ e s 4 ) reconstructed from the decoded information symbols. October 28, 2018 DRAFT 6 On figure 3 are represented th e frame error rates of the SISO, NAF and As ymmetric DF protocols as functi ons of th e SNR, for a spectral efficienc y o f 4 b its p cu. Even if the Asymmetric DF brings full di versity , there is a significant loss in p erformance (more th an 5 dB ) compared to the N AF protocol. This is due to the low rate (only 2 3 symb . pcu) of the DF . Due to this l oss, the Asymmetric DF protocol brings an adv antage on the non-cooperative prot ocol only for SNR greater than 35 dB, which makes it useless in most cases. I V . I N C O M P L E T E D F P ROT O C O L In order to solve the problem of low rates of the Asymmetric DF protocol, we define a ne w protocol named Incom plete DF . T o increase t he rate, th e first ph ase of t he t ransmission is shorten and the second phase is kept the same. The Incomplete DF proto col is also designed to be used with a 2 N × 2 N algebraic ST code, with N the number of relays. In the fol lowing, the general case is studied and two examples for the 1-relay and 2-relay ca ses are giv en. A. T ransmission s cheme Let’ s consider the same 2 N × 2 N algebraic ST code C to be implemented as in subsection III-A, which ca n b e for example a perf ect code or a T AST code. For the N -relay channel, the transmission frame i s defined as described in figure 4. It lasts 2 N × 2 N = 4 N 2 channel uses and is di v ided in tw o main phases: during t he first one, which lasts 2 N × N = 2 N 2 channel uses, the s ource sends the N first lines of the codew ord matri x in succession and the N relays listen. During the second phase, which also lasts 2 N × N = 2 N 2 channel uses, th e source s ends the N last li nes of the codew ord, while the N relays send the decoded version of the N first lines. Relay RS n , n ∈ { 1 , . . . , N } , sends th e decoded version of the n th line e l n of the code word while source sends the ( N + n ) th line l N + n . The destination keeps listening during the who le transm ission. The symbo l rate is t hen 4 N 2 4 N 2 = 1 symb . pcu. October 28, 2018 DRAFT 7 Recei ved signals at d estination can be e xpressed as i n a MIMO system:              y 1 . . . y N y N + 1 . . . y 2N              = √ ρ              g 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · g 0 0 · · · 0 g 1 √ 2 · · · 0 g 0 √ 2 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · g N √ 2 0 · · · g 0 √ 2                           l 1 . . . l N l N + 1 . . . l 2N              +              w 1 . . . w N w N + 1 . . . w 2N              , where ∀ k ∈ { 1 , . . . , 2 N } • y k is the k th array of leng th 2 N of the recei ved sign als, • l k is the k th line of t he considered codeword matrix as defined i n equatio n (1), • w k is an array of length 2 N of A WGN. The factor 1 √ 2 in the channel matrix comes from the power normalizatio n during the second transmissio n phase. As two terminals send i n each t ime slot, t hey ha ve to share t he resources. Reordering the receive d s ignals at desti nation we obtain the equi valent expression: Y eq = √ ρ H eq X eq + W eq (2) with H eq =        H 1 0 · · · 0 0 H 2 · · · 0 . . . . . . . . . . . . 0 0 · · · H N        (3) and ∀ n ∈ { 1 , . . . , N } H n =   g 0 0 g n √ 2 g 0 √ 2   . (4) Decoding at destinati on ca n b e performed by usin g ML lattice decoders such as a Schnorr- Euchner or a sphere decoder . B. P artial decoding at the r elays The challenge of the n e w transmi ssion scheme is decoding at relays. Indeed, the use of a full decode-and-forward strategy would mean that r elays hav e t o decode ever y information sy mbol s j , j ∈ 1 , 4 N 2 of our original cons tellation from only 2 N × N = 2 N 2 recei ved signals. October 28, 2018 DRAFT 8 The idea of the Incomplete DF i s t o estimate receiv ed signals as elements x k ∈ O K , k ∈ { 1 , . . . , 2 N } , witho ut s tating definitely about the information symbols s j , j ∈ { 1 , . . . , 4 N 2 } . Indeed, the knowledge of the s j is not necessary at relays, as soon as they know the signals x k that hav e to be forwar ded. P artial decoding at relays is suf ficient. The partial decoding will be more detailed and explained in the sequel by considering some examples. C. S election between the Incomp lete DF protocols and the non-cooperative case The same selection strategy as for the Asymm etric DF (described in sub section III-B) is used. Only the e x pressions of the o utage probabil ities of the so urce-relay lin ks change. Here the o utage probability of a source-relay RS n link, n ∈ { 1 , . . . , N } , is giv en by: P r O = P r  log(1 + ρ | h n | 2 ) < 2 R  (5) where R is the gl obal spectral effic iency . The spectral efficienc y of the source-relay link is twice since the sam e informati on is sent i n two times less channel uses. V . T H E O R E T I C A L S T U DY O F T H E I N C O M P L E T E D F P E R F O R M A N C E A. Outage pr obabil ity Outage probability is gi ven by the formula: P r out = P r { C ( H ) < R } with the instantaneous capacity C ( H ) = 1 T log det( I + ρ HH † ) where T is the number of tim e sl ots, ρ is the signal-t o-noise ratio, H is th e (equiv alent) channel matrix of the cons idered syst em and R is the spectral efficienc y . Theor em 1: The out age probability of t he Incompl ete DF is P r out = N X N u =0  N N u  P r out ,N u P r O ,N − N u . (6) where N is the number of relays in the network, N u is th number of relays whose source-relay link i s not in outage, P r out,N u is the out age probabi lity of the DF strategy with N u decoding October 28, 2018 DRAFT 9 relays and P r O , N N − N u is the probability that the source-relay links of the N − N u other relays are in outage. Pr oof : See Appendix A. W e plotted the outage probabilit ies thanks to Monte Carlo sim ulations. A relay selection has been added for all coooperative s chemes. For example, in the on e-relay case, the relay is chos en as the best of 3 reachable relays; in the two-relay case, relays are the two best ones between four reachable relays. In a first s tep the two best relays are s elected, and in a second step, the DF protocol determines which of these t wo relays can b e used (i.e. source-relay l ink not in outage) and chooses the corresponding strategy (SISO, Incomplete DF with 1 relay o r Incomplete DF with 2 re lays). The AF protocol alw ays us e both relays . Figure 5 represents the outage probabilities of the SISO, N AF and Incomplete DF protocols as functions of the SNR at spectral efficiencies of 2 and 4 bi ts per channel use i n the 1-relay case. W e can remark that the ne w DF brings a slight gain over the N AF . Moreove r and more interesting is the fact that due to selection it has good per formance at l ow SNR. Same remarks can be d one in t he 2-relay c ase. B. Diversity-Multiplexing gain T radeoff (DMT) The diversity-multiplexing gain tradeoff (DMT) has been introduced in [14] to e valuate the asymptotic performance of space-time codes. Definition 1: A di versity gain d ∗ ( r ) is achie ved at a multiplexing gain r i f lim ρ →∞ log P r out ( r log ρ ) log ρ = − d ∗ ( r ) . Theor em 2: The DM T of the Incom plete DF is d ∗ ( r ) = (1 − r ) + + N (1 − 2 r ) + , (7) where N i s the nu mber of relays i n the network. Pr oof : See Appendix B. On figure 6 is represented the DMT for the 2-relay case. One can remark that the DMT of the Incomplete DF protocol is exactly the same as the one of the N AF protocol, outperforming the ones of the L TW and NBK DF protocols. The DMT of the DD F protocol is still better , but Incomplete DF im plementation is m uch easier . October 28, 2018 DRAFT 10 V I . E X A M P L E S O F I N C O M P L E T E D F I M P L E M E N T A T I O N A N D S I M U L A T I O N R E S U LT S A. 1-r ela y channel with the Gol den code The Golden code is an algebraic code designed for a 2 × 2 MIMO system in [13] based on the cyclic division algebra of dimensi on 2, A = ( Q ( i, θ ) / Q ( i ) , σ , γ ) , where θ = 1+ √ 5 2 is the Golden number , σ : 1+ √ 5 2 7− → 1 − √ 5 2 and γ = i . A codew ord is gi ven by X =   α ( s 1 + θ s 2 ) α ( s 3 + θ s 4 ) iσ ( α )( s 3 + σ ( θ ) s 4 ) σ ( α )( s 1 + σ ( θ ) s 2 )   with t he s j , j ∈ { 1 , . . . , 4 } being the information symbols t aken in a QAM constellatio n and α = 1 + i − iθ . The elements of the code matrix are in O K the ring of integers of the numb er field mathb bK = Q ( i, θ ) . Let’ s note them x 1 = s 1 + θ s 2 and x 2 = s 3 + θ s 4 . The code word is then: X =   αx 1 αx 2 iσ ( α ) σ ( x 2 ) σ ( α ) σ ( x 1 )   . (8) W e propo se to imp lement this space-time code in a di stributed manner using th e new In- complete DF protocol with 1 relay as described in subsection IV -A. The transmi ssion frame is described in figure 7. Elements x 1 and x 2 both cont ain two informatio n symbols. T hey hav e to be recover ed respec- tiv ely from the rece ived signals y r 1 and y r 2 . The idea of ”incom plete decoding” is to decode x 1 and x 2 as elements O K without st ating definitely on the information symbols . W e consider in this paper two decodi ng methods: • an e xhaustive search, • a diophantine approximation. a) Exhaustive sear ch: Let’ s assume the inform ation sym bols s j , j ∈ { 1 , . . . , 4 } , bel ong to a constellation C (for example a 4-QAM constellatio n, see figure 8). W e can define a new constellation C ′ to wh ich th e coded symbols x k , k ∈ { 1 , . . . , 2 } , belong (see figure 8), which is a finite subset of O K . An exhausti ve search is performed in thi s new const ellation. e x 1 (resp. e x 2 ) is obt ained by look ing for the element x of C ′ that minimi zes the dist ance between y r 1 (resp. October 28, 2018 DRAFT 11 y r 2 ) and √ ρh 1 x . e x 1 = ar g min x ∈ C ′ {| y r 1 − √ ρh 1 x | 2 } e x 2 = ar g min x ∈ C ′ {| y r 2 − √ ρh 1 x | 2 } The compl exity of the exhaustiv e search grows with the size of the cons tellation. In the case of an M -QAM the comp lexity of the exhaustiv e search is of th e order M 2 . Howe ver , decomposing signals in t heir real a nd imagi nary parts, com plexity can be reduced to t he order M . Simulations have been r un for the one-relay cooperati ve scheme with the distrib uted Golden code for spectral ef ficiencies of 2 and 4 bits per channel use. The s ame relay selection as for the out age probabil ity has been appl ied. Figure 9 represents the frame error rates of the SISO, NAF and new DF protocols as functi ons of the SNR. The goo d performance for low and high SNR noticed in subsection V -A on the outage probabil ity curves are confirmed here by si mulation results . Usin g an exhausti ve decoding , we obtain s light asym ptotic gains ov er the N AF prot ocol. M oreover , we can see (especially for 4 bits pcu) that th e proposed DF prot ocol has better performance at low SNR. b) Diophanti ne approximation: In order to reduce relay decoding com plexity , we propose to use a diophantine approximation of t he x k , k ∈ { 1 , . . . , 2 } . There exist two types of diophantine approximation. Definition 2: A homogeneous di ophantine approxim ation of ζ ∈ R i s a fraction p q ∈ Q su ch that | ζ − p q | or D ( p, q ) = | q ζ − p | is s mall. Definition 3: An inhomog eneous d iophantine approxi mation of ζ ∈ R , giv en β ∈ R , is a fraction p q ∈ Q such t hat D ( p, q ) = | q ζ − p − β | i s small. Definition 4: A pair ( p, q ) ∈ N 2 is a best diophantine approximation if ∀ ( p ′ , q ′ ) 6 = ( p, q ) ∈ N 2 , we have : q ′ ≤ q ⇒ D ( p ′ , q ′ ) ≥ D ( p, q ) . Cassels’ algorithm has been proposed in [15] and explained in details in [16]. Giv en ζ , β ∈ R , this algorithm enumerates all best inhomog eneous app roximations. A simple modification of this algorithm provides ( p, q ) in a finite set { 1 , . . . , Z } that m inimizes D ( p, q ) . A change o f b asis provides ( p, q ) in a Z -P AM. The modified algorithm has a com plexity o f the order √ Z . Diophantine approximati on only deals with real n umbers. The probl em of decoding at the relay has then to be divided int o i ts real and im aginary parts. Let’ s note e y r 1 = y r 1 √ ρh 1 α . Given October 28, 2018 DRAFT 12 θ , Re ( e y r 1 ) ∈ R , we want to find ( Re ( s 1 ) , Re ( s 2 )) ∈ √ M -P AM su ch that | Re ( e y r 1 ) − Re ( s 1 ) − θ Re ( s 2 ) | is minimized. T o solve thi s minimization we can use the modified Cassels’ algorith m wit h the parameters β = − ( Re ( e y r 1 ) + ( √ M + 1)(1 + θ )) / 2 and ζ = − θ . The fi nal algorithm is gi ven in appendix C. The s ame p rocessing is done to decode the imaginary part of the signal. Finally , the decodin g complexity is only p √ M = 4 √ M . If θ = e i π 4 the decomposit ion in real and imagi nary p art i s more complex, but t he diophantine approximation s till can be us ed with a slight mo dification of the given algo rithm. Thus the diophantine approximati on can also be applied when usin g a distributed T AST code. Simulations have also been run with the Gold en code using a diophantine approxim ation at th e relay . W e can see on figure 9 that in this case performance are slight ly worse. This is explained by the non-optim al decoding at relays. Howe ver , this sligh t gap in performance (only 0.5 dB) is compensated by a much lower decoding complexity decreasing from M to 4 √ M . B. 2-r ela y channel with the T AST code For the 2-relay case, we propo se to us e the 4 × 4 T AST code in a di stributed manner , associated to the Incom plete DF prot ocol. After recalling th e structure of t he T AST code, we wi ll introduce two partial decoding methods, and so justify our choice of the T AST c ode. T AST codes, introdu ced in [9], are layered space-time codes. Here we use the 4 × 4 T AST code constructed using the cyclotomic field K = Q ( i, θ ) , where θ = e i π 8 , the generator of t he Gallois group σ : θ 7− → iθ and φ = e i π 8 . The codeword is X =        x 1 x 2 x 3 x 4 φσ ( x 4 ) σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) φσ 2 ( x 3 ) φσ 2 ( x 4 ) σ 2 ( x 1 ) σ 2 ( x 2 ) φσ 3 ( x 2 ) φσ 3 ( x 3 ) φσ 3 ( x 4 ) σ 3 ( x 1 )        , where, ∀ k ∈ { 1 , . . . , 4 } , x k = s 4 ∗ k − 3 + θ s 4 ∗ k − 2 + θ 2 s 4 ∗ k − 1 + θ 3 s 4 ∗ k . W e propose to use this code in a 2-relay cooperati ve system as described in subsection IV -A. The transmi ssion scheme is schemat ized in figure 10. October 28, 2018 DRAFT 13 Elements x 1 , x 2 , x 3 and x 4 ∈ O K and their conjugates ha ve to be recov ered from the signals y r j 1 to y r j 8 recei ved at t he relay RS j , j ∈ { 1 , . . . , 2 } . W e propose here two dif ferent methods for the partial d ecoding. c) E xhaustive sear ch (dim ension 4): x k , k ∈ { 1 , . . . , 4 } and t heir conju gates σ ( x n ) can be decoded a t relays by an exhaustive search as in the 1-relay case. The dif ference is that x n and σ ( x n ) cannot be decoded separately as the y are conjug ates. Let’ s assum e the s j , j ∈ { 1 , . . . , 16 } , belong to a const ellation C . W e can define a n e w constellation C 1 to which the x k belong, and a correspon ding constellatio n C 2 to which th eir conjugates σ ( x k ) belong. Decoded ver sions of the x k and their conjugates σ ( x k ) are obtained by looking for the elements x of const ellation C 1 and σ ( x ) o f constell ation C 2 minimizi ng the d istance between √ ρh 1 x and the recei ved signal corresponding t o x k and the d istance between √ ρh 1 σ ( x ) and the receiv ed signal corresponding to σ ( x k ) . W e decide to mini mize the sum of these two dis tances. For example: { e x 1 , ] σ ( x 1 ) } = a rg min x ∈ C 1 ,σ ( x ) ∈ C 2 n   y r j 1 − √ ρh 1 x   2 +   y r j 6 − √ ρh 1 σ ( x )   2 o Howe ver , this exhausti ve decoding can be quite complex if a high constellatio n size is con - sidered. Indeed, if the inform ation sy mbols s j belong to a M -QAM con stellation ( M elements), then, t he x k hav e to be decoded in a n e w constellation of M 4 elements. The comp lexity of decoding is of the order M 4 . Simulations hav e b een run for the two-relay cooperative s cheme with a 4 × 4 p erfect code for spectral ef ficiencies of 2 a nd 4 bit s per channel use. Figure 11 represents th e frame error rates of the SISO, N AF and ne w DF protocols as fun ctions of t he SNR. The same remarks than in the one-relay case can be do ne. The Incomplete DF and the NAF hav e nearly the s ame p erformance (nearly zero asy mptotic gain), but due t o selection t he p roposed DF protocol outperforms t he N AF at low SNR. d) T wo steps e xhaustive decoding (dimensio n 2): A slight modi fication can reduce this exhaustiv e decoding in a constellatio n of M 4 elements t o two exhaustive decodi ngs in a con- stellation of o nly M 2 elements. October 28, 2018 DRAFT 14 W e can notice that x 1 and its second conjugate σ 2 ( x 1 ) can be rewritten in the form : x 1 = ( s 1 + θ 2 s 3 ) + θ ( s 2 + θ 2 s 4 ) σ 2 ( x 1 ) = ( s 1 + θ 2 s 3 ) − θ ( s 2 + θ 2 s 4 )   x 1 σ 2 ( x 1 )   =   1 θ 1 − θ   | {z } M   z 1 z 2   , (9) where z 1 = ( s 1 + θ 2 s 3 ) and z 2 = ( s 2 + θ 2 s 4 ) are elements of t he ring of int egers of the field Q ( e i π 4 ) of di mension 2 ov er Q ( i ) . As 1 √ 2 M is a rotat ion m atrix, a simple multiplication by M † allows to obtain z 1 and z 2 from x 1 and σ 2 ( x 1 ) . In order to take advantage of this property , the i dea is that the s ource sends the first and third lines of the codew ord matrix du ring t he first phase o f the transmissio n and the second and fourth lines during t he second ph ase of t he transmissi on. The partial decoding at relays is then done in two st eps. First we compute the matrix product   z ′ 1 z ′ 2   = 1 2 M †   y r k 1 √ ρh 1 y r k 6 √ ρh 1   . Then we decode elements z 1 and z 2 of the ring of integers of Q ( e i π 4 ) in an e xhaustive way as in the example of the subsection VI-A. Finally x 1 and its conjug ate σ 2 ( x 1 ) can be easily d educed from equatio n (9). This second method all ows to decrease considerably the complexity . Indeed, the exhaustive search i s now performed in a constellat ion of M 2 elements inst ead of M 4 , which is quit e reasonable. This s econd decoding method cannot be applied to 4 × 4 perfect codes whose structure do not ha ve t he same p roperty . That is why we have chosen to use T AST cod e. The adv antages of the diophantin e approximation and this two-step decoding method could be combi ned. Ideally the Incomplete DF would be the least complex i f used wi th a distributed STBC of fering both a structure allowing the two-step decoding and θ ∈ R for a simple diophantine approx imation. Simulations have also been run for the two-relay cooperativ e scheme with a 4 × 4 T AST code and the two-step decoding at the relays. One can see on figure 11 t hat T AST codes provide slightly w orse performance than perfect codes. This is explained by the fa ct that perfect codes October 28, 2018 DRAFT 15 are NVD (non-v anishing-determinant ), on the contrary of T AST codes. Howe ver , when we use two or more relays, the partial decoding of the information at relays induces more complexity , as the t wo-step decoding method described in subsection VI-B cannot be used. That is why the use of tw o-step decodable STBC such as the T AST codes is necessary . V I I . C O N C L U S I O N In this paper , we define a new DF proto col using distributed space-tim e codes that provides both full di versity and full rate, as the best kno wn AF protocols , and unl ike the existing L TW and NBK DF protocols. This ne w protocol is based on an incomplete decoding of the signal at the relays. The receiv ed signals at relays are decoded as element s of the ring of i ntegers of th e considered number fi eld without decoding the in formation symbols. Sev eral decoding methods are proposed at relays: e xhaustive search, dioph antine approximation or a m ethod based on th e decoding decom position in two steps according to th e code structure. The t wo last met hods allow a considerable decrease of complexity . The dive rsity-mul tiplexing gain tradeof f is proved to be the same as the one of the NAF protocol wh ich is the best known AF prot ocol. Besides outage probabil ity and simulati on resul ts prove that the Incompl ete DF giv es slightly better performance th an t he N AF protocol in the high SNR re g ime, and selection p rovides an improvement for low SNR. In th is s tudy , we have only considered cooperative schem es wi th l ine-of-sight, but the use of DF protocols can be very important in a non-line-of-sight or mul tihop cooperati ve scheme. T o highlig ht the advantages of thi s ne w DF protocol over an AF s trategy , applications of the Incomplete DF to a multiho p sys tem will b e in vestigated in future works. A P P E N D I X A. Pr o of of Theor em 1 T wo cases hav e to be distingui shed: with or with out coo peration. When N u ≥ 1 source-relay links are no t in outage ( N − N u relays are in outage) , the Incompl ete DF cooperation schem e with N u relays is u sed. From equation (3) we can write det( I + ρ HH † ) = N u Y i =1 det( I + ρ H i H i † ) October 28, 2018 DRAFT 16 and usin g equation (4) the o utage probabil ity can be wri tten P r out ,N u = P r n 1 2 N u log  Q N u i =1  1 + ρ 2 (3 | g 0 | 2 + | g i | 2 ) + ρ 2 2 | g 0 | 4  < R o = P r n log Q N u i =1  1 + ρ 2 (3 | g 0 | 2 + | g i | 2 ) + ρ 2 2 | g 0 | 4  < 2 N u R o . (10) As the outage e vents of the s ource-relay links are independent, t he probability of h a ving o nly the last N − N u relays i n outage, is the prod uct of th e probabi lities of each N u first source-relay links not being in outage, and each of the l ast N − N u source-relay links being in outage. T hese probabilities are giv en by expression (5). So the out age probabilit y of the last N − N u relays only can be written P r O ,N − N u = N u Y i =1 P r  log  1 + ρ | h i | 2  > 2 R  N Y i = N u +1 P r  log  1 + ρ | h i | 2  < 2 R  . (11) When all source-relay links are i n out age, we use the non-cooperative scheme, whose o utage probability is P r out , 0 = P r  log  1 + ρ | g 0 | 2  < R  (12) As the ou tage events of a ll source-re lay links are independent, the probability of thi s case is giv en by the produ ct of the outage probabiliti es of each source-relay li nk. P r O ,N = N Y i =1 P r  log  1 + ρ | h i | 2  < 2 R  (13) Finally , as t here are  N N u  possible comb inations of N u relays in N , we can write in the general case P r out = N X N u =0  N N u  P r out ,N u P r O ,N − N u . (14) B. Pr o of of Theor em 2 1) Pr eliminari es: Definition 5: Let g follow a Ra yleigh distrib ution. The exponential order of 1 | g | 2 is u = − lim ρ →∞ log | g | 2 log ρ . W e can not e | g | 2 . = ρ − u where the notation . = denotes an asymptotic beha vior when ρ → ∞ . Lemma 1: The probabil ity densit y function of u is p u = lim ρ →∞ log( ρ ) ρ − u exp( − ρ − u ) , October 28, 2018 DRAFT 17 which satisfies p u . =    ρ −∞ , for u < 0 ρ − u , for u ≥ 0 Lemma 2: Let O be a certain set and P O = P r { ( u 1 , . . . , u N ) ∈ O } , th en P O . = ρ − d with d = inf ( u 1 ,...,u N ) ∈O + N X j =1 u j where O + = O ∩ R N + Pr oof : The proof is drawn in [17]. 2) Pr oof of t he theor em: The o utage probability of th e Incomplete DF is given i n equation (6). In order to com pute the DMT of th is cooperative st rategy , we hav e to s tudy the asym ptotic beha vior of this expression when ρ grows to infinit y . Let u 0 , u n and v n , n ∈ { 1 , . . . , N } be the exponential o rders of 1 | g 0 | 2 , 1 | g n | 2 and 1 | h n | 2 respectiv ely . In the case of signals being correctly decoded at N u relays P r out ,N u = P r ( log N u Y i =1  1 + ρ 2  3 | g 0 | 2 + | g i | 2  + ρ 2 2 | g 0 | 4  < 2 N u R ) which asymptot ically becomes P r out ,N u . = P r ( N u X i =1 log  ρ 1 − v 0 + ρ 1 − v i + ρ 2 − 2 v 0  < 2 N u r lo g ρ ) . = P r ( N u X i =1 max ( 1 − v i , 2 − 2 v 0 ) < 2 N u r ) . = ρ − d out,N u ( r ) As P N u i =1 (1 − v i ) < 2 N u r give s N u (1 − 2 r ) < P N u i =1 v i and P N u i =1 (2 − 2 v 0 ) < 2 N u r give s 1 − r < v 0 , we obtain the div ersity-multi plexing gain tradeoff d out,N u ( r ) = inf v 0 + N u X i =1 v i ! = (1 − r ) + N u (1 − 2 r ) + . (15) This case occurs when N − N u of the source-relay links are in outage and the others are not, October 28, 2018 DRAFT 18 with the probability: P r O ,N − N u = N u Y i =1 P r  log  1 + ρ | h i | 2  > 2 R  N Y i = N u +1 P r  log  1 + ρ | h i | 2  < 2 R  . = N u Y i =1  1 − P r  log ρ 1 − u i < 2 r log ρ  N Y i = N u +1 P r  log ρ 1 − u i < 2 r log ρ  . = N u Y i =1 (1 − P r { 1 − u i < 2 r } ) N Y i = N u +1 P r { 1 − u i < 2 r } . = N u Y i =1  1 − ρ − (1 − 2 r )  N Y i = N u +1 ρ − (1 − 2 r ) . = 1 × ρ − ( N − N u )(1 − 2 r ) so the d iv ersity-mult iplexing gain tradeoff is d O ,N − N u ( r ) = ( N − N u )(1 − 2 r ) + . (16) In the case of all source-relay li nks being in outage: P r out , 0 = P r  log  1 + ρ | g 0 | 2  < R  . = P r  log ρ 1 − v 0 < r lo g ρ  . = P r { 1 − v 0 < r } . = ρ − d out, 0 ( r ) with the di versity-mu ltiplexing g ain tradeoff d out, 0 ( r ) = 1 − r . (17) This case occurs with the probability: P r O ,N = N Y i =1 P r  log  1 + ρ | h i | 2  < 2 R  . = N Y i =1 P r  log ρ 1 − u i < 2 r log ρ  . = N Y i =1 P r { 1 − u i < 2 r } . = N Y i =1 ρ − (1 − 2 r ) . = ρ − N (1 − 2 r ) October 28, 2018 DRAFT 19 so the d iv ersity-mult iplexing gain tradeoff is d O ,N ( r ) = N (1 − 2 r ) + . (18) Finally we can write P r out = N X N u =0 C N N u P r out ,N u P r O ,N − N u . = N X N u =0 C N N u ρ − d out,N u ( r ) ρ − d O ,N − N u ( r ) . = ρ − max N u ∈{ 0 ,...,N } ( d out,N u ( r )+ d O ,N − N u ( r )) so the t otal diversity-multiplexing gain tradeoff is d ( r ) = max N u ∈{ 0 ,...,N } ( d out,N u ( r ) + d O ,N − N u ( r )) = ( 1 − r ) + N (1 − 2 r ) + . (19) C. Mo dified Cassels’ algorithm for decoding Z-P AM Input : y , θ , Z Output : P , Q β = − ( y + ( Z + 1)(1 + θ )) / 2 ; α = − θ ; D min = ∞ ; η 0 = α ; η 1 = − 1; ζ = − β ; p 0 = 0; p 1 = 1; P 1 = 0 ; q 0 = 1; q 1 = 0; Q 1 = 0 ; October 28, 2018 DRAFT 20 while η n − 1 6 = 0 ∧ ζ n − 1 6 = 0 ∧ Q n − 1 ≤ Z do a n = ⌊− η n − 2 η n − 1 ⌋ ; p n = p n − 2 + a n p n − 1 ; q n = q n − 2 + a n q n − 1 ; η n = η n − 2 + a n η n − 1 ; if Q n − 1 ≤ q n − 1 then b n = ⌊− ζ n − 1 − η n − 2 η n − 1 ⌋ ; P n = P n − 1 + p n − 2 + b n p n − 1 ; Q n = Q n − 1 + q n − 2 + b n q n − 1 ; ζ n = ζ n − 1 + η n − 2 + b n η n − 1 ; else P n = P n − 1 − p n − 1 ; Q n = Q n − 1 − q n − 1 ; ζ n = ζ n − 1 − η n − 1 ; end P ′ = 2 P n − ( Z + 1) ; Q ′ = 2 Q n − ( Z + 1) ; D cur r = ( y − P ′ − θ Q ′ ) 2 ; if D cur r ≤ D min then P = P ′ ; Q = Q ′ ; D min = D cur r ; end n = n + 1 ; end R E F E R E N C E S [1] A. S endonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity . Part I. 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Clarkson, “ Approximation of linear forms by lattice points with applications to signal processing, ” Ph.D. dissertation, Australien National Unive rsity , 1997. [17] L . Zheng and D. N. C. Tse, “Div ersity and multiple xing: a fundam ental tradeoff in multiple-antenna channels, ” IE EE T rans. Inform. T heory , vol. 49, no. 5, pp. 1073–1 096, May 2003. October 28, 2018 DRAFT 22 RS 1 RS 2 RS N D S h 1 g N g 0 g 2 g 1 h 2 h N Fig. 1. System model : r elay channel with one source, N relays and one destination listen listen listen listen listen listen listen listen listen listen listen listen listen listen S D listen first phase second phase 1 time slot 2 N time slots 2 N time slots 2 N time slots s 1 L N +1 L 1 RS N RS 2 RS 1 1 time slot s 2 N × 2 N s 2 L N +2 L 2 N L 2 L N 1 time slot · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . Fig. 2. T ransmission frame of the Asymmetric DF protocol October 28, 2018 DRAFT 23 0 10 20 30 40 SNR (dB) 10 -4 10 -3 10 -2 10 -1 10 0 Frame error rate SISO NAF ADF Fig. 3. Frame err or rate of the SISO, N AF and Asymmetric DF protocols (both implemented with the Golden code) for 1 relay , 4 bits pcu listen listen listen listen listen listen listen listen listen listen listen listen listen S D listen first phase second phase listen 2 N time slots 2 N time slots 2 N time slots L N +1 RS N RS 2 RS 1 2 N time slots L N L 2 L N +2 L 2 N f L 2 f L N · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . L 1 2 N time slots 2 N t ime slots f L 1 Fig. 4. T ransmission frame of the Incomplete DF protocol October 28, 2018 DRAFT 24 0 10 20 30 SNR (dB) 10 -4 10 -3 10 -2 10 -1 10 0 Outage probability SISO - 2 bpcu NAF - 2 bpcu IDF - 2 bpcu SISO - 4 bpcu NAF - 4 bpcu IDF - 4 bpcu Fig. 5. Outage probabilities of the S ISO, N AF and ne w DF protocols as functions of the SNR at spectral efficiencies of 2 and 4 bits per channel use in the 1-relay case 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 N=2 relays multiplexing gain r diversity gain d MISO bound NAF,IDF NBK LTW DDF Fig. 6. DMT of several cooperation protocols October 28, 2018 DRAFT 25 S D first phase second phase RS y r 1 y r 2 y 2 y 1 y 3 y 4 X 12 X 34 g X 12 g X 34 iσ ( X 34 ) σ ( X 12 ) Fig. 7. T ransmission frame of the Incomplete DF protocol in the 1-relay case implemented with a distributed Golden code −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (a) Constellation C −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (b) Constellation C ′ Fig. 8. 16-QAM and Golden constellation October 28, 2018 DRAFT 26 0 10 20 30 SNR (dB) 10 -4 10 -3 10 -2 10 -1 10 0 Frame error rate SISO - 2bpcu NAF (GC) - 2bpcu IDF (GC) (exhaust dec)- 2bpcu IDF (GC) (dioph approx) - 2 bpcu SISO - 4 bpcu NAF (GC) - 4 bpcu IDF (GC) (exhaust dec) - 4 bpcu IDF (GC) (dioph approx) - 4 bpcu Fig. 9. Frame error rates of the S ISO, NAF and new DF protocols as functions of the SNR at spectral efficiencies of 2 and 4 bits per channel use in the 1-relay case S D first phase second phase X 1 y r 1 1 y r 2 1 y 1 X 2 y r 1 2 y r 2 2 y 2 X 3 y r 1 3 y r 2 3 y 3 X 4 y r 1 4 y r 2 4 y 4 φσ ( X 4 ) y r 1 5 y r 2 5 y 5 σ ( X 1 ) y r 1 6 y r 2 6 y 6 σ ( X 2 ) y r 1 7 y r 2 7 y 7 σ ( X 3 ) y r 1 8 y r 2 8 y 8 φ 2 σ 2 ( X 3 ) f X 1 y 9 φσ 2 ( X 4 ) f X 2 y 10 σ 2 ( X 1 ) f X 3 y 11 σ 2 ( X 2 ) f X 4 y 12 φ 3 σ 3 ( X 2 ) ^ φσ ( X 4 ) y 13 φ 2 σ 3 ( X 3 ) ^ σ ( X 1 ) y 14 φσ 3 ( X 4 ) ^ σ ( X 2 ) y 15 σ 3 ( X 1 ) ^ σ ( X 3 ) y 16 RS 2 RS 1 Fig. 10. Transm ission frame of the Incomplete DF protocol in the 2-relay case implemented with a distributed 4 × 4 T AST code October 28, 2018 DRAFT 27 0 10 20 30 SNR (dB) 10 -4 10 -3 10 -2 10 -1 10 0 Frame error rate SISO - 2 bpcu NAF - 2 bpcu IDF (PC) - 2 bpcu IDF (TAST) - 2 bpcu SISO - 4 bpcu NAF - 4 bpcu IDF (PC) - 4 bpcu IDF (TAST) - 4 bpcu Fig. 11. Frame error rates of the SIS O, N AF and new DF protocols as functions of the SNR at spectral efficiencies of 2 and 4 bits per channel use in the 2-relay case October 28, 2018 DRAFT