Signal Processing. Non-coherent distributed space time coding techniques for two-way wireless relay networks $

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1 Signal Pocessing 93 (013) Contents lists available at ScienceDiect Signal Pocessing jounal homepage: Non-coheent distibuted space time coding techniques fo two-way wieless elay netwoks $ Same Alabed an1 Maius Pesavento a1 Anja Klein b a Communication Systems Goup echnische Univesität Damstadt Meckst. 5 D-6483 Damstadt Gemany b Communications Engineeing Lab. echnische Univesität Damstadt Meckst. 5 D-6483 Damstadt Gemany aticle info Aticle histoy: eceived 17 August 01 eceived in evised fom 30 Novembe 01 Accepted 4 Decembe 01 Available online 0 Decembe 01 Keywods: wo-way wieless elay netwoks Distibuted space time coding Diffeential space time coding Coopeative divesity abstact o ovecome the ovehead involved with channel estimation seveal non-coheent distibuted space time coding (DSC) stategies fo two-way wieless elay netwoks (WNs) using the amplify-and-fowad and the decode-and-fowad potocol have been ecently poposed that do not equie channel state infomation (CSI) at any node to decode the infomation symbols. In this pape novel diffeential DSC stategies fo WNs using the two- and thee-phase potocol ae poposed. In ou tansmission schemes the elays do not waste powe to tansmit infomation known at the espective destination nodes. his is achieved by combining the symbols fom both teminals eceived at the elays into a single symbol of the unalteed constellation. Futhemoe in ou stategies the diect link between the communicating teminals can be natually incopoated to futhe impove the divesity gain. Simulations show a substantially impoved pefomance in tems of bit eo ate (BE) of the poposed stategies as compaed to the existing stategies. & 01 Elsevie B.V. All ights eseved. 1. Intoduction In scatteing envionments coopeative divesity techniques employing multiple single-antenna elays can efficiently be applied to combat the effect of multi-path fading [1 3]. he basic setup of one-way distibuted elay netwoks consists of a souce teminal a destination teminal and multiple elay nodes. he signals tansmitted by the souce teminals ae pocessed at the elays and etansmitted emulating a vitual antenna aay and theefoe ceating multi-antenna tansmit divesity. $ his wok was suppoted by the Euopean eseach Council (EC) Advanced Investigato Gants pogam unde Gant 7477-OSE. n Coesponding autho. el.: þ ; fax: þ addesses: salabed@nt.tu-damstadt.de (S. Alabed) pesavento@nt.tu-damstadt.de (M. Pesavento) a.klein@nt.tu-damstadt.de (A. Klein). 1 el.: þ ; fax: þ el.: þ ; fax: þ It is widely established that using multiple elays can damatically impove the pefomance of wieless elay netwoks in tems of data ate and eo pefomance. heefoe elays have been identified as an integal pat of futue wieless communication netwoks [ 9]. Efficient one-way coopeative divesity techniques based on DSC have been ecently developed to exploit the spatial divesity povided by multiple single-antenna elay nodes located spatially distibuted in the aea in between the communicating teminals [4 11]. DSC techniques involve pocessing of the tansmitted symbols in the spatial dimension ove multiple antennas and in time dimension ove multiple time slots. By adding edundant infomation in space and time both the eliability and the thoughput can be impoved at no additional cost of bandwidth o tansmitted powe and without equiing CSI at the tansmitting nodes [3]. hese benefits have been widely ecognized in the intenational eseach community and standadization bodies. Depending on the functionalities of the elays seveal tansmission potocols have been poposed which specify /$ - see font matte & 01 Elsevie B.V. All ights eseved.

2 337 S. Alabed et al. / Signal Pocessing 93 (013) Fig. 1. WN with þ nodes. the pocess of pocessing and etansmission of the eceived signal at the elays [3]. Among them the most popula potocols ae (i) the amplify-and-fowad (AF) potocol whee each elay eceives a noisy vesion of the infomation signal which is amplified at the elays and etansmitted and (ii) the decode-and-fowad (DF) potocol whee symbols ae decoded and e-encoded at the elay and then fowaded to the destination. Based on the availability of CSI vaious coopeative divesity techniques have been consideed. Some coopeative divesity techniques ae based on the unealistic assumption of pefect CSI at all nodes [113]. Othe techniques such as the DSC techniques conside the case of pefect CSI available only at the eceiving nodes [614 16]. he ecently poposed non-coheent diffeential techniques have been designed based on the assumption of CSI available neithe at the teminals no at the elays. hese methods educe the system complexity by avoiding the ovehead involved with the tansmission and pocessing of pilot signals [ ]. In WNs the communicating teminals mutually exchange infomation via a goup of elays [16 4]. hese elay netwoks can be categoized accoding to the numbe of phases equied fo the infomation exchange. hee exist thee popula classes: the fou-phase WNs the thee-phase WNs and the two-phase WNs. In [1618 0] coheent and non-coheent DSC techniques using the AF and the DF potocol have been poposed. It has been shown that DSC techniques using the two- and thee-phase potocol outpefom the conventional DSC techniques using the fou-phase potocol due to the effective eduction in the symbol ate associated with the latte potocol [16190]. Howeve the two-phase potocol in [190] cannot exploit the diect link between the communicating teminals since both teminals tansmit thei infomation symbols simultaneously to the elay nodes and a half duplex constaint is assumed to apply i.e. devices can eithe tansmit o eceive in a specific time slot. In this pape we develop non-coheent two- and thee-phase DSC techniques fo two-way elay netwoks whee the diect link between the communicating teminals can be incopoated. he poposed techniques combine eceived symbols fom both teminals at the elays into a single symbol using a simple diffeential encoding scheme such that each teminal can decode the tansmitted symbol of the othe teminal using the infomation of its own tansmitted symbol. Inteestingly in contast to combination schemes that ely on extended modulations [19] the poposed diffeential encoding scheme pefomed at the elays is not associated with any waste in powe fo tansmitting infomation symbols known at eithe eceive esulting in impoved BE pefomance at both teminals.. Wieless elay netwok model We conside a wieless elay netwok with þ halfduplex single-antenna elay nodes as shown in Fig. 1 whee the two teminals and intend to exchange infomation and nodes ( 1... ) act as distibuted elays fo the signals tansmitted fom the teminals. We denote the channels fom to fom to the th elay and fom to the th elay as f 0 f and g espectively. We assume channel ecipocity fo the tansmission fom to and vice vesa. Futhe we conside the extended block fading channel model in the two-phase potocol fo which the channels ae assumed to emain appoximately constant ove consecutive time slots and to slowly evolve outside this time inteval whee denotes the block length. Similaly in theephase schemes the channels ae assumed to emain appoximately constant ove 3 time slots. We futhe conside that the elays ae pefectly synchonized and CSI is not available at any node. he nodes 1... have limited aveage tansmit powes P P P 1...P espectively. houghout this pape ðþ n J J ðþ ðþ H diagðaþ I e s ½aŠ i 0 and Efg denote the agument of a complex numbe the complex conjugate the Fobenius nom the matix tanspose the Hemitian tanspose the Hadamad (o Schu) poduct the diagonal matix whose diagonal elements ae the elements of the vecto a the identity matix the th column of I the noise vaiance the ith enty of a vecto a the matix with all zeo enties and the statistical expectation espectively.

3 S. Alabed et al. / Signal Pocessing 93 (013) Depending on the used context 9 9 denotes the absolute value o the cadinality of a set. 3. hee-phase two-way diffeential DSC techniques Let us assume that s ¼½½s 1 Š ½s Š 1 Š and s ¼½½s Š 1...½s Š Š denote the 1 vectos containing the kth block of infomation symbols of teminal and espectively whee ½s Š t i is taken fom a M-PSK constellation denoted by set S t and is the numbe of time slots in each phase which is assumed to be equal to the numbe of elays hence ¼. In ode to facilitate the pocessing at the elays and the destinations in the fist phase of the kth block teminal diffeentially encodes the infomation symbol vecto s as ¼ diagðx ðk 1Þ Þs ð1þ and tansmits this vecto afte powe scaling with 3P. At the beginning of tansmission in the fist block the ecusion in (1) is initialized with the symbol vecto x ð0þ which is known at the tansmitte and eceive. In the next block of symbols the last tansmitted symbol vecto of the pevious block can be used as a efeence in the decoding pocedue. Duing the second phase fom time slot þ1 to in the kth block teminal tansmits the diffeentially encoded 1 symbol vecto to the elays given by ¼ diagðx ðk 1Þ Þs ðþ whee x ð0þ ¼½11 1Š that is also scaled with 3P.In the fist phase of the kth block as shown in Fig. fom time slot 1 to the 1 vecto eceived at the th elay is given by 1 ¼ 3P f þn 1 ð3þ whee f denotes the channel fom teminal to the th elay in the kth block and n 1 denotes the 1 noise vecto of the kth block at the th elay in the fist phase. We assume that the noise vecto can be modeled as a spatially white independently and identically distibuted complex cicula Gaussian andom vaiable with zeo mean and covaiance s I. Similaly in the second phase of the kth block as shown in Fig. 3 fom time slot þ1 to the 1 vecto eceived at the th elay is given by ¼ 3P g þn whee g denotes the channel fom teminal to the th elay in the kth block and n denotes the 1 noise vecto at the th elay in the second phase of the kth block. In the following we popose an efficient encoding stategy at the elays that facilitates simple signal sepaation at the destinations without howeve involving decoding of the signals eceived at the elays. Duing the thid phase of the kth block as illustated in Figs. 4 and 5 the th elay combines the eceived signal vectos in (3) and (4) into a single 1 signal vecto 3 ¼ U y 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 3 P P f g U x þv 3 whee the combined signal pat is defined as ¼ x x 1 the additive noise tems ae combined in v 3 ¼ U 3P f n þ 3P þn 1 n g n 1 ð4þ ð5þ ð6þ ð7þ and U denotes a diagonal scaling matix that adjusts the tansmitted powe at the th elay. Let us conside without loss of geneality the simple choice of 0 " # U ¼ 1 9½ 1 Š 1½ Š A 9½ 1 Š ½ Š 9 ð8þ that tuns out to yield excellent decoding pefomance. Howeve anothe possible choice would be to select unifom constant scaling hence U ¼ I. Fig.. Block diagam of the fist phase.

4 3374 S. Alabed et al. / Signal Pocessing 93 (013) Fig. 3. Block diagam of the second phase. Fig. 4. Combination pocedue at the th elay. Fig. 5. Block diagam of the thid phase.

5 S. Alabed et al. / Signal Pocessing 93 (013) We define Š i ¼ 3 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P f g 9½ 1 Š i99½ Š i9 ½h we obseve fom (3) and (4) that fo sufficiently lage SN and with the scaling in (8) 9½h Š i 9 ¼ 3 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 9f 99g 9 ½ ½ 1: ð10þ 1 Š i Š i Making use of the appoximation in (10) Eq. (5) can be ewitten as 3 ejðf g Þ þv 3 ð9þ ð11þ which can be viewed as a scaled vesion of (6) that is coupted by the additive noise of the th elay defined in (7). Fom (5) and (6) we obseve that the th elay combines the eceived symbol vectos and x fom both teminals into a single symbol vecto using a specific type of diffeential encoding scheme that does not equie decoding of the symbols at the elays as illustated in Fig. 4. his combining scheme enables each teminal to decode the tansmitted symbols of the opposite teminal using the infomation of its own tansmitted symbols. It can also be obseved that the symbols in (6) belong to the same constellation as the symbols oiginally tansmitted fom both teminals. As a consequence using the poposed diffeential encoding stategy the elays do not waste powe to tansmit infomation that is aleady known at the individual eceives. his esults in an impoved oveall system pefomance in tems of BE as compaed to conventional combination in [19] that ely on extended constellations. Fom Eq. (11) we obseve that the combined symbol vectos 3 at the elays ae appoximately equal up to an unknown phase otation intoduced by the complex channels h and the additive noise. his means that the elay netwok can be consideed as a vitual centalized multi-antenna netwok with tansmittes in which CSI is not available at the tansmitte side. In this case space time block coding (SBC) techniques conventionally applied in centalized MISO systems can staightfowadly be applied. Hence the th elay encodes the combined symbol vecto in (5) using SBC pecoding scheme such that X 3 ¼ Xðy 3 Þ¼½A 1 3 þb 1ð 3 Þn...A 3 þb ð 3 Þn Š whee the code matix XðxÞ¼½A 1 xþb 1 x n...a xþb x n Š ð1þ ð13þ with pecoding matices A 1...A and B 1...B defines the applied SBC scheme. houghout this pape we assume that the SBC pecoding matices exhibit the mutual exclusivity popety that eithe A ¼ 0 o B ¼ 0. his popety is valid fo a lage numbe of commonly used SBCs. Fo instance let us conside a wieless elay netwok with two elay nodes ¼ ¼ fo which the popula Alamouti scheme can be applied [916187] whee A and B ae chosen as A 1 ¼ 1 0 B 1 ¼ A ¼ 0 B ¼ : ð14þ In the case of a wieless elay netwok with fou elay nodes i.e. ¼ ¼ 4 quasi-othogonal space time codes [95 7] can be applied whee A and B ae chosen e.g. as A 1 ¼ A ¼ A 3 ¼ 0 A 4 ¼ 0 B 1 ¼ 0 B ¼ B 3 ¼ B ¼ : ð15þ Consideing (6) the mutual exclusivity popety that eithe A ¼ 0 o B ¼ 0 and the appoximation in (11) Eq.(1) can be ewitten as X 3 ½A 1 h þb 1 ð Þn h n...a h þb ð Þn h n Š þ½a 1 v 3 þb 1ðv 3 Þn...A v 3 þb ðv 3 Þn Š ¼ X D f D g þv 3 ð16þ whee D f ¼ diagðf Þ and D g ¼ diagðg Þ fo f and g denoting a 1 vectos with the ith element defined as 8 < ½f Š i ¼ ejðf Þ if B i ¼ 0 ð17þ : jðf e Þ if A i ¼ 0 and 8 < ½g Š i ¼ ejðg Þ if B i ¼ 0 : e jðg Þ if A i ¼ 0 ð18þ espectively. Futhe X ¼ Xðx Þ¼½A 1 þb 1ð Þn...A þb ð Þn Š ð19þ and V 3 ¼ Xðv 3 Þ contain the combined signal and noise component in the SBC matix (16) espectively. he SBC matix in (19) descibes the SBC matix as it is used in a conventional i.e. centalized multi-antenna system with ¼ antennas tansmitting the combined symbol vecto in (6). Notethatwith(6) and (19) the code matix can also be expessed as X ¼ X X 1 ð0þ whee X t ¼ Xð Þ denotes the SBC matix in (19) coe- t sponding to the symbol vecto tansmitted by teminal t t in the tth phase. We obseve fom (16) that the th elay ecoves the SBC matix of the centalized multi-antenna system in (0) up to scaling of the columns that depends on the composite channels h (o its conjugate h n )in(9) and addition of a noise component obtained fom the combined

6 3376 S. Alabed et al. / Signal Pocessing 93 (013) elay noise v 3 in (7). Inthekth block of the thid phase fom time slot þ1 to3 the elays jointly tansmit a SBC matix whee the th elay tansmits the th column of its espective code matix X 3 in (1). Hence the vecto 3 ¼ X 3 e ð1þ is tansmitted by the th elay afte scaling with powe p coefficient b ¼ ffiffiffiffiffiffiffi P fo U defined accoding to (8) to satisfy the powe constaint. In the following we conside only the eceived signals at teminal. he signal eceived at teminal can be computed coespondingly. he eceived signal vecto at teminal in the kth block is given by ¼ X ¼ X ¼ 1 ¼ 1 ¼ X ¼ 1 b g 3 þn b g X 3 e þn b g ðx D f D g e þv 3 e Þþn ¼ðX X ¼ðS S 1 ÞD g D f Xðk 1Þ c g þv X ðk 1Þ ÞD g D f c g þv ðþ whee n denotes the 1 vecto containing the eceive noise of the kth block at teminal S ¼ Xðs t Þ and t v ¼ X ¼ 1 D ¼ X D f f e ¼ 1 D g ¼ X D g e ¼ 1 b g V 3 e þn ð3þ ð4þ ð5þ c g ¼½b 1 g 1...b g Š : ð6þ We emak that Eq. () also applies fo scaling matices diffeent fom the one chosen in (8). Howeve this equies coesponding modification in the definitions of the vectos in (17) and (18) aswellasthepowecoefficientb used fo the tansmission of the code vecto (1) at the th elay. Fo example if unifom constant scaling is used hence if U ¼ I then 8 < ½f Š i ¼ f if B i ¼ 0 ðf ð7þ : Þ n if A i ¼ 0 ( Š i ¼ g if B i ¼ 0 ½g ðg Þ n if A i ¼ 0 ð8þ ae consideed instead of (17) and (18) espectively and p b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P =ð9p P þ3p þ3p Þ. In the kth block the pevious diffeentially encoded symbol matix X ðk 1Þ at teminal the cuent symbol matix S of teminal and the estimate of the pevious diffeentially encoded matix X ðk 1Þ ðk 1Þ denoted by ^X can be consideed as known at teminal. Making use of the extended block fading assumption whee D g ¼ D ðk 1Þ g D f ¼ D ðk 1Þ and f c g ¼ c ðk 1Þ g the eceived signal vecto at teminal defined in () duing the thid phase of the ðk 1Þth and kth block can be expessed as ¼ X D gd f c g þv ¼ðS S X ðk 1Þ ÞD g D f c g þv ð9þ y ðk 1Þ ~y ðk 1Þ ¼ X ðk 1Þ D g D f c g þv ðk 1Þ ð30þ ¼ð^X ðk 1Þ Þ 1 y ðk 1Þ ¼ D g D f c g þð^x ðk 1Þ Þ 1 v ðk 1Þ : ð31þ Fom (9) to (31) the decode at teminal in the absence of a diect link between the communicating teminals and can be expessed as ag min whee J S ðs ðk 1Þ ^X ¼ S ^X ðk 1Þ ðk 1Þ ^X Þ ~y ðk 1Þ J ð3þ X ðk 1Þ. We emak that fo ou distibuted diffeential SBC coding scheme simila popeties as in conventional centalized systems apply. In paticula if othogonal o quasi-othogonal SBCs ae used in (13) then the decoding pocedue in (3) can be caied out symbol-wise o pai-wise espectively esulting in substantially educed decoding complexity at the teminals [697]. In the case that the diect link between the two teminals and is available the eceived signal vecto at teminal duing the fist tansmission phase is given by dl ¼ 3P ¼ 3P f 0 x þn f 0 S x ðk 1Þ þn ð33þ whee n denotes the 1 vecto containing the eceive noise of the kth block at teminal and is defined in (1). Making use of the symbol vecto defined in (33) dl the decode at teminal can be expessed as ag min J S ðs S ðk 1Þ ^X Þ ~y ðk 1Þ J þj dl S y ðk 1Þ dl J : ð34þ We obseve that in ou scheme each elay combines the eceived signal vectos fom both teminals into a single signal vecto without decoding them and boadcasts the esulting vecto to the destination. Each teminal can decode the tansmitted symbols of the othe teminal fom its eceived signals of the elays using the infomation of its own tansmitted symbols. On the othe hand the DF potocol can also be applied to the poposed technique whee each elay can diffeentially decode the infomation symbol vectos s and s of the fist and second teminal espectively defined in (1) and () without equiing CSI. In this case similaly as

7 S. Alabed et al. / Signal Pocessing 93 (013) in (6) the decoded and e-encoded symbol vectos could be combined at the th elay into a single symbol vecto. In the next phase the th elay encodes the combined symbol vecto and then boadcasts it to both teminals. We emak that ou poposed technique in combination with the DF potocol enjoys a substantially lowe elay decoding complexity as compaed to the peviously poposed two-phase DSC techniques which has the ode of 9S 99S 9 [0]. o apply DSC at the elay nodes using the DF potocol the elays have to ideally decode the eceived signal coectly i.e. ^ 1 ¼ ^ ¼ ^ 3 ¼¼. hus in ode to achieve the full divesity the elays should decode the infomation symbols coectly othewise the decode at the destination teminal suffes fom a poo eo pefomance. heefoe to achieve the full divesity the authos in [0] poposed the use of cyclic edundancy check (CC) at the elay nodes at the cost of educing the spectal efficiency. 4. wo-phase two-way diffeential DSC technique In the two-phase potocol e.g. poposed in [190] the teminals and simultaneously tansmit duing the fist phase fom time slot 1 to thei infomation symbol vectos to the elays. In the second phase fom time slot þ1 to the elays fowad the eceived signals to the destinations using eithe the AF o DF tansmission. Although the potocol poposed in [190] allows coopeation in two phases a majo dawback is that it can not exploit the diect link between the communicating teminals. Moeove the potocol poposed in [0] suffes fom a high elay decoding complexity and is often associated with poo eo pefomance. It was shown in [0] that due to the incease in the symbol ate the twophase potocols pefom bette than the thee-phase potocols specifically fo tansmitted symbols with high modulation odes. In this section we educe the thee-phase potocol to a two-phase potocol that exhibits a symbol ate equivalent to that of the two-phase potocols of [16190]. he poposed two-phase potocol enjoys impoved eo pefomance and low decoding complexity without equiing CSI at any node and allows the use of the diect link between the communicating teminals. Let us assume that Z is an even numbe and s ¼½½s t Š t 1...½s Š t Š denotes the kth infomation symbol vecto of dimension 1 coesponding to t whee the symbols s S t t. Evey two infomation symbols ½s Š t i 1 and ½s Š t i ae combined at the teminal t into a single supe-symbol ½d Š t i of constellation X t with M t ¼ 9X t 9 ¼ 9S t 9 using the mapping d ¼ G t t ðs Þ t with the combination function whee ½d t Š i ¼ G t ð½s t Š i 1½s t Š iþ¼exp j ð½s t Š i 1 þm t ½s M t t Š iþ! ð35þ t¼1 and d ¼½½d t Š t 1...½d Š t =Š denotes the kth supe-symbol vecto of dimension = 1. he teminals and encode thei supe-symbols d and d diffeentially into the symbols X t t to facilitate simple decoding at the elays without equiing CSI. In the fist peiod of the fist phase fom time slot 1 to = the teminals and simultaneously tansmit to the elays in the kth block the diffeentially encoded supe-symbol vectos P and P 1 espectively whee t ¼ diag d t x ðk 1Þ t ð36þ h i ¼ ½x t Š t 1...½ Š t = denotes the kth tansmitted diffeentially encoded supe-symbol vecto of dimension = 1 tansmitted by teminal t x ð0þ ¼½11...1Š t defines the initial tansmitted symbol vecto in the fist tansmission that can be used as a efeence at the eceive to stat the diffeential decoding pocedue and ½ Š t i X t. Duing the fist peiod of the fist phase fom time slot 1 to = the eceived signal vecto of the kth block at the th elay is given by 1 ¼ P f þ P g þn 1 ð37þ whee n 1 denotes the = 1 noise vecto of the kth block at the eceive of the th elay in the fist peiod of the fist phase. Similaly in the second peiod of the fist phase fom time slot =þ1 to the teminals and tansmit simultaneously P p and ffiffiffiffiffiffiffiffiffiffi P espectively. he only diffeence with espect to the fist peiod is that in the second peiod the teminal tansmits the same symbols multiplied by 1 to enable simple sepaation of the eceived symbols at each elay as illustated below. he eceived signal vecto at the th elay duing the second peiod of the fist phase is given by ¼ P f P g þn ð38þ whee n denotes the = 1 noise vecto of the kth block at the eceive of the th elay in the second peiod of the fist phase. Making use of the block fading assumption i.e. the channel emains constant ove time slots we can eplace in (37) and (38) the channel coefficients f and g by f and g espectively. Fom (37) and (38) the th elay obtains ~ 1 ¼ y 1 þy ð39þ ~ ¼ y 1 y ð40þ fom which soft estimates of the infomation symbol vectos ^s and ^s of the teminals and espectively can be computed by splitting the eceived signals that contain the tansmitted symbols as follows: ½^s t Š i 1 ¼ expðjp modfð½ ~ t Š i ½ ~y ðk 1Þ t Š iþm t pgþ ð41þ

8 3378 S. Alabed et al. / Signal Pocessing 93 (013) ½^s t Š i ¼ exp j ½ ~ t Š i ½ ~y ðk 1Þ t Š i ½^s t Š!! i 1 ð4þ M t whee i ¼ 1... Note that the sepaation pocedue caied out at the elays accoding to (41) and (4) is associated with negligible complexity as compaed to the had-decision elay decoding complexity of the two-phase coheent DSC using the DF potocol [0]. Simila to Eq. (5) theth elay combines the sepaated symbols ^s and ^s fom the two teminals into a single symbol as ^s ¼ ^s ^s : ð43þ Similaly as in Section 3 weobsevefom(43) that the poposed encoding stategy allows the elays to combine the sepaated symbols ^s and ^s fom the two teminals into a single symbol ^s belonging to the same constellation as the sepaated symbols. In this way the elays do not waste powe to tansmit known infomation to eithe side. Each teminal can then decode the tansmitted symbol of the opposite teminal using the infomation of its own tansmitted symbol esulting in oveall impoved the BE pefomance. In the second phase of the kth block fom time slot þ1 to the th elay pecodes the symbol vecto and its conjugate with the unitay matices A and B and scales the esulting vecto befoe boadcasting it to the teminals. Simila to (1) the th elay encodes the combined symbol vecto in (43) using a linea SBC pecoding scheme hence S ¼ Xð^s Þ¼½A 1 ^s þb 1ð^s Þn...A ^s þb ð^s Þn Š: ð44þ In the second phase of the kth block the elays jointly tansmit a diffeentially encoded SBC matix similaly as in Section 3 whee the th elay tansmits the th column of its diffeentially encoded SBC matix given by ¼ S Xðk 1Þ e 1 ¼ S xðk 1Þ ð45þ whee X ð0þ ¼ I x ð0þ ¼ e 1 and the stuctue of S depends on the used SBC pecoding matices defined in (14) and (15). Simila to () the eceived signal vecto at teminal is given by ¼ X ¼ 1 p whee b ¼ b g þn ffiffiffiffiffiffiffi P ð46þ and n denotes the noise vecto of the kth block at teminal. In the second phase of the kth block the eceived signal vecto at teminal can be expessed as ¼ X c g þn ð47þ whee c g ¼½b 1 g 1 b g...b g Š ð48þ X ¼ ^S Xðk 1Þ : ð49þ hen fom (49) we can ewite (47) as ¼ ^S Xðk 1Þ c g þn ¼ ^S ðyðk 1Þ n ðk 1Þ Þþn : ð50þ Fom (50) the decode at teminal in the absence of a diect link between the communicating teminals and can be expessed as ag min ^S yðk 1Þ ð51þ S whee the elements of the decoding matix ae a function of the tansmitted symbols i.e. ½s Š i ¼ ½s Š i ½s Š i. A simila decoding pocedue can be applied at teminal. In ode to allow the use of the diect link between the communicating teminals which povides a highe divesity ode an altenative tansmission scheme can be applied whee in the fist phase fom time slot 1 to = teminal tansmits 4P while teminal emains muted and then teminal tansmits 4P while teminal emains muted fom =þ1 to. In this case the teminal that emains muted can eceive the tansmission of the teminal that tansmits such that the diect link can be easily incopoated. We emak that in the latte scheme the espective teminals tansmit with twice the powe as compaed to the fome scheme such that the total tansmitted powe in both schemes is the same. Simila to (33) the eceived signal vecto at teminal duing the fist tansmission phase in the pesence of the diect link between the communicating teminals is given by dl ¼ 4P f 0 þn ¼ 4P f 0 diagðg ðs ÞÞx ðk 1Þ þn ^S ð5þ whee n denotes the = 1 vecto containing the eceive noise of the kth block at teminal and is defined in (36). Simila to the decode of (34) applied in the case whee the diect link between the communicating teminals is used with the altenative tansmission the decode at teminal can be expessed as ag min : s ^S yðk 1Þ : þ: dl diagðg ðs ÞÞy ðk 1Þ dl : : ð53þ Note that each element of the matix ¼ Xð^s Þ is a function of the elements of the infomation symbol vectos s and s i.e. ½s Š i ¼½s Š i ½s Š i and s is known at teminal p. he decode of (51) can be implemented using the sphee decode [8] o in the case of othogonal DSCs a symbol-wise decode can be applied to decode the eceived symbols at teminal [69]. he use of the diect link inceases the decoding complexity at the destination teminal exponentially with the incease of the constellation size o numbe of tansmitted symbols as geneally a full seach ove S is equied. Howeve this incease is accompanied by a highe divesity ode ^S

9 S. Alabed et al. / Signal Pocessing 93 (013) povided by the diect link. A simila decoding pocedue can be applied at. On the othe hand and similaly as in Section 3 thedf potocol can also be applied to the poposed two-phase technique. In this case each elay fist decodes the infomation symbols s and s of the fist and second teminals espectively defined in (41) and (4) without equiing CSI. hen similaly as in (43) the decoded infomation symbol vectos could be combined at the th elay into a single symbol vecto ^s. In the next phase the th elay diffeentially encodes the combined symbol vecto ^s using the pecoding matices A 1...A and B 1...B befoe boadcasting the esulting vecto to both teminals simila to (44) and (45). he poposed twophase technique using the DF potocol enjoys a substantially lowe elay decoding complexity as compaed to the peviously poposed two-phase DSC techniques [0]. Note that both tansmission stategies descibed above in this section i.e. the simultaneous and the altenating achieve exactly the same BE in the case that the diect link between the communicating teminals is not used. We emak howeve the simultaneous tansmission is beneficial in pactical hadwae implementations as the equiements on the powe amplifies can be elaxed as compaed to the altenating tansmission due to a educed peak-to-aveage powe ation (PAP). Howeve the diect link between the communicating teminals cannot be exploited. On the othe hand the altenative tansmission can be used to allow the diect link between the communicating teminals. 5. Simulation esults In ou simulations we have assumed a wieless elay netwok with two single-antenna elay nodes and independent flat ayleigh fading channels whee the powe is distibuted among the two teminals and elays as P ¼ P ¼ P ¼ 1 P similaly as in [160]. Fo fai compaison of the BE pefomance of all techniques the same total tansmitted powe (P ¼ P þp þ P ¼ 1 P P 1 ¼ P ¼ ¼P ) and bit ate ae used. In Fig. 6 the BE at teminal is displayed vesus the SN and the poposed two-phase non-coheent DSC technique using BPSK modulation is compaed with the fouphase non-coheent distibuted Alamouti space time coding technique using 4-PSK modulation [9] and the two-phase non-coheent DSC technique poposed in [19] using BPSK modulation fo a total ate of 0.5 bit pe channel use (bpcu). he fou-phase potocol is the conventional one way elaying scheme applied twice whee in the fist phase the elays eceive the signal fom the fist teminal pocess and fowad it in the second phase to the second teminal. Similaly in the thid phase the elays eceive the signal fom the second teminal pocess and fowad it in the fouth phase to the fist teminal. In the two-phase potocol both teminals tansmit thei infomation symbols simultaneously to the elays in the fist phase and then the elays fowad the eceived signal in the second phase to the destination teminals using eithe the AF o DF tansmission. Fom Fig. 6 it can be obseved that the poposed two-phase technique outpefoms the known two- and fou-phase technique and enables the teminals to use the diect link to impove the divesity gain whee the abbeviation DL stands fo the use of the diect link in the technique. In Fig. 7 the label Fist poposed 3-phase scheme stands fo the poposed thee-phase scheme using U ¼ I p and b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P =ð9p P þ3p þ3p Þ while the label Second poposed 3-phase scheme stands fo the poposed thee-phase scheme using U ¼ diagð½1= 9½ 1 Š 1½ Š =9½ 1 Š ½ Š p 9Š Þ and b ¼ ffiffiffiffiffiffiffi P. In Fig. 7 the BE at teminal is displayed vesus the SN and the poposed thee-phase non-coheent DSC technique using 8-PSK modulation is compaed with the poposed two-phase non-coheent DSC technique using Fig. 6. BE vesus SN fo seveal diffeential schemes with ¼ and a ate of 0.5 bpcu.

10 3380 S. Alabed et al. / Signal Pocessing 93 (013) BE SN (db) Fig. 7. BE vesus SN fo seveal diffeential schemes with ¼ and a ate of 1 bpcu. 4-PSK modulation the fou-phase non-coheent distibuted Alamouti space time coding technique using 16-PSK modulation [9] and the two-phase non-coheent DSC technique poposed in [19] using 4-PSK modulation fo a total ate of 1 bpcu. Fom Fig. 7 it can be obseved that the poposed two- and thee-phase techniques outpefom the existing two- and fou-phase techniques and also allow the communicating teminals to use the diect link between them to incease the divesity ode. In the absence of the diect link between the communicating teminals the BE pefomance achieved by the fist and the second poposed thee-phase scheme is diffeent. his is due to the fact that in the fist scheme the elays amplify the noise combined with the eceived symbol in the tansmission while in the second scheme the eceived signal is popely scaled to avoid noise amplification. Fom Fig. 7 the techniques that use the diect link between the communicating teminals outpefom those without diect link since the diect link inceases the divesity gain. Moeove in the pesence of the diect link between the communicating teminals the poposed thee-phase stategy outpefoms the poposed twophase stategy since duing the fist two phases the poposed thee-phase stategy tansmits symbols taken fom lowe constellation size than those tansmitted by the poposed two-phase stategy in the diect link. 6. Conclusion In this pape we popose novel diffeential DSC techniques fo two-way elay netwoks that do not equie CSI at the teminals o elays. Ou poposed techniques enjoy a substantially lowe elay decoding complexity and povide moe coding gain than the state-of-the at techniques by combining the eceived symbols fom both teminals at the elays into a single symbol of the same constellation. Futhemoe ou tansmission schemes allow the communicating teminals to use the diect link between them to incease the divesity ode which is not valid fo othe simultaneous bidiectional tansmission schemes. efeences [1] A. Amah A. Klein Non-egeneative multi-antenna multi-goup multi-way elaying EUASIP Jounal on Wieless Communications and Netwoking [] A. Sendonais E. Ekip B. Aazhang Use coopeation divesity. Pat I: system desciption IEEE ansactions on Communications 51 (Novembe (11)) (003) [3] A. Sendonais E. Ekip B. Aazhang Use coopeation divesity. Pat II: implementation aspects and pefomance analysis IEEE ansactions on Communications 51 (Novembe (11)) (003) [4]. Unge A. Klein On the pefomance of distibuted space time block codes IEEE Communications Lettes 11 (May (5)) (007) [5] W. Song Q. Liu M. Lee A simple asynchonous distibuted SBC netwok scheme with full divesity EUASIP Jounal on Advances in Signal Pocessing July [6] J.N. Laneman G.W. Wonell Distibuted space time-coded potocols fo exploiting coopeative divesity in wieless netwok IEEE ansactions on Infomation heoy 49 (Octobe (10)) (003) [7] S. Yiu. Schobe L. Lampe Distibuted space time block coding IEEE ansactions on Communications 54 (July (7)) (006) [8] B. Maham A. Hjoungnes Powe allocation stategies fo distibuted space time codes in amplify-and-fowad mode EUASIP Jounal in Signal Pocessing [9] Y. Jing H. Jafakhani Distibuted diffeential space time coding in wieless elay netwoks IEEE ansactions on Communications 56 (July (7)) (008) [10]. Wang Y. Yao G.B. Giannakis Non-coheent distibuted space time pocessing fo multiuse coopeative tansmissions IEEE ansactions on Wieless Communications 5 (Decembe (1)) (006) [11] F. Oggie B. Hassibi A coding stategy fo wieless netwoks with no channel infomation in: Poceedings of the Alleton Confeence on Communications Contol and Computing Monticello IL Septembe 006 pp [1] A.F. Dana B. Hassibi On the powe-efficiency of sensoy and ad hoc wieless netwoks IEEE ansactions on Infomation heoy 5 (July (7)) (006) [13] H. Boelcskei.U. Naba O. Oyman A.J. Paulaj Capacity scaling laws in MIMO elay netwoks IEEE ansactions on Wieless Communications 5 (Januay (6)) (006)

11 S. Alabed et al. / Signal Pocessing 93 (013) [14] M. Janani A. Hedayat.E. Hunte A. Nosatinia Coded coopeation in wieless communications: space-time tansmission and iteative decoding IEEE ansactions on Signal Pocessing 5 (Febuay ()) (004) [15] Y. Jing B. Hassibi Distibuted space time coding in wieless elay netwoks IEEE ansactions on Wieless Communications 5 (Decembe (1)) (006) [16] S.J. Alabed J.M. Paedes A.B. Geshman A simple distibuted space time coded stategy fo two-way elay channels IEEE ansactions on Wieless Communications 11 (Apil (4)) (01) [17] S.J. Alabed M. Pesavento A simple distibuted diffeential tansmit beamfoming technique fo two-way wieless elay netwoks in: he 16th Intenational IEEE/IG Wokshop on Smat Antennas (WSA 01) Desden Gemany Mach 01 pp [18] S.J. Alabed M. Pesavento A.B. Geshman Distibuted diffeential space time coding techniques fo two-way wieless elay netwoks in: Poceedings of the Fouth IEEE Intenational Wokshop on Computational Advances in Multi-Senso Adaptive Pocessing (CAMSAP 11) San Juan Pueto ico Decembe 011 pp [19] Z. Utkovski G. Yammine J. Lindne A distibuted diffeential space time coding scheme fo two-way wieless elay netwoks in: ISI009 Seoul Koea June 009 pp [0]. Cui F. Gao. Ho A. Nallanathan Distibuted space time coding fo two-way wieless elay netwoks IEEE ansactions on Signal Pocessing 57 (Febuay ()) (009) [1]. Unge A. Klein Applying elay stations with multiple antennas in the one- and two-way elay channel in: Poceedings of the Intenational Symposium on Pesonal Indoo and Mobile adio Communications Athens Geece Septembe 007. []. Unge A. Klein Duplex schemes in multiple antenna two-hop elaying EUASIP Jounal on Advances in Signal Pocessing [3]. Unge A. Klein On the pefomance of two-way elaying with multiple-antenna elay stations in: Poceedings of the IS Mobile and Wieless Communications Summit Budapest Hungay July 007. [4] S. Bege. Unge M. Kuhn A. Klein A. Wittneben ecent advances in amplify-and-fowad two-hop elaying IEEE Communications Magazines 47 (July) (009) [5] H. Jafakhani A quasi-othogonal space time block code IEEE ansactions on Communications 49 (Januay) (001) 1 4. [6] S.J. Alabed J.M. Paedes A.B. Geshman A low complexity decode fo quasi-othogonal space time block codes IEEE ansactions on Wieless Communications 10 (Mach (3)) (011). [7] Y. Jing H. Jafakhani Using othogonal and quasi-othogonal designs in wieless elay netwoks IEEE ansactions on Infomation heoy 53 (Novembe (11)) (007) [8] E. Agell. Eiksson A. Vady K. Zege Closest point seach in lattices IEEE ansactions on Infomation heoy 48 (August) (00)