Multihop MIMO Relay Networks with ARQ

Transcription

1 Multihop MIMO Relay Netwoks with ARQ Yao Xie Deniz Gündüz Andea Goldsmith Depatment of Electical Engineeing Stanfod Univesity Stanfod CA Depatment of Electical Engineeing Pinceton Univesity Pinceton NJ Abstact A multiple antenna multihop elay netwok consisting of a souce a elay and a destination node is consideed The divesity-multiplexing-delay tadeoffs (DMDT fo vaious multihop ARQ potocols ae obtained It is shown that the tadeoff egion is limited by the pefomance of the weakest link and hence the optimal ARQ potocol should balance the link pefomances by allocating the ARQ ounds among all links Based on this agument a Vaiable Block-Length ( potocol is poposed and its DMDT-optimality is shown I INTRODUCTION Multiple input-multiple output (MIMO systems can povide inceased data ates by ceating multiple paallel channels and obustness against channel vaiations by inceasing divesity Relaying povides simila divesity and/o multiplexing gains by exploiting the esouces of neaby teminals To obtain the coopeative divesity povided by elaying the destination combines the signals fom the souce and the elay to decode the undelying message [] [] In hop-by-hop elaying each teminal eceives the signal only fom the pevious teminal in the oute and hence the elays ae used fo coveage extension athe than inceasing divesity Hee we study a multihop MIMO elay system Anothe degee of feedom can be intoduced by an automatic epeat equest (ARQ potocol fo etansmissions With the multihop ARQ potocol the eceive at each hop feeds back to the tansmitte a one-bit indicato on whethe the message can be decoded o not In case of a failue the tansmitte sends additional paity bits until eithe successful eception o message expiation To chaacteize and compae the pefomances of vaious ARQ potocols we use the divesitymultiplexing-delay tadeoff (DMDT analysis The divesity-multiplexing tadeoff (DMT in a point-to-point MIMO system is intoduced in [] Consideing a thid dimension of delay in this high SNR analysis the DMDT analysis fo a point-to-point MIMO system with ARQ is studied in [] and the DMDT cuve is shown to be the scaled vesion of the coesponding DMT cuve without ARQ In this wok we extend the point-to-point DMDT analysis to multihop MIMO systems In ou model thee is no diect link between the souce and the destination and a elay enables the communication We constain the elay to half-duplex opeation that is it cannot tansmit and eceive at the same time Due to the half-duplex constaint an impotant issue is the allocation of the tansmit and listen times of the elay Fo coopeative elaying a dynamic decode-andfowad (DDF potocol is poposed and shown to dominate all existing potocols in tems of the DMT pefomance in [5] In DDF the elay listens to the souce tansmission until it can decode the message and then stats tansmitting jointly with the souce In a coopeative elay channel the DMT pefomance of DDF falls shot of the cut-set bound [5]; howeve it is shown to achieve the optimal DMT pefomance in multihop MIMO netwoks in [] In this wok we extend the DMT analysis in [] to include the delay dimension intoduced by ARQ A elated wok [7] consides the DMDT fo coopeative tansmission fo single antenna teminals Suppose that a maximum of L ARQ ounds is allowed fo the tansmission of each message fom the souce to the destination The DMDT analysis eveals that the system pefomance is limited by the weakest link Hence the optimal ARQ potocol should allocate the numbe of ARQ ounds among the hops to balance thei pefomances We conside two types of ARQ potocols: fixed and adaptive We futhe pesent two types of adaptive ARQ potocols: fixed block-length ( and vaiable-block-length ( We study the DMDT unde both the the long-tem static and the shottem static channel assumptions [] The DMDT has closed-fom expessions in some special cases and can be cast into a convex optimization poblem in geneal We also pove that the potocol achieves the optimal DMDT in multihop MIMO elay netwoks The may be a suboptimal appoximation when a single teminal is allowed to tansmit within each channel block While we pesent ou esults fo

2 ARQ ARQ M M M H H S R D Fig : The (M M M multihop MIMO elay netwok with ARQ a single elay extension to multiple elays is possible as in [] The est of the pape is oganized as follows Section II and Section III intoduce the system model and the ARQ potocol espectively Section IV pesents ou DMDT analysis fo vaious ARQ potocols Some illustative numeical examples ae shown in Section V Finally Section VI concludes the pape II SYSTEM MODEL We conside a thee node multihop elay netwok denoted by (M M M consisting of a souce a elay and a destination with M M and M antennas espectively (see Fig At the souce the message is encoded by a space-time encode and mapped into a sequence of L matices X l C M T : l = L} whee T is the block-length and L is the maximum numbe of ARQ ounds We conside using the DDF potocol fo tansmission which is descibed in moe detail below The decoded message at the elay is eencoded by a space-time encode into a sequence of L matices X l C M T : l = L} The souce-elay and the elay-destination channels ae given by: Y il = SNR M i H il X il + W il l L i ( fo i = espectively L i is the numbe of ARQ ounds used by link i and Y il C Mi+ T i = ae the eceived signals at the elay and the destination espectively in the lth ARQ ound Channels ae assumed to be fequency non-selective block Rayleigh fading and independent of each othe ie the enties of the channel matices H il C Mi+ Mi ae independent and identically distibuted (iid complex Gaussian with zeo mean and unit vaiance The additive noise tems W il ae also iid complex Gaussian with zeo mean and unit vaiance Thee is no diect link fom the souce to the destination and no feedback link fom the destination to the souce: fowad and feedback links only exist between the souce-elay and the elaydestination pais Seveal othe key assumptions in ou model ae as follows: (i The elay is half-duplex (ii Thee is a constaint on the total numbe of ARQ ounds L +L = L o equivalently a tansmission time constaint of t + t = LT whee t and t ae the numbe of channel uses of the two hops (iii We assume shot-tem powe constaints at the souce and the elay fo each block code given by Et(X il X il} M i T i = whee E } denotes expectation t is matix tace opeato and denotes the Hemitian tanspose (iv We conside both the long-tem static channel in which H il = H i fo all l ie the channel state independent fo diffeent i emains constant duing all the ARQ ounds; and the shot-tem static channel in which H il ae iid but not identical We emak that powe contol is not consideed in ou model as dictated by (iii The goal is to isolate the ARQ gain fom the powe contol gain We also note that the long-tem static channel assumption is wostcase in tems of the achievable divesity with a maximum of L i ARQ ounds [] because thee is no tempoal (o fequency inteleaving gain The shot-tem static channel model povides additional time-divesity hence highe obustness against fading III MULTIHOP ARQ PROTOCOLS We conside seveal ARQ potocols and compae thei DMDTs These multihop ARQ potocols all use standad point-to-point ARQ etansmissions and diffe in the way they allocate the total numbe of ARQ ounds among the hops To simplify the analysis the ACK/NACK feedback is assumed to be eo-fee and zeo-delay A Fixed ARQ Pocotols One can allocate a fixed numbe of ARQ ounds L and L fo each link subject to L + L = L The allocation is fixed egadless of the instantaneous link quality In the kth ound the souce tansmits the block X k to the elay Afte eceiving the whole block the elay ties to decode the message If the elay fails to decode it sends a negative acknowledgement (NACK message to the souce which tigges the souce to tansmit the next block of the cuent message X k+ If the souce eceives a NACK at the end of block L it moves on to the next message On the othe hand if the elay succeeds in decoding in the kth ound k L it sends an acknowledgement message (ACK to the souce and stats tansmitting X to the destination A simila ARQ scheme is applied to the elay-destination hop The

3 elay keeps fowading to the destination until eithe L ounds of ARQ has been eached o the destination sends an ACK Upon eceiving an ACK signal fom the destination the elay sends a second ACK signal to the souce Afte eceiving the fist ACK fom the elay the souce stops tansmitting and it stats the tansmission of a new message if eithe it eceives a second ACK fom the elay o L ARQ ounds have been eached afte the fist ACK B Adaptive ARQ Potocols One can also allocate a vaiable numbe of ARQ ounds fo each hop adapted to the instantaneous link quality The following two adaptive ARQ potocols ae designed depending on whethe the elay can stat tansmitting only at the beginning of a block o at any time within a block Fixed Block-Length ( Potocol: One type of adaptive ARQ potocol is the Fixed Block- Length ( potocol in which the elay can stat fowading only at the beginning of a channel block In the souce-elay link can use up to L ounds fo etansmission If the elay succeeds in decoding afte the τth ound τ L then it can use L τ ARQ ounds to tansmit the message to the destination Note that τ depends on the instantaneous souce-elay link quality To be moe specific in the fist hop the souce etansmits each time it eceives a NACK fom the elay Once the elay is able to decode it stats tansmitting at the beginning of the next channel block In the second hop the elay etansmits each time it eceives an NACK fom the destination Once the destination is able to decode (o fail to decode in L τ ounds it sends an ACK (o NACK to the elay and now the elay sends an ACK to the souce to initiate the tansmission of a new message Vaiable Block-Length ( Potocol: Anothe type of adaptive ARQ potocol is the Vaiable Block-Length ( potocol in which the elay can stat fowading at any time within a block IV DMDT ANALYSIS The DMDT chaacteizes the tadeoff among the data ate (the multiplexing gain the eliability (the divesity gain and the time divesity gain (the allowed amount of delay o total numbe of ARQ ounds We conside a family of space-time codes C(ρ indexed by thei opeating SNR ρ which has ate R(ρ and eo pobability P e (ρ Fo this family the multiplexing gain and the divesity gain d ae defined by = lim ρ R(ρ log ρ log P and d = lim e(ρ ρ log ρ It is shown in [] that the eo pobability is dominated by the infomation outage pobability P out (ρ when the block length is sufficiently long Fo each define f MM ( as the supemum of the divesity gain d ove all families of codes fo a MIMO system with M tansmit and M eceive antennas The chaacteization of the divesitymultiplexing tadeoff (DMT f MM ( fo a point-topoint MIMO system is given by the following theoem [] Theoem Fo a sufficiently long block-length L M + M the divesity-multiplexing tadeoff (DMT f MM ( fo a MIMO system with M tansmit and M eceive antennas is given by the piece-wise linea function connecting the points ( (M (M fo = min(m M A Long-Tem Static Channel Fist we analyze the DMDT of the afoementioned ARQ potocols fo the long-tem static channel model The poofs fo all the theoems below can be found in [8] Theoem The DMDT of the fixed ARQ potocol in a two-hop MIMO elay netwok with long-tem static channel L ARQ ounds on the fist hop and L L ARQ ounds on the second hop is given by d F ( L L = min f MM ( L f MM ( L L Theoem shows that the system pefomance is limited by the weakest link This implies that the optimal choice of ( the L i s should( equalize the DMTs such that f MM L = f MM L L When the solution is not an intege we conside the two integes closest to the solution and choose the one with the highe divesity gain Fo some special cases thee ae closed fom solutions captued in the following coollay Coollay Fo an (M M system letting A = M M and L ( = βl when A the optimal ARQ allocation is found by solving β = A +(A+± [A+(+A ]( A (A whee the sign is chosen so that β ( When A = we have β = If the solution is not an intege we check the two closest integes fo the highest divesity gain Similaly fo an (M M M system the optimal numbe of ounds fo the fixed ARQ potocol is given by L = L = L independent of the multiplexing gain If the solution is not an intege we check the two closest integes fo the highest divesity gain }

4 DMT of ( multi hop elay channel long tem static L = 5 5 Fixed ARQ L = L = L/ Fixed ARQ optimal L L f ( f ( DMDT of ( multi hop elay channel long tem staic L = 5 Fixed ARQ L = L = L/ Fixed ARQ optimal L L f 5 ( d( d( Fig : The DMDT of a ( system with L = Fig : The DMDT of a ( system with L = The DMDT of the potocol is also a piecewise linea function as stated in the next theoem Theoem The DMDT of the potocol in a two-hop MIMO elay netwok with a long-tem static channel is given by ( d F BL ( L = min f MM l= L L ( ( f MM + f MM l L l ( } f MM L In paticula we have a closed-fom DMDT expession fo an (M M system Coollay 5 The DMDT of the potocol fo an (M M system is given by ( d F BL ( L = min M + l= L L ( M + ( + M + l L l ( M } + L Next we chaacteize the DMDT of the potocol The DMDT of the potocol unde the long-tem static channel assumption is simila to the DMT of DDF without ARQ given in [] with pope scaling of the multiplexing gain Theoem The DMDT of a potocol fo an (M M M system in the long-tem static channel is given by d V BL ( L = inf (α α O M i (j + M i M i+ α ij i= j= whee Mi minm i M i+ } α ij ae the exponents of the channel matix eigenvalues and O (α α R M R M : α i α im i L > S } (α S (α S (α + S (α whee S i (α i M i j= ( α ij + Coollay 7 ( The DMDT of a potocol in an (M M system with a long-tem static channel is given by minm d V BL ( L = M } /L /L L/; othewise ( The DMDT of a potocol in a ( M two-hop MIMO elay netwok with a long-tem /L static channel is given by d V BL ( L = M /L fo L/ and othewise ( The DMDT of a potocol in a ( two-hop MIMO elay netwok with a long-tem static channel is given by ( 5/L /L L/; /L d V BL ( L = /L L/ /L; ( ( /L /L /L L B Shot-Tem Static Channel Compaed to the fixed ARQ the adaptive ARQ potocols ae moe flexible and hence achieve bette DMDT

5 d(l d(l DMDT of ( VBL DMDT of ( FBL Fig : DMDT of a ( system with the potocol (left and the potocol (ight pefomance The DMDT of the unde the shot-tem static channel model is given as follows Theoem 8 The DMDT of a potocol in an (M M M system with a shot-tem static channel is given by d F BL ( = min k= L (L f MM ( (k f MM ( k (L f MM ( L 5 L 5 L L ( + (L kf MM L k } Then one can deive a simila coollay to Coollay 5 fo Theoem 8 Compaing the above esult with that of the long-tem static channel model in Theoem we note that each pat of the piece-wise linea DMT function is multiplied by a gain facto due to the time divesity We do not have a closed-fom expession fo the DMDT of the potocol in the shot-tem static channel since a closed-fom solution fo the numbe of ARQ blocks (τ afte which the elay decodes is andom athe than deteministic as in the pevious case Hee the accumulated mutual infomation at the elay is a andom walk with a positive dift Hence the stopping time τ when this andom walk hits the theshold R is andom Theoem 9 The DMDT of the potocol in a two-hop MIMO elay netwok in the shot-tem static channel is d V BL ( L = inf (α α O M i L (j + M i M i+ αij l i= j= l= whee αij l s ae the exponents of the channel matix eigenvalues in the lth channel use The feasible domain is defined as O = O = (α α : L t l= S (α l + } ( γs (α L t < (α α : αi l αim l i } l ( whee t t = inf t R : S (α l + (t t S (α t + = i= and γ = t t ( ae detemined fom t l= S (α l + γs (α t + = fo a given while O (R M L R M L (O O ( and S i (α l i = M i j= ( αl ij + Theoem The achieves the optimal DMDT in the multihop MIMO elay netwok with ARQ in both the long-tem and shot-tem static channel models V NUMERICAL EXAMPLES We fist pesent a set of examples fo the long-tem static channel model Conside the ( system and a maximum of L = ARQ ounds In Fig we show the DMDT of the fixed ARQ scheme with L = L = and with the optimal L L given in Coollay as well as the DMDT of FBL and s The DMDT of the is a piece-wise appoximation to that of the which is the optimal DMDT Similaly the DMDTs of all the ARQ potocols in a ( system ae shown in Fig Fig shows the thee dimensional DMDT sufaces of the VBL and the s espectively fo a ( system Fig 5 shows the coss sections of the sufaces in Fig at L = and L = Note that the DMDT cuve not only stetches as L inceases like in the point-to-point 5

6 d( DMDT of ( System L = d( 8 DMDT of ( FBL L = Shot Tem Static Channel Long Tem Static Channel DMDT of ( System L = 8 DMDT of ( FBL L = 8 Shot Tem Static Channel Long Tem Static Channel d( d( Fig 5: The slices of the DMDT sufaces in Figue at L = (left and L = (ight 8 Fig : The DMDT fo a ( system in the long-tem static channel (left and in the shot-tem static channel (ight with L = MIMO system [] but also has moe pieces fo a lage L As L inceases the DMDT of becomes a bette appoximation to the DMDT of the The DMDT of a ( system unde the shot-tem static channel model fo the is shown in Fig Also note that the DMDT in the shot-tem static channel is not necessaily that of the long-tem static channel multiplied by L In this espect the esults in the multihop MIMO elay netwok diffes fom those in the point-to-point MIMO channel [] VI CONCLUSIONS We have deived the divesity-multiplexing-delay tadeoff (DMDT fo multihop MIMO elay netwoks We have consideed both the long-tem and shot-tem static channel models We have deived closed-fom expessions fo the DMDT in some special cases and showed some illustative numeical examples Finally we have poved that the potocol achieves the optimal DMDT unde both channel models Ou esults pesented hee fo a single elay case can be extended to the multiple elay scenaio as well REFERENCES [] A Sendonais E Ekip and B Aazhang Use coopeation divesity pat I: System desciption IEEE Tans on Comm vol 5 pp Nov [] J N Laneman D N C Tse and G W Wonell Coopeative divesity in wieless netwoks: Efficient potocols and outage behavio IEEE Tans Infom Theoy vol 5 pp 8 Dec [] L Zheng and D N C Tse Divesity and multiplexing: A fundamental tadeoff in multiple-antenna channels IEEE Tans Infom Theoy vol 9 pp 7 9 May [] H El Gamal G Caie and M O Damen The MIMO ARQ channel: Divesity-multiplexing-delay tadeoff IEEE Tans Infom Theoy vol 5 pp Aug [5] K Azaian H El Gamal and P Schnite On the achievable divesity-multiplexing tadeoffs in half-duplex coopeative channels IEEE Tans Infom Theoy vol 5 no 5 [] D Gündüz A Khojastepou A Goldsmith and H V Poo Multi-hop MIMO elay netwoks: Divesity-multiplexing tadeoff analysis submitted to IEEE Tansactions on Wieless Communications 9 [7] T Tabet S Dusad and R Knopp Divesity-multiplexing-delay tadeoff in half-duplex ARQ elay channels IEEE Tans Infom Theoy vol 5 pp Oct 7 [8] Y Xie D Gündüz and A Goldsmith Multihop MIMO elay netwoks with ARQ in pepaation 9