Space Diversity for Multi-antenna Multi-relay Channels

Transcription

1 Space Divesity fo Multi-antenna Multi-elay Channels Yijia Fan an John Thompson Institute fo Digital Communications School of Engineeing an Electonics Univesity of Einbugh Einbugh, EH9 JL, UK Abulaeem Ainoyi an Halim Yaniomeoglu Boaban Communications an Wieless Systems BCWS) cente Depatment of Systems an Compute Engineeing Caleton Univesity Ottawa, KS 5B6, Canaa Abstact In this pape we analyze the pefomance of multiple elay channels when multiple antennas ae eploye only at elays. We apply two antenna ivesity techniques at elays, namely maximum atio combining MRC) on eceive an tansmit beamfoming TB). We show that with K elays the netwo can be ecompose into K ivesity channels each with a iffeent channel gain, an that the signals can be effectively combine at the estination. We assume that the total numbe of antennas at all elays is fixe at. If the total tansmit powe fo all elays ae the same as fo the souce an equally istibute among all the elays, the netwo capacity will be lowe boune by that of elay channels each with single antenna, an uppe boune by that of single elay channels with antennas. I. ITRODUCTIO It is wiely believe that a hoc netwoing [] o multihop cellula netwos [] ae impotant new concepts fo futue geneation wieless systems [], whee eithe mobile o fixe noes often efee to as elays) ae use to help fowa the infomation to the esie use. One avantage of these stuctues ae that it is possible to unite multiple elays in the netwo as a vitual antenna aay to fowa the infomation coopeatively, while appopiate combining at the estination ealizes ivesity gain. The ivesity achieve in this way is often name as use coopeation ivesity o coopeative ivesity [4], as it mimics the pefomance avantages of multiple-input multiple-output MIMO) systems [5] in exploiting the spatial ivesity of the elay channels. The pefomance limits of space-time coes, which can exploit coopeative ivesity, ae iscusse in [6] [8] fo singleantenna elay netwos. Fo multiple-antenna elay channels whee evey teminal in the netwo can be eploye with multiple antennas, stuies ae mainly concentate on spatial multiplexing systems [9] []. In this pape we exploit the spatial ivesity of the elay channels in a iffeent way fom space-time coes base appoach. We apply two ins of antenna combining techniques at the elay, namely maximum atio combining MRC) [] fo eception an tansmit beamfoming TB) [] fo tansmission. Those techniques wee often use in point-topoint wieless lins to enhance signal-to-noise atio SR) at the output of the eceive by eploying multiple antennas at eithe tansmitte o eceive. In a elay context, we move the multiple antennas to the elays, while the souce an the estination ae only eploye with a single antenna. Ou investigation is base on igital elaying, whee the elays ecoe, e-encoe an e-tansmit the signals. We show that the netwo with K elays can be ecompose into K ivesity channels each with iffeent channel gain, an the signals fom all K banch can be effectively combine at the estination. We eive the capacity bouns fo this signal combining techniques. Ou analysis esults can be applie to both egoic capacity an outage capacity [4] pefomance. The est of this pape is oganize as follows. In Section II, the basic system moel an assumptions ae intouce. Section III intouces the signal combining techniques. The capacity pefomance analysis ae mae in section IV. Section V pesents an iscusses simulation esults an finally, conclusions ae awn in Section VI. II. SYSTEM MODEL We consie a two hop netwo moel with one souce, one estination an K elays. We ignoe the iect lin between souce an estination. We also assume that total tansmit powe of the souce an elays ae the same an that it is equally istibute among the elays. Each elay pocesses the eceive signals inepenently. We assume that the souce an estination ae eploye with single antennas, while elay is eploye with m antennas. We assume that the total numbe of antennas at all elays is fixe to. This can be expesse as m =. ) = We estict ou iscussion to the case whee the channels ae slow, fequency-flat faing. The ata tansmission is ove two times slots using two hops. In the fist tansmission time slot, the souce boacasts the signal to all the elay teminals. The input/output elation fo the souce to the th elay is given by = ηh s + n, )

2 whee is m eceive signal vecto. η enotes the tansmit powe at the souce. The s is the tansmit signal with covaiance an n is the m complex cicula aitive white Gaussian noise vecto at elay with ientity covaiance matix I m. The vecto h is the m channel tansfe matix fom souce to the th elay an can be futhe expesse as h = α h, ) whee each enty of h ae ientically inepenent istibute i.i.) complex Gaussian anom vaiables with unit vaiance. Each facto α contains the pathloss an can be witten as α = x γ, whee x is the istance between the souce an elay. The scala γ enotes the path loss exponent. In the secon hop, each elay pocesses its eceive signals an etansmits them to the estination. The signal eceive at the estination can be witten as: y = g + n, 4) whee the vecto g is the channel matix fom th elay to the estination, which might also be witten as: g = β g, 5) whee each enty of g is an i.i.. complex Gaussian anom vaiables with unit vaiance. The scala β also contains the pathloss fom the th elay to the estination. The scala n is the complex aitive white Gaussian noise at the estination with unit vaiance. The vecto is the tansmit signal vecto at elay, which shoul meet the total tansmit powe constaint: E [ ] F ηm, 6) whee F enotes the Fobenius nom. We assume a coheent elay channel configuation context whee the th elay can obtain full nowlege of both bacwa channel vecto h an fowa channel vecto g. Fo fai compaison, we also assume that fo each channel ealization, eithe bacwa o fowa channel coefficients fo all antennas emains the same egaless of the numbe of elays K. It will not be ifficult to see that the conclusions on eithe Egoic capacity o outage capacity in this pape also hol if we exten the iscuss to a moe geneal case whee each antenna is fixe in the netwo. Fig. gives a esciption fo the system moel. III. ATEA DIVERSITY TECHIQUES I RELAY CHAELS In this section we apply MRC an TB techniques to the system moel escibe in section II. We assume each elay pefoms MRC of the eceive signals, / by multiplying the eceive signal vectos by the vecto h H h F. The signals at output of the elay eceive is given by = η h i, s + m m h i, n i, h i, 7) Souce Relay Relay Relay K Destination Fig.. System moel fo a two hop netwo: Souce an estination ae each eploye with antenna. Totally antennas ae eploye at K elays. Fo each channel ealization, eithe bacwa o fowa channel coefficients fo all antennas emains the same egaless of the numbe of elays K. whee h i, enotes the ith antenna at elay, an n i, enotes the noise facto fo ith eceive input banch. The SR at the output of the eceive can be witten as: ρ m = η h i,. 8) Afte the elays ecoe the signals, each elay then pefoms TB of the tansmitte signals. If we enote the tansmitte signals as with unit vaiance, the tansmitte signal vecto fo elay can be witten as ηm = g H g F. 9) The estination eceive simply etects the combine signals fom all K elays. If we ajust the tansmission ata ate so that the signals ae coectly ecoe at all the elays i.e. = s), the output signal at the estination can be witten as: y = s = ηm g i, + n = s g + n 0) = It can be seen fom 0) that by applying antenna ivesity schemes at elays, the netwos can be ecompose to K ivesity channels each with channel gain g. The output SR at the estination eceive can theefoe be witten as: ρ m = = ηm g i,. ) When all the elays ae eploye with a single antenna, thee is no taitional maximum atio combining gain at the elays an the estination. Howeve, the estination still obseves a set of equal gain combine [5] amplitue signals fom all elays. Diffeent fom [5], the equal gain combining fo elay channels is applie at the tansmitte instea of the eceive.

3 Since we assume that the bacwa an fowa channel coefficients fo each antennas ae ept same fo iffeent numbe of K an m i. The output SR at the estination can be ewitten as ρ = η K m i g i, ) ; ) = when all the antenna ae eploye in one elay i.e. K = an m = ), full ivesity gain is achieve among all the antennas at the elay an also at the estination. The SR can be ewitten as m i ρ = η g i, ) = IV. CAPACITY PERFORMACE In this section we eive the capacity bouns fo the scheme popose in pevious section. The netwo capacity fo igital elaying can be witten as C m D = min C,m,C,m,,C K,m,C m ) 4) whee C,m = 0.5log + ρ m ) enoting the Shannon capacity fom souce to elay channel, an C m = 0.5log + ρ m ) enoting the Shannon capacity fom elays to estination channels. The facto 0.5 enotes the half banwith compae with non-elay channels. We fistly analyze channel capacity fom the elays to estination lin by bouning the ρ m,i.e. the output SR at the estination. Lemma : Fo any m, ρ ρm ρ. Poof: See Appenix. Fom Lemma, we can see that C C m C, 5) whee C enotes the capacity fo elays to estination channels when K =, an C enotes the capacity fo elay to estination channels when K =. ow also consieing the capacity fom the souce to elays lin an extening the analysis to the whole netwo scenaio, we have the following theoem: Theoem : If we enote the netwo capacity fo K = as CD an fo K = as C D, fo any m, CD Cm D C D. Poof: Consieing the SR ρ m fo the souce to the th elay lin, if we enote it as ρ n fo K = an ρ fo K =, it can be shown that min ρ n) min ρ m ) ρ. 6) Theefoe, we have the following: min C,,,C, ) min C,m,,C K,m ) C,. 7) Combining 7) an 5), we thus complete the poof. Fom the above analysis we have shown that fo the signal combining techniques iscusse in the pape, the netwo capacity will be lowe boune by that of elay channels each with single antenna, an uppe boune by that of a 0% outage capacity Egoic capacity K=6,m= K=,m= K=,m= K=,m= P K=6,m= K=,m= K=,m= K=,m=6 a) 0% outage capacity P b) Egoic capacity Fig.. Capacity of single MIMO elay channels fo iffeent numbe of elays K, while each elay is eploye with m antennas. a) 0% outage capacity. b) Egoic capacity. single elay channel with antennas. This means that even thee ae moe elays, the incease equal gain combining gain at the estination can not compensate fo the loss of maximum atio combining gain at the elay an the estination when numbes of antennas at each elay ae euce. V. SIMULATIO RESULTS We calculate the both egoic capacity an 0% outage capacity in bits pe channel use) fo 000 channel ealizations egaing iffeent values of η, enote as P in the figues. In this simulation example we assume that the istance between souce an estination is nomalize. The elays ae unifomly an anomly locate in the mile egion between the souce an elays. Theefoe x is set to 0.5. We assume the total numbe of antennas at elays ) is 6 an we also assume that all K elays have the same numbe of antennas m. Fig. shows the capacity pefomance. We can see that fo iffeent K, m), the capacity is always uppe boune by,6) an lowe boune by 6,). These esults veify the analysis mae in this pape. Futhemoe, we can see though the simulation that lage m an small K might give lage

4 benefit, since lage m allows moe feeom of coopeation among the antennas at each elay. Theefoe when m eaches K euces to ), full coopeation ae mae among all the antennas to give ise to the best pefomance. VI. COCLUSIOS In this pape we analyze the pefomance of multiple elay channels when multiple antennas ae eploye only at elays. We apply antenna ivesity techniques at elays which ae nown as maximum atio combining an tansmit beamfoming. We show that with K elays the netwo can be ecompose into K ivesity channels each with a iffeent channel gain, while the signals can be effectively combine at the estination. If we assume that the total numbe of antennas at all elays ae fixe to an total tansmit powe at all elays ae nomalize, the netwo capacity will be lowe boune by that of elay channels each with single antenna, an uppe boune by that of single elay channels with antennas. APPEDIX PROOF OF Lemma We fistly pove that ρ ρm. We wite the following ρ m K ρ = ηm m g i, η g i,. = }{{}}{{} A B 8) To compae A with B, we wite A B= η m ) m g i, g i, 9) ) = η m ) g i, ote that η m ) g i, = j=,j i m =0.5 g i, g j,. 0) So 0) can be futhe witten as: A B = η = η m g j, ) g i, + g j, ).) g i, g i, g j, + g j, ) g i, g j, ) 0. So A B an theefoe ρ ρm. ext we pove that ρ ρmi. Fo simplicity, we enote a = g i, ) in equation ) an ). Then ρ ρmi can be witten as ρ ρ mi = η m )a mi m j a i a j. = 4) ote the constaint by ) in section II, we have the following: m ) =,i m i. 5) Putting 5) into 4), we have the following: ρ ρ mi = η m i a mi a i m j a j. ote the following: =,i m i a = =,i =0.5 m i a j ) We can futhe wite 6) as follows: ρ ρ mi = η η + η = η Theefoe ρ ρmi m i a j ) + m i a j ) mi a j m j a i m j a i ) mi a j m j a i ). an ρ ρm ρ. REFERECES 6) m j a i ). [] A. J. Golsmith, S. B. Wice, Design challenges fo enegy-constaine a hoc wieless netwos Wieless Communications, IEEE Pes. Commun., Vol. 9, no. 4, pp. 8-7, Aug. 00. [] R. Pabst et al, Relay-base eployment concepts fo wieless an mobile boaban aio, IEEE Commun. Mag., Sept [] M. Foigh, S. Pavall, C. Roobol, P.Johansson, Lasson, Futuegeneation wieless netwos IEEE Pes. Commun., vol. 8, no. 5, pp. 0-7, Oct. 00. [4] J.. Laneman, G. W. Wonell an D.. C. Tse, An efficient potocol fo ealizing coopeative ivesity in wieless netwos, in Poc. IEEE Int. Symp. Infomation Theoy, Washington, DC, June 00. [5] D. Gesbet, M. Shafi, Shiu Da-shan, P.J.Smith, A.aguib, Fom theoy to pactice: an oveview of MIMO space-time coe wieless systems, IEEE Jounal on Selecte Aeas in Communications, vol., no., pp. 8-0, Apil 00.

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