Diversity and Spatial Multiplexing of MIMO Amplitude Detection Receivers

Transcription

1 Dvrst and Spatal Multplxng of MIMO mpltud Dtcton Rcvrs Gorgos K. Psaltopoulos and rmn Wttnbn Communcaton Tchnolog Laborator, ETH Zurch, CH-809 Zurch, Swtzrland Emal: {psaltopoulos, bstract W consdr nonlnar MIMO sstms that us ampltud-onl (nvlop) rcvrs. Such sstms ar ntrstng for low-powr, low-complxt applcatons lk snsor ntworks. W stud dffrnt modulaton schms and th rspctv dvrst ordr obtand b th maxmum lklhood dtctor n an uncodd sstm wth prfct channl stat nformaton. W show that nonlnar MIMO ampltud rcvrs loos half th avalabl rcv dvrst whn th uncodd transmsson rat ncrass abov 1 bt/s/hz, or whn spatal multplxng s prformd. I. INTRODUCTION Nonlnar MIMO sstms hav bn ntroducd n [1] and []. Th ar MIMO sstms that mplo ampltudonl or phas-onl rcvrs on ach rcvr antnna. Th complx-valud rcvd sgnal undrgos a nonlnar procssng, whr thr th ampltud or th phas s xtractd. Ths rcvr structur s sgnfcantl smplr and lss powrconsumng compard to th standard I/Q rcvr. Combnng an ampltud or phas rcvr wth multpl antnnas ams at takng advantag of both, th hgh data rat gans provdd b th spatal dmnson, and th low-complxt, low powrconsumpton proprts of ampltud or phas dtcton. Such sstms can b mplod n snsor ntworks or smlar applcatons, whr strct powr and cost constrants dmand smpl and ffcnt rcvrs. Howvr, nonlnar MIMO sstms hav not bn consdrd n ltratur t and lttl s known rgardng thr fundamntal proprts. Th nformaton thortc lmts of nonlnar MIMO sstms wr nvstgatd n [1] for th cas of prfct channl stat nformaton (CSI) at th rcvr, and n [] for nos CSI. It was shown that such sstms also xplot th spatal dmnson, howvr n a dffrnt wa compard to lnar MIMO sstms [3]. Spcfcall, an N N nonlnar MIMO sstm achvs N spatal multplxng gan at hgh SNR, whch mans that half th dgrs of frdom ar lost du to dmnsonalt loss nducd b th nonlnart. Howvr, nonlnar MIMO sstms ncras thr spatal multplxng gan bond N whn mor rcv antnnas ar mplod (N R N), contrar to a lnar MIMO sstm [1]. Nvrthlss, th rsultng spatal multplxng gan nvr rachs th lmt (optmum) st b th lnar MIMO sstm. Th nformaton thortc rsults prsntd n [1] and [] rval th potntal of ths sstms. In ths papr, w dal wth a mor ralstc sstm and consdr practcal modulaton schms and th prformanc of th maxmum lklhood (ML) dtctor. W xplor th dvrst ordr of th ML dtctor for uncodd transmsson and for dffrnt transmsson rats wth and wthout spatal multplxng. In ths framwork, w onl consdr MIMO ampltud dtcton rcvrs and assum that th rcvr has prfct knowldg of th fadng channl. W consdr on-off kng (OOK) and ts xtnson to multpl transmt antnnas as th basc modulaton schm, snc t s a straghtforward choc for ampltud dtcton. Contrar to convntonal OOK whch s alwas usd n conjuncton wth non-cohrnt communcaton n ltratur [4], n our cas w dal wth a cohrnt sstm, n th sns of channl knowldg avalablt at th rcvr. Fnall, w consdr modulaton across th transmt antnnas n on channl us (no STC) and nvstgat onl rcv dvrst. Th papr s structurd as follows. In Scton II w prsnt th sstm modl. In Scton III w consdr th parws rror probablt btwn an two transmt smbols, and analz th problm usng two tps of scalar dtcton problms. W show that th dvrst ordr of ths dtcton problms s 1 and 1, rspctvl. In Scton IV w show that th dvrst ordr of th corrspondng vctor dtcton problms s uppr boundd b N R and NR, rspctvl. Th dvrst loss of th scond dtcton problm occurs whnvr th uncodd transmsson rat ncrass bond 1 bt/s/hz, or whn spatal multplxng s prformd. W prsnt smulaton rsults n Scton V whch vrf our thortcal rsults. Notaton: Throughout th papr, bold-facd talc lowr and uppr cas lttrs stand for vctors and matrcs, rspctvl. b s th th lmnt of vctor b. Ô Õ H dnots complxconjugat transposton. I N s th N N dntt matrx. Th crcularl smmtrc complx Gaussan dstrbutd random varabl x wth man m and covaranc matrx R s dnotd b x CN Ôm,RÕ. δ,j s th Kronckr dlta. II. SYSTEM MODEL W consdr th MIMO sstm dpctd n Fg. 1, wth N T transmt and N R rcv antnnas. Th transmttr mts th sgnal s È C NT ovr th statonar mmorlss flat fadng channl H È C NR NT, wth tap-gan H j from th jth transmt to th th rcv antnna. W us a block fadng modl for th channl H. Th channl rmans constant for som (cohrnc) prod, long nough to allow for accurat stmaton, and changs to an ndpndnt ralzaton n th nxt block. W assum that th channl s prfctl known at th

2 s Fg. 1. H x w r MIMO Sstm Rfrnc Modl rcvr. Th rcvd vctor x È C NR s prturbd b a zroman crcularl smmtrc Gaussan nos vctor w È C NR, wth autocorrlaton functon EÖw k wl H σ w I N R δôk lõ, whr k and l ar smbol nstants. Th nvlop of th rcvd sgnal s xtractd on ach antnna, producng th obsrvaton x w on th th antnna. Th dtctor has accss onl to. Th common modulaton schm for th SISO cas s OOK,.., th transmttr mts on of th smbols s È Ø0, E b Ù wth qual probablt. OOK has bn studd xtnsvl n oldr publcatons for th WGN channl [4]. III. PIRWISE ERROR PROILITIES W ar ntrstd n th avrag parws rror probablt (PEP) btwn two quall lkl smbols s and s, whr s Y Ös Y 1,...,s Y N T T, Y È Ø,Ù. Furthrmor,E s Ðs Y Ð s th sgnal nrg. For th tm bng w do not dfn th structur of th smbols. Th corrspondng rcvd sgnal on th th antnna s Y N ô T j1 h j s Y j w Y f s s σw x Y w, Y È Ø,Ù. (1) Snc th rcvr has prfct knowldg of H, th dstrbuton fô s s Y Õ fô x Y Õ s Rcan, gvn b Å x Y σw, () I 0 x Y σ w whr I 0 Ô Õ s th zroth ordr modfd ssl functon [5]. If x Y 0 th dstrbuton rducs to a Ralgh dstrbuton. s can b sn, x Y appars n () onl through ts norm. Hnc, th sam dstrbuton also dscrbs th followng dtcton problm Y, (3) x Y w ½ whr w ½ CN Ø0,σwIÙ has th sam nos statstcs as w. That s, (1) and (3) ar quvalnt dtcton problms wth rspct to th ML dtctor. Eq. (3) s anothr manfstaton that th phas of th rcvd sgnal carrs no nformaton. W wll consdr dtcton problm (3) n th followng. Th dstrbuton of x Y gvn s Y s Gaussan x Y CN Ø0,σh E sù,. Furthrmor, x Y s ndpndnt across dffrnt antnnas E x Y x Y, j δ,j σ he s (4) for th sam smbol Y. Ths s n gnral not tru among dffrnt smbols at th th rcv antnna E x x, Nô T σh j1 s j s, j, (5) whr th corrlaton dpnds on th structur of th transmttd sgnals. Snc th nos s ndpndnt btwn dffrnt rcv antnnas, th dstrbuton of fôx Y Õ can b factord lk f x Y õn R 1 f x Y. (6) Consquntl, th log-lklhood rato can b wrttn as LLR ln f x f Ôx Õ Nô R 1 ln f x, f x (7) whch s th sum of th LLRs at ach rcv antnna. W wll trat th dtcton problm at ach antnna sparatl. Th ML dtctor wll optmall combn th lklhood ratos at ach antnna bfor takng th fnal dcson. Th prformanc of th vctor dtcton problm s closl connctd to th dtcton problm at ach antnna. Lt us now consdr th scalar dtcton problm at th th rcv antnna. W dstngush btwn two cass: thr x Y 0, whn s Y 0, or x Y 0 whn at last on ntr of s Y s non-zro. Ths sparaton lads to two dstnct scalar dtcton problms 1 : D1 : w ½, s 0, x w ½, s 0,x 0 D : x w ½, s 0,x 0 x w ½, s 0,x 0 Th rcvr has to dcd btwn a Ralgh and a Rc dstrbuton for th two hpothss n D1 (s ()), and btwn two Rc dstrbutons n D. Evaluatng th probablt of rror for D1 and D analtcall s not possbl to th bst of our knowldg. n approxmat analtc xprsson xsts for D1 for th WGN channl. For D, w wll us an auxlar dtcton problml whos rror probablt s a lowr bound to D.. Dvrst ordr of D1 W wll prform an approxmat analss for D1. Snc n ths cas s 0, w st Ðs Ð E s to kp th total nrg of both smbols normalzd. Consquntl th varanc of x s σh E s. W us th normalzd varabl x x ß E s, whr x CN Ô0,σh Õ. dcson thrshold for th ML dtctor can not b computd analtcall du to th non nvrtblt of th ssl functon. Furthrmor, th optmum thrshold s a functon of th SNR. good approxmaton for th optmum thrshold s gvn n [4]. For hgh SNR, th thrshold convrgs to th man of th two 1 W dnot wth D 1 and D th rspctv vctor dtcton problms. (8) (9)

3 x Y L w ½ D L Fg.. Dtcton Problms D and L. noslss rcvd smbols, x ß. n approxmaton to th rror probablt for hgh SNR s gvn n [6] P D 1 x 1 x Ô4π x γõ 1ß4 γ, (10) b substtutng γ wth x γ n ordr to tak fadng nto account, and γ s th SNR pr rcv antnna γ Es σ. w vragng ovr x, whch ncorporats th fadng channl coffcnts, w obtan th avrag probablt of rror P D 1 f x x P D1 x d 1ß4Γ 3 x π 4 0 ÔγÔ γõ 3 Õ 1ß4, (11) whr f x Ô x Õ s a Ralgh dstrbuton and w usd σh 1. Th Talor srs xpanson of (11) for hgh SNR lds π P D 1 1ß4Γ 3 4 γ whch provs that th dvrst ordr of D1 s 1.. Dvrst ordr of D Lt us dfn th dtcton problm D Å 1 O γ, (1) L : x w½, s 0,x 0 x w½, s 0,x 0 (13) D s obtand from L b takng th norm of L as dpctd n Fg.. Snc D s a procssd vrson of L, th avrag probablt of rror of L s alwas lowr or qual to that of D. P L P D. (14) In fact procssng wll thr lad to a suffcnt statstc, and hnc to dntcal rror prformanc, or wll dstro nformaton, and thus dtrorat rror prformanc. From (14) w obsrv that th dvrst ordr of D can not xcd th dvrst ordr of L and hnc L provds an uppr bound. W can prform th sam lowr boundng to th rror probablt of D1 too, howvr ths s not ncssar snc (1) provds th xact dvrst ordr. Th probablt of rror of L s radl obtand, snc t s a standard ral-valud dtcton problm n Gaussan nos. W us agan th normalzd varabls x Y x Y ß E s, Y Ø,Ù, such that x Y CN Ô0,σh Õ, and obtan [7]: P L x, x Q x x γ Å. (15) t hgh SNR, th Rc dstrbutons nd can b approxmatd b Gaussan dstrbutons (cf. [8]) and th probablt of rror of D s wll approxmatd b (15). Hnc, th bound s tght at hgh SNR. In ths rgm, th dtcton thrshold s th man of x and x. Furthrmor, th norm xy s mportant, manng that th dstrbuton of th powr of s Y across th transmt antnnas dos not chang th probablt of rror, whch s onl affctd b th total powr of s Y. Th avrag probablt of rror s computd b avragng ovr x and x P L ø 0 f x, x x, x P L x, x d x d x. (16) s w saw arlr, x and x ar ndpndnt onl f NT j1 s j s, j 0. In ths cas, thr jont pdf s dcomposd n th product of two Ralgh dstrbutons, and (16) can b computd analtcall n (17) at th bottom of ths pag. Th Talor srs xpanson for hgh SNR lds 1 P L γ 1ß O 1 γ 3ß Å, (18) whr w usd σh 1. Thus, th dvrst ordr of L s onl 1, whn x and x ar ndpndnt. Othrws, th jont pdf s a two dmnsonal Ralgh dstrbuton, whch s known analtcall [9]. Howvr, th avrag probablt of rror can not b computd analtcall anmor. Smulatons show that th dvrst ordr n ths cas s 1, manng that th corrlaton dos not affct th dvrst. IV. DIVERSITY ND SPTIL MULTIPLEXING So far w consdrd th scalar dtcton problm at ach antnna sparatl. Howvr, th ML dcodr jontl dtcts th transmttd sgnal usng th whol rcvd vctor and (7). Unlk for lnar MIMO rcvrs, comng up wth a combnaton schm,.g. maxmum rato combnng, s not possbl n ths cas. Ths would sgnfcantl smplf th ML xprsson. Howvr, w know that th ML, or an combnaton schm for that mattr, wll achv at most a dvrst qual to th numbr of rcv antnnas (ndpndnt Th Gaussan approxmaton s n ths cas tghtl concntratd around th man, and th ngatv part of th dstrbuton can b nglctd. P L 8γ γ γ 3ß 4 γ 1 γ 4 γ Ô4 8 γõ γ 6 4γ 4 γ 8 γ Ô γõ 4 γô (17) 4 γõ

4 scalar dt. problm D 10 scalar dt. problm L EbßN Lnar PSK, R= bts/smbol mpl. Dtcton, R=1 bt/smbol mpl. Dtcton, R= bts/smbol PEP btwn Ö0,1 T and Ö1,0 T PEP btwn Ö0,1 T and Ö1,1 T PEP btwn Ö1,0 T and Ö1,1 T Fg. 3. of scalar D and L. Fg. 4. plot for a MIMO sstm. branchs) tms th dvrst ordr of th ndvdual dtcton problms. Ths mans, that for th vctor dtcton problm D 1, th ML dtctor wll achv a dvrst of N R, whl for D, th dvrst wll not b largr than NR. Notc that (7) s th sam n th cas of a 1 N rcv dvrst schm. Th achvabl dvrst n that cas quals N tms th dvrst of th dtcton problm on ach rcv branch. Whn t coms to valuatng th dvrst of a modulaton schm wth mor than two smbols, th poorst PEPs domnat th prformanc. s w saw, th fundamntal dffrnc btwn problms D 1 and D ls n th achvd dvrst. Howvr, D 1 corrsponds to s 0, whch mans that th dvrst of th PEP btwn an two othr smbols, not qual to 0, wll b at most NR. Ths occurs whnvr w prform spatal multplxng, snc n ths cas w wll hav at last NT smbols, and NT 1 of thm wll not b qual to th all-zro smbol. Hnc, prformng spatal multplxng wth a MIMO ampltud dtcton rcvr rducs dvrst to on half of th maxmum avalabl dvrst. Th onl wa to achv th maxmum dvrst s b constranng th rat to onl two smbols, s 0 and s 0,.. to a rat of 1 bt/s/hz. In ths cas, th multpl antnnas at th transmttr ar of no us, and a sngl-nput multpl-output sstm suffcs. Exampl: Lt us consdr th xtnson of OOK to multpl transmt antnnas. W us NT smbols of th form s È Ø0, E s ßN T Ù NT (s th ppndx). Th uncodd rat of th modulaton alphabt s N T bts/s/hz, achvd through spatal multplxng. Th PEP btwn th all-zro smbol and an othr smbol has th form of D 1, and dvrst ordr qual to N R. Howvr, th PEP of all othr combnatons of smbols has th form of D. Thos combnatons whr at last on smbol has a zro for vr j 1,...,N T, fulfll NT j1 s j s, j 0 and th corrspondng uppr bound has bn computd n L. Th rst of th smbols, whr at last for on j both smbols qual E s ßN T, do not fulfll th abov condton. In an cas, th achvabl dvrst can not b hghr than NR. Snc th dvrst loss occurs whnvr w hav mor than smbols, thr s no modulaton alphabt wth rat hghr than 1 bt/s/hz that can achv full rcv dvrst. V. SIMULTION RESULTS In ths scton w prsnt rsults for ampltud dtcton MIMO rcvrs and compar thm to a lnar MIMO rcvr that uss uncodd PSK and ML dtcton. Th rfrnc sstm uss spatal multplxng, hnc transmttng N T bts/s/hz, and xplots th full dvrst of th MIMO channl, qual to N R. For th ampltud dtcton MIMO sstm, w us two modulaton schms: OOK wthout spatal multplxng (rat R 1 bt/s/hz) and xtndd OOK wth spatal multplxng (rat R N T bts/s/hz). Th frst modulaton schm consst of th smbols Ö0,...0 T and Ö Es ßN T,..., E s ßN T T and s qual to usng onl on transmt antnna and th smbols Ø0, E s Ù. For a far comparson w us th SNR pr bt, gvn b γßr. Th channl has..d. Gaussan ntrs wth unt varanc. Fg. 3 dpcts th avrag rror rat of th scalar dtcton problmd and th corrspondng lowr boundl. Th lowr bound xhbts bttr rror prformanc at low SNR, and th prformanc s practcall dntcal bond 15 d. Smulaton shows that th dvrst ordr of D s n practc qual to that of th lowr bound, naml 1. Fg. 4 dpcts th avrag for a sstm. mpltud dtcton xhbts dvrst qual to N R whn R 1, and dvrst qual to NR 1 whn R. In th cas of spatal multplxng, th dvrst s dstrod b th pars: s,s È,,,,,, whch corrspond to dtcton problms of tp D. Th frst par xhbts th proprt that x and x ar ndpndnt, whl ths s not th cas for th othr two pars. Th avrag PEPs of ths pars s also shown n Fg. 4. W s that

5 Lnar PSK x3, R= bts/sm mpl. Dt. x3, R= bts/sm mpl. Dt. x3, R=1 bt/sm Lnar PSK x4, R= bts/sm mpl. Dt. x4, R= bts/sm mpl. Dt. x4, R=1 bt/sm Ln. PSK 3x3, R=3 bts/sm mpl. Dt. 3x3, R=1 bt/sm mpl. Dt. 3x3, R= bts/sm mpl. Dt. 3x3, R=3 bts/sm Ln. PSK 3x4, R=3 bts/sm mpl. Dt. 3x4, R=1 bt/sm mpl. Dt. 3x4, R= bts/sm mpl. Dt. 3x4, R=3 bts/sm Fg. 5. plot for a 3 and 4 MIMO sstm. Fg. 6. plot for a 3 3 and 3 4 MIMO sstm. corrlaton has no mpact on th dvrst ordr. Th small SNR gan of th scond and thrd par s du to th hghr transmttd nrg of th rspctv smbols, compard to th frst par. Furthrmor, th uppr bound to th dvrst of D s actuall achvd n practc. In both cass, th prformanc s clarl nfror to th lnar MIMO rcvr, whch howvr rqurs hghr powr consumpton and hardwar complxt. Fg. 5 dpcts th avrag of a 3 and 4 sstm, wth and wthout spatal multplxng. s prdctd b th analss of th PEPs, th dvrst ordr wth spatal multplxng s onl NR 1.5 and NR, rspctvl. Wthout spatal multplxng, th full dvrst of N R 3 and N R 4, rspctvl, s achvd. Onc agan, th uppr bound to th dvrst of D s achvd n practc for both sstms. Fg. 6 fnall dpcts th sam comparson for a 3 3 and 3 4 sstm. Ths tm, w hav thr lvls of spatal multplxng, ldng th uncodd rats R 1,,3 bts/s/hz. Th obsrvaton for th R 1 and R 3 cass ar as xpctd th dvrst ordr quals N R and NR, rspctvl. Th modulaton for th R cas has th form s Öa 1,a 1,a T, a È Ø0, E s ßN T Ù, such that two ndpndnt data strams ar transmttd. Th rsultng dvrst ordr s agan NR. Not that th dvrst loss occurs mmdatl whn th rat ncrass from 1 to bts/s/hz. Howvr, furthr ncrasng th spatal multplxng and th rat hav no mpact on th dvrst, whch rmans at NR. VI. CONCLUSIONS W analzd th prformanc of th ML dtctor of a MIMO rcvr wth ampltud-onl dtcton. Contrar to lnar MIMO sstms, th nonlnar MIMO sstm looss half th avalabl rcv dvrst whn th uncodd rat s hghr than 1 bt/s/hz, or whn spatal multplxng s prformd. Ths rsult dos not mpl that multpl transmt antnnas ar uslss for such a sstm, but t rathr shows that a prformanc pnalt s nducd b th nonlnart of th rcvr. Futur works ncluds th xtnson to MIMO rcvrs wth phas-onl dtcton, whr th sam bhavor can bn obsrvd. PPENDIX VERGE ENERGY OF EXTENDED OOK ssum th st of quprobabl transmt smbols s È Ø0,αÙ NT. W comput th avrag nrg of th smbols. Thr s on all-zro smbol, N T smbols wth a sngl ntr N qual to α, T smbols wth two ntrs qual to α, tc. Hnc, on smbol wll hav zro nrg, N T smbols wll hav nrg qual to α N, T nrg α, tc. Thus, N 1 ô T NT E s Åkα N Tα Nô T 1 Å NT 1 k k NT k1 NT k0 N Tα NT 1 α N T NT, (19) whr w usd dntt (3.1.6) from [10]. Fnall, w obtan α Es ßN T. REFERENCES [1] G. K. Psaltopoulos, F. Trösch, and. Wttnbn, On chvabl Rats of MIMO Sstms wth Nonlnar Rcvrs, n IEEE Intrnatonal Smposum on Informaton Thor, Jun 007, pp [] G. K. Psaltopoulos and. Wttnbn, chvabl rats of nonlnar MIMO sstms wth nos channl stat nformaton, n IEEE Intrnatonal Smposum on Informaton Thor, Jul 008, pp [3] I. E. Tlatar, Capact of Mult-antnna Gaussan Channls, n Europan Transactons on Tlcommuncatons, Novmbr 1999, pp [4] M. Schwartz, W. R. nntt, and S. Stn, Communcaton Sstms and Tchnqus. IEEE Prss, [5] J. G. Proaks, Dgtal Communcatons, 4th d. McGraw Hll, 000. [6] J. Gst, smptotc rror rat bhavor for noncohrnt on-off kng, IEEE Transactons on Communcatons, vol. 4, p. 5, Jan [7] D. Ts and P. Vswanath, Fundamntals of Wrlss Communcatons. Cambrdg Unvrst Prss, 005. [8] E. glr, Codng for Wrlss Channls. Sprngr, 005. [9] K. S. Mllr, Multdmnsonal Gaussan Dstrbutons. Wl, [10] M. bramowtz and I. Stgun, Handbook of mathmatcal functons wth formulas, graphs, and mathmatcal tabl, 1965.