Optimal Spacing Design for Pilots in OFDM Systems over Multipath Fading Channels

Transcription

1 Communication and twork doi:10.436/cn Publihd Onlin ovmbr 010 ( Optimal Spacing Dign or Pilot in OFDM Sytm ovr Multipath Fading Channl Abtract Youi My Abdlkadr El Abbadi Jamal aboratoir d Elctroniqu t Communication EMI-Rabat Morocco {ab.youi j.labbadi}@gmail.com Rcivd July 1 010; rvid Augut ; accptd Sptmbr In wirl orthogonal rquncy diviion multiplxing (OFDM) ytm th tim-varying channl i otn timatd by algorithm bad on pilot ymbol. Such an timator howvr rquir tatitical prior knowldg that i not aily obtaind. Thror th pilot ton hav to b clo nough to ulill th ampling thorm. In thi ca th tatitical knowldg o th channl i not rquird to rcontruct corrctly th channl impul rpon (CIR). Thi papr xplor th optimal placmnt and numbr o pilot ymbol w invtigat optimal training qunc in OFDM ytm and w analyz th numbr o pilot ymbol rquird to ulill th ampling thorm. Uing a gnral modl or a multipath lowly ading channl th approach i bad on th S a a critrion o channl timation whil th channl intrpolation i don uing th picwi-contant intrpolation compromiing btwn complxity and prormanc. Simulation rult dmontrat th good prormanc o our approach. Kyword: OFDM Multipath Channl Channl Etimation Pilot Symbol 1. Introduction OFDM (Orthogonal Frquncy Diviion Multiplxing) ha bn widly applid in wirl communication ytm du to it high data rat tranmiion and it robutn to multipath channl dlay [1]. Th channl charactritic mut b known at th rcivr to rcovr th tranmittd ignal atr dmodulation o channl timation i ndd. Two cla o mthod ar availabl or th rcivr to acquir channl tat inormation (CSI): On i bad on training ymbol that ar a priori known to th rcivr whra th othr rli only on th rcivd ymbol to acquir CSI blindly. Rlativ to training blind chm typically rquir longr data rcord and ntail highr complxity. In OFDM ytm channl timation i uually prormd by nding training pilot ymbol on ub-carrir known at th rcivr. Th quality o th timation dpnd on th pilot arrangmnt. Th channl timation can b prormd by ithr inrting pilot ton into all o th ubcarrir o OFDM ymbol with a pciic priod (Block-typ) or inrting pilot ton into ach OFDM ymbol (Comb-typ) [3]: 1) Block-typ Arrangmnt (how in Figur1a): Th block typ pilot channl timation ha bn dvlopd undr th aumption o low ading channl thi aum that th channl tranr unction i not changing vry rapidly. In block-typ pilot bad channl timation OFDM channl timation ymbol ar tranmittd priodically in which all ub-carrir ar ud a pilot. I th channl i contant during th block thr will b no channl timation rror inc th pilot ar nt at all carrir. Th timation can b prormd by uing ithr S or MMSE. ) Comb-typ Arrangmnt (how in Figur1b): th comb-typ pilot channl timation ha bn introducd to atiy th nd or qualizing whn th channl chang vn in on OFDM block. Th comb-typ pilot channl timation conit o algorithm to timat th channl at pilot rqunci and to intrpolat th channl. Thi work i bad on a Comb-typ arrangmnt Comb-typ pilot pattrn i mor appropriat or th timation o tim varying channl [4]. Th channl i conidrd to b rquncy-lctiv invariant ovr ach rcivd block but i allowd to vary rom block-toblock. Th dign o a channl timator i bad on two undamntal problm: Copyright 010 SciR. C

2 Y. M. ABDEKADER ET A. (a) Block-typ pilot arrangmnt Th amount o pilot ymbol to b tranmittd. Th complxity o th timator. Th MMSE timator ha good prormanc but high complxity. Th S timator ha low complxity but it prormanc i not a good a that o th MMSE timator [5]. Thi work i bad on th S a a critrion o channl timation bcau S and MMSE critria xhibit imilar prormanc in high SR rgim. Th OFDM tranmiion tchniqu i robut to multi-path channl dlay cyclic prix xtnion i ud to liminat intr-ymbol intrrnc (ISI) but th Intr-carrir intrrnc (ICI) rmain a problm bcau o Dopplr rquncy. In at ading channl whn th channl chang vn in on OFDM block orthogonality o th ton can t b prrvd bcau o Dopplr Ect [6]. Studying th optimum pacing o th pilot in th gnral ca how that th optimum tim-rquncy pacing dpnd upon th channl paramtr (th Dopplr rquncy and th powr dlay o th channl) [7]. Thror th OFDM tchniqu i xtrmly attractiv or lowly ading multipath channl a that uually ud in WA ytm or in indoor nvironmnt. In thi papr w tudy th ca o multipath lowly ading channl whil th orthogonality o ton i prrvd. Th optimal pacing o pilot ymbol or OFDM ytm ha bn invtigatd by vral tudi ovr th pat tn yar. In litratur th rult o work bad on channl capacity invtigating thi optimization problm [8] conirm th rult o th tudy bad on th OFDM linar ytm (Y = HX+) th tudi how that in a multipath channl with indpndnt path channl timation rquir at lat pilot ymbol qually- pacd in th OFDM block. In thi work w xplor th optimization o pilot ubcarrir rquird or channl timation and optimal poition o pilot. W how that th application o ampling thorm conirm th rult o prviou work. In Sction w dcrib th channl timation bad on a Comb-typ arrangmnt. Sction 3 dicu th optimal placmnt and numbr o pilot ymbol in an OFDM ram. Simulation rult ar prntd in Sction 4.. OFDM Channl Etimation Bad on Comb-Typ Pilot Arrangmnt.1. otation (b) Comb-typ pilot arrangmnt Figur 1. Pilot arrangmnt Standard notation ar ud in thi papr.. t Tranpo (.) Hrmitian.. pl Etimat o. Vctor with pilot ton ubcript E. Expctation oprator tr. Trac oprator. Abolut valu. -norm I n n x n Idntity matrix umbr o OFDM ton T Th ampling intrval T Th OFDM block duration (T=.T ) Maximum lngth o channl K Frquncy ubcript (ton numbr) m Tim ubcript (OFDM ymbol numbr) Y km Rcivd ignal X km Tranmittd data h m Channl impul rpon H m Channl matrix km AWG noi F d Dopplr rquncy j / W.. Channl Etimation Bad on Comb-Typ Pilot Arrangmnt In an OFDM ytm th rcivd ignal at th k th ton can b xprd a Yk Hk X k Ikk k=01-1 whr I (k) th intr-carrir intrrnc (ICI) bcau o Dopplr rquncy. Copyright 010 SciR. C

3 Y. M. ABDEKADER ET A. 3 In th ca o multipath lowly ading channl whn th channl i rquncy-lctiv invariant ovr ach rcivd block OFDM ymbol th orthogonality btwn ubcarrir can b ully prrvd and w can writ Y H X km km km km Thr ar vral critria ud or channl timation or raon o complxity th timation can b prormd by uing ithr linar MMSE critrion (Winr iltring) [910] and S critrion. Howvr in high SR rgim and whn th noi lvl i low S critrion or a good compromiing prormanc / complxity. 1) Th timat o th channl at pilot ub-carrir bad on S timation i givn a ollow Aum that ach OFDM ymbol ha ubcarrir whr pilot occupy p ubcarrir. By S timation th channl tat inormation (CSI) at pilot ton in th k th OFDM ymbol can b obtaind a ˆ Ykm H km X km whr k i th pilot ton k k1 k... kp ) Intrpolation In comb-typ pilot bad channl timation an intrpolation tchniqu i ncary in ordr to timat channl at data ub-carrir by uing th channl inormation at pilot ub-carrir [11]. Th channl intrpolation i don uing th picwi-contant intrpolation linar intrpolation cond ordr intrpolation low-pa intrpolation plin cubic intrpolation In th ollowing Figur thr a comparion btwn thr typ o rquncy channl rpon Indoor A Pdtrian B and Vhicular B [1]: Th pilot ymbol hav to b clo nough to ulill th ampling thorm and avoid aliaing. Thror with th picwi-contant intrpolation w can rcontruct corrctly th channl impul rpon a it will b hown in thi papr. Whra whn w hav nough inormation about channl charactritic w can choo a typ o intrpolation to timat th channl (vn i pilot ymbol arn t clo nough to ulill th ampling thorm). Th channl timation H k at th data-carrir k( = 01-1) i givn a ollow a) H k at pilot ton i ud to ind th timatd tranmittd ignal X k w can gt Yk X k H k k k 1 k... kp Figur 3 X k i mappd to th contllation and thn obtaind X pur k. b) Th timatd channl X k at pilot ton i updatd: Yk H k k=01-1 X k pur H Th channl timation k at th data-carrir k k i givn uing an icint intrpolation tchniqu in ordr to timat channl at data ub-carrir by uing th channl inormation at pilot ub-carrir. 3. Th Minimum umbr o Pilot Ton In thi ction w prnt om anwr to th qution o minimum numbr o pilot ndd or channl timation in OFDM ytm. Thi problm ha bn th ubjct o vral invtigation th thortical tudy how that th channl timation rquir at lat pilot ymbol and w can prov thi rult by two dirnt mthod. On i bad on th ampling thorm whra th othr rli only on th tudy o th OFDM linar ytm Y = HX Th Application o Shannon Sampling Thorm Th channl timation with a Comb-typ pilot arrangmnt i bad on an inrtion o contant valu calld pilot in om ub-carrir o th OFDM ram. Figur 4. Atr th ignal dmodulation an S timator giv Figur. Rayligh modl GHz ETSI TR Figur 3. Contllation. Copyright 010 SciR. C

4 4 Y. M. ABDEKADER ET A. Figur 4. Pilot inrtion. Figur 5. Sampling proc. channl timation at pilot ub-carrir. Th picwicontant intrpolation i ud in ordr to timat channl at th valu btwn th pilot ampl a it hown in Figur 5. Thror th proc o channl timation bad on pilot ymbol with picwi contant intrpolation i imilar to th ampling proc. Th BER can b lowrd by placing th pilot ymbol clor than that pciid by th ampling thorm. Multipath channl modling Th complxity o an timator dpnd on th numbr o th propagation path (dtrmind by th ratio btwn th maximum dlay prad τ max and th ampling priod T). For xampl in BRA A channl thr 8 path (th propagation channl or an oic nvironmnt a part o th tandard HiprA-5Ghz). In th analogu domain th multipath channl i dcribd by th ollowing complx-valud impul rpon [13]: l( ) l ht t t l 1 whr α l i th complx gain o th lth path τ l i th propagation dlay or th lth path. In digital tranmiion ytm ymbol ar otn tranmittd in rgular tim intrval T calld th ampling priod. Thi priod i otn mall in comparion with th maximum dlay tim τ max. Th dicrt multipath modl i rgardd a a rrnc or mot o th contmporary widband trrtrial wirl ytm undr a raonabl approximation o contant numbr o multipath componnt and low variation o th dlay valu l t. Th multipath channl i thn modld a l l l1. ht t T Th path gain ar givn a ampl takn at dirnt priod T ( Figur 5). Th application o ampling thorm Th pctrum o a dicrt ignal i a priodic ignal ( Figur 6). W will xplor th duality o th Fourir tranorm thn w can apply th ampling thorm on th rquncy rpon o th channl. Sinc th intr-carrir pacing i Δ and th duration o 1 1 an OFDM ymbol i T=.T w hav T T I w aum that pilot ar placd ubcarrir apart in vry OFDM ymbol (Comb-typ pilot arrangmnt) and th pilot ymbol ar qually-pacd and th pacing Figur 6. Typical multipath channl rpon. Copyright 010 SciR. C

5 Y. M. ABDEKADER ET A. 5 btwn two concutiv pilot ymbol i qual to w hav T Figur 7. Structur o pilot ymbol or =4 t H () th rquncy rpon o th channl contructd rom th pilot ymbol with a picwi contant-intrpolation. H H n n n H n n Th invr Fourir tranorm o thi ditribution i h (t): h t h t TF n n 1 n ht * t n 1 n ht 1 n T h tnt n Th invr Fourir tranorm o thi ditribution i h (t): h t h t TF n n 1 1 ht * 1 n n n n ht T h tnt n t Thn w can rcontruct h (t) rom h (t). W nd a tim-domain windowing to rcontruct th tim-domain channl impul rpon (CIR). Th ampl hav to b clo nough to ulill th ampling thorm and avoid aliaing. So w mut choo th ampling priod T gratr than th channl impul rpon lngth. t h m =[h 0m h 1m h -1m ] t th channl impul rpon. Th lngth o th multipath channl impul rpon i.t. To avoid aliaing w mut rpct th ollowing inquality T T.. Knowing that T. T w hav T howvr w hav th ollowing inquality T. T. W will how latr that th quipacd pilot ymbol ar optimal in th n o MMSE. Bcau quipacd placmnt o pilot ymbol i impoibl whn / i not an intgr w will conidr only th ca whn / i an intgr. For thi pcial ca w hav umbr _ o _ Pilot _ Ton knowing that i th numbr o pilot ymbol in th OFDM ram thn th channl timation rquir at lat pilot ymbol. 3.. OFDM inar Sytm Y = HX+ In th OFDM ytm whn orthogonality i prrvd th ytm quation can b xprd a Y H X (1) km km km km whr km AWG noi. t k... 1 k k b th t o ton ud or tranmitting pilot ymbol. In th abnc o noi rom (1) w can writ: whr H pl pl pl pl Y H X () k k 1 1( 1) 1 W... W k k ( 1) 1 1 W... W h k k ( 1) W W 1... Sinc Q i a Vandrmond matrix with paramtr ditinct h can b ound xactly by invrting Q. With th ytm quation (1) w hav unknown h 1... h - I th numbr o pilot ymbol i qual to thn th Q (3) Copyright 010 SciR. C

6 6 Y. M. ABDEKADER ET A. numbr o quation qual th numbr o unknown thn th ytm can b olvd. - I th numbr o pilot ymbol i l than th numbr o quation i l than th numbr o unknown thn w hav an undr-dtrmind ytm o linar quation [14] and hnc a non-uniqu olution h. Thror in both ca (abnc or prnc o noi) pilot ymbol ar th minimum numbr rquird or xact channl timation Optimum St o Pilot Ton It ha bn ound in [1415] that in th prnc o AWG noi th MMSE timat o h occur whn th pilot ymbol ar qually-pacd in th OFDM ram. Proo: From (1) w can writ: Y S R H k (4) km km km km X km X km km whr Skm jarg X km rom (3) w hav pl H Qh. (5) rom (45) th MMSE timat o h i givn by: pl1 pl h Q R (6) Knowing that ach pilot X km carri data o contant modulu X km x rom (4) and (6) w hav: pl1 1 pl h hq S x t ( pl) k k k1 k pl pl t R R R S S S Th matrix Q dpnd on th choic o th t o pilot ton. Thu th dign problm i to choo th t that minimiz th MSE E h h.thi can b obtaind a ollow 1 E h h trq pl* pl x ( pl ) 1 tr( Q Q ) x I ( pl)* ( pl) 1 Q Q can b xprd a whr tr Q pl* pl Q t λ 1 λ λ th ignvalu o Q ( pl ) 1* pl* pl Q Q 1 thn th ignvalu o pl* pl Q Q 1 ar 1/ λ1 1/ λ 1/ λ. Sinc th trac o a matrix i th um o it ignvalu hnc th rlaxd optimization problm i minimiz p (7) p1 1 Whr. Sinc th matrix Q Q i non-ngativdinit all th ignvalu ar non-ngativ. So th minimization occur i all th ignvalu λi (i=1.. ) ar idntical. Thi occur i: p1 p pl* pl 1 pl* pl Q Q I Thror th optimal t o ton ar 1 ii... i whr i Simulation Rult 4.1. Ect o Multipath Channl To validat our analyi w tt an OFDM 16 QAM tranmiion ovr a multipath channl in th abnc o noi (4 path 6 path and 10 path) with =56 ubcarrir and cyclic prix lngth qual to CP=3. In all th ollowing imulation w hav chon th cyclic prix to b gratr than th maximum dlay prad in ordr to avoid intr-ymbol intrrnc (how in Figur 7). In Figur. 9 w can clarly th ct o multipath channl; w notic that a th multipath channl lngth incra th rror rat incra. Channl timation i ncary to corrct th OFDM ymbol. Figur QAM contllation. Copyright 010 SciR. C

7 Y. M. ABDEKADER ET A Optimal Spacing o Pilot Symbol W conidr a typical OFDM ytm whr =56 ubcarrir and cyclic prix lngth qual to CP=3 and 64 QAM tranmiion ovr a multipath channl. W alo Figur 9. Ect o multipath channl. 4.. Channl Etimation Bad on Pilot Symbol W conidr a typical OFDM ytm whr =56 ubcarrir and cyclic prix lngth qual to CP=3 and 16 QAM tranmiion ovr a multipath channl in th abnc o noi (8 path). W alo inrt qually-pacd pilot ymbol (1 + j) th rcivr u an S timator and intrpolation. W dal with two ca: 1) W inrt 16 pilot ymbol in vry OFDM ram thn th numbr o pilot ymbol i gratr than th channl lngth (=8 path). ) W inrt 8 pilot ymbol in vry OFDM ram thn th numbr o pilot i qual to th channl lngth (=8 path). Th imulation rult ar hown in Figur 10 and Figur 11. W notic that th channl timation with 16 pilot ymbol /OFDM ram hav good prormanc a it i hown in th Figur11. Othrwi uing 8 pilot/ofdm Figur 10. Signal contllation bor corrction. Figur 11. Contllation atr corrction. Figur 1. BER vru pilot pacing (4 path). Copyright 010 SciR. C

8 8 Y. M. ABDEKADER ET A. Figur 13. BER vru pilot pacing (8 path). ram thi numbr i not nough or th channl timation (w hav to choo th numbr o pilot ymbol gratr than th multipath channl lngth). inrt pilot ymbol (1 + j) th pilot ymbol ar not alway qually-pacd in th OFDM ram. W vary th numbr o pilot ymbol (1 + j) in OFDM block. Th rcivr u an S timator and intrpolation. W dal with two ca: 1) Th multipath channl hav 4 indpndnt path (impul rpon = [ ]) Figur 1 how th bit rror rat (BER) vru pilot pacing in th rquncy grid. W notic in Figur 1 that whn w incra th numbr o pilot ymbol (dcraing ) th BER dcra and bcom almot zro at on lvl o. Whn th numbr o pilot ymbol i qual to th numbr o indpndnt channl path (pilot pacing = 6 numbr o pilot/ofdm ram = 4) th BER i about 0.08 which i accptabl. Whn th numbr o pilot ymbol i l than 4 ( > 6) w can at BER incra which man that th quantity o pilot ymbol i not nough to timat th channl. ) Th multipath channl hav 8 indpndnt path (impul rpon = [ ]) Figur 13 how th bit rror rat (BER) vru pilot pacing in th rquncy grid. Thortical rult how that th qually-pacd pilot ymbol ar optimal in th n o MMSE and thi optimal placmnt o pilot ymbol i impoibl whn / i not an intgr. In Figur 1 and Figur 13 w can clarly that whn pilot ymbol ar qually-pacd in th OFDM ram ( / i an intgr) th curv ha a local minimum ( = ). Othrwi th bit rror rat tak th mallt valu in a givn nighborhood. 5. Concluion In thi papr w hav tudid th optimum pacing o OFDM pilot ymbol in th multipath lowly ading channl. It wa hown that channl timation rquir at lat pilot ymbol qually-pacd. W provd that th ampling thorm giv th minimum numbr o pilot ton and conirm th rult o prviou invtigation and with imulation w validatd th thortical rult. To approach th thortical rult o th timation problm w hav to rduc th noi ct on rcivd pilot ymbol with high SR rgim. W hav tratd th ca o invariant multipath channl ovr ach rcivd OFDM block. But thi work or th opportunity to tudy th optimization problm in at ading channl with high mobility which rmain an opn rarch ara. 6. Acknowldgmnt Th author would lik to thank all th anonymou rviwr o th papr. Th critical commnt by all th rviwr hav hlpd u to improv th quality o our Copyright 010 SciR. C

9 Y. M. ABDEKADER ET A. 9 papr. 7. Rrnc [1] M. Engl Wirl OFDM Sytm Kluwr Acadmic Publihr w York 00. [] R. S. B. Ahmad and R. S. Burton Multi-Carrir_Digital_Communication: Thory and Application o OFDM Kluwr Acadmic Publihr w York 00. [3] M. S. Akram Pilot-bad Channl Etimation in OFDM Sytm Matr Dirtation Tchnical Univrity o Dnmark Dnmark 007. [4] S. Colri M. Ergn A. Puri and A. Bahai Channl Etimation Tchniqu Bad on Pilot Arrangmnt in OFDM Sytm IEEE Tranaction on Vol. 48 o. 3 Sptmbr 00 pp [5] J. J. van d Bk O. Edor M. Sandll S. K. Wilon W. Pr and P. O. Börjon On Channl Etimation in OFDM Sytm Vhicular Tchnology Conrnc IEEE 45th Vol pp [6] H. Hijazi and. Ro Rayligh Tim-varying Channl Complx Gain Etimation and ICI Cancllation in OFDM Sytm Europan Tranaction on Tlcommunication Vol. 0 o pp [7] R. Gonzalz and F. badr Tim-Frquncy Spacing Dign or PACE in OFDM Sytm 14th IST Mobil and Wirl Communication Summit Drdn 005. [8] S. Ohno and G. B. Giannaki Capacity Maximizing MMSE-Optimal Pilot or Wirl OFDM ovr Frquncy-Slctiv Block Rayligh Fading Channl IEEE Tranaction on Inormation Thory Vol. 50 o pp [9] P. Hohr S. Kair and P. Robrton Pilot-Symbol-Aidd Channl Etimation in Tim and Frquncy Kluwr Acadmic Publihr USA [10] M. Sandll and O. Edor A Comparativ Study o Pilot-Bad Channl Etimator or Wirl OFDM Diviion o Signal Procing ulå Univrity o Tchnology ulå Swdn Sptmbr [11] P. Y. Tai and T. D. Chiuh Frquncy-Domain Intrpolation Bad Channl Etimation in Pilot-Aidd OFDM Sytm Vhicular Tchnology Conrnc IEEE 59th Vol pp [1]. Ro Récption multi-captur pour un trminal radio-mobil dan un ytèm d accè multipl à répartition par cod IP Grnobl 001. [13] D. T and P. Viwanath Fundamntal o Wirl Communication Cambrig Univrity Pr UK 005. [14] R. gi and J. Cioi Pilot Ton Slction or Channl Etimation in a Mobil OFDM Sytm Conumr Elctronic. ICCE Stanord 1998 pp [15] X. D. Cai and G. B. Giannaki Error Probability Minimizing Pilot or OFDM with M-PSK Modulation ovr Rayligh-Fading Channl IEEE Tranaction Vol. 53 o pp Copyright 010 SciR. C