Chaptr 2 Fundamntal of OFDM 2. OFDM Baic [9] Th baic principl of OFDM i to divid th high-rat data tram into many low rat tram that ach i tranmittd imultanouly ovr it own ubcarrir orthogonal to all th othr. Du to narrowband proprty, thy xprinc motly flat fading, which ma channl qualization vry impl. In ordr to liminat intrymbol intrfrnc (ISI and intrcarrir intrfrnc (ICI a much a poibl, it i a good ida to add a trailing portion of ach ymbol to th had of itlf, which i calld cyclic prfix xtnion a hown in Figur 2.. Th guard intrval i chon largr than th xpctd dlay prad, uch that multipath componnt from on ymbol will not intrfr with it uccding ymbol. Aftr a ignal pa through th tim-dipriv channl, orthogonality of it ubcarrir componnt can b maintaind by th introduction of cyclic prfix. 3
Cyclic prfix Guard intrval Uful ymbol priod Figur 2. Guard intrval and cyclic prfix tructur of an OFDM ymbol 2.. Continuou-tim OFDM Modl Th continuou-tim OFDM modl prntd blow i an idal OFDM ytm, which in practic i digitally ynthizd. Th baband modl i hown in Figur 2.2. ( i ( i ( t ( t ( t ( t ( i ( i ( i 2 ( t 2 Channl 2 ( t 2 i ( ( i ( t ( t Y ( i Figur 2.2 Continuou-tim OFDM ytm modl In Figur 2.2, data (i n i modulatd by QPSK or QAM and thn tranmittd by uing a t of inuoidal ocillator. Hr, w aum that th OFDM ytm ha ubcarrir and total ymbol duration of T cond, of which T cond i th g lngth of th cyclic prfix, and T u i th uful ymbol duration. Th tranmittr aum th following wavform at th -th ubcarrir 4
j ( t 2π ( ( t T T g, T g t T, othrwi (2. Thn aftr ummation of th ubcarrir, th baband ignal of th i -th OFDM ymbol i S ( t i ( i ( t T (2.2 Whn an infinit qunc of OFDM ymbol i tranmittd, th output of th tranmittr can b rprntd a S( t i S ( t i i d ( i ( t it (2.3 Th pctrum of ach ubchannl will b a hown in Figur 2.3. Th cntr ubcarrir frqunci of ubchannl pctrum ar not ovrlapping with ach othr, which i th y fatur of th OFDM ymbol tchniqu. Figur 2.3 Spctrum of an OFDM ymbol 2..2 Dicrt-tim OFDM Modl In ordr to analyz th prformanc of ynchronization tchniqu aily, w nd to undrtand dicrt-tim OFDM modl a wll. Compard with th continuou-tim modl xprd bfor, modulation and dmodulation in a dicrt-tim OFDM ytm ar prformd by IDFT and DFT, rpctivly. In practic, DFT/IDFT can b implmntd vry fficintly by FFT/IFFT. A dicrt-tim OFDM ytm can b rprntd a: 5
Y ˆ( i DFT ( IDFT ( ( i (2.4 A a rult, w hav th following dicrt-tim OFDM modl a hown in Figur 2.4. ( i ( i IDFT P/S S/P DFT i ( ( i ( i 2 Channl 2 i ( ( i Y ( i Figur 2.4 Dicrt-tim OFDM ytm modl 2.2 Effct of th Synchronization Error on OFDM [] On of th argumnt againt OFDM i that it i highly nitiv to ynchronization rror. Synchronization of an OFDM ignal rquir finding th ymbol timing, cloc timing and th carrir frquncy offt. Hr, w will dicu th ffct of th carrir frquncy and th timing offt to an OFDM ytm. 2.2. Effct of th Carrir Frquncy Offt OFDM ytm ar vry nitiv to carrir frquncy offt, bcau thy can only tolrat offt which ar a fraction of th ubcarrir pacing without a larg dgradation in ytm prformanc [9]. Frquncy offt ar cratd by diffrnc in ocillator in tranmittr and rcivr and Dopplr hift. If th frquncy rror i an intgr multipl n of th ubcarrir pacing f, thn th rcivd frquncy-domain data ar hiftd by n ubcarrir poition. ot that th 6
ubcarrir ar till mutually orthogonal but th rcivd data ymbol, which ar mappd to th OFDM pctrum ar now in th wrong poition in th dmodulatd pctrum. If th ubcarrir frquncy rror i not an intgr multipl of th ubcarrir pacing, thn w will ampl tho non-orthogonal frquncy point aftr th FFT opration. Intrfrnc i thn obrvd btwn th ubcarrir, which dtriorat th ytm prformanc. In ordr to invtigat th ffct of th frquncy offt, w conidr on baband ymbol from th tranmittr givn by [2] ( t ( t (2.5 j2πf t, t [, Tg + Tu, othrwi (2.6 whr i a complx data for th -th carrir; Tu i th uful ymbol tim, Tg i th guard intrval tim, and f / T, u W aumd th channl and th ampling cloc ar idal, and th rcivd ignal i only affctd by th carrir frquncy offt f, a formulatd by r( t Th ampld ignal with th ampling priod whr T f z m j2πm ft 2πm( + j j2π ft j2πf mt ; ( t (2.7 T Tu / i m,,... (2.8 i th ratio of th frquncy offt to th carrir pacing. Thu, th rcivd ignal ampl on th -th ubchannl i 7
ˆ ' m z m m 2πm ' j 2πm( + j 2πm ' j (2.9 ' in( π in( π / jπ ( + I ' Th firt trm how th rcivd ' amplitud dgradation and pha hift du to th frquncy offt. Th cond trm I ' i th ICI caud by th frquncy offt. In [], if dgradation D in SR caud by a frquncy offt that i mall rlativ to th ubcarrir pacing, thn it can b approximatd a whr D E 2 ( π ftu (2. 3ln E i th avrag rcivd nrgy, and 2 i th powr pctral dnity of th AWG. 2.2.2 Effct of th Timing Offt Finding th ymbol timing for an OFDM ymbol i quivalnt to finding th ymbol tart. Thr uually allow om tolranc for ymbol timing rror whn thr i a cyclic prfix in an OFDM ymbol. If th rcivr FFT window i hiftd in th rcivd ampling tram by m ampl, thn thr will introduc a pha rror of 2π m /, whr i th FFT lngth. Howvr, th pha rror can b mitigatd by channl timation. If th timing rror ar o high that th FFT window covr ampl from two concutiv OFDM ymbol, both ISI and ICI will occur and will vrly affct th ytm prformanc. 8