Buletnul Ştnţfc al Unverstăţ "Poltehnca" dn Tmşoara era ELECTRONICĂ ş TELECOMUNICAŢII TRANACTION on ELECTRONIC and COMMUNICATION Tom 49(63), Fasccola, 4 BER Performances of a Dfferental OFDM ystem n Fadng Channels Marus Oltean, Eugen Mârza, Mranda Nafornţă 3 Abstract In an OFDM system, varous modulaton methods can be used n order to encode the bnary nformaton. If a dfferental phase modulaton scheme s chosen, data can be encoded n the relatve phase of consecutve symbols n each subchannel or n the relatve phase of symbols n the adjacent subchannels. The two methods exhbt two essentally dfferent behavors n fadng condtons. In ths paper, we shall nvestgate the BER performances of both modulaton types. The performance evaluaton s based on computer smulaton. We wll consder a multpath fadng channel, as met n moble communcaton systems. Keywords: OFDM, dfferental, fadng I. INTRODUCTION OFDM (Orthogonal Frequency Dvson Multplexng) s one of the most promsng modulaton technques that were proposed for beng used n the 4 th generaton wreless systems. In a typcal moble rado channel the transmtted sgnal s subjected to multpath fadng whch generally exhbts tme selectvty (commonly referred to as Doppler effect) and frequency selectvty [], []. The nfluence of the II (Inter-ymbol Interference) can be reduced by ncreasng the duraton of the transmtted symbol. Usng OFDM, the hgh-rate data sequence to be transmtted s splt nto a large number of lower speed symbol streams, each of them modulatng a dfferent carrer. The carrer spacng s selected such that all carrers used are orthogonal each other over a symbol nterval. As t s well known, the orthogonal sgnals can be separated at recever by correlaton technques. In addton, a cyclc prefx (a copy of the last several samples of an OFDM frame ) s nserted at the begnnng of each OFDM symbol, n order to counteract the nherent tmedspersve nature of the channel, preventng two or more symbols to nterfere each other [3], thus nducng II. The cyclc prefx gves also an appearance of perodcty or crcularty to the sgnal, facltatng the equalzaton process to the recever. The lengthenng of the symbol duraton, ntroduced n order to combat the frequencyselectvty s however lmted by the tme-varant nature of the channel that generates the Doppler effect. Larger the symbol duraton, hgher the probablty that the channel parameters vary durng the transmsson of an OFDM frame gvng rse to frequency offsets of the carrers, thus destroyng ther orthogonalty and generatng nter-carrer nterference (ICI). The transmtter and recever for OFDM can be effcently mplemented usng Fast Fourer Transform (FFT), a rapd mathematcal algorthm of processng Dscrete Fourer Transform (DFT). The data symbols that modulate multple orthogonal carrers n OFDM are obtaned usng a classcal dgtal modulaton scheme. Varous modulaton methods could be employed such as BPK, QPK (also wth ther dfferental form) and QAM wth several dfferent sgnal constellatons. If a dfferental phase modulaton s chosen (the partcular case of an OFDM-DBPK system) there are two optons to perform t. Thus, data can be encoded n the relatve phase of consecutve symbols n each subchannel (correspondng samples n adjacent OFDM symbols, or frames), obtanng an nter-frame dfferental modulaton. On the other hand, data can be encoded n the relatve phase of symbols n adjacent subchannels (consecutve samples of an OFDM symbol), achevng an n-frame dfferental modulaton. The two methods exhbt two essental dfferent behavors n fadng condtons. In ths paper we realze a performance comparson of the two methods, rarely reported n the lterature, focusng on the dfferences between them. The performance evaluaton of both methods s based on computer smulaton. We wll consder a multpath fadng channel, as met n moble communcatons systems. Facultatea de Electroncă ş Telecomuncaţ, Departamentul Comuncaţ Bd. V. Pârvan Nr., 33 Tmşoara, e-mal marus.oltean@etc.utt.ro Facultatea de Electroncă ş Telecomuncaţ, Departamentul Comuncaţ Bd. V. Pârvan Nr., 33 Tmşoara, e-mal eugen.marza@etc.utt.ro 3 Facultatea de Electroncă ş Telecomuncaţ, Departamentul Comuncaţ Bd. V. Pârvan Nr., 33 Tmşoara, e-mal mranda.nafornta@etc.utt.ro 389
II. YTEM DECRIPTION AND FADING CHANNEL MODEL In OFDM, the avalable bandwdth s parttoned nto N subchannels. The desred hgh-rate symbol stream s acheved by smultaneously transmttng N slower rate substreams usng N orthogonal subcarrers. The bnary data to be transmtted s dfferentally encoded usng a DBPK modulaton scheme, obtanng a sequence of complex data symbols (fg. ). DBPK Bnary Modulaton Informaton Decson Estmated Informaton Fg.: The OFDM transcever model There are two possbltes to perform dfferental modulaton n the presented OFDM scheme. Data can be encoded n the relatve phase of adjacent symbols n each subchannel (correspondent samples n two consecutve OFDM symbols) or n the relatve phase of samples transmtted n adjacent subchannels, that s consecutve samples of an OFDM symbol (see fg. ). nce the IFFT block accepts N parallel samples to ts entry, the whole dfference of the two methods can be thought as follows: f the phase modulaton s separately acheved on each of the N parallel streams that consttute the entry to the IFFT block, then we are n the case of the frst presented modulaton type, namely an nter-frame modulaton s performed (see fg. a). If the modulaton s made on the seral stream, pror to the parallel converson requred by IFFT, then an n-frame modulaton s chosen, snce N consecutve seral samples wll smultaneously modulate N orthogonal carrers, formng an OFDM symbol (fg b). eral stream, consecutve frames s, n s, n Correspondent samples a) Relatve phase codng FFT IFFT /P converson /P converson b) Cyclc prefx nserton Cyclc prefx removal T k N IFFT Channel OFDM symbol Relatve phase codng Both methods have an rreducble error rate because of the random change of the relatve phase, caused by the fadng channel. In the frst method the dstorton I F F T OFDM symbol Fg.: (a) Inter-frame modulaton (b) In-frame modulaton caused by the multpath fadng s ndcated by the Doppler spectrum. In the second case, the channel multpath ntensty profle and the length of the OFDM symbol ndcate the phase change rate due to fadng condtons. Therefore, the two methods exhbt essentally dfferent behavors although both encode the data dfferentally. Another major dfference between the two methods (that gves them ther specfc name) s that the consecutve OFDM symbols are nterconnected through dfferental encodng n the frst case (nter-frame dfferental modulaton), whle no successve OFDM symbols connecton s realzed by the second method (n-frame dfferental modulaton), where the dfferental encodng s performed on the samples belongng to the same frame (or to the same future OFDM symbol), as llustrated n the fg. b. After the dfferental encodng of the bnary message usng one of the two methods presented above, the sequence s,n s obtaned, s,n denotng the n- th symbol of the -th frame, where ( n N, < < ). The n-th carrer s modulated by the samples { s, n, < < } and the modulated carrers (orthogonal one-another) are added together to form the OFDM symbol to be transmtted. In a practcal mplementaton, the N samples of the OFDM symbol correspondng to the - th frame are generated by processng {s,n }stream usng the fast mplementaton of the Inverse Dscrete Fourer Transform (IDFT) (see fg. ). In order to combat the nter-symbol and nter-carrer nterference ntroduced by the frequency selectvty and the tme selectvty of the rado channel, each OFDM symbol s preceded by a cyclc prefx of L samples. The cyclc prefx s a crcular extenson of the tme doman samples, beng obtaned by copyng the last L samples of the OFDM symbol n the front of t. The - th transmtted symbol (ncludng the prefx) contans N+L tme doman samples, of whch the m-th sample s gven by the equaton below: ( ) = E N,, =,..., + j π N g m s e m L N N L n = n () Assumng the data symbols are statstcally ndependent and havng a unt average energy, the transmtted average energy per symbol equals E. The transmtted sgnals can be expressed n complex form as: nm s(t)= p ( t T ) g ( t) () = where g (t) represents the analogcal waveform correspondng to the OFDM symbol, obtaned after a DAC converson of the sequence {g (m)}, m=,,,n-. p(t) s the pulse-shapng waveform of each symbol, defned as: 39
, for t ts p( t) = (3), otherwse T = +t s stands for the total duraton of an OFDM symbol, composed by the cyclc prefx perod ( ) and by the observaton perod (t s ). The fadng channel (assumng Rce condtons) can be modeled as a 3-ray tapped delay lne wth one lne-of-sght (LO) path and two multpath components. If h(τ) denotes the channel mpulse response at tme t-τ, t can be expressed as: h( τ ) = P δ ( τ ) + P a ( t) δ ( τ τ ) P a ( t) δ ( τ τ ) + (4) power of the two multpath components are consdered for channel smulaton. The BER computaton was averaged over transmtted OFDM symbols. Nether channel codng nor further equalzaton to the recever were consdered at ths stage. A comparson of the two methods s made, studyng the nfluence of the block length N, of the channel multpath delay spread and of the Doppler shft ntroduced by the tme-varant character of the channel on the BER performance n both n-frame and nter-frame DBPK-OFDM system. We emphasze the essental dfferent behavors of the two methods wth respect to the parameters presented above. The BER performances of the DBPK-OFDM system n a Raylegh fadng channel, as a functon of the normalzed delay of the second multpath τ /T s llustrated n the fgures 3,4. where P s the power of LO sgnals, P and P are the powers of multpath replcas. The channel mpulse response above can be vewed as a summaton of the LO determnstc sgnal and two attenuated (a (t) and a (t) are ndependent tme-varyng complex Gaussan random processes wth maxmum Doppler shft f d, accountng for these attenuatons) and delayed replcas (τ and τ represent the delays of the two multpath components). An mportant parameter characterzng the Rcan fadng channel s the Rce factor, defned as the rato of the determnstc LO component power P s and the multpath components power P m =P +P,.e. K=P s /P m. As a specal case, the channel s AWGN when K, whle Raylegh fadng condtons are met for K=. The receved sgnal can be wrtten as [4]: r( t) = s( t τ ) h( τ ) dτ + n( t) (5) where n(t) s a complex Gaussan nose and h(τ) s the mpulse response of the multpath fadng channel at the tme t-τ. At the recever, after dong FFT to the sgnal, the output of each m-th subchannel can be obtaned as: Fg. 3: BER performances of nter-frame and n-frame modulaton n a Raylegh fadng channel, NR=4dB, P /P =db, N=3 In the fgure 3, one can observe that the nter-frame modulaton s sgnfcantly more senstve to the Doppler shft than the n-frame modulaton. Thus, at the two normalzed Doppler shfts (f D *T ) taken nto accoun the performance of the n-frame DBPK system s almost dentcal, especally for an mportant multpath delay of the channel. On the other hand, nter-frame modulaton performs sgnfcantly better for low values of the maxmum Doppler shf provng + = t T j f D ( t T ) m { s( t τ ) h( τ ) dτ + n( t) e, t } π T r (6) Fnally, the dfferental detector decdes what symbol was transmtted. III. IMULATION REULT AND DICUION dt The BER performance of an OFDM system wth both DBPK n-frame and nter-frame modulaton was studed by the means of computer smulaton. To smplfy the computer mplementaton, the cyclc prefx duraton s consdered to be equal to the seral symbol duraton,.e. T= for all the smulatons. Two-ray Raylegh fadng condtons, wth equal Fg. 4: BER performances of nter-frame and n-frame modulaton n a Raylegh fadng channel, NR=4dB, P /P =db, N=3 39
a senstvty of ths method to the tme-varant channel character. The same observaton becomes more obvous regardng the fgure 4, where another dfference between the two Doppler shfts s taken nto account. The BER performance of the nter-frame DBPK-OFDM system for three dfferent Doppler shfts s plotted n the fgure 5, n order to stress the effectveness of ths parameter. It s shown that the maxmum Doppler shft has a sgnfcant nfluence on the BER, especally when the delay of the second multpath s small. For large delays of the second multpath the man amount of errors s brought by the II ntroduced by the multpath components, whch confrms the concluson n [5], respectvely n [4] for an n-frame DBPK modulaton. IV. ABOUT REFERENCE References should be numbered n a smple form [], [], [3], and quoted accordngly []. References are not allowed n footnotes. It s recommended to menton all authors; et al. should be used only for more than 6 authors. Table Parameter Value Unt I.4 A U. V method to the varaton of the maxmum Doppler shft parameter comparng to the nter-frame modulaton (whose performance s ndcated by the two outer curves). If at low maxmum Doppler shft nter-frame modulaton performs better, once the value of ths parameter grows, the n-frame modulaton method becomes more effcent. It can also be observed that at sgnfcant Doppler shfts, the two methods exhbts a very poor mprovement of BER performance, wth respect to NR, especally for the nter-frame modulaton. The nfluence of the block length N on the BER performance n both modulaton types s llustrated n the fgure 7, where maxmum Doppler shft s consdered to be constant. As stressed n [6], the nframe OFDM-DBPK system sgnfcantly mproves ts performance when the block (or, equvalently, the OFDM symbol) length ncreases, consderng the same multpath delay spread of the channel mpulse response. On the other hand, t turns out that the BER performance of nter-frame modulaton s almost dentcal for the dfferent values chosen for the parameter N. In the three smulated stuatons (N=6,3,64), the system performs to wthn -db spread of the results. It can be asserted that the performance obtaned usng ths modulaton type s very lttle senstve to the OFDM symbol duraton. Fg. 5: BER performances of nter-frame DBPK modulaton n a Raylegh fadng channel, NR=4dB, P /P =db, N=3 In the fgure 6, we plotted the BER performance of both nter-frame and n-frame OFDM-DBPK systems wth respect to the average NR per bt and the parameter used s stll the normalzed maxmum Doppler shft (f D *T ). Fg. 6: BER performances of nter-frame and n-frame DBPK modulaton n a Raylegh fadng channel, τ /T=, P /P =db, N=3 The two nner curves correspond to n-frame modulaton, showng the lttle senstvty of the Fg. 7: BER performances of nter-frame and n-frame DBPK modulaton n a Raylegh fadng channel, τ /T=, P /P =db, f D=. Fgure 8 shows the BER performance evaluated over a range of normalzed delay spreads of the channel mpulse response. Unlke the prevous evaluaton, nter-frame modulaton mproves ts performance sgnfcantly when the OFDM symbol length N ncreases. The same expected pattern s exhbted by the n-frame modulaton method. A comparatve analyss llustrates that n-frame modulaton s slghtly more senstve to the parameter N than the nter-frame modulaton, when the performance s evaluated aganst the normalzed delay of the second multpath. At small values for the delay of the second reflected path (comparable wth the seral symbol duraton) no correlaton between the 39
frame length and the BER performance can be asserted, snce the system performs better for N=6 than for N=64. Fg. 8: BER performances of nter-frame and n-frame DBPK modulaton n a Raylegh fadng channel, NR=4dB, P /P =db, f D =. In the fgure 9, the effect of multpath delay spread on the both methods s studed. The maxmum Doppler shft was kept constant whle the BER performance aganst sgnal-to-nose (NR) rato was plotted for two dfferent values of the normalzed delay of the second multpath component. Consderng a small value of the mentoned parameter, the n-frame OFDM-DBPK system performs to about 8dB better than the nter-frame system. To le n such a stuaton, a correspondent value was chosen for the normalzed Doppler shft (f D *T =.5). When a three tmes bgger value was consdered for the normalzed delay of the second multpath component nter-frame modulaton performed better, despte the sgnfcant value of the Doppler shft. One can conclude that n-frame modulaton s more senstve to the multpath delay spread ntroduced by the nherent dspersve nature of the rado channel. IV. CONCLUION In ths paper we have studed the BER performance of an OFDM-DBPK system wth two dstnct phase modulaton types. The prncples of both n-frame and nter-frame modulaton n an OFDM transmsson scheme were brefly exposed, accentuatng on ther dfferences. The essental dfferent behavor n multpath fadng condtons was emphaszed by means of computer smulaton. The nter-frame modulaton system, whle generally performng better has though shown to be more senstve at the varaton of the Doppler shft parameter. The n-frame modulaton method allows sgnfcant performance mprovement by ncreasng the data-block length. The multpath delay spread degrades the BER performance of both studed modulaton types. Even f at delay spreads that sgnfcantly exceed the cyclc prefx duraton the performance s only slghtly mproved ncreasng the sgnal-to-nose rato, the nter-frame modulaton proved to be more resstant to the nter-symbol nterference ntroduced by the multpath delayed components. REFERENCE [] B. klar, Raylegh Fadng Channels n Moble Dgtal Communcaton ystems- Part I: Characterzaton, IEEE Commun. Mag., July 997; [] B. klar, Raylegh Fadng Channels n Moble Dgtal Communcaton ystems- Part II: Mtgaton, IEEE Commun. Mag., July 997; [3] J.A.C. Bngham, Multcarrer Modulaton for Data Transmsson, An Idea Whose Tme Has Come, IEEE Communcaton Magazne, Vol. 3, No. 5, May 99 [4] Lu, J., Tjhung, T.T., Adach, F., BER performance of OFDM system n frequency-selectve Rcan fadng wth dversty recepton, avalable onlne: http://www.cwc.nus.edu.sg/~cwcpub/zfles/ct97_.pdf. [5]Lupea, E., Banu, M., ălăgean, M., Oltean, M., Nafornţă, M., BER Performance of Frequency electve Channels wth Cyclc Prefx Based Equalzers, Buletnul Ştnţfc al Unverstăţ Poltehnca Tmşoara, Tom 47 (6), Fasccola -, [6]Chn, A., Analyss and mulaton of Multcarrer Modulaton n Frequency electve Fadng Channels,Ph. D. Thess, 994, Chapter 3 Fg. 9: BER performances of nter-frame and n-frame DBPK modulaton n a Raylegh fadng channel, P /P =db, f D*T =.5, N=3 393