OFDM is a good candidate for wireless multimedia communication

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1 Detecton of OFDM-CPM Sgnals over Multpath Channels Imran A. Tasadduq and Raveendra K. Rao Department of Electrcal & Computer Engneerng Elborn College, 11 Western Road The Unversty of Western Ontaro London, ON N6G 1H1, Canada. Abstract A class of Orthogonal Frequency Dvson Multplexng - Contnuous Phase Modulaton (OFDM-CPM) sgnals s ntroduced n whch bnary data sequence s mapped to complex symbols usng the concept of correlated phase states of a CPM sgnal. Canoncal optmum and suboptmum multple-symbolobservaton OFDM-CPM recevers are derved. Multpath channel wth AWGN s assumed. The recevers are analyzed for bterror-rate (BER) performance n terms of hgh- and low-snr bounds. These bounds are llustrated as a functon of parameter h, tme delay and attenuaton level. It s shown through numercal computaton that OFDM-CPM system outperforms a smlar OFDM-PSK system n a two-ray multpath model. Moreover, an OFDM-CPM system gves the best error performance when the parameter h s.5 wth observaton nterval of. I. INTRODUCTION OFDM s a good canddate for wreless multmeda communcaton by vrtue of ts excellent propertes n frequency-selectve fadng envronment [1], []. In OFDM, we transmt data over several parallel low data rate channels. Ths provdes data ntegrty due to fadng, relatve to modulaton methods that employ sngle channel for hgh data rate transmsson. Among other benefts of OFDM s that t fully explots the advantages of dgtal sgnal processng concepts []. A typcal OFDM transmtter wors as follows: Seral encoded data s sent to a mapper that outputs a complex number. A seral-to-parallel converter serally taes n the complex numbers and forms a parallel stream by ncreasng ther tme perod. Ths stream has as many complex numbers as the number of subchannels. Inverse Fast Fourer Transform (IFFT) s then appled to the parallel stream that results n orthogonal sgnals on the subchannels. The orthogonal sgnals are then converted bac nto a seral stream and after up convertng the sgnal to desred carrer frequency the sgnal s transmtted. Whle n the lterature OFDM-PSK, QAM, DPSK and DAPSK have been consdered [4]-[7], OFDM-CPM sgnals that use the concept of correlated phase states of a CPM sgnal have not been consdered to-date. One of the advantages of OFDM- CPM sgnals s that we can systematcally ntroduce correlaton amongst adjacent OFDM symbols by an approprate choce of parameter h (n typcal CPM sgnals h s modulaton ndex). Furthermore, ths correlaton can be exploted n order to reduce the BER n such a system. Thus n the paper we address the detecton problem of observng n OFDM-CPM symbols and arrvng at an optmum decson on one of the symbols. The channel s modeled as multpath and AWGN. The paper s organzed as follows: In secton II we descrbe OFDM-CPM sgnalng scheme. We address the detecton problem n secton III and arrve at the recever structure. In secton IV hgh and low SNR bounds on the performance of the optmum OFDM-CPM recever are derved. These bounds are llustrated through numercal computaton n secton V. The paper s concluded n secton VI. II. OFDM-CPM SIGNALING SCHEME As shown n Fg. 1, seral bt stream b, =1 :::wth bt duraton of T b seconds s converted nto blocs of N bts represented by a p =1 :::and p =1 :::N1 N denotes the number of carrers and a p = 1. For example, a p would denote the frst bloc of N bts and a 1p the second bloc of N bts and so on. The CPM mappers transform the ncomng a p nto approprate complex numbers c p gven by c p = cos ( p )+jsn ( p ) (1) wth p = a p h + h 1 q= a qp + () h, < h < 1, s a parameter and represents the ntal mappng pont that s assumed zero wthout loss of generalty. It s evdent from () that the angles p depend not only on the current data but also on the past data. Fg. shows all the possble values of p as a functon of tme when h = :5. Current value of s determned by addng +h (f data bt s a +1) orh (f data bt s a 1) to the prevous value of. Fg. 1. OFDM-CPM Transmtter and Channel //$17. IEEE 1651

2 The correspondng complex numbers le on a crcle. In Fg. we show the constellaton dagram of CPM mapper for h =:5 and h = :5. The correspondng complex numbers of a 4- carrer OFDM-CPM sgnal wth h = :5 for two consecutve blocs of data sequences are shown below: 6 4 [a p a +1p ] =) [c p c +1p ] =) 6 4 +j 1 +j 1 j 1 +j : The complex numbers from the output of CPM mappers are passed through pulse shapng flters g(t), then modulated by orthogonal carrers and fnally summed to gve the transmtted OFDM symbol whch s mathematcally represented as x(t) = p g(t) = c p g(t T )e j T pt t<1 () 1 pt t T else: In (), T (= NT b ) s the OFDM symbol duraton. III. THE DETECTION PROBLEM Wth reference to Fg. 1, we can model the receved sgnal as r(t) = x(t) h(t) +n(t) = s(t) +n(t) t (4) h(t) s the channel mpulse response, n(t) s AWGN wth a double sded power spectral densty of No and * denotes convoluton. s(t) s gven by [8] n1 N1 s(t) = Fg.. = p= c p p(t T ) (5) Phase trells for OFDM-CPM sgnalng Fg.. Constellaton dagram of CPM mapper for (a) h =:5 and (b) h = :5 p(t) =g(t)e j T pt h(t): (6) s(t) can be wrtten as s (t A [A 1 :::A n1 ]) A s a vector [c c 1 :::c N1 ] T of complex numbers whch n turn relates to the nput data sequence [a a 1 :::a N1 ] T. Thus there are N possble A s. The detecton problem s to observe r(t) over seconds and arrve at an optmum estmate of A (gven by ^A ) transmtted durng t T seconds. For the case of N carrers and n observaton ntervals, the detecton problem s the composte hypothess testng problem stated below [9]: H : r(t) =s t A [A 1:::A n1 ] + n(t) (7) =1 ::: N 1 and t. A denotes the th of the N possble A s. The soluton to ths problem s the lelhood rato test (LRT) whch can be expressed as B = B Pr (R j H B) Pr (B) db (8) B = [A 1 A :::A n1 ] (9) db = A 1 A ::: A n1 da 1 da :::da n1 (1) Pr (B) = Pr (A 1 ) Pr (A ) :::Pr (A n1 ) (11) Pr(A l ) = 1 v v1 = A l A l (1) and v = N. Substtutng (9)-(1) n (8) t can be shown that the lelhood functons can be wrtten as = z1 1 N o exp4 N o d= r(t)s t A Bd dt s t A Bd dt 5 (1) z = N(n1) and B d s the dth of the z possble realzatons. The correspondng recever structure s shown n Fg. 4 K d represents the porton n (1) that s ndependent of the receved sgnal r(t). 165

3 Snce s are Gaussan, n order to determne the error performance of the recever we need to fnd the condtonal means and varances of s. Then the probablty of symbol error wll be gven by x Ps = 1 z v1 Y = 6=j 8 z1 1 >: 1 = 1 1 p exp 1 Q y u + y u 9 du> (17) Fg. 4. OFDM-CPM recever IV. PERFORMANCE OF OFDM-CPM RECEIVER It s too complex to analytcally compute the performance of the recever shown n Fg. 4. Therefore, we derve performance bounds at low and hgh SNRs. These bounds taen as a sngle bound wll be a good performance bound at all values of SNR [1]. A. Low SNR Approxmaton At low SNR the lelhood functon can be approxmated by 1+ N o 1 N o z1 ( s t A Bd) dt z1 r(t) d= s t A Bd dt (14) d= the recever dctated by (14) s gven as = ( z1 r(t) z1 s t A Bd) dt d= s t A Bd dt: (15) d= Let hypothess j and a data sequence B be true, then the receved sgnal r(t) s gven by r(t) =s t A j B + n(t): (16) y = [ j j H j ] and y,, y and are the condtonal means and varances of y and gven by y = = " z1 n1 n1 d= = = z1 n1 n1 C jdh d= = = " z1 n1 n1 d= = = z1 n1 n1 d= = = z1 n1 z1 y = N o C dh n1 d= = d = = z1 n1 z1 n1 = N o d= = d = = C jdh R( )C j R( )C jd C dh R ( )C j R( )C d C jdh # # R ( )C jd C dh R ( )C d : In these expressons R s a matrx of cross correlatons whose ath row and bth column s gven by C d R ab ( ) = b(t T ) are vectors of complex numbers defned as C j = C d = and C H =(C ) T. B. Hgh SNR Approxmaton h c j cj 1 h :::cj c d cd 1 :::cd N1 a (t T )dt: (18) T N1 T The optmum recever of (1) can be wrtten as = z1 d= exp (J d ) (19) 165

4 1 N o J d = N o At hgh SNR t can be shown that d= r(t)s t A Bd dt s t A Bd dt: () z1 exp (J d ) exp ej (1) ej = max fj d d = 1 :::z 1g : Snce the functon exp(.) s monotonc, the hgh SNR suboptmum recever computes J d = 1 :::v 1 d = 1 :::z 1 and a decson s made based on the largest of these. Upper bound on the error rate of such a recever can be unon bounded whch s gven by Ps 1 vz v1 z1 Pr J l J d js t A Bl () = l= d= Pr J l J d js t A Bl can be shown to be s Pr J l J d js t A Bl = Q4 and E l = E d = n1 d l = n1 = = n1 n1 = = ( n1 C lh C dh n1 = = E l + E d d l 5 N o R( )C l R( )C d C dh R( )C l V. NUMERICAL RESULTS ) : () We now demonstrate some of the results based on the theory presented n prevous sectons. For all the results a two path channel s assumed havng mpulse response gven by h(t) = 1 (t) + (t ) (4) 1 and are the levels of drect and delayed paths and s the delay. In all the results that we present, observaton nterval s two symbols and ntal mappng ponts are assumed zero. Further, n Fgs. 5-7, 1 = =4dB and =T =:1. The performance of OFDM-CPM system s evaluated for three values of h whch are.5,.5 and.715. Constellaton dagrams for h =:5 and h =:5 are shown n Fg.. h =:715 s chosen because t gves optmum error performance n a sngle-carrer CPM system for a 5-bt observaton nterval. In the paper we derved expressons for OFDM symbol error rate from whch BER can be approxmated by usng the approach gven n [11]. Fg. 5 shows the BER performance of an 8-carrer OFDM- CPM system wth h =:5. The composte upper bound s constructed by tang the smaller of the hgh SNR bound and the low SNR bound. The BER performance of an 8-carrer OFDM- BPSK system wth symbol by symbol detecton s also shown n ths fgure. For the case of OFDM-CPM, sgnfcant mprovement n performance s observed because OFDM-CPM symbols are correlated and multple-symbol-observaton recever s employed. Ths recever explots the extra nformaton and as a result BER mproves. Fg. 6 shows a plot of composte upper bound of an 8-carrer OFDM-CPM system for varous values of h. Best performance s acheved when h = :5. Ths s due to the fact that sgnal constellaton has four ponts and the dstance between sgnals s the maxmum. When h =:5, there are eght constellaton ponts and as a result the dstance between sgnals decreases whch deterorates the performance. BER for h = :715 falls some n between h = :5 and h = :5. BER performance of an OFDM-CPM system as a functon of =T s shown n Fg. 7. SNR s fxed at 1 db and 1 = =4 db. Performance s senstve to delay for =T less than.1. However, BER stablzes when =T goes beyond.1. Ths s because delayed and drect waves are synchronzed at some pont addng the overall power. Ths eeps the BER stable. However, we antcpate that the performance wll deterorate for hgher values of =T when the rato 1 = s larger because ICI and ISI wll domnate. It s seen that the proposed system performs better than OFDM-PSK and the best performance s gven by h =:5. Fg. 8 shows a plot of composte bound as a functon of 1 = for an 8-carrer OFDM-CPM system wth h =:5 :5 and :715 and SNR=1 db. For low levels of 1 =, BER s hgh because of the ICI and ISI ntroduced by the delayed wave. As 1 = ncreases, system performance mproves because the BER Composte bound Low SNR bound Hgh SNR bound SNR (db) Fg. 5. BER of an 8-carrer OFDM-CPM system wth h =:5, 1 = =4 db and =T =:1 1654

5 Composte Upper Bound h=.5 h=.5 h= SNR (db) Fg. 6. Composte bound of an 8-carrer OFDM-CPM system for varous values of h wth 1 = =4dB and =T =:1 Composte Upper Bound h=.5 h=.5 h= Rato of powers of drect and delayed waves, α /α db Fg. 8. Composte bound of an 8-carrer OFDM-CPM system as a functon of 1 = wth SNR = 1 db and =T =:1 errors are predomnantly due to Gaussan nose. VI. CONCLUSIONS We ntroduced a new class of OFDM-CPM sgnals and derved optmum and suboptmum multple-symbol-observaton recevers assumng multpath and AWGN. The performance was compared wth a smlar OFDM-PSK system and t was shown that OFDM-CPM outperformed OFDM-PSK. The mplementaton and performance evaluaton of the proposed recevers requres very hgh computatonal power and that s why we could not go beyond 8 carrers and two observaton ntervals. However, ther performance s useful as a reference on what s possble to attan. The recever complexty can be reduced by choosng ratonal values for h that guarantees fnte number of ponts on constellaton. Suboptmum algorthms such as MLSE (Maxmum Lelhood Sequence Estmaton) usng Vterb algorthm can Composte Upper Bound h=.5 h=.5 h= Normalzed tme delay, τ/t Fg. 7. Composte bound of an 8-carrer OFDM-CPM system for varous values of =T wth 1 = =4dB and SNR = 1 db also be nvestgated for detecton of OFDM-CPM sgnals. Furthermore, we are nvestgatng the performance of an FFT-based recever for an OFDM-CPM system. Ths wll allow us to go for hgher number of carrers. REFERENCES [1] J.A.C. Bngham, Multcarrer modulaton for data transmsson: An dea whose tme has come, IEEE Communcatons Magazne, pp. 5 14, May 199. [] Leonard J. Cmn Jr., Analyss and smulaton of a dgtal moble channel usng orthogonal frequency dvson multplexng, IEEE Trans. on Comm., vol., no. 7, pp , July [] S.B. Wensten and Paul M. Ebert, Data transmsson by frequencydvson multplexng usng the dscrete Fourer transform, IEEE Trans. on Comm. Tech., vol. 19, no. 5, pp , October [4] Mnoru Oada, Shnsue Hara, and Norho Mornaga, Bt error rate performances of orthogonal multcarrer modulaton rado transmsson systems, IEICE Trans. Commun., vol. E76-B, no., pp , February 199. [5] Yun Hee Km, Icho Song, Hong Gl Km, Taejoo Chang, and Hyung Myung Km, Performance analyss of a coded OFDM system n tme-varyng multpath Raylegh fadng channels, IEEE Transactons on Vehcular Technology, vol. 48, no. 5, pp , September [6] Thomas May, Hermann Rohlng, and Voler Engels, Performance analyss of Vterb decodng for 64-DAPSK and 64-QAM modulated OFDM sgnals, IEEE Trans. on Comm., vol. 46, no., pp , February [7] Jun Lu, Tjeng Thang Tjhung, Fumyu Adach, and Cheng L Huang, BER performance of OFDM-MDPSK system n frequency-selectve Rcan fadng wth dversty recepton, IEEE Transactons on Vehcular Technology, vol. 49, no. 4, pp , July. [8] Anders Vahln and Nls Holte, Maxmum-lelhood sequence estmaton for OFDM, Sgnal Processng n Telecommuncatons: Proc. of the 7th Int l Thyrrhenan Worshop on Dgtal Communcatons, Vareggo, Italy/Ezo Bgler and Marco Luse, eds., pp. 7 19, September [9] Anthony D. Whalen, Detecton of sgnals n nose, Academc Press, [1] Wllam P. Osborne and Mchael B. Luntz, Coherent and noncoherent detecton of CPFSK, IEEE Trans. on Comm., vol., no. 8, pp. 1 16, August [11] John G. Proas, Dgtal Communcatons, McGraw Hll Inc.,