Generalized OFDM. Myungsup Kim and Do Young Kwak Dept. of Mathematical Sciences, KAIST, Deajeon, Korea {myungsup,

Transcription

1 / 6 eneralized OFDM Myungsup Kim and Do Young Kwa Dep. of Mahemaical Sciences, KAIST, Deajeon, Korea {myungsup, dy}@ais.ac.r Absrac In his paper, a generalized OFDM (-OFDM) which proecs he ou of band (OOB) leaage and saisfies he orhogonaliy beween subcarriers is proposed for nex generaion wireless communicaions. This scheme maps many daa symbols o many subchannels hrough filer marix, unlie he convenional OFDM ha maps a daa symbol o a subchannel in a one-o-one fashion. In -OFDM, he filer marix is used in order o limi he OOB leaage of he specrums of inpu daa, which is generaed hrough processes such as jump-removing, filering, and orhogonalizing a specially chosen iniial marix. -OFDM is compared wih OFDM in view of OOB leaage, cres facor, complemenary cumulaive disribuion funcion, and bi error rae (BER) performance. Index Terms Marix filer; pilo vecor; jump-removing marix; -OFDM; frequency; muliplexing; OFDM; BER; cres facor; CCDF. I. Inroducion OFDM plays a significan role in modern elecommunicaions due o is wide ranging applicaions from is use in home communicaion echnologies o wireless local area newor (WLAN) [] and mobile communicaion sysems. This is because OFDM uses specral resources effecively hrough overlapping he subcarriers while mainaining orhogonaliy, and i can esimae he communicaion channel easily using pilo symbols []. Furhermore, OFDM can be used o increase he channel capaciy significanly using muliple inpu muliple oupu (MIMO) echnology hrough which several users can communicae simulaneously hrough muliple channels on he same bandwidh using space diversiy [3]. Nowadays, OFDM has been uilized in almos all areas such as mobiles, local area newors, broadcasing, and saellie communicaions; i will coninue o be a major echnology in he fuure. owever, he specral environmen for communicaion will become increasingly worse because numerous sysems will be conneced o inerne hrough fixed and mobile wireless communicaion newors. Therefore, i will be a significan problem o solve he inerference beween communicaion sysems. Recenly, many effors have been focused on sudies of new ransmission schemes o enhance he specral characerisics of OFDM for nex generaion mobile communicaions [4-7]. Filer ban muli-carrier (FBMC), universal filered muli-carrier (UFMC), and generalized frequency division muliplexing (FDM) are prominen oucomes in reducing he specral leaage of OFDM. FBMC has he meri of being able o saisfacorily reduce he specral leaage o neighbor channels hrough filering in he frequency domain, bu i uses samples wo or four imes per symbol, which may be a drawbac in high-speed ransmission because he frame lenghs increase in doing so. UFMC is a scheme ha miigaes he specral leaage hrough filering using Dolph-Chebyshev filers. owever, many IFFT processors and filers are required for a symbol ransmission, and he use of he FFT processor wih wice he lengh of he IFFT used in he ransmier in a receiver is a facor ha increases he complexiy of implemenaion. In addiion, since he response of he filer of he ransmier becomes long, he inerval beween he symbols may be very close o each oher, so here may be a possibiliy of overlapping beween symbols in a mulipah environmen. FDM is also an effecive scheme o reduce he specral leaage. owever, since he frequency bands of subcarrier are designed o overlap, i causes self-inerference, which may increase he complexiy of he receiver because i mus implemen he means o remove i [8]. In his paper, we inroduce -OFDM ha can effecively reduce he specral spread ha occurs in OFDM wihou changing he srucure of OFDM. Because -OFDM performs filering hrough filer marices in he frequency domain wih minimal operaional complexiy and i does no increase he lengh of he OFDM symbol, i is no only a specral and power efficien modulaion scheme, bu also i can be operaed a high speed and is compaible wih he exising sysems using OFDM. In secion II, we inroduce he concep of -OFDM and he generaion of filer marix ha consiues -OFDM. Unlie OFDM ha one subcarrier is creaed using one frequency, one subcarrier in -OFDM is creaed by synhesizing several frequencies. By performing he filering operaion in he frequency domain, -OFDM does no increase he lengh of he OFDM symbol in he ime domain. In secion III, we inroduce a mehod for insering pilo vecors ino he filer marix o esimae he communicaion channel. In secion IV, we analyze he characerisic of -OFDM ha can reduce he OOB leaage. In secion V, we evaluae he upper bound of he cres facor and complemenary CDF for -OFDM. In secion VI, we obain he BER performance of -OFDM over AWN (Addiive Whie aussian Noise) channel. In secion VII, we summarize he feaures of -OFDM described in all previous secions. II. -OFDM In -OFDM as shown in Fig., a daa vecor d is divided ino J shorer vecors d, d,, d J-, each of which is muliplied by he filer marix, which are denoed as v, v,, v J-. The -h daa symbol in he l-h daa vecor is carried on he -h column of he filer marix. The condiion for orhogonaliy o /7/$3. 7 IEEE 65 ICTC 7

2 / 6 recover he daa symbols wihou inerference beween symbols is I. In -OFDM as shown in Fig., a daa vecor d is divided ino J shorer vecors d, d,, d J-, each of which is muliplied by he filer marix, which are denoed as v, v,, v J-. The -h daa symbol in he l-h daa vecor is carried on he -h column of he filer marix. d d d J Fig.. -OFDM I is no simple o mae a filer marix wih good specral and orhogonal characerisics a once, so we generae he filer marix from an iniial marix hrough he procedure depiced in he signal flow diagrams in Figs.. We can obain a filer marix where... v v v J T f WFΨΦF W, () f K K K K. () The wo marices Φ and Ψ remove he jump componens and filer all he columns of he incoming daa marix, respecively so ha heir specrums are forced o be zeros a he beginning and end of he frequency band. Fig.. Signal flow diagram for filer marix generaion A. Iniial Marix IFFT Channel W F Φ Ψ F A P Q R B s An iniial marix is consruced so ha all he columns have wo minimum nonzero enries, are uni vecors, and are orhogonal o each oher. For an odd N, we define an iniial marix. (3) r FFT T W c c c J K... f ˆd ˆ ˆd d J B. Zero Padding and Cyclic Shifing We have an enlarged marix A W, (4) where W is he zero-padding and cyclically shifing marix N / W L N. N / C. Inverse DiscreeFourier Transform Convering A from he frequency domain o he ime domain, we can obain he following marix: and F is he DFT marix (5) P F A F W, (6) F e e e where / L. j e j e jl e j j4 jl jl jl jl L e e e D. Jump Removing In order o raise a pulse from a small value a he beginning poin and o lower he pulse o a small value a end, he firs row should be a zero vecor hrough muliplying he jump-removing marix Γ in he ime domain [,] as where, (7) Q Γ P ΓF W, (8) Γ. (9) The resulan jump-removed marix Q + becomes a marix of L N as size 65

3 3 / 6 p p Q. p p L () Since all he enries of he firs row of his marix Q are zero, he row can be discarded in order o shoren he column lengh. L L reduced marix Thus, we have a where he marix Q WQ ΦF W WΓF W, () he firs row as given by: W is a L L marix ha discards z z. (6) Since he response of he subcarrier in he frequency domain is muliplied by his funcion, i becomes zero a boh ends of he frequency band of he subcarrier. F. Discree Fourier Transform and Truncaion Applying DFT o each column of R in order o ransform i from he ime domain o he frequency domain, we have B FR FΨΦF W, (7) From (), we have E. Inernal Filering Marix W. () Φ W Γ. (3) The inernally filered marix R using a filering operaor Ψ can be represened as where Ψ is he LL R ΨQ ΨΦF W, (4) filering marix. For example, he filering marix wih -ap moving average (MA) can be expressed as Ψ. (5) where B is a L N marix.. Downward Cyclic Shifing and Truncaing This process is cyclic shifing downwards and eliminaes he rows wih zero enries in order o undo zero padding and cyclic shifing by he operaor W. Reversely cyclically shifing and runcaing he marix B, we have. Neares Orhogonal Marix T T K WBWFΨΦF W. (8) The marix K is generaed hrough jump removing and inernal filering using Φ and Ψ in he ime domain, and Ω and S in he frequency domain. owever, even if an iniial marix is orhogonal, he marix K can be no longer guaraneed o be orhogonal. Therefore, he marix K should be ransformed o be orhogonal while reaining is original properies. We can obain a filer marix which is neares o K and whose columns are orhogonal o each oher hrough he following formula [9]. f K K K K (9) Since he magniude of all he columns of he marix obained hrough he funcion f is, is a special filer marix having a consan gain in he occupied frequency band. The marix Ψ operaes in all he columns of he marix Q as a filer, and i filers each column of he inpu marix. Noe ha he lengh of he filered subcarriers in he ime domain should be equal or less han he IFFT size L, which means ha he lengh of he symbol should be no increased by filering. In an exended form, he z-ransform of he MA filer wih a parameer may be wrien as III. FILTER MATRIX WIT PILOT VECTOR We will show how o inser a pilo vecor nex o he 4 h column of a 9 8 jump-removed marix below. 653

4 4 / 6 U. () Firs, we inser an empy column vecor afer he 4 h column of he marix U and le all he remaining columns from he 5 h column be one enry away from he cener row as T. () Second, we inser and - in he fifh column so ha hey do no overlap wih he oher rows on he horizonal line, as shown in (). The vecor in he brace of he marix T p is he pilo vecor. The size of he generaed marix is hus 9 and as can be seen, all he columns are orhogonal o each oher, so he ran becomes 9. () T p. IV. FREQUENCY RESPONSES OF -OFDM (3) U p. From (9), we obain a real filer marix 75 p, which is given in Table I. TABLE I The firs column can be a pilo vecor since i is no affeced by 75 oher columns, and he enries g, and g 6, of he marix p can be used as he pilo symbols I can be seen ha all enries of his marix excep he second and sixh enries of he firs column are zeros, bu all enries in he second and sixh rows excep he firs column are zeros. Since he firs column is no affeced by remaining columns, i can be used as he pilo vecor. Example : (73, 7)-OFDM according o he Long Term Evoluion (LTE) Specificaions Fig. 3 illusraes he frequency responses of OFDM and -OFDM wih = 4 according o he LTE specificaions wih 7 subcarriers and he frame lengh of L = 8. The magniudes of he frequency responses of -OFDM decrease significanly and vanish enirely a he beginning and end. This indicaes ha -OFDM is less liely o have a specral influence on neighbor channels han OFDM. Magniude[dB] Example : (7, 5)-OFDM The filer marix used in (7, 5) -OFDM wih = has he size of 7 5. A jump-removed marix including a pilo vecor is illusraed as Frequency[Kz] (a) 654

5 5 / 6 - C OFDM N / q. N (8) Magniude[dB] (b) Fig. 3. Frequency response of (73, 7)-OFDM. (a) OFDM, (b) -OFDM V. Cres Facor and Complemenary CDF A. Cres Facor (CF) The signal of -OFDM is ransmied hrough he IFFT afer he inpu daa symbols are band-limied hrough he filer marix. The -h column can be represened as N / n cos cos L L l, l L r n z y x for,,3,, N /. For each column of s ln Φ, is IDFT can be expressed as B n (4) j n sin. L L (5) The ransmied signal for -OFDM can be represened as where Frequency[Kz] N / wn d r n d s n, (6) d is a daa symbol, and r n and s n are real and imaginary subcarriers, respecively. We can obain he average power where E d d Wn Cn Dn (7) d is he symbol power. Fig. 4 shows he average power of 6 subcarriers when N = 7, d and L = 4. CF can be wrien as where we used ha Power Power x -6 x -6 d P. d (a) (b) n (c) Fig.4. Signal power of -OFDM wih =. (a) Real power C(n) (b) Imaginary power D(n) (c) Toal power W(n). The gain for CF of -OFDM compared wih OFDM becomes N ain log. (9) N / q Specifically, when N = 4p + where p is a posiive ineger, he CFs of OFDM and -OFDM become COFDM C OFDM p, p. A his ime, he CF gain for OFDM of -OFDM is always as follows. Pr(PAPR > PAPR ) Power x -6 p ain log log 3[ db]. p OFDM -OFDM (3) (3) PAPR [db] 655

6 6 / 6 Fig. 5. CCDFs of OFDM and -OFDM wih =. B. Complemenary CDF (CCDF) wn is composed of a sum of many random signals, Since i can be approximaed by a aussian disribuion as f wn * where n w exp, wn (3) N E wnw Wn n. CCDF can be expressed as L x PX x exp. n wn (33) Fig. 5 shows he CCDFs of OFDM and -OFDM. VI. BER Performance The BER performance of -OFDM wih QPSK daa symbols can be expressed as p erfc b E N b o r r, (34) where E b is he energy per bi and N o / is he noise power specral densiy. BER Eb/No (db) Fig. 6. BER performances of heoreical QPSK and (73,7) -OFDM by simulaion. VII. CONCLUSION Theory Simulaion Approximaion We proposed a -OFDM ha could reduce he OOB leaage ha was an inrinsic drawbac of OFDM. -OFDM includes filer marices ha have good specral characerisics as well as orhogonaliy, which are generaed hrough such processes as jump-removing, signal filering, and orhogonalizing. A filer marix consiss of pilo column vecors in order o esimae mulipah fading channels and remaining column vecors in order o carry daa symbols. Through simulaion, i was found ha -OFDM is an excellen scheme for reducing he OOB leaage effecively. Since -OFDM filers he inpu symbols in he frequency domain, i is a mehod ha can grealy reduce he amoun of compuaion compared wih any oher scheme of performing convoluion in he ime domain. We invesigaed he cres facor and he complemenary CDF of -OFDM and found ha he CF of -OFDM has a gain of 3 [db] wihou any PAPR reducion scheme compared wih ha of OFDM and he CCDF of -OFDM has he same performance as OFDM. Through heoreical analysis and simulaion, i was found ha he BER performance of -OFDM was close o he ideal BER performance, which means ha he orhogonaliy of filer marix does no suffer from channel noise. The proposed scheme can be compaible wih such as exising LTE or Wi-Fi sysems using OFDM because i can suppress he OOB leaage wihou changing he lengh of he OFDM symbol. REFERENCES [] Sephen B. Weinsein, The isory of Orhogonal Frequency-Division Muliplexing, IEEE Communicaions Magazine, vol. 47, no., pp. 6 35, Nov. 9. [] O. Edfors, M. Sandell, J.-J. van de Bee, and S. K. Wilson, OFDM Channel Esimaion by Singular Value Decomposiion, IEEE Trans. Commun., vol. 46, no. 7, pp , July 998. [3] ordon L. Süber, Seve W. Mclaughlin, Ye Li, Mary Ann Ingram, and Thomas. Pra, Broadband MIMO-OFDM Wireless Communicaions, Proc. IEEE, vol. 9, no., pp. 7-94, Feb. 4. [4] 4 Americas, 5 Specrum recommendaions, Aug. 5. [5] Fran Schaich, Thorsen Wild, and Yejian Chen, Waveform Conenders for 5 - Suiabiliy for Shor Pace and Low Laency Transmissions, in VTC Spring, Seoul, Korea, May 4, pp. -5. [6] Fran Schaich and Thorsen Wild, Waveform Conenders for 5 - OFDM vs. FBMC vs. UFMC, in 6h ISCCSP, May 4, pp [7] Myungsup Kim and Do Young Kwa, "eneralized OFDM for 5h eneraion Mobile Communicaion." Acceped paper, VTC7-Spring, 7. [8] Nicola Michailow, Maximilian Mahé, Ivan Simões aspar, Ainoa Navarro Caldevilla, Luciano Leonel Mendes, Andreas Fesag, and erhard Feweis, eneralized Frequency Division Muliplexing for 5h eneraion Cellular Newors, IEEE Trans. Commun., vol. 6, no. 9, pp , Sep. 4. [9] Nicholas J. igham, Compuing he Polar Decomposiion - Wih Applicaions*, SIAM J. SCI. STAT. COMPUT. vol. 7, no. 4, Oc. 986 [] Myungsup Kim and Do Young Kwa, A Pilo Included Column Mean Vanishing Marix, Journal of Mahemaics Research, Vol. 9, No., pp. 8 33, April 7. [] Myungsup Kim and Do Young Kwa, Column Mean Vanishing Marices, Inernaional Journal of Pure and Applied Mahemaics, Vol. 4, No. 3, pp , May