A Spectrally Efficient Frequency Division Multiplexing Based Communications System M. R. D. Rodrigues and I. Darwazeh

Size: px

Start display at page:

Download "A Spectrally Efficient Frequency Division Multiplexing Based Communications System M. R. D. Rodrigues and I. Darwazeh"

Delilah Lester
6 years ago
Views:

1 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) A Spectrally Effcet Frequecy Dvso Multplexg Based Commucatos System M. R. D. Rodrgues ad I. Darwazeh Laboratory for Commucatos Egeerg elecommucatos Research Group Dept. Egeerg, Uversty of Cambrdge Dept. E&EE, Uversty College Lodo mrdr3@eg.cam.ac.uk.darwazeh@ee.ucl.ac.uk Abstract-- I ths paper we propose a spectrally effcet frequecy dvso multplexg (FDM) based commucatos system. hs FDM system hadles tercarrer frequecy separatos smaller tha those of orthogoal frequecy dvso multplexg (OFDM) systems. he FDM trasmtter cossts of a bak of modulators that geerate the dfferet formato coveyg sub-carrers that compose the trasmtted FDM sgal. he FDM recever cossts of a bak correlators followed by a maxmum lkelhood (ML) detector. he demodulators extracts a set of suffcet statstcs from the receved FDM sgal ad the ML detector uses the suffcet statstcs to estmate the trasmtted symbols. o allevate the complexty of ML detecto alteratve detecto methods are examed cludg lear detectors ad geetc algorthms (GA) based detectors. Fally, FDM system fudametal lmtatos are dscussed. Idex erms Frequecy dvso multplexg, Orthogoal frequecy dvso multplexg, Badwdth effcecy, Spectral effcecy, Maxmum lkelhood detecto, Lear detecto, Geetc algorthm based detecto I. INRODUCION Orthogoal frequecy dvso multplexg (OFDM) based commucatos systems have bee adopted /proposed for a umber of wreless applcatos such as dgtal audo broadcastg (DAB), dgtal vdeo broadcastg (DVB), door wreless etworks (e.g. IEEE a, IEEE g ad HIPERLAN2) ad fxed broadbad wreless access (e.g. IEEE ). OFDM offers may advatages for wreless applcatos such as good performace the presece of frequecy selectve fadg wthout the eed for complex tmedoma equalsato techques, effcet usage of the avalable badwdth ad effcet dgtal sgal processor based trascever mplemetatos [1,2]. At a OFDM trasmtter, a hgh rate data stream s parttoed to N lower rate data sub-streams; each substream modulates oe of N sub-carrers; ad fally, the modulated sub-carrers are added to geerate the trasmtted OFDM sgal. he OFDM recever cossts of a bak of flters matched to the dfferet OFDM subcarrers followed by a bak of detectors. he tercarrer frequecy separato s set to be equal to the verse of the sgallg terval thereby esurg that the formato coveyed by a partcular sub-carrer ca be recovered depedetly from the formato coveyed by other sub-carrers. hs paper vestgates frequecy dvso multplexg (FDM) systems wth tercarrer frequecy separatos smaller tha those of OFDM systems. Note that FDM based systems wth tercarrer frequecy separatos equal to half the verse of the sgallg terval have already bee proposed elsewhere [3,4]. However, such proposals oly hadle BPSK or M-ASK modulato of the FDM sub-carrers whereas ths ew proposal also hadles M-PSK or M-QAM modulato of the FDM sub-carrers. II. FDM SYSEM BASICS he FDM based commucatos system proposed ths paper works as follows: At the trasmtter, a hgh data rate stream s also parttoed to N lower rate data sub-streams; each sub-stream modulates oe of N subcarrers; ad fally, the modulated sub-carrers are added to geerate the trasmtted FDM sgal. However, ths case the tercarrer frequecy separato ca be smaller tha the verse of the sgallg terval ad hece the spectral effcecy of a FDM system s hgher tha that of a equvalet OFDM system. We wrte the trasmtted FDM sgal as follows g s N 1 ( t) S g ( t k) ( t) k 0 1 j 2πΔft e 0, ( k,, t t [ 0, ] [ 0, ] where S k, s the symbol trasmtted tme slot k ad sub-chael, N s the umber of FDM sub-chaels, s the FDM symbol durato ad Δf s the tercarrer frequecy separato. We cosder for smplcty (2)

2 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) symbols trasmtted dfferet tme slots ad subchaels to be depedet. We wrte the receved FDM sgal as follows ( t) s( t) ( t) r + (3) where s(t) s the trasmtted FDM sgal ad (t) s addtve whte complex Gaussa ose (wth power spectral desty N 0 ). We cosder for smplcty commucato addtve whte Gaussa ose chaels. he recever for FDM sgals cossts of a bak correlators followed by a detector. he correlators extracts a set of suffcet statstcs R k,, 0,,N 1, from a receved FDM symbol r(t), k<t<(k+, ad the detector uses the suffcet statstcs R k,, 0,,N 1, to estmate the trasmtted symbols S k,, 0,,N 1. he goal of ths process s to mmse the probablty of error for the estmate 1. Accordgly, the frst step of recepto the receved FDM symbol s projected oto a complete set of orthoormal fuctos that spa the FDM symbol space gve by b 1 1 ( t) g ( t) g ( t) b ( t) dtb ( t) ξ 0 where the value of ξ s chose so that the eergy of b (t) s equal to oe. Note that b (t), 0,,N 1, are determed from g (t), 0,,N 1, usg the Gram- Schmdt orthoormalsato procedure [5]. he suffcet statstcs are the gve by R k, k k Sʹ r s k, ( t) b ( t k ) ( t) b ( t k ) + N k, dt + k ( t) b ( t k ) Here N k, s a crcularly symmetrc complex Gaussa radom varable wth mea zero ad varace N 0. Note that by vrtue of orthoormalty of the bass fuctos N k,, 0,,N 1 are depedet. I the secod step of recepto the suffcet statstcs R k,, 0,,N 1, are used to estmate the trasmtted 1 Uder the assumpto that the formato coveyed by dfferet OFDM symbols s depedet the t suffces to carry out decsos o a OFDM symbol by OFDM symbol bass. dt (4) (5) symbols S k,, 0,,N 1, so as to mmse the probablty of error for the estmate expressed as P N M 1 ( ) ( ) ( e) 1 p p R S S 0 Rego dr where R[R k,0 R k,1 R k,n 1], S[S k,0 S k,1 S k,n 1], p, 0,,M N 1, deotes the probablty of each possble trasmtted vector S (), 0,,M N 1, ad Rego deotes a partto of the mult-dmesoal vector space for whch the trasmtted vector S () s assumed to have bee trasmtted. Note that N deotes the umber of FDM sub-chaels ad M deotes the sze of the M-PSK or the M-QAM costellato used each FDM subchael. he probablty desty fucto p(r SS () ) s expressed as p ( ) ( R S S ) ( 2πN ) ( ) ( ) ( R Sʹ ) ( R Sʹ ) 1 2N0 e N 1 0 where Sʹ[Sʹk,0 Sʹk,1 Sʹk,N 1]. he probablty of error for S s mmsed f ad oly f we choose as a estmate for the trasmtted vector ŜS (j) such that j ( j) ( ) ( R S S ) p R S S ( ) (6) (7) p p max (8) or, the case of equally lkely trasmtted vectors (p 1/M N, 0,,M N, we choose as a estmate for the trasmtted vector Ŝ S (j) such that ( j) ( ) ( R S S ) RS S ( ) p max (9) he decso rule (8) s the maxmum a posteror (MAP) crtero ad the decso rule (9) s the maxmum lkelhood (ML) crtero. Note that for equally lkely trasmtted vectors mmsato of the probablty of error for S wll also result mmsato of the probablty of symbol error or, the bary case, probablty of bt error. I ths paper, we shall oly cosder equally lkely trasmtted symbols ad cosequetly ML detectors. Fg. 1 shows the structure of a FDM based commucato system mpared by ose. Note that the FDM recever structure reduces to the OFDM recever structure whe the tercarrer frequecy separato s set to be equal to the verse of the sgallg terval.

3 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) S k,0 Itegrate ad Dump Flter R k,0 Ŝ k,0 S k,1 g 0 (t-k) g 1 (t-k) s(t) (t) r(t) b 0 * (t-k) b 1 * (t-k) Itegrate ad Dump Flter R k,1 ML Detector Ŝ k,1 S k,n-1 Itegrate ad Dump Flter R k,n-1 Ŝ k,n-1 g N-1 (t-k) FDM rasmtter b N-1 * (t-k) FDM Recever Fgure 1: Structure of a FDM based commucatos system mpared by ose. III. FDM SYSEM PERFORMANCE Here we exame the error probablty performace of the FDM based commucatos system the presece of addtve whte Gaussa ose. We cosder FDM sgals wth N2,4,8 ad 16, Δf0.1,,1.0 ad BPSK ad QPSK modulato of the sub-carrers. Fgs. 1 ad 2 show the smulated BER versus ormalsed tercarrer frequecy spacg for varous umbers of sub-chaels for the FDM/BPSK system ad the FDM/QPSK system, respectvely. I these smulatos E b /N 0 was set to be equal to 5 db. Observato of these fgures reveals that a FDM system ca exhbt a hgher spectral effcecy tha a equvalet OFDM system wthout ay pealty the BER. However, the creased recever complexty assocated wth ML detecto has to be addressed for practcal purposes. IV. COMPLEXIY CONSIDERAIONS he FDM based commucatos system proposed ths paper employs ML detecto. Hece, ts complexty grows expoetally wth the umber of sub-carrers ad the cardalty of the costellato. Subsequetly, we exame two possbltes to reduce detecto complexty. A. Lear Detecto Here we explot lear techques to detect the trasmtted symbols a FDM sgal. he suffcet statstcs are related to the trasmtted symbols a FDM based commucatos system as follows R MS + N (10) where R[R k,0 R k,1 R k,n 1 ], S[S k,0 S k,1 S k,n 1 ], N[N k,0 N k,1 N k,n 1 ] ad the matrx M s gve by where m m1,1 m1,2! m1, N m2,1 m2,2! m2, N M (1 " " # " mn,1 mn,2! mn, N ( t k) b ( t k), j 1 j 1 (12) k g he matrx M exhbts the followg propertes: () m, 1 ad m, >m j,j for >j ad () m,j 0 for >j (the matrx s upper tragular). Accordgly, we have two detecto optos. he frst opto s based o the verso of the matrx M so that a estmate Ŝ[Ŝ k,0 Ŝ k,1 Ŝ k,n 1 ] for the trasmtted symbols S[S k,0 S k,1 S k,n 1 ] s gve by 1 [ M R] dt Sˆ HD (13) he secod opto s based o the upper tragular propertes of the matrx M so that a estmate Ŝ k, for the trasmtted symbol S k, s gve by N 1 1 S ˆ ˆ k, HD Rk, m+ 1, S, (14) k m+ 1,

4 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) hs s a teratve method where frst S k,n 1 s estmated, secod S k,n 2 s estmated usg Ŝ k,n 1,, ad fally S k,0 s estmated usg Ŝ k,, 0,,N 2. Note that the operator HD[ ] (13) ad (14) performs hard detecto. verted. Moreover, m N,N rapdly approaches zero as the umber of sub-chaels s creased or the tercarrer frequecy separato s decreased whch prevets lear detecto the secod case for a hgh error propagato would occur otherwse. Accordgly, alteratve detecto procedures have to be sought. able I: BER for varous lear detecto methods for a FDM/BPSK system wth varous umbers of sub-chaels (E b/n o5 db). Lear Number of Sub-Chaels Method Iverso Detecto Iteratve Detecto B. GA Detecto Fgure 2: BER versus ormalsed tercarrer frequecy separato for varous umbers of sub-chaels for the FDM/BPSK system. Fgure 3: BER versus ormalsed tercarrer frequecy separato for varous umbers of sub-chaels for the FDM/QPSK system. Ufortuately, lear detecto yelds a very hgh BER as show able I. Lear detecto also yelds dfferet error rates dfferet sub-chaels. Fally, we ote that the propertes of the matrx M are such that lear detecto may ot always be feasble. he matrx determat rapdly approaches zero as the umber of sub-chaels s creased or the tercarrer frequecy spacg s decreased whch prevets lear detecto the frst case for the matrx caot be Here we explot geetc algorthms (GA) based techques to detect the trasmtted symbols a FDM sgal. A GA s a guded radom techque whch mmcs the process of atural evoluto. GAs have bee frequetly used to allevate the complexty of combatoral optmsato problems commucatos systems [6-8]. A smple GA cotas the followg steps: 1. Geerate a tal populato P(0). 2. Evaluate the ftess values of the dfferet elemets of the tal populato P(0). 3. Create a ew populato P(r) usg crossover ad mutato. 4. Evaluate the ftess values of the dfferet elemets of the ew populato P(r). 5. If the termato crtero s ot satsfed set rr+1 ad go to step 3 otherwse output soluto. he populato comprses a set of the possble trasmtted symbols (the chromosomes). he ftess value of the trasmtted symbols (the chromosomes) are gve by the probablty desty fucto (7). Based o the ftess values paret chromosomes are selected from the populato P(r) to reproduce thereby geeratg offsprg chromosomes for the populato P(r+. Here we employ ftess proportoate selecto whereby the expected umber of tmes a dvdual s selected to reproduce s equal to the dvdual s ftess dvded by the average ftess of the populato. Crossover ad mutato are two types of geetc operators. he crossover operato combes sub-strgs of two paret chromosomes to geerate two ew

5 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) offsprg chromosomes. Here we employ sgle-pot crossover whereby a sgle crossover posto s chose at radom ad the parts of the two paret chromosomes after the crossover posto are exchaged to geerate the two offsprg chromosomes. he mutato operato troduces a radom varato to every elemet of a offsprg chromosome wth a gve probablty. Note that crossover combes parts of hghly ft chromosomes to geerate possbly eve ftter chromosomes whereas mutato avods the permaet fxato at ay partcular locally optmum soluto. I ths work, we use a populato sze of 50, a sglepot crossover rate of 0.85 ad a mutato rate of Moreover, we copy 20% of the fttest chromosomes populato P(r) to populato P(r+. hs operato s kow as eltsm. Fg. 4 shows the average BER vs. umber of teratos for FDM/BPSK sgals wth N8,16 ad 32 ad Δf0.75. I these smulatos E b /N 0 was set to be equal to 5 db. Note that to acheve the deal ML detector BER 10, 20 ad 30 teratos are requred for 8, 16 ad 32 sub-chaels, respectvely. akg the umber of tmes the probablty desty fucto (7) s evaluated as a rudmetar complexty measure t follows that the mprovemet complexty that a GA based detector offers over a ML detector s 2 8 /(50 10) , 2 16 /(50 20) ad 2 32 /(50 30) for 8, 16 ad 32 sub-chaels, respectvely. For a small umber of sub-chaels the GA detector offers o mprovemet complexty. However, for a large umber of subchaels the GA offers a substatal complexty mprovemet. Accordgly, a GA based detector costtutes a vable opto to detect the trasmtted symbols a FDM sgal. V. CONCLUSIONS AND FUURE WORK I ths paper we have proposed a spectrally effcet FDM based commucatos system. hs FDM system hadles tercarrer frequecy separatos smaller tha those of OFDM systems. he FDM trasmtter cossts of a bak of modulators that geerate the dfferet formato coveyg sub-carrers that compose the trasmtted FDM sgal. he FDM recever cossts of a bak correlators followed by a ML detector. he demodulators extracts a set of suffcet statstcs from the receved FDM sgal ad the ML detector uses the suffcet statstcs to estmate the trasmtted symbols. Lear detecto ad GA based detecto have bee proposed to allevate the complexty assocated wth ML detecto. Lear detectors were show to yeld very poor BER whereas GA detectors were show to yeld very good BER wth reasoable complexty. Future work should cocetrate o evaluatg the performace of FDM based commucatos systems the presece of varous degradato factors such as multpath fadg chaels wth delay ad Doppler spread ad tmg ad frequecy offsets. Fgure 4: Average BER vs. umber of teratos for FDM/BPSK sgals wth N8,16 ad 32 ad Δf0.75. Fally, we ote that the fact that the determat of the matrx M approaches zero rapdly may fact costtute a fudametal lmtato for proper operato of the system for small tercarrer frequecy spacgs ad large umbers of sub-chaels. hs topc should be vestgated further. VI. ACKNOWLEDGEMENS M.R.D. Rodrgues gratefully ackowledges the facal support from the Portuguese Foudato for Scece ad echology through a postdoctoral fellowshp. VII. REFERENCES [1] J. A. C. Bgham, Multcarrer modulato for data trasmsso: A dea whose tme has come, IEEE Commucatos Magaze, vol. 28, pp. 5-14, May [2] W. Y. Zou ad Y. Wu, COFDM: A overvew, IEEE rasactos o Broadcastg, vol. 41, pp. 1-8, March [3] J. Oh ad M. Lm, he badwdth effcet creasg method of multcarrer CDMA ad ts performace evaluato comparso wth DS-CDMA wth rake recevers, Proceedgs of the IEEE Vehcular echology Coferece, vol. 1, pp , May [4] M. R. D. Rodrgues ad I. Darwazeh, Fast OFDM: A proposal for doublg the data rate of OFDM schemes, Proceedgs of the Iteratoal Coferece o elecommucatos, vol. 3, pp , Jue 2002.

6 IOWo'03, 8th Iteratoal OFDM-Workshop, Proceedgs, Hamburg, DE, Sep 24-25, 2003 (Prepublcato draft) [5] J. G. Proaks. Dgtal Commucatos, 3rd ed. New York: McGraw- Hll, [6] M. Mtchell. A Itroducto to Geetc Algorthms. Cambrdge, MA: he MI Press, [7] M. J. Jutt,. Schlosser ad J. O. Llleberg, Geetc algorthms for multuser detecto sychroous CDMA, Proceedgs of the IEEE Iteratoal Symposum o Iformato heory, p. 492, Jue-July [8] X. F. Wag, W.-S. Lu ad A. Atoou, A geetc-algorthm-based multuser detector for multple-access commucatos, Proceedgs of the IEEE Iteratoal Symposum o Crcuts ad Systems, vol. 4, pp , May-Jue 1998.

Module 6. Channel Coding. Version 2 ECE IIT, Kharagpur

Module 6. Channel Coding. Version 2 ECE IIT, Kharagpur Module 6 Chael Codg Lesso 36 Coded Modulato Schemes After readg ths lesso, you wll lear about Trells Code Modulato; Set parttog TCM; Decodg TCM; The modulated waveform a covetoal ucoded carrer modulato