Bayesian channel estimation in chaos based multicarrier CDMA system under slowly varying frequency selective channel

Transcription

1 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, 06 Bayesan cannel estmaton n caos based multcarrer CDMA system under slowly varyng frequency selectve cannel Meer Krsna Patel, Stevan M. Berber, Senor Member, IEEE, and Kevn W. Sowerby, Senor Member, IEEE Abstract Ts paper presents te teoretcal analyss of te multcarrer-code dvson multple access MC-CDMA system n slowly tme varyng frequency selectve cannel. Caotc sequences are used as spreadng codes of CDMA system wt BPSK modulaton sceme. Performance mprovement n Bayesan estmator n te presence of caotc sequence s nvestgated. Under perfect syncronzaton assumpton, bt error rate BER n closed form s derved under mperfect cannel estmaton for downlnk communcaton system. Smulaton results sow tat tere s sgnfcant performance mprovement n MC-CDMA system as compared to CDMA system. Index Terms Bayesan estmaton, Caotc sequence, CDMA, Frequency selectve cannel, Multcarrer Communcaton I. INTRODUCTION Fadng s te penomena wc makes wreless communcaton more dffcult as compare to oter communcaton systems e.g. optcal fber communcaton and wred communcaton etc. Terefore for many wreless systems, ndependent of weter tme dvson multple access TDMA or code dvson multple access CDMA s employed, estmaton of cannel fadng coeffcent s necessary for g speed communcaton. Cannel estmates can be updated frame by frame for slower fadng rate as compare to frame rate. If cannel coeffcents cange sgnfcantly wtn te frame ten t s necessary to update coeffcents teratvely based on symbol by symbol bass [], []. Varous estmaton metods ave been studed n last few decades and eac metod as ts own advantages and dsadvantages. Mnmum mean square estmators MMSE [3], [4], [5] are easy to mplement and perform well n flat fadng envronment. But tese estmators requre correlaton computaton and ave poor performance for tme varyng cannel estmaton. Bayesan estmators [6], [7], [8], [9] used pror knowledge of data to generate posteror analyss. Terefore estmator performance extensvely depends on pror nformatons. On te oter and, neural networks [0], [], [] do not requre pror knowledge of cannel statstc, but tere s uge computatonal burden for tranng process. Fnally, partcle flters [3], [4], [5], [6] use te sequental Monte Carlo samplng metod to mplement recursve Bayesan flter. But tese flters ave very g computatonal load for correctng eac partcle, wc results n ger energy consumpton. Terefore ardware mplementaton of tese flters are dffcult. Drect sequence-cdma system as receved muc attenton n wreless communcaton system due to ts ger capacty, robustness aganst fadng and ant-nterference capablty [7]. All users sare complete avalable spectrum and dstngus from eac oter by ter spreadng code at te recever. Furter, frequency selectve cannel severely degrades te performance of te CDMA system due to t s ntersymbol nterference ISI effect. To overcome ts degradaton RAKE recevers n CDMA systems are wdely nvestgated [8], [9], [0], [], [], [3], [4], [3], [5], [6]. On te oter and multcarrer systems are also drawn sgnfcant attenton n wreless communcaton system [7]. In ts system te transmtted data s dvded nto a number of slow data streams and eac stream s modulated by one subcarrer. At te recever eac subcarrer data s demodulated separately. Fnally te demodulated data are reassembled. Terefore transmtted data rate s reduced wereas overall receved data rate s same as te sngle carrer system. Snce te transmsson data rate s reduced terefore cannel can be consdered as flat fadng cannel and ISI effect due to frequency selectve cannel does not occur n ts case. Hg data rate and bandwdt effcent dgtal communcaton tecnologes urge te need to develop new transcever structures. Terefore MC-CDMA system s bandwdt effcent system and can be used for g data rate communcaton system. In ts system spreadng code s seral to parallel converted and ten eac cp s modulated by dfferent carrer frequency. MC-CDMA system are wdely nvestgated for bnary spreadng sequences [7], [8], [4], [9], [30]. Few papers are presented n te feld of caos based MC- CDMA system [3], [3], [33] but to our best knowledge, no paper as been publsed on BER performance of caos based MC-CDMA system under complex frequency selectve fadng cannel coeffcents and nose, even toug caos based CDMA system ave ganed sgnfcant nterest among te researcers n last decade [34], [35], [3], [], [36], [37], [38], [39], [], [3], [33]. For ts reason te objectve of ts researc work s to study te Bayesan cannel estmator for caos based MC- CDMA system under frequency selectve cannel for downlnk communcaton. Maxmum a pror MAP estmator equaton s derved for tese systems, wc needs a pror knowledge of cannel statstcs. Furter, we ave also derved te ML estmaton equaton for consderng te case were te mean and varance of te cannel s unknown at te recever. Two algortms are derved to consder te multplexed plot-data case and added plot-data case. In multplexed plot-data case, after demultplexng, cannel estmaton can be performed ISSN:

2 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, 06 drectly on te extracted plot sgnal. Wereas for added plot-data case, plot needs to be extracted by multplyng correspondng caotc sequence, before cannel estmaton process. Performance dfference n tese two metods ave been sown usng smulaton results. Furter, BER n closed form s derved for mperfect cannel estmaton case. Fnally, we ave sown tat te results n [40] s a specal case of ts paper. Ts paper s organzed as follows. In secton II caos based MC-CDMA system s sown. Analytcal performance of ts system s presented n secton III. MAP and ML estmaton algortms are derved n secton IV. Smulaton results are sown n secton V. Fnally some concludng remarks are gven n secton VI. II. SYSTEM MODEL Fg. sows te baseband representaton of te caos based MC-CDMA system. Te baseband sgnal s g k for gt user at k t cp nstant, s gven by s g k = γg xg k were γ g s te gt user symbol at t tme nstant, x g k s kt cp of te g t user caotc spreadng sequence or cp wtn an nformaton bt.e. k =,, β and β s te spreadng factor. After addng all te users data, eac cp of te resultant sgnal s modulated by dfferent carrers. Fnally all te modulated cps are added togeter before transmsson. Terefore transmtted sgnal S s gven by S = n= j= J γ j xj k,n cosπf nt were N and J are te number of subcarrers and users respectvely, f n s te frequency of n t subcarrer and x j k = [x j k,,, xj k,n ] s te caotc sequence of jt user. For L multpat cannels, te receved symbol r of g t user at t tme nstant s gven by L r =,l S τl + ξ 3 l=0 were ξ denotes te complex addtve wte Gaussan nose at t tme nstant wt power spectral densty equal to N 0 and τ l s te delay for l t pat wt respect to frst pat. Delay of frst pat s assumed to be zero.e. τ 0 = 0 and oter delays are more tan one cp duratons.,l s te complex cannel coeffcents for l t pat at t tme nstant, wc s assumed to be known at te recever. III. PERFORMANCE ANALYSIS OF MC-CDMA SYSTEM In fg., snce caotc cps are spread to bt level usng multcarrers terefore te small cp delay caused by te cannel n equaton 3 can be neglected.e. S τl S. Hence receved sgnal can be wrtten as L r =,l S + ξ = α e jφ S + ξ 4 l=0 were L,l = α e jφ 5 l=0 In above equaton α and φ are ampltude and pase component of te resultant cannel coeffcents. Estmated fadng coeffcents ˆα e j ˆφ are multpled wt ts receved sgnal. Te resultant sgnal s multpled by subcarrers and ten passes troug low pass flter to remove ger frequency terms as sown n Fg.. Snce subcarrers are ortogonal to eac oter terefore tere s no cosne term n te low pass flter output. Ts output s ten multpled to g t user caotc sequence and te decson varable s gven by or were Z g = Re{α e jφ ˆα e j ˆφ J γ j xj k,n xg k,n + n= j= 6 ˆα e j ˆφ ξ cosπf n tx g k,n } n= Z g = Zag + Z bg Z ag = α ˆα cosφ ˆφ + Z cg 7 n= γ g xg k,n xg k,n } {{ } Desred Sgnal Z bg = α ˆα cosφ ˆφ J n= j g,j= γ j xj k,n xg k,n } {{ } Interuser Interference Z cg = ˆα n= cosπf n tx g k,n {Reξ cos ˆφ + Imξ sn ˆφ } }{{} Nose Te probablty of error P g a for te t bt of g t user s gven by [35] P g a = P r Z g < 0 γ g = + P r Z g 0 γ g = [ ] [ ] = E erfc Z g γ g = / var Z g γ g = were Pr and erfc are te probablty operatons and complementary error functon respectvely. Solvng above equaton see Appendx A for dervaton and omttng te subscrpt tat s related to bt under nvestgaton, we ave fnal equaton of P g a as Terefore fnal BER equaton s gven by P g a [ = { erfc Ψ N + J N + α cos φ ˆφ } ] / E b N 0 ISSN:

3 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, 06 Transmtter Cannel Recever User Data User Data User J Data J x k Caotc Sequence... J x k x k s k s k J s k Seral to Parallel Converson cos ft cos fnt k, L L,0 Fadng Pats AWGN cos ft LPF Estmated Coeffcent ĥ cos fnt LPF g x k, g x k, N DC ˆg Fg.. Block Dagram of Proposed System were Ψ = var{x g k,n }/P c Puttng N = β ten above equaton becomes P g a = erfc [ { Ψ β + J β + α cos φ ˆφ } ] / E b N 0 3 For perfect flat fadng cannel estmaton case.e. α e jφ = ˆα e j ˆφ, BER equaton derved n [40] s te specal case of equaton 3. Comparng equaton 3 wt results n [40], we can see tat performance of MC-CDMA system s same as DS-CDMA system for flat fadng cannel. But effect of ISI s removed by MC-CDMA system n frequency selectve fadng cannel. IV. MAP AND ML ESTIMATOR In ts secton we ave derved Bayesan estmator equatons for two cases,.e. for multplexed plot-data and added plotdata case. Fg. sows te baseband representaton of Bayesan cannel estmator. In ts fgure cannel estmaton s performed after multplyng te caotc sgnal to receved plot symbols. Te wreless cannel s assumed to be quas-statc fadng cannel.e. pat gans are constant over a symbol duraton. At t tme nstant, te receved plot sgnal at any user can be descrbed as: y = + ξ 4 j= s T j,c j, were s = [s, s,, s L+ ] T s te transmtted plot symbols, = [ ] T 0,,,,, L, s te quasstatc tme varyng cannel for j t user and ξ s te zero mean Wte Gaussan nose wt varance of w. L and N represent te total number of pats and users respectvely. C = dag[c, c,, c L+ ] s te dagonal matrx wt elements c of lengt β known as spreadng factor. c s te spreadng sequence for plot symbols. Subscrpt j denotes tat te symbol s related to j t user. Cannel coeffcents are assumed to be Gaussan dstrbuted [4].e. Nm, were m and are te mean and varance of te cannel respectvely. A. MAP and ML estmaton wt multplexed plot and user data If plot s multplexed wt user data ten t can be extracted usng demultplexer at te recever and can be processed by cannel estmator. Here we ave to assume tat fadng and nose ave same effect on plot and user data symbols. In ts case, te condton dstrbuton functon p y for j t user s defned as p y = exp π w y st j, Cj, w 5 Snce mean m and varance of Gaussan dstrbuted cannel s known at recever, terefore MAP estmaton algortm s gven by [6] y st j, Cj, w m m T + constants = ĥ = 0 6 Above dervatve reduces to followng equaton see appendx B for dervaton ĥ MAP, = m + w + C T w j, s j,s T j, C j, C T j, s j y s T j, C j,m 7 ISSN:

4 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, 06 Transmtter Cannel Recever c c H n / c n Plot Sgnal s + y sc z s c c H Estmaton Rule ˆ arg maxln p y ĥ Fg.. Block Dagram of Bayesan Estmator If we do not ave a pror knowledge of te cannel statstc, ten we remove te second term n equaton 6 and resultant algortm s known as ML estmaton [6].e. y s T j, C j, + constants w =ĥ = 0 8 After solvng dervatve, we ave followng ML estmaton equaton ĥ ML, = s T j,c j, y 9 B. MAP and ML algortm for added plot and user data For multplexed plot and user data, we ave to assume same fadng effects on bot te sgnals. If we add plot symbols to user symbols ten fadng ave same effect on bot te sgnals. Terefore same fadng and nose effect assumptons can be removed. Furter, n ts case plot as to be extracted from data for te cannel estmaton process. Snce te caotc sequences are ortogonal to eac oter terefore plot symbols can be extracted by multplyng te receved sgnal wt caotc sequence of plot symbols. Multplyng receved sgnal.e. equaton 4 wt caotc sgnal of j t user we ave C H j, z = y C j, C H j, 0 In ts case MAP and ML estmaton equatons are gven by see appendx C for dervaton and ĥ MAP, = m + w + s w j, sj, T s j z s T j, m ĥ ML, = s j, s T j, sj, z V. SIMULATION RESULTS In te smulaton we compare te performance of estmators for tree cases. In frst case, te plot s multplexed wt data wtout multplyng wt caotc sequences at te transmtter. We represent ts case as Wtout Caotc Multplcaton n te smulaton results. Smlarly oter two cases.e. multplexed plot-data and added plot-data wt caotc sequences multplcaton at transmtter, are denoted by Before Caotc Multplcaton and After Caotc Multplcaton respectvely n te results. Frst order Markov process s used to model te fadng process of cannel [4], wc s descrbed as +,l = α l,l + v,l l = 0,, L 3 were v,l s te complex Gaussan process for l t pat at tme. In te smulaton, varance of Gaussan process s set to. α l s te correlaton coeffcent tat depends on maxmum Doppler frequency f d and defned as α l = J 0 πf d T s 4 were J 0 s te Bessel functon of frst knd and zerot order, and T s s te sgnalng rate. f d T s s set to 0. n te smulaton. Te value of te spreadng factor β s 50. Followng Cebysev polynomal functon s used to generate te caotc sequence [43]. x j = x j 5 were x j denotes te j t cp value caotc sequence. Fg. 3 and Fg. 4 sow te cannel trackng performance of te tree estmators at 0dB and 0dB SN R condtons respectvely. Caotc sequences spread te data over entre bandwdt durng transmsson and despreadng takes place durng recepton. Furter nose s spread by te caotc sequence multplcaton at te recever. Due to ts spreadng of nose, performance of te estmator n te presence of caotc sequence s better tan te wtout caotc spreadng case, as sown n Fg. 3 and Fg. 4. From tese fgures t s clear tat performance of all te estmators are mproved wt ncrease n te SNR. Snce caotc sequences are drectly used for cannel estmaton ISSN:

5 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, Cannle Coeffcent Value True Wtout Caotc Sequence After Caotc Multplcaton Before Caotc Multplcaton Number of Iteratons Fg. 3. MAP cannel estmators performance, SNR = 0dB wtout caotc sequence after caotc multplcaton before caotc multplcaton perfect estmaton Fg. 5. BER performance of caos based CDMA system usng MAP, β = Cannle Coeffcent Value True Wtout Caotc Multplcaton After Caotc Multplcaton Before Caotc Multplcaton Number of Iteratons Fg. 4. MAP cannel estmators performance, SNR = 0dB BER L= MC CDMA L=3 MC CDMA L=3 CDMA L= CDMA Eb/No [db] Fg. 6. Smulated BER comparson of dfferent multpat components n one user and plot system wt added plot-data case, β = 50 as well as nose spreadng n multplexed plot-data case terefore ts performance s better tan te added plot-data case, for lower SNR condtons. For ger SNR condtons performance of bot te caotc estmators are practcally same as sown n Fg. 4. In Fg. 5, te BER performances are sown. 0dB performance mprovement can be seen at SNR = 0dB wt caotc spreadng sequence over wtout spreadng case. Performance mprovement can be seen n Before Multplcaton Case over After Multplcaton case at lower SNR condtons. Furter, performance of Before multplcaton case s same as te te perfect estmaton case. Dfferent smulaton results ave been sown to compare te performance of te CDMA and MC-CDMA systems n fg. 6. In multpat cannel smulatons, second cannel s delayed by one cp wt respect to frst cannel. Smlarly trd cannel s delayed by one cp wt respect to second cannel. It can be seen tat performance of CDMA system degrades wt ncrease n number of cannels. Performance of ts system deterorates because of ISI effect due to multpat cannels. However, te multpat frequency selectve cannels ISSN: are converted nto flat fadng cannels due to slow data transmsson rate n MC-CDMA. Furter, tese multpat flat fadng cannels are added togeter, wc s provdng a vrtual equal gan combnng EGC effect wtn cannels. Terefore te performance of MC-CDMA system as mproved n multpat envronment wt ncrease n number of cannels. Fg. 7 sows te performance degradaton wt ncrease n number of users n te system. Obvously, wen te number of users ncreases te BER ncreases due to te ncrease of IUI. Furter, from equaton 3 t s clear tat te performance degradaton wt ncreased number of users can be mproved by usng g values of spreadng factor. Fg. 8 sows ts performance mprovement wt ncrease n spreadng factor value. VI. CONCLUSION In ts paper, performance of caos based MC-CDMA system s nvestgated and compared wt te performance of classcal CDMA system. Performance analyss s evaluated under frequency selectve fadng cannel. Under perfect syncronzaton, analytcal expresson for BER n close form s

6 BER INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 0, 06 Users = 50 Users = 5 Users = 0 Varance of te frst term Z ag var[z ag ] = {α ˆα cosφ ˆφ } N can be wrtten as [ varx g k ] 8 Cannel s assume to slowly varyng terefore we can assume tat τl. Under ts assumpton, snce caotc sequences are uncorrelated wt t s sfted verson terefore var[z bg ] = {α ˆα cosφ ˆφ } NJ P c 9 Real and Imagnary part of complex AWGN ave power spectral densty equal to N 0 / and ndependent to caotc sequences, terefore varance of Z cg can be derved as Eb/No [db] Fg. 7. Smulated BER comparson of dfferent user n sngle pat communcaton for β = 50 var[z cg ] = ˆα N N 0 P c 30 Puttng equatons 6, 8, 9 and 30 n equaton and after rearrangng te equaton we ave equaton. BER beta = 0 beta = 50 beta = 50 APPENDIX B DERIVATION OF EQUATION 7 Rewrtng equaton 6 after solvng dervatve, we ave ĥ = + C T w j, s j,s T j, j, C m + CT j, sj,y w Eb/No [db] Fg. 8. Smulated BER comparson of dfferent values of spreadng factor beta n sngle pat communcaton for 0 users derved for mperfect cannel estmaton. Smulaton results sow tat te performance of Bayesan cannel estmator mproves n te presence of spreadng sequences. Furter, analytcal and smulaton results sow tat te performance of MC-CDMA and CDMA system s equvalent for flat fadng cannel, but tere s a sgnfcant performance mprovement n MC-CDMA system under frequency selectve cannel. APPENDIX A DERIVATION OF EQUATION Snce te caotc spreadng sequences and AWGN nose are zero mean uncorrelated processes. Hence E [Z g γ g = ] = α ˆα cosφ ˆφ gk NE [x ] = α ˆα cosφ ˆφ 6 NP c All te tree terms n 7 are uncorrelated to eac oter, ence var [Z g γ g = ] s gven by var[z g γ g = ] = var[zag ] + var[z bg ] + var[z cg ] 7 Let + w C T j,s j, s T j,c j, = T 3 Hence equaton 3 becomes ĥ = T m + CT j, s j,y w 33 Put te value of from equaton 3 to 33, we get equaton 7 APPENDIX C DERIVATION OF EQUATION AND Puttng te value of y from equaton 4 n equaton 0, we ave z = s T j, + C H j, C j,c H j, + ξch j, C j,c H j, j=,k j s T j, C j, 34 Snce te cross-correlaton of two dfferent caotc sgnals s very small, ence we can neglect te second term.e. z s T j, + ξ C H j, C j, C H j, 35 Now, followng te same steps as n secton IV-A, we get equaton and equaton. ISSN:

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