UNIVERSITY OF CINCINNATI

Transcription

1 UNIVERSITY OF CINCINNATI DATE: Marc st, 004 I, Bn Xu ereby submt ts as part of te requrements for te degree of: Master of Scence n: Electrcal Engneerng It s enttled: A Blnd Space-Tme Decorrelatng RAKE Recever n a DS- CDMA system n Multpat Cannels, Approved by: Dr. James Caffery, Jr. Dr. oward Fan Dr. Qng An Zeng

2 A Blnd Space-Tme Decorrelatng RAKE Recever n a DS- CDMA System n Multpat Cannels A tess submtted to te Dvson of Researc and Advanced Studes of te Unversty of Cncnnat n partal fulfllment of te requrements for te degree of MASTER OF SCIENCE n te Department of Electrcal & Computer Engneerng and Computer Scence of te College of Engneerng 004 by Bn Xu B.S., Tsngua Unversty, Beng, Cna, 99 Commttee Car: Dr. James Caffery, Jr. Dr. oward Fan Dr. Qng An Zeng

3 Abstract Ts paper addresses te problem of blnd multple access nterference MAI and ntersymbol nterference ISI mtgaton n drect sequence code dvson multpat access DS/CDMA systems. Studes sow tat multuser detecton can be performed wtout te knowledge of users cannel parameters n a frequency-selectve fadng cannel, and tese approaces are of g computatonal complexty. In ts tess, a space-tme decorrelatng RAKE recever s developed n te AGN cannel. Ts metod s blnd because t only needs to know te desred user s sgnature sequence and tmng nformaton. A tme-varyng multpat cannel tat as te desred temporal-spatal correlaton propertes s smulated, and s used to evaluate te performance of te recever. e also develop an adaptve algortm for te proposed recever. Smulatons sow tat te proposed recever s near-far resstant, able to elmnate MAI and ISI effectvely, and as g performance wt low complexty.

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5 Acknowledgements I am sncerely grateful to Dr. James Caffery, Jr. for beng my advsor and provdng gudance for ts tess work. e ntroduced me to CDMA systems, and s nsgt and motvaton were nvaluable and nsprable. I would also lke to express my apprecaton to my commttee members for ter correctons and comments on ts work. I am grateful to all members of te reless Systems Researc Lab wo are very elpful to me. Fnally, most of all, I tank my wfe for er uncondtonal love and support

6 Table of Contents Table of Contents... Lst of Fgures... 3 Capter Introducton Researc Background Organzaton of Tess... 9 Capter Spread Spectrum and CDMA Sgnal Detecton Rado Propagaton Envronment Pat Loss..... Fadng Slow Fadng Sadowng Fast Fadng Doppler Spread: Tme-Selectve Fadng Delay Spread: Frequency-Selectve Fadng Angle Spread: Space-Selectve Fadng Space-Tme Processng Temporal Processng Spatal Processng Space-tme processng....3 Smulaton Examples... 5 Capter 3 Tme-varyng Multpat Vector Cannel Smulator Tme-varyng Multpat Vector Cannel Second Order Caracterzaton... 35

7 3.3 Vector Cannel Smulator Complex Pat Vector Generator Tme-correlaton Sapng Flter Spatal Transformaton Interpolator Smulaton Examples Capter 4 Space-Tme Decorrelatng RAKE Recever for a DS/CDMA System Introducton System Model Space-Tme Decorrelatng Detector Smulaton Results Capter 5 Adaptve Implementaton LMS-based Algortm Smulaton Results Capter 6 Conclusons and Future ork... 77

8 Lst of Fgures FIGURE. MULTIPAT MODEL IT BASE ANTENNA ARRAY... FIGURE. TEMPORAL TIME-ONLY PROCESSING FIGURE.3 EQUALIZER... 7 FIGURE.4 BEAMFORMER FIGURE.5 SPACE-TIME RECEIVER... FIGURE.6 QPSK MODULATED SIGNAL CONSTELLATION... 8 FIGURE.7 RECEIVED SIGNAL CONSTELLATION AT TE EQUALIZER INPUT FIGURE.8 SIGNAL CONSTELLATION AFTER TIME-ONLY PROCESSING FIGURE.9 SIGNAL CONSTELLATION AFTER SPACE-TIME PROCESSING... 9 FIGURE.0 BEAM PATTERN FOR TE SPACE-TIME PROCESSING FIGURE 3. UNIFORM LINEAR ARRAY IT TREE ELEMENTS... 3 FIGURE 3. TIME-VARYING MULTIPAT VECTOR CANNEL SIMULATOR.. 38 FIGURE.3 TE COMPLEX PAT VECTOR GENERATOR FOR TE IT PAT39 FIGURE 3.4 SPACE-CORRELATION TRANSFORMATION FIGURE 3.5 INTERPOLATOR FIGURE 3.6 LINEAR INTERPOLATION ILLUSTRATION FIGURE 3.7 CUBIC SPLINE INTERPOLATION ILLUSTRATION FIGURE 3.8 DESIRED AND OBTAINED TEMPORAL CORRELATION FIGURE 3.9 CANNEL COEFFICIENT MAGNITUDES VS. TIME FOR TE FIRST ANTENNA

9 FIGURE 3.0 CANNEL COEFFICIENT MAGNITUDES BEFORE AND AFTER INTERPOLATION FOR PAT OF TE FIRST ANTENNA ELEMENT FIGURE. STRUCTURES OF SPACE-TIME RECEIVERS FIGURE 4. RECEIVER STRUCTURE FIGURE 4.3 BER PERFORMANCE FIGURE 4.4 NEAR-FAR RESISTANCE FIGURE 4.5 BER PERFORMANCES AT f d = 50z AND f d = 00z FIGURE 5. LMS ALGORITM STRUCTURE... 7 FIGURE 5. PERFORMANCE OF TE LMS BLIND ADAPTIVE ALGORITM FIGURE 5.3 STEADY-STATE BER PERFORMANCE FIGURE 5.4 PERFORMANCE OF ADAPTIVE ALGORITM FOR EIGENVECTOR ESTIMATION

10 Capter Introducton. Researc Background reless Communcatons for moble telepone and data transmsson s undergong very rapd development. Many of te emergng wreless systems wll ncorporate consderable sgnal-processng ntellgence n order to provde advanced servces suc as multmeda transmsson. In order to make optmal use of avalable bandwdt and to provde maxmal flexblty, many wreless systems operate as multple access systems n wc cannel bandwdt s sared by many users on a random-access bass. As demand for wreless communcatons contnues to grow, trd-generaton cellular communcatons systems are beng standardzed to provde better voce and data servces, and drectsequence Code-dvson multple access DS-CDMA tecnques are beng used as ts bass. In Nort Amerca, te second generaton DS-CDMA standard, IS-95, s te bass for a trd-generaton system CDMA000. In Japan and Europe, a trd-generaton wdeband CDMA CDMA system s also beng developed [], []. An effort s beng made to merge tese systems nto a common, globe trd generaton CDMA standard. le CDMA possesses many ntrnsc advantages over te earler access tecnques suc as tme-dvson multple access TDMA and frequency-dvson multple access FDMA [], te capacty of current practcal CDMA systems s lmted by te multple access nterference MAI caused by code non-ortogonalty due to dverse penomena suc as asyncronous transmsson, mult-pat propagaton, or lmted bandwdt. Moreover, te presence of nter-symbol nterference ISI due to te tme-dspersve nature of wreless cannels s often neglected n low rate CDMA systems, but t becomes a maor problem n wdeband CDMA systems. In addton, a maor tecnologcal urdle 5

11 of CDMA systems s te near-far problem: te bt-error-rate BER of conventonal recever a matced flter for te user of nterest s so senstve to dfferences between te receved energes of te desred user and nterferng users tat relable demodulaton s mpossble unless strngent power control s exercsed. Te optmum multuser recever for asyncronous mult-access Gaussan cannels sows tat te near-far problem s overcome by a more sopstcated recever tat accounts for te presence of oter nterferers n te cannel. Ts recever optmum recever was sown [4] to attan essentally sngle-user performance assumng tat te recever knows te followng: Te sgnature waveform of te desred user; Te sgnature waveform of te nterferng users; 3 Te tmng bt-epoc and carrer pase of te desred user; 4 Te tmng bt-epoc and carrer pase of eac of te nterferng users; 5 Te receved ampltudes of te nterferng users relatve to tat of te desred user. Dfferent tecnques ave been proposed to suppress MAI as well as ISI usng lnear flterng. Te conventonal recever only requres and 3, but t s severely lmted by te near-far problem, and even n te presence of perfect power control, ts BER s orders of magntude far from optmal. Te optmum detector for te asyncronous multpleaccess Gaussan cannel [4] sows near-far resstance and sgnfcant performance mprovement over tat of te conventonal recevers by ontly detectng all of te users sgnals. owever, te optmum recever needs to know,, 3, 4, and 5, and te computatonal complexty of te optmum recever grows exponentally wt te number 6

12 of actve users. Decorrelatng recevers [5] are suffcent n order to aceve optmum resstance aganst te near-far problem, and, at te expense of slgt ncrease over te mnmum BER, te decorrelatng recever avods te exponental complexty n te number of actve users of te optmum multuser detector. But decorrelatng recevers requre,, 3, and 4. Conventonal lnear mnmum mean square error MMSE recevers [5], [6] offer superor performance wt lnear complexty. owever, tey need to know,, 3, 4, and 5. Te adaptve MMSE detectors n [6] and [7] substtute te need to know, 4, and 5 by te need to ave tranng data sequences for every actve user. Te typcal operaton of tose adaptve multuser detectors requres eac transmtter to send a tranng sequence at start-up wc te recever uses for ntal adaptaton. After te tranng pase ends, adaptaton durng actual data transmsson occurs n decsondrected mode. owever, any tme tere s a drastc cange n te nterference envronment, decson-drected adaptaton becomes unrelable, and te data transmsson of te desred user must be suspended and a fres tranng sequence s requred. Tus, te relance on tranng sequence s cumbersome n most CDMA systems, were one of te most mportant advantages s te ablty to ave completely asyncronous and uncoordnated transmssons tat swtc on and off autonomously. So tat mples te need for blnd adaptve recevers, and many papers ave talked about t [0]. owever, most proposed blnd estmaton algortms nvolve computatonally ntensve operatons suc as Sngular Value Decomposton SVD and terefore may be probtve n practce. Tus, several recevers based on te lnear constraned mnmum varance LCMV ave been proposed to decrease te computatonal complexty. Te autors n [0] and [] ntroduce blnd adaptve recevers wc mnmze te mean output energy 7

13 MOE wt te knowledge of te desred user s sgnature sequences and ts tmng, and tey also sow tat te canoncal MOE detector s, n fact, te MMSE detector. Te autors n [6] and [35] ntroduce constraned MOE recevers wc are used n a sngle cannel envronment. Based on a LCMV crteron smlar to tat used n [6] and [35], te autors n [8] develop a decorrelatng recever n a multpat envronment for a syncronous CDMA system by usng a partcular constrant. All of tese recevers are developed for a sngle antenna at te recever. Anoter approac to nterference suppresson n wreless systems s troug space-tme processng usng an antenna array at te recever. Ts approac apples varous dversty tecnques to mprove te detected data qualty so as to ncrease te system capacty. Among tese tecnques, te spatal and temporal combner suc as te beamformer and te RAKE recever as been proposed and ts superor performance as already been verfed n many papers [8], [9]. To empasze te specfed user s sgnal, te temporal combner, suc as te RAKE recever, performs a coerent combnng of te temporally spread multpat components wereas te beamformer combnes te spatally spread components. Te goal of ts tess s to nvestgate and obtan a blnd recever, wc does not requre tranng sequences and requres knowledge of only and 3, tat s, te same knowledge as te conventonal recever, wle ts recever s near-far resstant. In te meantme, ts recever also apples space-tme processng tecnques to mprove ts performance n a frequency-selectve, asyncronous cannel. 8

14 . Organzaton of Tess Te rest of te tess s structured as follows. Capter begns wt a bref ntroducton to rado propagaton because ts s mportant to sgnal detecton. Ten we ntroduce varous CDMA recever structures: tme-only, space-only, and space-tme processng, and compare ter performances n detecton. Capter 3 ntroduces a tme-varyng multpat vector cannel smulator tat wll be used to analyze te performance of te space-tme recever. Te smulator generates cannel coeffcents tat smulate a tmevaryng multpat cannel and as te desred temporal and spatal correlatons. Capter 4 models and proposes te space-tme decorrelatng recever. Performances based on computer smulaton are gven. In Capter 5, an LMS-based adaptve algortm s developed. Capter 6 provdes concluson and future work. 9

15 Capter Spread Spectrum and CDMA Sgnal Detecton Ts capter wll present some of te caracterstcs of spread spectrum systems. Tere are several ways to spread te sgnal spectrum, eac wt ts own partcular set of advantages [5]. Te common measure s called Processng Gan tat reflects te degree of spectral spreadng. e wll focus on drect-sequence spread spectrum systems. Frst, we wll revew te propagaton envronment n wc te rado sgnals are passed. By dong so, we wll understand te problems tat varous recevers ave tred to solve. Ten, we wll brefly dscuss several sgnal detecton metods: tme-only, space-only, and spacetme, and compare ter performance by ter BER beavor.. Rado Propagaton Envronment Rado sgnals generally propagate accordng to tree mecansms: reflecton, dffracton, and scatterng. en sgnals arrve at te recever, te orgnal sgnal arrves from several dfferent pats, called multpat, as Fgure. sows. Every pat as ts own delay, attenuaton, and drect of arrval DOA. Ts secton ntroduces te rado cannel and ts effects. 0

16 Delay Pat Ampltude Fadng Delay L Antenna Array Pat Ampltude L Fadng L Fgure. Multpat Model wt base antenna array... Pat Loss It s well known tat te rado sgnals propagate n te free space n te form of an electromagnetc wave, and te ntensty decays wt te square of te rado pat lengt. Tus te receved power s [] P r = P λ t 4 π d g t g r. were P t and P r are transmtted and receved sgnal power, respectvely, λ s wavelengt, gt and g r are te gans of transmtter antenna and recever antenna, respectvely, and d s te dstance between transmtter and recever. owever, free space propagaton does not apply n a moble rado envronment and te propagaton pat loss depends not only on te dstance and wavelengt, but also on te antenna egts of te moble statons MS and base statons BS, and te local terran caracterstcs suc as buldngs and lls. Te ste specfc nature of rado propagaton makes te teoretcal predcton of pat loss

17 dffcult and tere are no easy solutons. In practce, we can approxmate te receved sgnal power as P r = t r P t g t g r d. were t and r stand for te effectve egts of transmtter antenna and recever antenna, respectvely. Te effectve pat loss follows an nverse fourt power law exponent equal to four tat result n a loss of 40dB/decade []... Fadng In addton to pat loss wc causes sgnal attenuaton, receved sgnal power s also attenuated or vared because of te envronmental dsturbance, called fadng, wc s represented by [3] γ t = γ. γ.3 s r were γ s s called slow fadng and represents long-term tme varatons of te receved sgnal, wereas γ r s called fast fadng, sort-term fadng or multpat fadng. Te slow fadng γ s s te envelope of te sgnal γ t...3 Slow Fadng Sadowng Slow fadng s usually caused by te sadowng effect of mountans or buldngs, and s affected by antenna egt, transmsson frequences and envronments. From [], slow fadng can be descrbed by a log-normal dstrbuton. Its probablty densty functon pdf s

18 log x µ exp p x = πσ x σ 0 x > 0 x < 0.4 were x s a random varable, representng sgnal level, and µ and σ are te mean and standard devaton, respectvely, all n db...4 Fast Fadng In a fast fadng cannel, te cannel mpulse response canges rapdly wtn te symbol duraton. Suppose tat a scattered sgnal as random ampltude wt angle of arrval unformly dstrbuted n [0, π]. Ten te receved sgnal envelope as a Rayleg dstrbuton wt pdf [] exp y / σ y 0 p y = σ. 0 y < 0.5 If tere exsts lne of sgt LOS between transmtter and recever, ten te dstrbuton s Rcean wt pdf [3] exp p y = σ 0 { y + s / σ } ys J σ 0 y 0 y < 0.6 were s s te average power, J p s te modfed pt-order Bessel functon of te frst knd...5 Doppler Spread: Tme-Selectve Fadng en a cannel s tme-varant t s referred to as possessng tme-selectve fadng. Te moble staton ntroduces a Doppler, or frequency, sft nto te receved sgnal. Doppler spreadng results n te sgnal bandwdt beng stretced so tat te receved sgnal s 3

19 bandwdt s dfferent greater or less from tat of te transmtted sgnal. Takng te Doppler power spectrum to be te Fourer transform of te tme autocorrelaton of te cannel mpulse response, te Doppler spread s defned as te support of te Doppler power spectrum [3]. In a smple case tat assumes scatters unformly dstrbuted n angle [0, π] around te moble staton, te baseband power spectrum densty psd of te vertcal electrcal feld as te followng form [3] = 3σ f f c S f f c f m < f < f c + πf m f m / f m.7 were f m v = s te maxmum Doppler sft, λ c s te wavelengt of te carrer, and σ λ c s te total average receved power from all multpat components. e can estmate at wat transmtted sgnal duraton dstorton becomes notceable by referrng to te cannel s coerence tme. Te cannel s coerence tme s approxmately nversely proportonal to te maxmum Doppler sft experenced by te sgnal [3]...6 Delay Spread: Frequency-Selectve Fadng A moble rado cannel causes bot delay and Doppler spreadng. Delay spreadng results n two effects: tme dsperson and frequency-selectve fadng. Tme dsperson s a result of te sgnals takng dfferent tmes to cross te cannel by travelng dfferent propagaton pats. Frequency-selectve fadng occurs because te electrcal lengt of eac propagaton pat can expressed as a functon of frequency. A measure of te transmsson bandwdt at wc dstorton becomes apprecable s often based on te cannel s coerence bandwdt. Te coerence bandwdt ndcates te frequency 4

20 separaton at wc te attenuaton of te ampltudes of two frequency components becomes decorrelated suc tat te envelope correlaton coeffcent reaces a desgnated value. Tat s, wen te transmsson bandwdt s greater tan te coerence bandwdt, te cannel s called frequency-selectve fadng; wen te transmsson bandwdt s less tan coerence bandwdt, te cannel s called non-frequency-selectve, or flat fadng. Coerence bandwdt s approxmately nversely proportonal to te delay spread [3]...7 Angle Spread: Space-Selectve Fadng In te presence of multpat, te sgnal transmtted by eac user arrves at te recever not from one drecton, but from a contnuum. Te angle spread gves te range of angle values for wc sgnfcant energy s receved. At te recever, angle spread refers to te spread of angles of arrval of te multpats at te antenna array; lkewse, at te transmtter, angle spread refers to te spread of departure angles of te multpats. Angle spread causes space-selectve fadng, wc means tat sgnal magntude depends on te spatal locaton of te antenna. Coerence dstance represents te maxmum spatal separaton for wc te cannel responses at two antennas reman strongly correlated. Coerent dstance s used to caracterze te space-selectve fadng. Te larger te angle spread, te sorter te coerence dstance.. Space-Tme Processng In ts secton, we dscuss te caracterstc of temporal, spatal processng tecnques, and ten dscuss temporal-spatal processng. Temporal processng corresponds to 5

21 equalzers tat use a wegted sum of sgnal samples and spatal processng corresponds to smple beamformng tat uses a wegted sum of antenna outputs... Temporal Processng In a frequency selectve cannel, tere are multple replcas tat are resolvable n tme of te transmtted sgnal at te recever, traversng dfferent multpat. Tese multple copes can be combned to mprove te sgnal to nose rato SNR at te recever. Snce te sgnals are comng from dfferent pats, tey encounter ndependent fadng. Ts means tat f one of te pats s undergong a deep fade, t s very unlkely tat te sgnals from te oter pats are also encounterng fadng. As a result te recever stll as a good cance to attan acceptable fdelty. In a CDMA system, te recever can employ multple correlators to separate te multple copes of te sgnal and mtgate fadng. Ts recever, commonly known as a Rake recever [4], s sown n Fgure.. Fnger delay d spreadng waveform correlator Fnger rt delay d spreadng waveform correlator RAKE Combner decson Fnger L dealy dl spreadng waveform correlator Fgure. Temporal tme-only processng. 6

22 Te structure sown conssts of a bank of L RAKE fngers, eac correlatng to a dfferent delay of te receved sgnal. Te fnger outputs are ten combned to form a decson statstc. Te structure s equvalent to more practcal forms n wc te receved sgnal s frst fltered wt a cp pulse sapng matced flter and despreadng s performed usng receved cp samples and te spreadng code sequence. Temporal processng by te RAKE recever lets te CDMA system explot multpat dversty and make t nerently resstant to fadng. If te cannel s descrbed as a tapped-delay-lne TDL model, temporal processng can be mplemented by an equalzer, as sown n Fgure.3. Te desred sgnal st s transmtted troug te cannel, and receved at te recever antenna as xt. Te receved sgnal xt s contnuous, and sampled to obtan te dscrete sgnal xk wc s ten fltered by a lnear flter to get yk. Te equalzer s used to elmnate nter-symbol nterference ISI and mult-access nterference MAI [6]. Cannel SK XK X k- X k-m+ X k-m+ Z - Z - Z - * * M- * M * Σ Y k Fgure.3 Equalzer. 7

23 Assumng tere are Q users and te cannel s nose-free, we obtan te receved dscrete sgnal as x k = Q Σ q= q s k q.8 were =,, L, ] s te dscrete cannel gan coeffcents correspondng to q [ q q ql te multpat components and L s te number of resolvable pats. ere, we model te cannel as a tme-nvarant fnte mpulse response FIR cannel. s q = [ s k, s k, L, s k L + ] q q q T s te data sequence, and te dmenson of te equalzer s wegt vector s Mx, so te output of te equalzer s gven by y k = w x.9 k were x k = [ x k, x k, L, x k M + ] T s gven by Q x = q s q k, k q=.0 q s a Toepltz matrx wt dmenson M x M+L- gven by q q L 0 L 0 0 ql = 0 q q L ql 0 L 0 q, M M O O O O M 0 0 L 0 q q L ql. and T s q k = [ sq k, sq k, L, sq k L M + ] s a vector wt dmenson M+L- x. e rewrte te output of te equalzer n.9 as y k = w S k. 8

24 were = [,,..., Q ] wt dmenson beng M x QM+L- and T T T S k = [ s k, s k, L, sq k] s a vector wt dmenson QM+L- x. If we want to recover te sgnal s k from te receved sgnal xk were s k s te desred sgnal and cancel nterference, te followng zero-forcng ZF requrement must be satsfed n order to elmnate MAI and ISI: w = [,0, L,0]..3 Equaton.3 requres tat must be of full-column rank [5], but t s clear tat as te dmenson of M x QL+M-, and usually M < QL+M-. Ts mples tat ISI and MAI cannot be elmnated smultaneously f only temporal-processng s used. One soluton to ts problem s to oversample te receved sgnal by a factor P so tat as te dmenson of MP x QL+M- were MP > QL+M-. By oversamplng, can reac full-column rank, but te dsadvantages are less effcency of te system and nose enancement... Spatal Processng Te adaptve antenna array can aceve spatal dversty and mtgate multpat fadng. Ts s n addton to te nterference cancellaton attaned from steerng beams towards te desred user and/or steerng nulls n te drecton of nterferers. Te sgnal envelopes observed across te elements of an antenna array sould ave low cross-correlaton n order to aceve dversty gan. As a result, f te sgnal at one of te elements s gong troug a deep fade, t s gly unlkely tat te sgnals at te oter elements are encounterng smlar fades at te same tme. So tere s nearly always good sgnal 9

25 recepton on one of te antenna elements. Terefore combnng te sgnals from varous elements wll ncrease te SNR and te fdelty of te receved sgnal. Spatal processng can be mplemented by a beamformer, as sown n Fgure.4. w * s x k x k w * Σ y k s Q Q x Jk w J * Fgure.4 Beamformer. Assumng tere are Q actve users, and te cannel as no multpat effect, te receved sgnal s gven by x k = Q Σ q= q s k q.4 were q s a J x vector tat represents space-only cannel n te absence of delay spread, and terefore s q k s a scalar. J s te number of antenna array elements. Let s k be te desred sgnal. Te output of te beamformer s gven by y k = w x k.5 were w s te J x wegt vector. Te above equaton can be rewrtten n matrx form y k = w sk.6 0

26 were = [,,..., Q ] J x Q and sk = [s k, s k,..., s Q k] T Q x. Te ZF condton to retreve te frst user and cancel all MAI wll ten be w = [,0, L,0]..7 Te above equaton requres must be a full-column rank matrx. Ts requrement mples J Q,.e., te number of antenna array elements must be greater or equal to te number of actve users. Furtermore, consderng multpat effects, te requrement becomes J Q L q q=.8 were L q s te number of multpats for eac user. en L q and Q are large numbers, te number of antenna array elements s requred to be large, wc s not practcal. But unlke tme-only processng, space-only processng can be qute effectve aganst MAI. Te space-only and tme-only confguratons dscussed above use a ZF structure and dd not balance te effect of nose. As stated n [5], te sngle user matced flter recever s optmzed to fgt te background wte nose exclusvely, wereas te conventonal decorrelatng detector elmnated te multuser nterference dsregardng te background nose, and n te contrast, te MMSE mnmum mean-square error recever can be seen as a compromse soluton tat takes nto account te relatve mportance of eac nterferng user and te background nose. Moreover, te relatve performance of te two structures wll depend eavly on te cannel parameters [7]...3 Space-tme processng A Beamformer-RAKE cascades a beamformer wt RAKE recepton to process te sgnal bot n te spatal and te temporal domans. For eac fnger of te temporal

27 RAKE processor, tere s a beamformer to mprove te fdelty of te sgnal of tat partcular branc. At te front end of te recever s an antenna array. Te sgnals from te array are fed nto a set of spatal combners tat perform beamformng for dfferent multpat and eac wegt vector accentuates te sgnal from a partcular multpat component of te desred user. A temporal combner follows te spatal combner were te contrbuton from dfferent multpat from ter correspondng spatal combner s combned to explot te multpat dversty. Te structure s sown n Fgure.5. Te receved sgnal at eac antenna array element s fed nto an equalzer. Te wegt matrx acts partally as a beamformer. # w * x x x M- x M z - z - z - * * * w w M- w M #J w J * x J x J x JM- x JM z - z - z - * * * w J w JM- w JM Σ yk Fgure.5 Space-tme recever. e use te system model descrbed n Secton... e assume Q s te number of actve users, M s te number of taps n te equalzer, and J s te number of antenna array elements. Te antenna array response sould be taken nto account n te cannel model. tout loss of generalty and to smplfy te analyss, we assume te drecton of

28 3 arrval DOA for all multpats of a user be te same, θ q, tat s, te DOA at te antenna array for te qt user, or at least tey all fall nto te same beam. Te array response s a θ wc s a J x vector and s also known as a steerng vector. e get ],,, [ JxL ql q q q q q T q q θ θ θ θ a a a a L =..9 Substtutng te above nto., we get a modfed cannel matrx = ql q q q q q ql q q q q q ql q q q q q q θ θ θ θ θ θ θ θ θ a a a a a a a a a M L O O L L L O O M M L L Te dmenson of te space-tme cannel matrx for te qt user s JM x L+M-. Te receved sgnal at te antenna array s gven by n s x + + = = = + + Q q q q x M L M L k k k x JM JM M J J s s s x x x x x x x k,,, M M M M. were s s te transmtted nformaton, and we assume te nose n s spatally and temporally wte and Gaussan. From above, as dmenson JM x L+M-, were JM L+M-, so s full column rank and satsfes te ZF condton, and terefore, cancels bot ISI and MAI. In te case of nose, we no longer te consder ZF condton, but nstead dscuss ow to optmze te soluton by usng spatal-temporal tecnques.

29 A popular optmalty crteron s ST-MMSE space-tme mnmum mean-square error. In ST-MMSE, we obtan an estmate of te transmtted sgnal as a space-tme wegted sum of te receved sgnal and seek to mnmze te mean square error between te estmate and te true sgnal at any tme nstant. Te recever structure s sown n Fgure.5. In a space-tme flter, te wegt as te followng form w k L wm k k = M O M. w J k L wjm k. e defne te operator vec. as v vec[ v L v ] = M M. v M Terefore, w k = vec k JMx..3 Tus, we obtan a convenent formulaton for te space-tme flter output: y k = w k x k..4 Te ST-MMSE flter cooses te space-tme flter wegts to aceve te MMSE mnmum mean square error,.e., arg mn { w E w x k s k..5 Te soluton to ts LS least squares problem s gven by [6] w = R z.6 xx 4

30 were R xx = E[xx ] s te auto-correlaton matrx of te receved sgnal at te recever, or * te nput sgnal to te flter. Te vector z = E[xs ] s te cross-correlaton vector between te receved sgnal and te desred sgnal [6]. ST-MMSE combnes te strengts of tme-only and space-only processng, and trades MAI and ISI reducton aganst nose enancement. It wll cancel te MAI prmarly n te spatal dmenson, and ISI n te temporal dmenson..3 Smulaton Examples In ts secton, we compare te performance dfference between te temporal processng and space-tme processng by computer smulaton. Frst we consder temporal processng. Assume te number of users Q=3, te number of multpats L=4, te number of te equalzer taps M=4, and te sgnal-to-nose rato SNR=0dB. Te receved sgnal s gven by x xk xk xk k 3 = s, 0 s, k 0 M 4 s, k k 6 + s + 3s3 + n..7 Te equalzer s output s gven by s y k = w [ ] 3 s. s 3.8 In te deal case, te followng sould be satsfed: [ 3 ] = [ 0 L 0] x L+ M w..9 Te smulaton result s gven below. Te result s a x column vector. ss = w [ ] 3 5

31 ss = [ ] T. From te result above, we can see te frst element s muc larger tan oters. Te rato of -norms of te frst element and te rest s sown below : Ts result s far from te deal case :0, and t proves tat te tme-only processng cannot provde a satsfactory performance. Te Fgure.6 sows te QPSK constellaton of te desred user. Te modulated sgnal s presented by a+b. Fgure.7 sows te receved sgnal at te equalzer nput. Due to te cannel effects, multpat effects, nterference, and nose, te receved sgnals are corrupted and no longer old QPSK caracterstcs. Te recever must process te receved sgnal to elmnate te MAI, ISI and nose n order to recover te transmtted sgnal. Fgure.8 sows te sgnal constellaton at te equalzer output. Te sgnals ave smlar ampltudes to te orgnal transmtted sgnal, but cannot be effectvely recovered. Te performance of tme-only processng s poor. Next, we consder space-tme processng. A unform lnear antenna array wt fve elements and alf-wavelengt spacng s used. Te DOAs are [ ], and 0 s te DOA of te desred sgnal. Te smulaton result s gven below. Te result s a x column vector: 6

32 ss = w [ ] 3 ss = [ ] T. Te rato of -norms of te frst element and te rest s : Te result s very close to :0. Fgure.9 sows te sgnal constellaton after space-tme processng. It sows te recovered sgnals ave smlar ampltudes wt te orgnal transmtted sgnals, and are dstrbuted around eac center of te orgnal QPSK modulated sgnals. Te receved sgnals can be effectvely recovered. Te beam pattern for te above space-tme processng s llustrated n Fgure.0. As we can clearly observe, te antenna gan reaces te largest magntude at te DOA of te desred user,.e., [ 0 ], and as a null at te drectons of te nterferers,.e., [ ]. Terefore, MAI can be effectvely elmnated n te space-tme processng. In ts capter, we presented a descrpton of a wreless cannel and dscussed dfferent pyscal effects suc as pat lass, fadng, delay spread, angle spread, and Doppler spread tat make te receved sgnal muc weaker tan te transmtted sgnal. e also dscussed tree processng tecnques tat are used at te recever: tme-only, space-only and space-tme processng. As sad, tme-only and space-only processng tecnques ave fundamental drawbacks tat make te recever very dffcult to elmnate MAI and ISI smultaneously. In te contrast, space-tme processng ave overcome te drawbacks and 7

33 ave sgnfcant advantages over te oter two processng tecnques, and te recever s performance ave mproved dramatcally Fgure.6 QPSK modulated sgnal constellaton Fgure.7 Receved sgnal constellaton at te equalzer nput. 8

34 Fgure.8 Sgnal constellaton after tme-only processng Fgure.9 Sgnal constellaton after space-tme processng. 9

35 Antenna Gan DOA n Degree Fgure.0 Beam pattern for te space-tme processng. 30

36 Capter 3 Tme-varyng Multpat Vector Cannel Smulator In a wreless cannel, te cange of surroundng structures and te movement of moble users make te cannel tme-varyng. For te purpose of analyzng te performance of communcaton systems, tere s a need for effectvely smulatng te rado cannel. t te employment of an antenna array, te sgnals are receved at te recever by multple antennas, and terefore, te cannel model becomes a vector as opposed to a scalar were only a sngle antenna s employed. In ts capter, we ntroduce a tme-varyng multpat fast fadng vector cannel smulator tat can be used to evaluate te performance of antenna array recever. 3. Tme-varyng Multpat Vector Cannel In [8], a multpat cannel smulator was ntroduced wc s a mult-cannel generalzaton of te scalar cannel presented n [3]. Takng nto account an antenna array, an expanson s made to smulate te new cannels n ts capter. e consder te transmsson from a moble staton to a base staton. Te moble s surrounded by reflectng structures wtout lne of sgt between te moble and base staton and causes multpat effects. Te moble staton s movement also causes Doppler sft. Te faster te moble staton moves, te faster te cannel response canges. In order to present te cannel model, te followng parameters are defned: J number of antenna array elements f c carrer frequency ω c = πf c 3

37 B te transmtted sgnal bandwdt λ c carrer wavelengt v speed of te moble θ drecton of arrval of te receved sgnal; a plane wave s assumed d te dstance between moble staton and base staton τ = d / c, te propagaton delay were c s te velocty of lgt. τ mn and τ max are te mnmum and maxmum propagaton delay, respectvely. D array spacng L number of multpats Te antenna array used ere s a Unform Lnear Array ULA,.e., te spacng between te elements of a lnear array s equal, as llustrated n Fgure 3. Incdent plane wave θ Dsnθ #3 # D # Antenna Fgure 3. Unform Lnear Array wt tree elements. e assume: 3

38 . Te sgnals orgnate far away from te array and te plane wave assocated wt te sgnal advances troug a non-dspersve medum tat only ntroduces a propagaton delay. Under suc crcumstances, te sgnal at any oter element can be represented by a tme-delayed verson of te sgnal at te frst element.. D << c/b,.e., te bandwdt of te mpngng sgnal s muc smaller tan te recprocal of te propagaton tme across te array. Ts s commonly known as narrowband assumpton. Ts assumpton makes t possble to represent te propagaton delay wtn te elements of te array by pase sfts n te sgnal. t te assumptons mentoned above, te receved sgnal at te antenna array s gven by x t = J m= 0 s t a m θ 3. were st s te nformaton and a m θ s te array response. Te above equaton can be wrtten n a vector form as x t = s t a θ 3. were eac element of xt contans te receved sgnal at te correspondng array element and ωcρ θ ωcρ J θ [, e, L, e ] a θ = =, e D π sn θ λc, L, e T D π J sn θ λc T 3.3 were ρ s te propagaton delay relatve to te frst array element. 33

39 Te vector a θ s known as te array response vector or te steerng vector of a ULA, wc s a functon of DOA under suc crcumstances. It can be sown tat te baseband equvalent of J x vector cannel mpulse response s gven by [] L g t; τ = δ τ τ = t 3.4 were t s te observaton tme, τ s te propagaton delay of te t pat, L s te number of tme dfferentable pats wc are composed of a large number of tme nondfferental subpats wt te smlar DOA. Te DOA of eac tme dfferental pat s te mean of angles of all subpats. Te spread of angles of subpats s defned as angle spread. Smlarly, τ s te mean of te delays of te subpats. Te larger te τ, te less correlaton between tose pats. Te parameter s te complex pat vector for te t pat and s gven by [ ] T = L M were te elements of are te cannel coeffcents wc are functons of tme and frequency. If we assume all propagaton pats le n te same plane as te array elevaton angle ψ =0, te t complex pat vector s gven by ωdt ωcτ t = α θ, τ a θ e dθdτ 3.6 τ θ were τ ranges n τ + / B, τ + + / B], θ ranges [ θ /, θ + / ], θ s te [ mn mn mean angle of arrval, s te angle spread, α θ, τ s te ampltude ntensty functon for eac pat, and aθ s te array response. Te Doppler sft and pat delay result n pase varaton, ωdt ωcτ e, were Doppler sft s 34

40 f d = f c v cosψ c 3.7 wt ω = πf and we wll assume ψ=0. d d Usually, all tme dfferentable pats are composed of a large number of nondfferentable subpats. By te central lmt teorem, we fnd te cannel coeffcents to be well approxmated by complex Gaussan varables [6]. Snce Gaussan varables are entrely caracterzed by ter frst and second order statstcs, we can smulate te cannel smulator coeffcents by generatng Gaussan varables tat ave some predefned approprate frst and second order statstcs. Next, we ntroduce te frst and second caracterstcs of te cannel coeffcents. 3. Second Order Caracterzaton Snce te moble s surrounded by local reflectng structures, we ave many ndrect transmsson subpats eac of wc exbts dfferent propertes. t suc a scenaro, we assume:. Subpats correspondng to dfferent delays or angles of arrval ave uncorrelated ampltudes,.e., * ' ' ' ' E[ α θ, τ α θ, τ ] = δ θ θ, τ τ f θ, τ 3.8 were f θ, τ s te ont pdf of θ and τ, and * te complex conugate. Ts s also referred to as te wde sense statonary uncorrelated scatterng SSUS assumpton [0] [36].. For a gven subpat, te delay, angles of arrval are mutually ndependent,.e., 35

41 f θ, τ = f a θ f τ 3.9 b were f θ and f τ are te power densty functon of angles of arrval, θ, and a b transmsson delay τ, respectvely. 3. Te power densty functon of te transmsson delay τ s gven by [0] f b τ = e τ τ τ 3.0 were τ s te mean delay. Integratng f b τ wt respect to te delay doman for te t pat, we get τ F = τ mn mn + + / B + / B f τ dτ b 3. = [ e / Bτ ] e / Bτ were F s smply te power fracton assocated wt te t pat. Because of te random pase assocated wt eac ndvdual tme ndfferentable subpat n a gven tme dfferentable pat, te cannel coeffcents are well modeled by a zero mean complex varable [9]. Under te above assumptons, te cross-correlaton matrx of te complex pat vectors s gven by [] R t, t = E[ t d = δ J ω t F R 0 t ] a, 3. were, represent te t and t pat, δ = 0 wen wc mples te tme dfferentable pats are ndependent from eac oter, t = t t s te tme lag, and J 0. s te Bessel functon of te frst knd and of order 0. R a, s te spatal correlaton matrx for te t pat, and s gven by 36

42 = f θ a θ θ dθ 3.3 R a, a a θ If we assume te power densty functon wt respect to te angles of arrval, f θ, s unform n θ ± /, 3.3 can be wrtten as a θ / R a a +, = a θ θ dθ. θ / 3.4 Tus, te smaller te angle spread, te ger te spatal correlaton between te sgnals at te antenna elements. Te spatal correlaton s also a functon of antenna element spacng D. Te larger D, te lower te spatal correlaton. In macrocellular moble communcatons, snce te base s usually far from te moble and te local reflectors surroundng t, te angle spread nduced by te local reflectors s, terefore, often relatvely small [7], and te spatal correlaton can reman relatvely g even f te antenna elements are spaced by many λ c. 3.3 Vector Cannel Smulator Te vector cannel smulator structure s sown n Fgure 3.. It s sngle-nput multpleoutput SIMO dscrete-tme FIR system wt tme-varyng coeffcents, based on a TDL tapped delay lne model wt taps evenly spaced one sample apart. Te smulator as J brances correspondng to eac antenna array element. Eac branc as L wegts wt eac of wc represents a resolvable pat coeffcent. Te samplng nterval s T c.e., /B. Te power of te nose sgnal s cosen accordng to te requred SNR. Te nput to te smulator s a baseband transmtted sgnal, st, and t s multpled by te correspondng wegt,, wc s te cannel coeffcent correspondng to te t pat 37

43 of te t antenna element. AGN s added at te output and te receved sgnal at te t antenna element s y t. Fnally, we smulate te sgnals receved at te antenna travelng troug te tme-varyng multpat cannel. Next, we descrbe ow to generate te cannel coeffcents. st Tc Tc Tc 00 0 L-0 L-0 AGN y 0 t Tc Tc Tc 0 L- L- AGN y t Tc Tc Tc 0J- J- L-J- L-J- AGN y J- t Fgure 3. Tme-varyng multpat vector cannel smulator. 3.4 Complex Pat Vector Generator e ave known tat te tme dfferentable pats are ndependent from eac oter and te cannel coeffcents can be generated ndependently and approxmated by Gaussan varables. Te cannel coeffcents ave zero mean and second order caracterstcs 38

44 dscussed above, ncludng spatal-temporal correlaton propertes. Terefore, we can apply some knd of space-tme correlaton sapng transformaton to uncorrelated Gaussan wte nose sequences n order to obtan tme-varyng pat vectors tat exbt te approprate spatal-temporal correlaton propertes. Te tme-varyng cannel s mostly due to moble staton moton. Te extent of ow te cannel vares n tme s measured by te Doppler sft. As long as te complex pat vectors are obtaned at a rate ger tan twce te Doppler frequency, ter temporal correlaton structure s preserved. Fgure 3.3 sows te structure of te t pat vector generator. Zero-mean Gaussan Varable Generator _ n 0 m _ n m z z y 0 m y m Space correlaton transformaton 0 m m T/T c T/T c 0 k k _ n J- m z y J- m J- m T/T c J- k Tme-correlaton transformaton flter Interpolators Fgure.3 Te complex pat vector generator for te t pat 39

45 e take te samplng nterval of T were = 3 f, ten feed te samples nto te d tmecorrelaton transformer and space-correlaton transformer, and fnally te smulated T complex pat vectors are nterpolated n tme to te desred samplng rate, /T c = B. Equaton 3. can be rewrtten n te followng way for te case =: R t = J 0 ω t F R a, 3.5 d were F s ndependent of t, and terefore, we defne C = FR a,. 3.6 Ts suggests tat n order to obtan te approprate space-tme correlaton caracterstcs for a gven pat, one can use a tme transformaton followed by a spatal transformaton. Te tme ndex n Fgure 3.3 refers to te samplng nterval T of te complex pat vectors pror to te nterpolator. Te nose generator produces zero-mean complex Gaussan vectors: T n m = [ n0 m n m L n M m ]. 3.7 Te tme-correlaton sapng flter denoted by z s desgned so tat te temporal correlaton of ts output, y m = z n m, s approxmately equal to temporal component n 3.5,.e., J ω. Te space-correlaton transformaton takes care of 0 d t te spatal component of te correlaton,.e., C. Ts transformaton s appled to te vector y m. Fnally te smulated complex pat vectors are nterpolated n tme to te desred samplng rate. 40

46 3.4. Tme-correlaton Sapng Flter A tme correlaton sapng flter for obtanng te desred temporal correlaton s wdely used. Te desred power spectral densty functon s te Fourer transform of te Bessel functon n 3.5, and s gven by [8] s ω = f ω ω d oterwse ω d ω ω For te tme correlaton sapng flter, we must ave e = s ω. Snce s ω as sngulartes at ω = ±ω d, te desgn of ts flter s unrealzable. In practce, te sngulartes at ω = ±ω d are replaced by sarp peak. If FIR s selected to mplement ts flter, te order of te flter must be g to obtan te sarp frequency response. To reduce te computatonal complexty, a low order IIR flter s used. ω e desgn an analog flter tat satsfes e = s ω, and ten apply AD converson. Te samplng rate s T 3 = x f d = 3 f d 3.9.e., te samplng rate s 3/ tmes te Nyqust frequency. Fnally, we get a 4t-order IIR flter [] z + 05z +.53z z z = z +.334z +.343z z 3.0 e wll smulate te result later by comparng te correlaton of te cannel coeffcents after te flter wt Bessel functon. 4

47 3.4. Spatal Transformaton Te spatal transformaton used to obtan te complex pat vector takes te form of a smple matrx operaton to y : = M y 3. Te matrx M must satsfy M M = C,.e., te correlaton matrx of te spatal transformaton matrx must be equal to C. Te development of M s based on Karunen-Loeve expanson for vectoral random process. Snce te C s a correlaton matrx of a dscrete tme random process, t s nonnegatve defnte ermtan matrx [6]. A ermtan matrx s dagonalzable, so we can wrte C = Q Λ Q 3. were Q s te matrx wose columns are ortonormalzed egenvectors of C, denoted by q = 0,..., J- and Λ s te dagonal matrx wose entres correspond egenvalues, λ. From q =, we get q q. Te fact tat C s nonnegatve defnte mples q = tat q Cq 0. By defnton Cq = λq, so we ave q Cq = λ q = λ 0, so te egenvalues of C are real and nonnegatve. e can terefore get C = Q Λ Λ Q 3.3 = M M = [ Q Λ / / / ][ Λ / Q ] were M = Q Λ s te desred spatal transformaton matrx. / Premultplcaton of te spatally uncorrelated sgnal vector y by M = Q Λ gves te / complex pat vector suc tat 4

48 E[ m k] = Q Λ [ Q Λ / d d / E[ y m y J ω T k m C d J ω T k m Q Λ Q 0 0 k] Λ / ] J ω T k m[ Λ 0 Q / Q ] 3.4 wc s te desred result. Fgure 3.4 llustrates te space-correlaton transformaton for te t pat. Eac element of y s frst scaled by te square root of te egenvalue. Te resultant vector s ten multpled by te egenvector matrx Q. Te output of te pat vector generator as te approprate tme-space correlaton propertes. y 0 m 0 m λ 0 y m λ Premultplcaton by Q m y J- m λ J J- m Fgure 3.4 Space-correlaton transformaton Interpolator As mentoned before, te samplng rate to sample te Gaussan varable generator s /T=3f d, and te samples are appled to te tme-space correlaton transformaton. Te frequency assocated wt te pat vector generator s muc smaller tan te one assocated wt cannel flterng /T c = B. e terefore need to obtan te g rate 43

49 cannel coeffcents requred for cannel flterng va nterpolaton. en a sgnal wt a samplng frequency muc ger tan te Nyqust frequency as to be nterpolated, smple metods suc as lnear or cubc nterpolaton can be used wtout compromsng te precson [33]. Snce /T c = B s muc larger tan /T, t s possble to decompose te nterpolaton process to reduce te computatonal requrements, as llustrated n Fgure 3.5. Te system s composed of a lnear nterpolator followed by a cubc nterpolator. Te lnear nterpolator ncreases by a factor I 30 te samplng rate to become muc ger tan te Nyqust frequency, and ten a cubc nterpolator s used to furter ncrease te samplng frequency to ts desred value of /T c. m Samplng perod T Lnear nterpolator Samplng perod T/I Cubc nterpolator k Samplng perod Tc Fgure 3.5 Interpolator. Fgure 3.6 llustrates te lnear nterpolaton: Fgure 3.6A represents te orgnal sgnals, Fgure 3.6B te sgnals after addng I- zeros between samples zero-paddng, and ten te sgnals are fed nto a low-pass flter wt a radal cutoff frequency π / I. Fgure 3.6C represents te sgnals after lnear nterpolaton. Lnear nterpolaton s computatonally complex, but as more precson. en lnear nterpolaton s completed, a cubc nterpolator s used. 44

50 6 A Ampltude B Ampltude Ampltude C Tmesec Fgure 3.6 Lnear nterpolaton llustraton. Fgure 3.7 llustrates te cubc splne nterpolaton, between te sample nterval [ T T ], were te sgnal s represented by a trd-order polynomal: p t + t 3 = a0 + at + at a Terefore, samples can be obtaned by samplng te above functon at te requred sample tme nstant so tat te rate s ncreased. 45

51 xt T- T T+ t Fgure 3.7 Cubc splne nterpolaton llustraton. 3.5 Smulaton Examples e smulate and examne te cannel realzaton va te pat vector generator. Consder a DS/CDMA system wt carrer frequency beng f c = Gz, transmtted sgnal bandwdt s B =.88Mz Tc = / B, te number of antenna array elements s M=3, te mean delay s τ = / B = T, te speed of te moble s ν =30m/s, te Doppler c frequency s f d = f c. ν = 00z, and terefore, te samplng nterval before te c nterpolator s T = = 4096Tc. Te lnear nterpolator decreases te nterval to 8Tc 3 f d I=3, and ten te cubc nterpolaton reduces t from 8T c to T c. By passng a wte Gaussan nose sequence troug te IIR flter z developed prevously, we get fltered outputs wose temporal correlaton s approxmately equal to te temporal component n 3.5, as llustrated n Fgure 3.8 wc plots te desred and obtaned temporal correlaton. Te smaller te tme lag, te more tey are correlated. 46

52 obtaned desred Temporal correlaton Normalzed tme lag Fgure 3.8 Desred and obtaned temporal correlaton. After passng troug te tme correlaton sapng flter, te smulated pat vector needs to be spatally transformed to get an approprate spatal correlaton. Frst of all, assume tere are tree tme dfferentable pats, L = 3, wose DOAs are [ 60,0,50 ], respectvely. Te angle spreads are [0,,5 ], respectvely. Fgure 3.9 llustrates te plot of te magntude of te cannel coeffcents at te frst antenna element. Snce tere s g correlaton of te cannel coeffcents between antenna elements, te cannel coeffcents are very smlar for dfferent antennas, and te plots of te cannel coeffcents versus tme are nearly te same. Te tme nterval of te plot along x-axs s /B, correspondng to a cp n DS-CDMA. As llustrated, te cannel coeffcents are rapdly tme-varyng. 47

53 Fgure 3.9 Cannel coeffcent magntudes vs. tme for te frst antenna. Te nterpolator s used to nterpolate te cannel coeffcents to ncrease te fnal sample rate to te transmtted sgnal bandwdt B. Fgure 3.0 llustrates te plot of te cannel coeffcent magntudes vs. tme for te frst pat of te frst antenna element before and after te nterpolator. As llustrated, te curve s not smoot before te nterpolator, and as sarp corners. After te nterpolator, te curve s very smoot, and very smlar to tat of a typcal wreless cannel [0]. 48

54 Cannel coeffcent magntudes before/after nterpolaton for Pat of te st antenna element after nterpolator before nterpolator 0-5 Gan db tme sec Fgure 3.0 Cannel coeffcent magntudes before and after nterpolaton for Pat of te frst antenna element. In ts capter, a new multpat cannel smulator s ntroduced. Te smulaton results agree wt te dscusson above, and later, we wll use ts tme-varyng multpat vector cannel smulator to generate te desred cannel coeffcents to analyze te recever s performance. 49