Pulse Shaping in Non-Coherent DLL Tracking of CDMA Signals

Transcription

1 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai Pulse Shapig i -Cheret DLL Trakig f CDMA Sigals Mhamed Ada Ladlsi ad Gaiyu B. Hussai Eletrial Egieerig Departmet Kig Fahd Uiversity f Petrleum & Mierals P.O. Bx 43, Dhahra 36 Saudi Arabia {adalusi,ghussai}@kfupm.edu.sa Abstrat: - The paper aalyses the impat f pulse shapig de trakig fr diret-sequee de divisi multiple aess (DS-CDMA) sigal syhrizati usig -heret delay-lk lps (C-DLL). The depedey f the residual timig errr variae the hip wavefrm is emphasized, ad the perfrmae f several -retagular pulse shapes is preseted. The desig f ptimized hip wavefrms fr miimizig the DLL trakig errr variae is the frmulated usig a st futi expressed i frequey-dmai ad highlightig the depedey f the trakig jitter the hip wavefrm Furier trasfrm. A sluti methdlgy is itrdued based the use f Prlate Spheridal Wave Futis. Sme illustrative desig examples are preseted, establishig theretial limits fr the best ahievable perfrmae (miimum jitter). I partiular, it is shw that the half-sie ad Hammig pulses are ear-ptimum whe siderig pulses with 99% ad 99.9% i-bad pwer badwidth upaies, respetively. Key-Wrds: - CDMA, DLL, Cde trakig, Trakig jitter, Chip wavefrm shape. Itrduti Fr DS-CDMA systems, hip timig aligmet betwee the reeived ad the lally geerated sigature sequees (r spreadig des) must be ahieved prir t data demdulati. Cde timig syhrizati is typially ahieved i tw steps: de aquisiti ad de trakig [,]. First, durig aquisiti, the reeiver btais the relative delay betwee the reeived ad the lally geerated des t withi a hip iterval. The, i the subsequet trakig phase, fier timig adjustmet is perfrmed i rder t brig the residual timig errr as lse t zer as pssible. I this paper, we fus de trakig whih is typially implemeted by meas f the lassial early-late delay-lked lp (DLL) that has bee thrughly disussed i the spread spetrum literature (see [] ad referees therei). Liear ad liear DLL mdels have bee preseted, ad the trakig errr jitter ad mea-time-t-lse are derived as key perfrmae idiatrs. I ur ase, we adpt the liear DLL mdel (appliable fr pratial SR rages), fr whih the DLL perfrmae is essetially determied by the harateristis f the disrimiatr S-urve [,], whih deped the rrelati prperties f the early-late spread-spetrum wavefrms. Quite imprtatly, i additi t the rrelati prperties f the spreadig des, it is als fud that ather imprtat fatr that affets several perfrmae aspets f DS-CDMA systems is the atual shape f the hip wavefrm pulse. The impat f hip wavefrm shapig has bee addressed frm the perspetive f multiple-aess iterferee ad errr perfrmae aalysis (e.g., [3,4,5]). There have als bee sme limited studies f this impat de syhrizati. Fr example, i [6], aquisiti ad trakig aspets f differet hip wavefrms are mpared, ad ther similar mpariss appear i [7,8]. I [9], it is shw that the use f umathed hip pulses a redue trakig errr variae. Mst f these wrks have maily fused mparig differet vetial pulse shapes, ad did t udertake ay ptimizati f the hip wavefrms i rder t maximize perfrmae. Reetly, i [], suh ptimizati is disussed i the text f timig aquisiti. O the ther had, fr de trakig, a heret DLL is sidered i [], ad sme ptimized hip wavefrms are preseted. Hwever, i pratie, the -heret DLL (C-DLL) arhiteture is preferable beause f its isesitivity t data mdulati ad arrier phase syhrizati. I this paper, we exted the wrk i [] t C- DLL de trakig systems. First, we preset a mprehesive perfrmae aalysis ad mparis f the impat f several mm (retagular) hip wavefrms de trakig jitter. I a sed part, a straied ptimizati prblem fr miimizig the residual errr variae is defied ad slved usig Prlate Spheridal Wave Futis []. ISS: Issue, Vlume 7, Deember 8

2 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai The rest f the paper is rgaized as fllws. I Seti, the system mdel is preseted. I Seti 3, the perfrmae aalysis f the C-DLL is disussed usig a frequey dmai frmulati ad mparative results fr sme vetial hip pulses are preseted. I Seti 4, the ptimized pulse shapig predure is disussed, ad several desig examples are the give i Seti 5, fllwed by fial lusis i Seti 6. System Mdel We sider a geeral CDMA system mdel where the sigal frm a give iteded user is reeived after sme trasmissi delay, ad is give by: rt () = Pt ( τ)s ( ωt+ φ) + () t + MAI th () t () where P is the sigal pwer, τ is the prpagati delay,φ is the arrier phase (that iludes the time delay effet), th() t is the additive white gaussia ise with tw-sided pwer spetral desity, ad th() t is the multiple-aess iterferee term whih a als be mdeled as additive gaussia ise (by ivkig the etral limit therem, fr a large umber f -users). The spread-spetrum wavefrm (t) is assumed t use a lg radm P de, ad is give by: T = ( t) = h( t ) () with{ } represetig a biary ± hip sequee, ad h (t) the hip wavefrm pulse defied ver the hip iterval [, T ]. The mdel used is appliable fr a pure pilt sigal withut data mdulati sie we maily fus de trakig usig a -heret delay-lked lp (t be disussed ext). We als te that, fr simpliity, multipath fadig effets are t expliitly iluded, but this will t impat the mai theme f the paper, whih is the explrati f the impat f pulse shapig P de trakig. A -heret delay lk lp (C-DLL) is assumed fr trakig the timig eph f the arrivig sigal. The C-DLL is based the earlylate sheme where the reeived sigal is rrelated with tw lally geerated, delayed de replias, the early ( t τˆ + ) ad late t ( τˆ ) lal des, where ˆ τ is the timig estimate, ad is the early-late spaig typially set t T. The utput wavefrms are filtered ad squared, ad the differee is used t drive the vltage trlled sillatr (VCO) t adjust the lal de timig. The lp disrimiatr utput due t the desired sigal mpet is btaied as []: Z ( ε) = R ( ε ) R ( ε + ) (3) where ε = τ τˆ is the timig errr, ad R(τ ) is the de sequee autrrelati futi give fr a rrelati perid f hips by: T R( τ ) = ( t) ( t + τ ) dt (4) T I the sequel, we resrt t a frequey-dmai frmulati that is fud t simplify the aalysis f the impat f pulse shapig DLL trakig perfrmae, as will be highlighted ext. Fr lg P des mdeled as radm sequees, the hip autrrelati futi a be expressed as []: R( τ) = H( f) s( π fτ) df (5) The disrimiatr utput i (3) is kw as the DLL S-urve, ad a be rmalized t yield: G() s = R ( s δ ) R ( s+ δ ) (6) where s = ε / T ad δ = / T dete rmalized timig errr ad early-late spaig variables. A blk diagram illustratig the perati f the C-DLL is shw belw. r(t) ( t τ ) ( t τ + ) P GEERATOR ( t τ ) BPF (. ) BPF (. ) VCO + z (τ ) LOOP FILTER Fig. : -heret delay-lked lp (C-DLL). 3 Perfrmae Aalysis 3. DLL Trakig Errr A mm perfrmae measure f the DLL is the trakig errr variae (r jitter). At high SR, the trakig errr will usually be small eugh suh that the lp disrimiatr utput a be take as a liear futi f the relative timig errr. Usig liear lp aalysis, it a be shw that a upper bud the trakig errr variae fr first-rder lps is apprximated by []: V Var( ε ) Eκ (7) where is the rrelati perid legth (i hips), ISS: Issue, Vlume 7, Deember 8

3 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai E = P/ T is the eergy per hip, κ is the S-urve slpe at the rigi ( s = ). The V term aptures the ttal iterferee variae give by: V = + I (8) where is the AWG tributi ad I is the variae f the multiple-aess iterferee (MAI) due t ther users. Sie CDMA systems are maily iterferee-limited, the AWG fatr a be drpped t fus exlusively the MAI term. With equal reeived pwer P fr all users, the MAI term variae a be expressed i frequeydmai as []: I P( Ku ) M (9) where Ku is the umber f users, P is the reeived pwer, ad M the pulse-depedet MAI fatr give by []: 4 M = H( f) df () with H ( f ) detig the Furier trasfrm f the hip wavefrm pulse h (t). It is als desirable t express the S-urve slpeκ i a tratable frm that emphasizes its depedey the hip pulse shapig filter. This is readily ahieved by bservig that: dg() s κ = ds s= () = R ( δ ) R( δ) R ( δ) R( δ) where the aut-rrelati futi R(.) a be substituted frm (5). I typial DLL desigs, the valueδ =. 5 is mmly used (half-a-hip earlylate brah spaig), ad it fllws that: ( ) R.5 = H( f) s( π f) df () O the ther had, ( ) R.5 = π f H( f) si( π f) df (3) Substitutig () ad (3) it () ad bservig that R(δ ) is eve ad R' ( δ ) is dd, we fially get: κ = 4 H( f) s( π f) df (4) π f H( f) si( π f) df 3. Chip Wavefrm Cmpariss Gig bak t the expressi f the trakig errr variae i (7), the de trakig perfrmae a be diretly estimated by evaluatig the MAI variae ad the C-DLL S-urve slpe κ. Mre speifially, it is see that, aside frm stat fatrs, the perfrmae fr a give pulse shape will deped the rati M / κ, whih i tur depeds the pulse spetrum H ( f ). This will frm the basis fr the relative perfrmae assessmet f several pulse shapig shemes, ad the subsequet ptimized desig f these spreadig wavefrms, as will be disussed i the ext seti. Fr mparative purpses, Table gives a list f mmly used pulses, startig with the uit retagular pulse pt () t defied ver the iterval [ T, T ]. Table shws the rmalized trakig errr fatr fr the differet pulse shapes. Retagular p () t Half-Sie si( π t/ T ) p () t T T Triagular ( t/ T ) p () t T Raised-Csie [.5.5s( πt/ )] p ( t) T T Hammig [.54.46s( πt/ )] p ( t) T T Table : Cvetial Pulse Shapes. Pulse Shape M κ M / κ Retagular Half-Sie Triagular Raised-Csie Hammig Table : Trakig Perfrmae Cmpariss. Frm the results i Table, it is see that the best perfrmae is ahieved by the half-sie pulse. ISS: Issue, Vlume 7, Deember 8

4 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai Hwever, it shuld be pited ut that ther fatrs (suh as badwidth fiemet) play a imprtat rle as well, ad this will be further disussed subsequetly. 4 Perfrmae Optimizati 4. Prblem Frmulati The wavefrm-depedet term i the DLL trakig errr is a futi f the rati M / κ as shw i the previus seti. Therefre, t ptimize the DLL trakig perfrmae, it is imprtat t desig pulse shapig filters H ( f ) that miimize this quatity. The ptimizati shuld als be subjet t additial sigal straits iludig fixed sigal eergy ad limited badwidth upay. Earlier wrk the miimizati f timig jitter i the heret delay lk lp was de i []. We exted the apprah preseted i [] ad apply it t the heret delay lk trakig lp C- DLL sidered i this wrk. As disussed previusly, the trakig errr perfrmae is ptimized by miimizig a st futial F (give by the rati M / κ ) ver the hip filter respse H ( f ) : F = H ( f ) H ( f ) s( πf ) df 4 df πf H ( f ) si( πf ) df (5) The miimizati f this futial is subjet t additial straits iludig fixed pulse eergy ad limited badwidth upay. The fixed rmalized eergy strait is expressed by: H ( f ) df = (6) I additi, it is als desired t imprve pulse badwidth fiemet s that ly a small frati η (e.g., %) f eergy spills ver a give bad[ B, B]. I CDMA, a badwidth effiiet hip wavefrm allws fr the use f a higher spreadig fatr fr a give allated spetrum, thereby imprvig verall system perfrmae. The strait fr badwidth fiemet is: B B H ( f ) df = η (7) A diret, lsed-frm sluti t this straied ptimizati prblem is t tratable. Istead, we prpse a apprah that verts the prblem t a equivalet disrete frmulati with redued mplexity. This is ahieved by expadig the hip pulse Furier trasfrm H ( f ) usig a adequate set f basis futis, ad the slvig fr the expasi effiiets that miimize the bjetive futi F, subjet t the speified straits. Speial futis kw as Prlate Spheridal Wave Futis (PSWF) [] are partiularly well suited fr this ad have bee suessfully used fr similar ptimizati prblems [,]. 4. Prlate Spharidal Wave Futis Csiderig the spae f square-itegrable (i.e., fiite-eergy) futis, the PSWF s are speial futis that a be struted either i time- r frequey-dmai []. We adpt a frequeydmai struti, well i lie with ur ptimizati frmulati. The PSWF s deted by { ( f )}, with assiated eigevalues { λ }, are struted with fiite time supprt (limited fr symmetry t [-T /,T /]) ad speifi badwidth fiemet t a rage [-B,B]. These futis B satisfy ( u)si( f u) du Tλ ( f) ad = B have the fllwig tw prperties: i) they frm a mplete, rthrmal set ver the spae f fiiteeergy, time-limited pulses ver [-T /,T /], ad ii) they frm a mplete, rthgal set ver the spae f fiite-eergy futis ver the iterval [-B,B]: ( i) ( ii) B B ( f ) ( f ) m m ( f ) df = δ m ( f ) df = λ δ m (8) whereδ m = fr m = ad therwise, ad λ is the eigevalue assiated with the futi ( f ), ad represets the frati f eergy f ( f ) fallig i the bad [-B,B]. A imprtat feature f PSWF s is that, amg all time-limited futis with give durati T, they ahieve, i dereasig rder, the highest eergy etrati i the bad [-B,B]. This eergy fiemet is represeted by the mtially deayig ISS: Issue, Vlume 7, Deember 8

5 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai eigevalues, whih beme essetially egligible whe the rder exeeds the time-badwidth prdut BT []. 4.3 Prblem Simplifiati The PSWF prperties lead t a sigifiat reduti f the mplexity f ur ptimizati prblem. Ideed, sie the { ( f )} frm a mplete set fr the spae f futis H ( f ) f iterest, the fllwig liear expasi is btaied: where = H ( f ) = x ( f ) (9) x is give by the ier prdut < H, >= H( f) ( f) df () Beause the eigevalues { λ } deay rapidly t zer, a truated series with a limited umber f PSWF s may be used. I additi, assumig eve symmetry fr the hip pulses f iterest, we ly eed t keep the eve-idexed ( f ) (sie the eve-idexed futis are eve ad the ddidexed es are dd, as shw i []). The truated series bemes: H ( f ) x ( f ) () = Therefre, withut sigifiat lss f auray, we a redue the prblem frmulated i (5)-(7) t a fiite-dimesial lw-mplexity e, i terms f the expasi effiiets { x }. This is mpatly expressed by the fllwig equatis: Miimize... γ(,..., m) x = m= = F = γ(, m) xxm γ3(, m) xxm = m= = m= x = = subjet t λ x = η = () where the effiiets f the st futi a be heked t satisfy: γ (,..., m) = (4;,,... ) γ (, m) = γ (, m) = 3 πf ( f ) m ( f ) = ( f ) df ( f )s( πf ) df m ( f )si( πf ) df (3) with ( 4;,,..., ) detig the multimial effiiet equal t 4!/(!!...!). Based this apprah, stadard umerial ptimizati ruties a be used t slve the prblem frmulati i (). As will be shw i the ext seti, it is als fud that fr a pratial rage f badwidth upaies, it suffiiet t use mre tha fur r five PSWF s i the expasi (), whih makes the umerial sluti f the ptimizati prblem mputatially effiiet. 5 umerial Results T illustrate the C-DLL ptimized pulse desig methdlgy, we w preset sme vel hip wavefrms ad mpare their perfrmae t the vetial pulses disussed i Seti 3. The desig apprah fr btaiig these ptimized hip pulses is based maitaiig a give badwidth upay that is typially the same as the best perfrmig vetial pulse ad the slvig the ptimizati prblem () util a feasible sluti that miimizes the st futi F is fud. Fr the umerial results, tw typial values fr ibad pwer badwidth measures are adpted, 99% ad 99.9%, whih rrespd t ut-f-bad η fatrs f % ad.%, respetively. The badwidth figures are take i a rage f /T t 3/T (whih is quite typial i pratie). It shuld als be ted that retagular pulse shape has a partiularly pr perfrmae with respet t the 99% r 99.9% badwidth (i exess f /T ) due t the slw deay i its pwer spetral desity []. As shw i Table 3, with respet t the 99% badwidth, the Half-Sie pulse is fud t have the lwest upay f apprximately. / T, s this value is used t desig the OPT pulse as highlighted abve. Similarly, fr the 99.9% badwidth, the Hammig pulse displays the lwest badwidth f.6 /T ad this is used t desig a sed ptimized pulse OPT. A illustrati f ISS: Issue, Vlume 7, Deember 8

6 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai these pulses is give i Figures ad 3. The expasi effiiets i terms f PSWF s are als give i Table 4. The results f Table 3 shw that the ptimized pulses OPT ad OPT ahieve sme small reduti ( the rder f -5%) i the rmalized trakig errr fatr M / κ. This idiates that, with -heret DLL trakig, the half-sie pulse is quasi-ptimal whe takig it siderati the 99% badwidth upay. O the ther had, the Hammig pulse is lsest t ptimal with respet t the 99.9% measure. Similar lusis were als reprted i [] fr heret DLL trakig shemes. These bservatis are further firmed by the results illustrated i Figs. 4 & 5 whih shw the ptimal perfrmae limit (i terms f miimum ahievable st fatr F = M / κ ) as a futi f ireasig 99% ad 99.9% badwidth upaies, ragig frm /T t /T, ad it a be see that the prximity f the Half-sie ad Hammig pulses t the lwer buds demstrates the quasi-ptimality f these pulses. 6 Clusi The paper addressed the de trakig perfrmae f several hip wavefrms i diret-sequee CDMA sigallig usig -heret delay-lked lps. Based the liear C-DLL mdel, it was shw that the trakig errr variae is a futi f the DLL S-urve slpe at the rigi ad the multiple-aess iterferee pwer, bth f whih have a strg depedey the hip wavefrm pulse shape. Sme mparative results were the give t illustrate the trakig perfrmae f several vetial hip wavefrms, shwig that -retagular pulses utperfrm the retagular e, as was bserved i previus wrk as well. These bservatis mtivated the wrk preseted i the sed part f the paper, where the prblem f desigig pulse shapes that miimize the residual trakig errr was thrughly addressed. A ptimizati apprah with eergy ad badwidth straits, was preseted i frequeydmai usig a expasi f the hip wavefrm i terms f Prlate Spheridal Wave Futis, whih lead t a redued mplexity disrete ptimizati prblem. Examples f ptimized pulses were preseted, ad the results prvided theretial limits the miimum ahievable trakig jitter, shwig that the Half-Sie pulse is quasi-ptimum whe siderig 99% badwidth, while the Hammig pulse is lsest t ptimum whe siderig the 99.9% badwidth measure. Akwledgmet This wrk was supprted by Kig Fahd Uiversity f Petrleum & Mierals, Dhahra, Saudi Arabia. Referees [] M.K. Sim, J.K. Omura, R.A Shltz ad B.K. Levitt, Spread Spetrum Cmmuiatis Hadbk, Rev. Ed., MGraw-Hill, [] A.J. Viterbi, CDMA: Priiples f Spread Spetrum Cmmuiati, Addis-Wesley, 995. [3] R. Ajaria ad R. Wyrwas, The Effet f hip wavefrms CDMA systems i multipath fadig isy haels, Pr. IEEE VTS, pp , May 99. [4] P.I. Dallas ad F.. Pavlidu, Ivative hip wavefrms i mirellular DS/CDMA paket mbile radi, IEEE Tras. Cmm., Vl.44,., pp.43-46, v [5] Y. Huag ad T-S. g, A DS-CDMA system usig dispreadig sequees weighted by adjustable hip wavefrms, IEEE Tras. Cmm., Vl.47, pp , De [6] M. El-Tarhui ad M. Ladlsi, "The effet f hip wavefrm shapig the syhrizati perfrmae f DS-CDMA sigals," Pr. IEEE VTC, Vl. 3, pp , May 999. [7] S. Thayapara, T.-S. g, ad J. Wag, "Halfsie ad triagular despreadig hip wavefrms fr heret delay-lked trakig i DS/SS systems," IEEE Tra Cmm, vl. 48, pp ,. [8] C. Lee et al, Aalysis f deteti perfrmae f a eergy detetr fr several hip wavefrms i DS/SS mmuiati,'' IEEE MILCOM, Vl., pp. 3-7,, [9] X. Wu, C. Lig, ad H. Xiag, "Despreadig hip wavefrm desig fr heret delaylked trakig i DS/SS systems," Pr. IEEE ICC, Vl., pp , May. [] M. A. Ladlsi, Miimisati f timig jitter i CDMA de trakig, IEE Eletris Letters, vl. 4, pp , 4. [] M.A. Ladlsi, Perfrmae Limits i DS- CDMA Timig Aquisiti, IEEE Tras. Wireless Cmm. Vl. 6,. 9, pp , Sept. 7 [] D. Splepia ad H. Pllak, Prlate spheridal wave futis, Furier aalysis ad uertaity, Bell Sys. Teh. Jur., Vl. 4,., pp , Jauary 96. ISS: Issue, Vlume 7, Deember 8

7 WSEAS TRASACTIOS COMMUICATIOS Mhamed Ada Ladlsi, Gaiyu B. Hussai H(f) Optimized Chip Wavefrm 99% Badwidth OPT Half Sie Cst Futi F Optimal Perfrmae limit 99.9% Badwidth Measure Optimal Hammig Raised Csie /T Fig. : Illustrati f Optimized Chip Pulse OPT rmalized 99.9% Badwidth BT Fig. 5: Relative perfrmae w.r.t miimum ahievable limit (fr 99.9% badwidth measure) H(f) Optimized Chip Wavefrm 99.9% Badwidth OPT Hammig PULSE 99% BT 99.9% BT M / κ Half-Sie Triagular Raised-Csie Hammig /T Fig. 3: Illustrati f ptimized hip pulse OPT OPT OPT..6.6 Table 3: Cmparative trakig perfrmae f ptimized hip pulses. Cst Futi F Optimal Perfrmae limit 99% Badwidth Measure Optimal Half Sie Triagular Hammig Raised Csie PULSE x x x x 4 6 OPT.997E -.654E-.9E-.33E- OPT.999E -.445E-.663E-3.873E-4 Table 4: PSWF expasi effiiets f the ptimized wavefrm pulses OPT ad OPT rmalized 99% Badwidth BT Fig. 4: Relative perfrmae w.r.t miimum ahievable limit (fr 99% badwidth measure) ISS: Issue, Vlume 7, Deember 8