An Optimization of Gaussian UWB Pulses

Transcription

1 0 h Ieraioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romaia, May 7-9, 00 A Opimizaio of Gaussia UWB Pulses Adria POPA "Gheorghe Asachi" Techical Uiversiy of Iaşi Bd.Carol No., RO0.506 Iaşi, Romaia padirovs@gmail.com Absrac UWB is a ew ieresig echology for wireless commuicaios. I ca replace radiioally carrier-based radio rasmissio by pulse-based rasmissio usig ulrawide bad frequecy bu a a very low eergy. A impora aspec of research i his domai is o fid a pulse wih a opimal shape, whose power specral desiy respecs ad bes fis emissio limiaio mask imposed by FCC. I his paper we review commo used Gaussia pulses ad is derivaives ad he ifluece of shape facor, fidig a opimal specific value for each derivaive. Nex, we search o obai possible beer pulse shapes as liear combiaios of Gaussia derivaives. Older sudies refer i oe case o he same shape facor for all derivaives ad i oher case o higher facor for firs derivaive ad smaller shape facors for subseque derivaives. Our ew idea is o use Gaussia derivaives, each wih is specific opimal shape facor ad o use a "rial ad error" algorihm o obai a liear combiaio pulse wih beer performace. because o i deped he performaces of a UWB sysem, like efficiely use of permied emissio power, coexisece wih oher radio commuicaio sysems ad a simple circui implemeaio. Oe of he fudameal challeges is o maximize he radiaed eergy of he pulse while he power specral desiy complies wih he specral FCC mask. The FCC regulaed he use of UWB devices respecig emissio limi values as depiced i Fig. [4]. Due o he exremely low emissio level currely allowed, comparable wih uieded emissio (FCC Par 5), UWB sysems are suied for shor-rage ad idoor applicaios. UWB operaes bes over shor disace of abou -3 meers deliverig daa speeds of 480 Mbps. As disace icreases, speed decreases, bu a 0 meers sill reaches or exceeds 00 Mbps. Idex Terms Gaussia Moocycle; Shape facor; Trial ad error; UWB; Wireless LAN I. INTRODUCTION UWB (Ulra-Wide-Bad) wireless rasmissio is based o impulse radio ad ca provide very high daa raes over shor disaces. Is radiioal applicaio was i ocooperaive radar. UWB device by defiiio has a badwih equal or greaer ha 0% of he ceer frequecy or a badwidh equal or more ha 500MHz. Sice FCC (Federal Commuicaios Commissio) auhorized i 00 he ulicesed use i he domai GHz, UWB became very ieresig for commercial developme. High daa rae UWB ca eable wireless moiors, efficie rasfer of daa bewee compuer i a Persoal Wireless Area Nework, from digial camcorders, rasfer of files amog cell phoe ad home mulimedia devices, ad oher radio daa commuicaio over shor disaces [7]. Tradiioal wireless echologies use radio sie waves ha provide "coiual" rasmissio a a specific frequecy. UWB radio sysem is associaed wih is impulse-based, carrier-free, ime-domai radio sysem forma ha rasmis very shor UWB pulse sigal rais (sub-aosecod pulse widh) wihou usig ay coiuous siusoidal wave carriers. For pulse-based UWB sysem, a exremely shor pulse spreads is sigal power over a very wide frequecy specrum ( GHz) where he duy cycle of UWB pulse rai ca be as low as % [3]. The pulses are emied i a rhyhm uique o each rasmier. The receiver mus kow he rasmier's rhyhm sigaure or pulse sequece o "kow how o lise" for he daa beig rasmied [5]. The sudy of he shape of base pulse is fudameal Figure UWB idoor emissio mask reglemeed by FCC. Sice he ulra-shor pulses are relaively easy o geerae oly wih aalog compoes, he Gaussia Moocycle ad his derivaives are commoly used for UWB. I his paper, we sudy he specral properies of UWB pulses. Secio II aalyzes he power specral desiy of he Gaussia moocycle ad he i Secio III oe exeds he sudy o higher-order derivaives of he Gaussia pulse ad he ifluece of shape facor σ ad his opimal values. I Secio IV, we sudy he pulses obaied by liear combiaios of Gaussia derivaives ad a algorihm o fid opimal coefficies. We propose a ew se of basic pulses, havig a shape facor specific for each derivaive. The coclusio regardig he performace of he pulse combiaio we obaied is preseed i Secio V. 56

2 0 h Ieraioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romaia, May 7-9, 00 II. GAUSSIAN MONOCYCLE By far he mos popular pulse shapes discussed i UWB commuicaio lieraure are he Gaussia pulse ad is derivaives, as hey are easy o describe ad work wih. The basic Gaussia pulse is described aalyically as: σ A = e () π σ If a Gaussia pulse is rasmied, due o he derivaive characerisics of he aea, he oupu of he rasmier aea ca be modeled by he firs derivaive of he Gaussia pulse []. Therefore, he pulse radiaed is give by he firs derivaive of he Gaussia pulse, called a moocycle : x () σ A ( = e () 3 π σ The waveform of he pulse is preseed i Fig. for a value of shape facor σ = 0.s, ad is correspodig specrum i Fig. 3. Ampliude [V] Time [s] Figure The waveform of a Gaussia moocycle Figure 3 Power specral desiy of Gaussia moocycle f peak=.6ghz B[0dB]=4.3GHz Sricly speakig, he duraio of he Gaussia pulse ad all of is derivaives is ifiie. Here, we defie he pulse widh, Tp, as he ierval i which 99.99% of he eergy of he pulse is coaied. Usig his defiiio, i ca be show ha Tp 7σ for he firs derivaive of he Gaussia pulse.[] The possibiliy for uig PSD specrum i order o respec ad fi he mask FCC is he choice of he shape facor σ. We observe ha decreasig he value of σ i ime domai he duraio of pulse Tp is shorer, ad leads i frequecy domai he specrum o migrae o higher frequecies. For example, whe value of σ is 0. s, Tp=0.84 s ad frequecy a maxim PSD is f_peak=.4 GHz (coiuous curve i Fig.4); whe σ decreases a s, Tp become shorer, 0.56 s ad f_peak = GHz, higher (dashed curve) ; for σ=0.04 s, Tp=0.8s ad f_peak moves o higher frequecies, f_peak=4ghz (doed curve). y g sigma =0.04s f peak =4GHz --- sigma =0.08s f peak =GHz - sigma =0.s f peak =.34GHz Figure 4 PSD of Gaussia moocycle for hree values of σ As oe ca see, i is clear ha he PSD of he firs derivaive pulse does o mee he FCC requiremes, o maer wha value of he pulse widh is used [9]. Therefore, a ew pulse shape mus be foud ha saisfies he FCC emissio requiremes. Oe possibiliy is o shif he ceer frequecy ad adjus he badwidh, so ha he requiremes are me. This could be doe by modulaig he moocycle wih a siusoid o shif he ceer frequecy ad by varyig he values of σ. For example, for a pulse widh Tp = 0.3 s, by shifig he ceer frequecy of he moocycles by 3 GHz, he PSD will fall compleely wihi he specral mask. Impulse UWB, however, is a carrierless sysem; modulaio will icrease he cos ad complexiy. Therefore, aleraive approaches are required for obaiig a pulse shape which saisfies he FCC mask []. I he ime domai, he high-order derivaives of he Gaussia pulse resemble siusoids modulaed by a Gaussia pulse-shaped evelope. As he order of he derivaive icreases, he umber of zero crossigs i ime also icreases; more zero crossigs i he same pulse widh correspod o a higher carrier frequecy siusoid modulaed by a equivale Gaussia evelope. A Gaussia moocycle has a sigle zero crossig. Furher derivaives yield addiioal zero crossigs, oe addiioal zero crossig for each addiioal derivaive [9]. These observaios lead o cosiderig higher-order derivaives of he Gaussia pulse as cadidaes for UWB rasmissio. Specifically, by choosig he order of he derivaive ad a suiable pulse widh, we ca fid a pulse ha saisfies he FCC s mask []. 57

3 0 h Ieraioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romaia, May 7-9, 00 III. HIGHER ORDER DERIVATIVES We ivesigae i he sequel he pulses obaied as derivaives of he basic Gaussia pulse. The equaio of -h derivaive pulse is give by: d A σ x ( = e (3) d π σ I ime domai, we observe ha he duraio of pulse remais he same for various high-order derivaives, bu oe ca cosider Tp = 0σ [] for all pulses. The waveforms illusraig several of hese pulses ad heir PSD are preseed i Fig.5 ad 6, respecively. Ampliude [V] h --- 7h -.-. h Time [s] Figure 5 Waveforms for hree derivaives of Gaussia pulse (σ=0.s) PSD [dbm/mhz] - f peak 5 =.6GHz B [0dB] = 4.3GHz -- f peak 7 =.6GHz B [0dB] =.93GHz.-.f peak =.6GHz B [0dB] = 3.4GHz Figure 6 PSD for hree derivaives of Gaussia pulse (σ=0.s) Ieresig, i frequecy domai he Fourier rasforms of hose pulses have a relaively simple expressio, ad heir specrum has a ampliude give by: X (π f σ ) ( f ) = A(π f ) e (4) So, oe ca sudy he specrum by easily compuig he frequecy peak ad he badwidh. The frequecy peak of specrum is give by eq.(5) ' ) ( X ( f ) ) = 0 f M = (5) π σ ( For every derivaive, we chose a paricular value for σ i order o obai a pulse ha maches he FCC s PSD mask as closely as possible. For example, Fig.7 illusraes he resuls of simulaio for he 5h derivaive of he Gaussia pulse for several values of σ. The ploed curve wih a hicker lie is he resul for he opimal value σ 5op = 0.05 s. sigma opim = 0.05 s B[0dB] = 6.58 GHz Figure 7 Opimizaio of σ for he 5 h derivaive pulse I Table I, we summarize he values of he opimal parameer σ, he peak emissio frequecy, ad he 0dB badwidh obaied for firs 5 order derivaives of Gaussia pulses. - order TABLE I. OPTIMAL VALUES FOR DERIVATIVES GAUSSIAN PULSE σ op [ s] f M [Ghz] B 0dB [Ghz] These resuls show ha he pulse widh will be less ha aosecod for all cases, ad he 0dB badwidh is 4.5 GHz or greaer; peak frequecy icreases wih square roo of order of derivaives.the maximum PSD ca be corolled by chagig he value of he ampliude A of he pulse [8]. Noe ha he derivaive operaio could be implemeed as highpass filerig; o rasmi he fifh derivaive over he air, he Gaussia pulse mus be filered o he fourh-order derivaive []. 58

4 0 h Ieraioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romaia, May 7-9, 00 IV. OPTIMAL COMBINATION OF PULSES I order o obai UWB pulse wih a beer performace, we ivesigaed liear combiaios of Gaussia derivaives pulse. Le us cosider a base pulses composed of firs 5 derivaives of Gaussia pulse, each wih a idividual shape facor. The combiaio pulse has he expressio give by eq.(6). 5 = c x ( σ, (6) = The problem is o fid he opimal se of coefficies S={c },so ha he resulig combiaio pulse respecs ad bes fis FCC requiremes []. We propose a compuerbased mehod by meas of sarig wih radom ses of coefficies ad esig wih a rial-ad-error procedure described as follows: Sep. Iiialize he radom umber geeraor. Sep. Geerae a radom se of coefficies S. Sep 3.Check if he PSD of he liear combiaio obaied wih coefficies ad base pulses saisfies he emissio limis. Sep 4. If he emissio limis are o me, go o sep ad geerae aoher combiaio. Sep 5. If he emissio mask are me ad his is he firs se verifyig he limis, iiiae he opimal se BS = S Sep 6. If he emissio mask is me ad already exiss a opimal se, compare acual valid se S wih opimal se BS. If S have a beer fi of a mask, i.e. he sum of PSD (expressed i mw/hz) a all frequecies is greaer, redefie BS=S. Sep 7. Repea his procedure goig o sep for some umber of "rial" cycles, i order o obai a ew possible improved pulse. Sep 8. Afer ha umber of idepede radom searches (because i sep he radom geeraor is reiiialized), algorihm sops ad he curre BS is he opimal foud. If oe rus his algorihm for a sufficiely big umber of "rials", he resuls coverge ad oe obais a bes resul possible. Our resuls are obaied by ruig he algorihm wih a 000 umber of idepede "rials". We ru his algorihm for hree cases, as a fucio of choice of he se of shape facors σ of each based pulse for combiaio. A. The case of he same shape facor σ for all derivaives. I his case we cosider he pulses derivaives havig same shape facor σ. The resul for a value of σ = 0.s is preseed i Fig.8. PS= 7.50e-06 mw Figure 8 PSD of opimal combiaio for same σ =0.s As we observe, FCC limis are respeced a all frequecies, bu he fi of he mask is o beer, excep abou 4 GHz, herefore ouside his bad power is o efficiely used. The compliace wih mask guaraeed by he rial-aderror procedure is obaied a he price of a reduced efficiecy[]. B. The case of differe shape facors. Improved performace ca be achieved by adopig differe values for compoe derivaive pulses. Cosider a secod se of a values characerized by a higher value of σ (0.4 s) for he firs derivaive ad smaller values (0.08 s) for he higher derivaives []. The ew values improve he performace of he rialad-error procedure, leadig o a PSD ha is quie close o he arge for frequecies up o abou 8 GHz, as is depiced i Fig.9. PS=.99e-04 mw Figure 9 PSD of combiaio σ=0.4s ad σ-5=0.08s This PSD achieves a beer approximaio of he mask a higher frequecies ha he coiuous oe, haks o he larger badwidh of higher derivaives. Noe ha he selecio of a relaively high a for he firs derivaive improves efficiecy i power uilizaio a low frequecies. A upper boud for a is give, however, by waveform duraio, as deermied by he chip ime []. 59

5 0 h Ieraioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romaia, May 7-9, 00 C. The case of opimized shape facors Regardig he discussio i Secio, we propose he idea o cosider as base pulses for he combiaio of Gaussia derivaives, pulses havig is specific opimized shape facor σ op (cf. Table ). The PSD for hose opimized pulses is depiced below i Fig Figure 0 PSD of σ-opimal Gaussia derivaives pulses Wih hose based pulses, we cosider a combiaio pulse havig he expressio deermied by eq.(7). 5 = c x ( σ op, (7) = Applyig opimizaio algorihm will resul a opimal pulse combiaio wih PSD preseed i Fig Figure Opimal Pulse Combiaio PS=.73e-04 mw As we see, his combiaio pulse has very good properies i fiig he mask ad herefore usig maximum allowed emissio power. For a objecive compariso ad correc performace evaluaio of his pulse, i each figure is displayed he parameer used i algorihm "rial ad error", PS, wha is "Specral Power" obaied summarizig umerical values of PSD (expressed i mw/hz), a all sampled frequecies. The value obaied here is he bes, PS = mw. I he case of usig he same values for σ, he obaied value is poor PS= mw ad i case of higher σ for firs derivaive ad smaller for higher order derivaives, PS is beer, PS= mw. V. CONCLUSION UWB echology is ew ad subjec o more improvemes. Sigal pulses wih duraio of he order of s are easy o geerae usig a aalogue circui. Gaussia moocycle ad his derivaives is a commoly used waveform for pulses i UWB. I is easy o geerae, bu does o have eough good performace. Oe of he problems is o obai a PSD ha beer fis limied emissio mask. Higher order derivaives have bee sudied ad proved o brig beer performace, bu o eough. For idoor applicaio he fifh-order derivaive is currely a choice for implemeaio. Now, we ry a algorihm o obai a pulse from a liear combiaio of firs 5 derivaives of he Gaussia pulse. Usig his mehod wih same shape facor for all derivaives, ad ex a old implemeaio wih small values for moocycle ad high values for higher derivaives leads o UWB pulses wih PSD ha respec ad show improved fiig of he mask, bu o eough good above 4, respecively 7 GHz. We proposed he idea o use as base pulses for combiaio he Gaussia derivaives wih σ values idividually opimized by he crierio o obai maximum badwidh. Wih hese pulses, he opimal combiaio obaied by a "rial ad error" algorihm leads o a pulse wih beer performace ha respecs ad beer fi he mask. The diagram illusraes ha heses pulses have a PSD well approximaig he mask a all frequecies, herefore hey efficiely use he available badwidh ad power. This resul from he diagram is argued by he compued parameer PS (Specral Power), as he sum of values of power specral desiy, he combiaio pulse havig i his case he bes value. Numerical resuls ad diagrams are obaied by meas of a simulaio program wrie by us i MATLAB. For already discussed pulses i echical lieraure we obai he same values ad diagrams as for hose foud i he lieraure. This proves he accuracy of our simulaio program. The resuls for he ew proposed pulses are ieresig ad subjec o furher heoreical aalysis ad pracical implemeaios. REFERENCES [] Maria-Gabriella Di Beedeo, Luca De Nardis, "Tuig UWB sigals by pulse shapig: Towards coex-aware wireless eworks ", i Sigal Processig 86 (006) [] D. K Hogsa Sheg, Philip Orlik, Alexader M. Haimovich, "O he Specral ad Power Requiremes for Ulra-Widebad Trasmissio " [3] Haolu Xie, Xi Wag, Alber Wag, Bi Zhao, Yumei Zhou "Varyig Pulse Widh 5h-Derivaive Gaussia Pulse Geeraor for UWB Trasceivers i CMOS " [4] Federal Commuicaios Commissio, Revisio of Par 5 of he commissio s rules regardig ulra-widebad rasmissio sysems, Firs Repor ad Order. February 4, 00." [5] hp:// [6] hp:// [7] B. Hu, N. C. Beaulieu, Pulse Shapes for Ulrawidebad Commuicaio Sysems, IEEE Tras., July 005 [8] Opperma I., Hämäläie M., Iiai J.: UWB Theory ad Applicaios, Wiley, 004 [9] M. Ghavami, L. Michael, R. Koho, "Ulra Widebad Sigals ad Sysems i Commuicaio Egieerig", Wiley,