On the Probability Density Function and Stability Properties for a Cross-Product Frequency-Locked Loop

Transcription

1 O he Probabiliy Desiy Fucio ad Sabiliy Proeries for a Cross-Produc Frequecy-Locked Loo sug-yu Chiou Saford Uiversiy Palo lo Califoria BOGRPHY sug-yu Chiou is a Ph.D. cadidae i he eroauics ad sroauics Dearme a Saford Uiversiy. He received his B.S. i erosace Egieerig i 998 from amkag Uiversiy aiwa ad his M.S. from Saford i. His research currely focuses o he erformace aalysis ad validaio of erial-aided GPS carrier-rackig loos. He is also lookig io he soluios o he roblem of GPS/WS erformace degradaio caused by ioosheric sciillaio. BSRC he frequecy locked-loo FLL has received ew aeio for moder Global avigaio Saellie Sysems GSS receivers esecially for is erformace uder severe oise ierferece ad high dyamic eviromes. has bee show ha a FLL is more robus o ierferece ad dyamics ha a hase locked-loo PLL. herefore i is beeficial o use a FLL as a fallback rackig loo whe he rimary PLL is uable o maiai carrier rackig i hosile eviromes. Besides rackig i is o also crucial o reserve he Bi Error Rae BER of he daa demodulaio ad robabiliy of losig locks whe receivers oerae a FLL. hese characerisics rely o he robabiliy desiy fucio PDF of he frequecy esimae made by he frequecy discrimiaor i he FLL. However he PDF has o bee deermied as of ye. he urose of his aer is o solve he PDF ad evaluae he sabiliy regio which is esseial for deermiig he saisics of loss of lock. his work a oliear model was develoed ad he corresodig Fokker-Plack equaio FPE for he cross-roduc FLL was derived. he PDF of he frequecy esimae was he solved by alyig he Crak-icolso umerical mehod.. RODUCO he use of a frequecy-locked loo FLL ca be raced back o he 93s [] while a FLL used o be called a auomaic frequecy corol FC loo. he fucio of a FLL i a GSS receiver is o seer he frequecy of he relica carrier close eough o he frequecy of he received carrier such ha a furher daa demodulaio is alicable. sead of a cohere rackig meaig ha he sigal hase is racked by a hase-locked loo PLL a FLL erforms a o-cohere rackig oly. For examle oly he frequecy i he carrier is racked while assumig ha he hase is uiformly disribued over [ ]. hus a FLL ca ull i from a frequecy error much larger ha a PLL does. is believed ha a FLL is more robus i he resece of oise ierferece uder high dyamic eviromes ha a PLL []. he alicaios of a FLL geerally ca be divided i wo caegories i which a FLL ca be used o imrove he iiial rackig of a PLL [4] or used as a backu rackig loo whe he received sigal ower is weak durig a shor eriod [5]. Hece ivesigaig he erformace of a FLL used i a GSS receiver is sigifica ad imora. Research o he liear model aalysis of various yes of FLL s has already bee accomlished [ 3]. he oise erformace of he FLL i he resece of Gaussia oise has bee give i [ 3]. However he robabiliy desiy fucio PDF of a FLL has o bee solved. Obaiig he PDF is crucial o evaluae he bi error rae BER i he demodulaio rocess. Moreover esimaig he robabiliy of losig lock also relies o he PDF of he FLL. he obecive of he rese aer is o develo he oliear model of he cross-roduc FLL evaluae he oliear sabiliy characerisics i he absece of oise ad solve for he PDF i he resece of oise.. BSC OPERO OF HE CROSS- PRODUC FLL

2 Vf his secio would like o review he oeraio of he cross-roduc FLL wihou givig deails of he mahemaical derivaio. For he deails refereces [ 3] are recommeded. Figure shows he cofiguraio of he cross-roduc FLL. he esseial characerisic of his FLL is he fac ha i emloys a cross-roduc frequecy discrimiaor which is a coveioal FLL desig i moder digial basebad imlemeaio []. Exce he cross-roduc discrimiaor he fucioaliy show i Figure is he well kow Cosas receiver [6]. he iu sigal V s is assumed o be a carrier a frequecy modulaed wih differeially ecoded daa ad luses a addiive whie Gaussia oise wih secral desiy. V s is he mulilied by he local relica carrier a frequecy geeraed by he umerically corolled oscillaor CO. he mulilicaios were erformed by he i-hase ad quadraure relica carrier o roduce ad Q chaels resecively. he ad Q chaels are he assed hrough he iegrae-ad-dum filers o furher reec he iu oise rior o erformig he frequecy error deermiaio i he followig se. Before he deails of he frequecy discrimiaor we defie a comlex sigal comosed by he ouus of he iegrae-ad-dum filers k ad Q k as he followig form V k Q. s k k is sigal amliude; ; is he eriod of he iegraio; si x sic x ; x k is he oise erm. Obviously he sig of V f k deeds o he sig of he symbol roduc d k d k. he sig chages ca be removed by erformig he do roduc of curre samle V s k ad he revious samle of V s k [ 3]. he ouu of he do roduc is he used as a decisio feedback o remove he sig chages i Eq.. Here we assume ha he daa has bee wied off by he meas of do roduc oeraio. his assumio is reasoable sice we are solvig for he geeric erformace of a carrier rackig loo. Wihou loss of geeraliies oe ca furher assume ha he iu sigal is a ure siusoidal carrier. he lo of he frequecy discrimiaor assumig oise free is show i Figure. oe ha he frequecy error show i Figure has bee ormalied by mulilyig f by he iegraio ime V s Q 9 o egrae k ad Dum egrae Q k ad Dum CO o Delay Delay Loo Filer F Figure : Cross Produc FLL + V f k sum f Figure : Cross Produc FLL discrimiaor curve he oeraio of he cross-roduc frequecy discrimiaor is he cross roduc of he curre samle of V s k ad he revious samle of V s k. his oeraio is rereseed as [3] V f k Vs k Vs k k Qk Qk k dkdk sic si k d is he daa symbol a he k h iegrae-ad-dum se; k From Figure we see ha ulike he eriodic roery of he hase discrimiaor i a PLL [8] here is o exac eriodiciy i he frequecy discrimiaor. he mai lock oi is a he origi. Oce he frequecy error deviaes far away from he firs ero crossig oi he loo likely sars o lose lock. s ca be see i Figure he characerisic of he discrimiaor is close o liear whe he value f is small. rule-of-humb hreshold o reserve his lieariy assumio is whe [7] f.83. 3

3 lhough he iu sigal is corrued by he Gaussia oise k i Eq. i fac is o a Gaussia disribued oise because of he oliear rocess of he discrimiaor. is show ha k is ero mea ad ucorrelaed i successive samles [3]. o advace he model aalysis oe usually assumes ha k is Gaussia. s a resul give ha he oise a he iu sigal is whie Gaussia oise wih wo-sided secral desiy ad he re-deecio basebad badwidh oe ca obai he mea variace or he secod mome ad ower secral desiy PSD of k as follows [3]. E [ k]. 4 E[ k] S raio. C. 5 C f wo-sided 6 C is he sigal ower o oise ower desiy Wih he assumio of Eq. 3 a secral aalysis ca be erformed i he liear model of he FLL. hus he ormalied rackig error variace of he cross-roduc FLL is [9] age 38 4B f dimesioless 7 4 C C B i H is he oe-sided oise badwidh of he closed loo FLL.. OLER MODEL D HE SBLY OF HE FLL HE BSECE OF OSE -. OLER MODEL Give he block diagram i Figure ad he characerisic of he cross-roduc discrimiaor i Eq. he oliear model rereseig he cross-roduc FLL is show i Figure 3. he addiive oise has he roeries described i Eqs Fs sic si V f + Figure 3: oliear Model of he Cross Produc FLL + Fs is he loo filer rereseed i he Lalace domai ad is he loo gai. oe ha he carrier CO i a FLL does o ac as a iegraor. is simly a meas of coverig a frequecy umber o sie ad cosie of frequecy [9] age 384. herefore he order of he closed-loo is he same as he order of he loo filer. For a oliear aalysis ivesigaig he erformace of he firs order loo suffices he required characerisics of he FLL. herefore i he firs order loo F s s. We ca obai he followig sae equaio describig he closed loo show i Figure 3 as he followig equaio. d d d d. sic si We cosider rimarily frequecy-ram siusoidal ius d so ha which is a cosa i uis of d 8 rad. sec f is ero we say ha loo is usressed. he usressed case he disribuio of he rackig error would be ero mea. f is oero he FLL is dyamically sressed ad he mea of he rackig error would be biased from ero. Le f ad we have ad d d. Perform he chage of variables i ad for Eq. 8 we have he followig goverig equaio of he crossroduc FLL i he domai of he ormalied frequecy error. d d sic si.. 9 Eq. 9 is a sochasic firs order ordiary differeial equaio which fully describes he behavior of he crossroduc FLL. he soluio of i Eq. 9 is of ieres. he ex secio we will show he saisical soluio o

4 d/d d/d Eq. 9. he res of his secio would like o discuss he sabiliy erformace i he absece of oise which is esseial o obai he boudary of he FLL loses lock. -. HE SBLY OF HE FLL HE BSECE OF OSE ivesigae he sysem raecory described i Eq. 9 wihou cosiderig he oise erm. For he firs order loo he loo oise badwidh is [9] B. 4 he iu frequecy ram ca be wrie as f i f i is he frequecy ram iu i H sec. Subsiue Eqs. ad io Eq. 9 wih igorig he oise erm we have d 4B fi sic d si. he sysem raecory of Eq. is show i Figure 4 for he case of a usressed FLL. Oe ca fid ha he sysem is sable whe i reaches a value of for which d d. here are mulile sable ois for he sysem. However he FLL is allowed o be locked oly wihi he mai lobe for a hysical meas of rackig. Oe ca fid ha he sysem moves oward he righ whe d d ad vice versa. herefore here are dyamically sable ois for = - he sysem will reur o he sable ois afer ay erurbaio of i eiher direcio. However for oher sables ois ay erurbaio of i eiher direcio will cause he sysem o move uil i reaches he ex dyamically sable oi Figure 4: Sysem raecory o-sressed lock f i loss 4B We call he dyamically sable oi i he mai lobe lock ad he ex o-dyamically sable oi loss. Oce is larger ha loss he sysem moves farher away from he mai lobe ad sos whe i reaches he ex sable oi. However whe he sysem is dyamically sressed here may be o sable ois beyod loss. Figure 5 illusraes he issue of havig oly wo sable ois lock loss Figure 5: Sysem raecory Dyamically Sressed f.5 i 4B Wih a osiive frequecy ram iu oe ca fid ha oce he sysem will migrae oward ifiiy ad loss ever reurs o ay sable ois. hus loss is he hreshold he FLL sars o lose lock. f he iu frequecy ram icreases agai he whole curve shifs uward such ha lock ad loss move furher oward each oher. his meas ha he FLL is more likely o lose lock uder sroger dyamics. he frequecy ram for he

5 case ha lock ad loss coicide is he maximum allowable frequecy ram for he firs order FLL. y iu beyod he maximum allowable ram frequecy will cause he FLL o be usable. owig he values of loss is crucial for evaluaig he robabiliy of loss of lock i he resece of oise. he robabiliy of loss of lock is defied as P loss B fi. 3 Deermiig he value of Eq. 3 relies o he PDF of which will be rovided i he ex secio. V. HE FOER-PLC EQUO D HE PDF OF HE CROSS-PRODUC FLL his secio a saisical aroach o Eq. 9 will be discussed. Eq. 9 is a sochasic ordiary differeial equaio drive by a Gaussia oise. herefore give he PDF of d d is Gaussia oo. s a resul he comlee soluio of is deermied by is PDF. Sice is Gaussia he rocess described i Eq. 9 is a Markov rocess ad he relaios goverig a PDF of a Markov rocess is give by [8]! wih iiial codiio is he PDF of ; 4 = he i of he h mome of he icreme of he rocess give ha i sared a some value a ime ormalied by he ime icreme as he laer aroaches ero; is Dirac dela fucio. is kow ha for a firs-order sochasic ordiary differeial equaio wih a whie Gaussia drivig fucio he quaiies vaish for greaer ha [8]. ccordigly Eq. 4 becomes he well kow Fokker-Plack equaio FPE as follows 5 order o obai he corresodig FPE for he crossroduc FLL we mus deermie he quaiies ad accordig o Eq. 9. he derivaio is give i he edix ad ad are give i Eqs. -3 ad -4 resecively. addiio o he iiial codiio give i Eq. 5 solvig he FPE eeds wo boudary codiios. We kow ha he oal area uder a PDF should be ad accordigly he wo ails of he PDF aroach ero as he ideede variable aroaches lus ad mius ifiiies. he wo boudary codiios are he defied as follows. he ormaliaio codiio is d ad 6 he symmeric codiio is for all. 7 oe ha he symmeric codiio is a heriage of he ormalied codiio sice he oal area has o be fiie ad herefore he wo ails have o vaish a ifiiies. o furher advace he soluio of he FPE we would like o arameerie he FPE i erms of B ormalied iiial frequecy offse frequecy ram iu f i f i C ad ormalied. Eq. 8 gives he exressio of he fial FPE o be solved. he deails for obaiig Eq. 8 are give i edix B. D ; d ; f ; he ormalied frequecy error; 8 he PDF of he FLL i he domai of he ormalied frequecy error; ; B f

6 ime D sic si ; ; f ; fi 4B he dimesioless dyamic sress; f i he ormalied iiial frequecy offse. Because of he olieariy of he discrimiaor show i he erm D he closed form rereseaio of he seady sae PDF is o achievable ye. umerical mehod o solve he FPE will be discussed i he ex secio. V. UMERCL RESULS BY HE CR- COLSO MEHOD o solve he arial differeial equaio of Eq. 8 coduced he Crak-icolso mehod []. he corresodig differece equaio o Eq. 8 is herefore i i D i D i i i D D i i i i i i i i i 9 he subscris i deoe saial domai of ad he suerscris deoe ime. oe ha D is ime ideede ad here is o suerscri for D. Figure 6 shows he ime ad saial meshes for Eq. 9. Give i a ime solvig Eq. 9 will give he PDF a ime for all. he iiial codiio for he umerical soluio is he roecker dela fucio which saisfies boh of he ormalied codiio ad he boudary codiios a he iiial sae. he symmeric boudary codiio is aroximaed by usig a absorbig boudary codiio. is reasoable o assume ha he ails of he PDF aroach ero wihi a fiie rage of. Oce he accuracy of he soluio is me give a absorbig boudary codiio he soluio was claimed o be valid. he aforemeioed accuracy was calculaed from he differece of he oal area uder he solved PDF o. oher words he ormalied boudary codiio i is used as a meric for claimig a successful soluio. his aer he accuracy requireme was se o be e-. he umber of he grids i direcio was 7 or more. uiively as more grids used for he mesh more accurae resuls will be. However he amou of memory o he comuer is he grid umber. he required umber of grids also deeds o C. s execed he PDF for a lower C has a wider rage i direcio. herefore o reserve he required accuracy more grids are eeded o cover he wider rage of. he ime se sie also deeds o he grid sie i. he deails of he rade off bewee he grid sie i ad he ime se sie are give i []. For calculaios of D oe should kow ha he defiiio of he sic fucio i Malab differs from wha was defied i Eq.. Malab he sic fucio is defied si x sic x. x Figure 7 shows oe examle of he PDF soluio. Figure 7 here is a iiial frequecy offse assumed. he iiial imulse is a. 5. he gree curve rereses he PDF a he half of he evolvig ime ad he red curve is he PDF a he ed of he evolvig ime. he coicidig of he gree ad red curves shows ha he FLL has reached he seady sae. Figure 7 also shows ha he seady sae error is ero as execed for he erformace of a firs order loo wih a iiial imulse iu. Figure 8 shows aoher examle for he case ha he PDF is dyamically sressed. he fial PDF is ceered a he seady sae value of he frequecy error. iiial codiios absorbig boudary codiios sace Figure 6: he ime ad saial meshes for umerical soluio of FPE

7 Probabiliy desiy fucio Probabiliy desiy fucio Probabiliy desiy fucio = = half = ed.5 x Figure 7: PDF of he FLL o-sressed C / db H B H f.5 f i i Figure 9: PDF of he FLL Weak Sigal C / db H B H fi fi H = = half = ed Sice here is a seady sae soluio of he PDF for he FLL we ca eiae he ime deedece of he PDF i Eq. 8. O he oher had he lef had side of Eq. 8 becomes ero ad he PDF o he righ had side of Eq. 8 does o have he variable. s a resul Eq. 8 becomes a secod order oliear ordiary differeial equaio ODE show i Eq Figure 8: PDF of he FLL Dyamically-sressed V. DSCUSSOS C / db H B H fi.5 fi 3H Oe may o obviously see he correlaio bewee he sysem raecory show i Figure 4 ad he PDF show i Figure 7. o illusrae his correlaio a PDF wih C = db-h is loed i Figure 9. From he case of weak sigal show i Figure 9 we see wo dees i he PDF for =.5. his agai reveals he fac ha he loss ois i Figure 4 are o dyamically sable ois. he sysem has he lowes ossibiliy o say a hese wo ois. he wavele behavior i Figure 9 also rereses he same behavior i Figure 4. he sysem has relaive lower ossibiliies comarig o is adace ois o say a hose o-dyamically sable ois. d ; D ; he soluio of Eq. is he fial sasho of he imedeede soluios obaied from solvig Eq. 8. simle fiie differece mehod was alied o solve Eq. for he seady sae PDF. Figure shows he soluios from boh of he ODE Eq. ad he PDE Eq.8. he soluio of he ODE furher verified he soluio of he PDE. herefore i is rue ha he seady sae PDF of he FLL ca be direcly solved from he ODE i Eq. wihou solvig he PDE i Eq. 8 by he Crak-icolso mehod. However wihou he rocess of ivesigaig he soluio of he PDE we ca o coclude he fac ha he seady sae soluio exiss. f oe is ieresed i he seady sae PDF of he FLL oe ca direcly solve he ODE i Eq.. However if he rasie roeries are of ieres solvig he PDE is ecessary.

8 Probabiliy desiy fucio Probabiliy ODE Soluio PDE Soluio robabiliy of exceedig he liear hreshold would be larger ha wha i is show i Figure. - Probabiliy of exceedig he liear hreshold FLL PLL ormalied frequecy error df*i Figure : PDF of he FLL ODE ad PDE Soluios C / db H B H fi fi H Oce he PDF of he FLL is available oe ca for examle esimae he robabiliy of exceedig he liear hreshold give i Eq. 3. Figure shows he curve of his robabiliy versus differe C wih fixed oise badwidh ad he iegraio ime. he curve was obaied by calculaig he ail area of he PDF beyod he liear hreshold defied i Eq. 3 for he FLL. Figure also shows he robabiliy for a Cosas PLL. Sice he PDF of he PLL has bee solved [8 ] his robabiliy ca be evaluaed by usig he closed form soluio. oe ha he PDF of he PLL give i [8 ] mus be modified o accou for he use of a Cosas loo. he modified PDF of he PLL ca be foud i [5] or [] age 74. he liear hreshold of he PLL was se o be 5 degrees [7]. herefore he red curve i Figure was he ail area beyod he 5 degree hreshold. oe ha he hase error variace used i he PDF of he PLL cosidered oly he hermal oise wihou a squarig loss erm [3] age 9 bu o cosidered he hase error variace wih he squarig loss erm [9] age 37. s see i Figure he robabiliy of exceedig he liear hreshold for he FLL is smaller ha ha for he Cosas PLL. Figure suggess ha if he PLL does o maiai lock due o low C oe may swich o use FLL wih accuracy degradaios i carrier measuremes. Figure emhasies he relaive differece bewee he wo rackig loos bu o he absolue values read from he figure. he reaso is ha he curves i Figure cosidered rackig error due o hermal oise oly. However he rackig error is also affeced by receiver clock dyamics saellie clock dyamics laform vibraio imacs o he local oscillaor ad he acceleraio sesiiviy of he local oscillaor [5]. Wih cosiderig all of he above error sources give he same oise badwidh iegraio ime ad C he acual Figure : Probabiliy of Exceedig he Liear hreshold V. COCLUSOS B H.sec C/ db-h he oliear model of he cross-roduc FLL has bee develoed ad he sabiliy aalysis as well as he PDF for he FLL has bee solved for he firs ime. he PDF of he FLL was solved umerically i boh PDE ad ODE aroaches. Wih he PDF oe ca esimae a more accurae Bi Error Rae BER due o imerfec frequecy esimaig of he FLL. Give he sysem raecories of lock ad loss show i Figure 4 5 ad he soluio of he Fokker-Plack equaio oe ca esimae he robabiliy of loss of lock defied i Eq. 3. coclusio his aer successfully solves he robabiliy desiy fucio of he cross-roduc FLL ad rovides he sabiliy aalysis for evaluaig he robabiliy of loss of lock for a GSS receiver usig he FLL. COWLEDGMES he auhor graefully ackowledges he sosor of his research a Saford Uiversiy he F saellie avigaio roduc eam. am also sicerely graeful o Dr. Fracis aali for his advice o his aer ad esecially for his comrehesive commes o he aalysis of he frequecy-locked loo. Secial haks o Professor Per Ege ad Dr. odd Waler of Saford Uiversiy for heir kowledgeable commeary o he aalysis of his aer. For Je-Der Lee i he erosace Comuig Lab a Saford Uiversiy much areciaio is due for his cosrucive commes ad advice o he umerical aroach o a arial differeial equaio.

9 PPEDX From he defiiio of i Eq. 4 he exressio of ca be wrie as [8] E d - By iegraig boh sides of Eq. 9 over he ifiiesimal ierval from o we have d si sic - Recallig ha is whie Gaussia oise of ero mea ad he wo-sided secral desiy give i Eq. 6 we fid ha he firs wo ormalied momes of Eq. - are E si sic -3 C dudv v u C dudv v u E E -4 PPEDX B Rewrie Eq. 5 B- ad are give i Eqs. -3 ad -4. Le. Give he oe-sided oise badwidh i Eq. ad he variace of he ormalied frequecy error i Eq. 7 we ca fid ha f B. B- Defie C ad C. Eq. B- he ca be wrie as ; ; ; d D B-3 D si sic. For he firs order FLL here he dimesioless dyamic sress ca be wrie as [9] age 389 i B f B 4 4 dimesioless. B-4 Wih defied i Eq. B-4 ad f i Eq. 7 ad ca be furher rereseed as f ad B-5. B-6 Fially Eqs. B-3 B-4 B-5 B-6 ad 7 comleely defie he FPE for he cross-roduc FLL i he domai of he ormalied frequecy error. Furhermore he FPE is i erms of B C ormalied iiial frequecy offse f i ad ormalied frequecy ram iu f i.

10 REFERECES. ravis C. uomaic Frequecy Corol Proc. he siue of Radio Egieers Vol. 3 o. Oc aali F.D. FC rackig lgorihms EEE rasacios Commuicaios Vol. COM-3 o ugus aali F.D. oise Performace of a Cross-Produc FC wih Decisio Feedback for DPS Sigals EEE rasacios Commuicaios Vol. COM-34 o March Cah C.R. mrovig Frequecy cquisiio of a Cosas Loo EEE rasacios Commuicaios Vol. COM-5 o December Chiou.Y. Gebre-Egiabher D. ad e al Model alysis o he Performace for a erial ided FLL-ssised-PLL Carrier-rackig Loo i he Presece of oosheric Sciillaio Proc. O M Hayki S. Commuicaio Sysems 4 h Ediio Joh Wiley & Sos c.. 7. Ward P.W. Saellie Sigal cquisiio rackig ad Daa Demodulaio i Udersadig GPS Priciles ad licaios Secod Ediio rech House Washigo DC Vierbi.J. Priciles of Cohere Commuicaio McGraw-Hill c Va Dieredock.J. GPS Receivers i Global Posiioig Sysem: heory ad licaios Vol. Washigo DC Gerald C.F. ad Whealey P.O. lied umerical alysis 5 h ediio ddiso-wesley ikhoov V.. he Oeraio of Phase ad uomaic Frequecy Corol i he Presece of oise uomaio ad Remoe Corol Vol. o Holmes J.. Cohere Sread Secrum Sysems Joh Wiley & Sos c Holmes J.. Sread Secrum Sysems for GSS ad Wireless Commuicaios rech House 7.