Fast Simulation of Digital Phase Detectors Using Importance Sampling

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1 Fat Simulatio of Digital Phae Detector Uig Importace Samplig Fracico A. S. Silva Joé M. N. Leitão Itituto de Telecomuicaçõe - Itituto Superior Técico Liboa, PORTUGAL fea, jleitaog@lx.it.pt Abtract Thi paper addree the developmet of importace amplig techique for the evaluatio of a cla of ope-loop digital phae modulatio receiver. We aume additive white Gauia oie chael, ymbol-by-ymbol detectio ad carrier phae trackig withi the ymbol iterval. For the irreducible error oor aalyi Importace Samplig baed o large deviatio theory wa ued. I the geeral oiy operatig coditio, a adaptive importace amplig approach i adopted. The reultig imulatio algorithm yield practically the ame reult of covetioal Mote Carlo, while exhibitig remarkable imulatio time gai. I. INTRODUCTION Referece [] coidered the developmet of a cla of opeloop receiver uitable for operatio with frequecy at-fadig chael diturbed by fat-fadig coditio ad large carrier phae variatio. The propoed olutio, coitig of a bak of matched tochatic oliear filter (NLF) ad a deciio proceor drive by the filter iovatio procee, perform ymbol-by-ymbol detectio while keepig track of carrier amplitude ad phae withi the ymbol iterval. Performace evaluatio wa carried out by covetioal, time coumig, Mote Carlo (MC) imulatio. To further tet ad refie the approach ad the algorithm developed i [] a adequate ad fat imulatio tool ha to be devied. Thi i the mai purpoe of the work herei reported. Specifically we addre the applicatio of importace amplig (IS) techique to thi cla of problem by focuig o the additive white Gauia oie (AWGN) chael with radom carrier phae drift, which i a particular ceario of []. We how how to derive the receiver error et due to the irreducible error oor iduced by the radom phae drift; from thi iformatio we deig a IS procedure baed o large deviatio theory (LDT). Our referece, for thi purpoe are [], [3] ad [4]. Whe departig from the aymptotic error oor a adaptive importace amplig (AIS) approach, adapted from [5], i adopted. The paper i orgaized a follow: Sectio preet the adopted commuicatio model ad ome IS apect relevat to the imulatio deig. I Sectio 3 we derive the error et for deity biaig uig LDT, ad preet the mai apect behid the AIS techique applied. Implemetatio apect are icluded i thi ectio. I ectio 4 we how the reult of our IS aalyi. Some cocludig remark are preeted i ectio 5. Thi work wa upported by Portuguee program Praxi XXI, uder project /./TIT/583/95. A. Model II. PROBLEM FORMULATION Coider the dicrete bae-bad model of a digitally phaemodulated carrier with a radom phae drift due to ocillator itability or Doppler effect, ad tramitted through a AWGN chael. Thi igal, i for the ymbol iterval [kt ; (k +)T ], repreeted by it N ample give by = exp h j () + ffi i + v; =;:::;N () where () i the digital phae modulatio equece aociated with tramitted ymbol ff, ffi i a dicrete Browia motio 0;ff ffi decribed by ffi = ffi + ffi, with ffi οn, ad v i a complex zero mea white Gauia equece with variace ffv. B. Receiver e -j The receiver tak i to detect the k th tramitted ymbol while keepig track of the radom phae ffi ; = ;:::;N. Thi detector/etimator tructure coit of a bak of matched NLF drive by the ame iput. Matchig to ymbol ff coit of elimiatig the modulatig equece from the obervatio vector (rotatig block i Fig. ). The NLF propa z NLF ( ^ P P, ) ( ^ P() P(), ) Deciio algorithm mi{ } m ^ Figure : Diagram of receiver brach matched to ymbol ff gate recurively probability deitie of phae ffi coditioed o the obervatio z (). Followig [], deitie are repreeted, for thi calar phae proce, a Tikhoov fuctio with mea ^ffi ad cocetratio parameter. Propagatio i accomplihed i two tep. A filterig tep ^ffi F = arcta z() ; + P ff v i ^ffi P z () ; + P ff v co ^ffi P ()

2 F =» z () + P ff v followed by a predictio tep =ffv 4 + z ; co ^ffi P + z ; i ^ffi P = + P (3) ^ffi P + = ^ffi F (4) P + = P F ;ff ffi Deciio i favorable to the brach m that yield the miimum value of ß - the Euclidea metric for the iovatio proce of the repective NLF. All the M NLF are the iitialized with the parameter iterval. See [] for detail. ^ffi P N+ ;P N+ (5) for ext ymbol C. Performace Aemet ad Importace Samplig Fig. i a chematic repreetatio of the commuicatio model coidered i the previou ubectio. I thi figure N k V Nk A k Y Nk S Nk Symbol k MOD Chael detector Figure : Model for the commuicatio ytem A k = ff i the tramitted ymbol with ff fff ; ;ff M g, ad Y Nk ad S Nk are the tramitted ad received igal vector repectively cotaiig N ample each withi ymbol iterval [kt ; (k +)T ]. V Nk i the AWGN vector V Nk = [v ;:::;v N ] k, ad Nk = [ffi ;:::;ffi N ] the phae icremet vector. Sice the ymbol detector propagate deity parameter from the (k ) th ymbol iterval to the k th oe, a obervatio record of ize L S L = [S k ;S k ;:::;S k L+ ] ; L> (6) S = S k ; L = (7) iuece etimate ^A k. Accordigly, we defie the aociated tramitted igal ad diturbace record Y L ad U L = L;V LΛ repectively. The ubiaed MC error probability etimator i where g A k ; ^A k X ^P e = N MC N MC k= g A k ; ^A k i oe if ^A k 6= A k ad zero otherwie, N MC beig the umber of imulatio ru. For i.i.d. error ^P e o var = ffmc = P e ( P e ) =N MC. No-idepedet error icreae variace (ee [6]), with a higher N MC beig required to obtai the oppoite effect. IS imulatio i iteded to reduce high value of ff MC =P e aociated with low P e. For thi, a biaed imulatio deity p Λ ( ) i ued to geerate the record [Y;U] Λ (* mea that [Y;U] Λ i geerated accordig to p Λ ( )) i order to obtai more frequet error - the importat ad otherwie rare evet. Although biaed imulatio deitie lead aturally to biaed error rate etimate, IS provide appropriate correctio for each error evet by mea of the likelihood ratio W ([Y;U] Λ i ) = p ([Y;U] Λ i ) =pλ ([Y;U] Λ i ). Thi yield the ubiaed error rate IS etimator X ^P e Λ = N IS N IS i= E ([Y;U] Λ i ) W ([Y;U]Λ i ) (8) where E ( ) i the idicator fuctio for the error et E. Note that i (8) each imulatio ample i a record. Thi i ot the cae with ^P e uder covetioal MC, ad the differece betwee the approache relie o the type of imulatio - tream imulatio for MC, ad error evet imulatio for IS. Error evet imulatio itroduced i [7] i coidered a the method epecially appropriate for IS i the preece of memory effect. It coit of geeratig idepedet realizatio of U L for a give iformatio patter A L while tetig A k for error occurrece. Bia doe i the imulatio deity will be coditioed o each patter A L belogig to a fiite deumerable et of cofiguratio. Whe miimizig the IS etimator variace for a give N IS, we mut act o W ( ) through p Λ ( ). I our aalyi, we ued two ditict tool for thi purpoe: Large deviatio theory reult - for the error oor aalyi, ff ffi 6=0;ff v =0 ad adaptive importace amplig for the oiy chael - ffv 6=0;ff ffi 6=0. LDT tudie the rate of covergece to zero of certai probabilitie ad it relatiohip with the propertie of the error et E. See for example [], [4] ad [3]. We derive the error et tructure i the pace of radom phae icremet, deoted by D. Outide the error oor limit aalyi, ffffi 6=0;ff v 6= 0, for which the error et i ukow, we proceed with a tochatic miimizatio techique for ffis - AIS - uig iformatio from the deity biaig i D. III. IMPORTANCE SAMPLING DESIGN From model aalyi, we foud that evet imulatio with L = i a adequate procedure. Thi require appropriate imulatio of the receiver iitializatio i order to model the impact of the (k ) th deciio o etimate ^A k. A. Error et aalyi No-liear recurio i equatio () to (5) alog with the deciio algorithm, preclude i geeral the error et aalyi. If we retrict our aalyi to D, (ffffi 6=0, ff v =0), we obtai a impler model that i aalytically tractable. For implicity we derive here the error et for the biary cae (M =). The filter equatio i the error oor are: ^ffi F =arg z () = ffi F = ^ffi P + = ^ffi F P + = P ff ffi : I the receiver, brach t deciio metric coditioed o tramitted ff become (idex t refer to target - the ymbol

3 ff t which i to be detected itead of ff ). ß tj = e j( () (t) +ffi) P P (t) j ^ffi P(t) e (9) = 8 6 Q Q where P P (t) = exp ff ffi = 4 C ^ffi P (t) = () (t) + ffi ; > (0) ^ffi P (t) = I ijj + ffi 0 : () ^ffi P (t) Recall that i the receiver iitializatio for all brache. P (t) I the error oor, ^ffi i the um of the radom phae ffi 0 (previou to ffi ) with a error term I ijj = (j) N (i) N reultig from the (k ) th deciio feedback - electio of brach i give the tramitted ymbol ff j. From (9) we get the probability of error coditioed o tramiio of ymbol ff ad iitializatio error I ijj P Φ ß j >ß tj Ψ = P (co ffi I ijj + where < co f tj + ffi I ijj + f tj = () (t) = = co ffi () co f tj ) + ffi (3) () (t) ;=;:::;N f tj = () (t) : (4) We ow defie Etj;I D ijj = Φ N Ψ R N : ß j >ß tj a the error et i D coditioed o tramiio of ymbol ff ad iitializatio error I ijj. Equatio ß j = ß tj defiig the D ijj ca ot be olved i geeral. However we may a the ifiite et of deumerable olutio atifyig co ffi I ijj =co f tj + ffi I ijj (5) co ffi =co f tj + ffi ; =;:::;N: (6) There i a fiite collectio of ß cotaiig oly N elemet obtaied by the ß f N [ ß; ß] [ ß; ß]g. Ay igle elemet ß allow the derivatio of the remaiig N elemet ad alo of the ymmetry ceter of the error et, which we deigate by C N. We are able to idetify a poit ß for each oe of the N quadrat Q i wrt to C N. I geeral C N doe ot coicide with the origi of D. A a example of uch a error et, we how i Fig. 3 a diagram for N =obtaied by radom geeratio of ample i R. The olutio ß = f ; ; 3 ; 4 g i alo repreeted. The error regio preet a periodic tructure geerally o-coected ad extedig all over D Q Figure 3: Error et for N=. I =0, f ==3ß ad f = ß=3 B. Large Deviatio ad Biaig Strategy Havig idetified Etj;I D ijj D, with N D ditributed accordig to the imulatio deity ψ! N ψ! p ( N )= p exp ffi =(ff ffi ) ; ßffffi = we mut ow check the coditio for the optimal biaig of p ( N ). Refer to [] - Defiitio, of miimum rate poit ad domiatig poit. LDT aure the aymptotic efficiecy of the expoetial twited imulatio ditributio uig the uique domiatig poit ν of E. Sice p ( N ) i Gauia, thi would reult i the tralatio of it mea to the domiatig poit ν, iteded to eure a appropriate coverage of E tj;i D ijj. I thi cae it i ot poible to obtai a poit D ijj ad a half plae H(ν) (taget to the level of p(ν) i ν) uch that Etj;I D ijj ρ H(ν), due to the periodicity of the olutio ß j = ß tj. Whe there i o domiatig poit, Theorem i [] tate the coditio for the ue of a et of poit fν ;:::ν m g for the mea tralatio of a fiite umber of term that will cotitute the biaed deity. Thi reult i p Λ ( ) beig a mixture of Gauia term for appropriate coverage of Etj;I D ijj. The et fν ;:::ν m g mut cotai at leat all the miimum rate poit of E (ee alo Def i []) ad other poit, which, whe ued a bia vector, will improve the coverage of E with the biaed deity p Λ ( ). The miimum rate poit defiitio relie i the LD rate fuctio of p ( N ) Q 4 N = I ( N )=P ffi ffffi : (7) It i the poit ν for which I(ν) =I(E) = if N E I( N ), where I(E) i the Cramér traform (ee []) of E. The miimum rate poit jut miimize the Euclidea orm of I our cae, uig the miimum rate poit, would ot be ufficiet to cover E by the ame reao that determied the oexitece of the domiatig poit. However, ice E

4 how a ymmetric patter wrt C N,weueE = [ i= N E Q i, with E Qi = Etj;I D ijj Q i, ad the we earch for the importat poit o each Qi. Thi approximate the eparate etimatio of the probability mae of iteret for each error ubet E Qi ρ Q i followed by additio to fid the ma o E. The miimum rate poit wrt each et E Qi may eve be domiatig poit of the elemetary et E Qi. Poit i the olutio ß are ued to tart a quadratwie miimizatio of the Euclidea orm of N ubject to the cotrait N E Qi. At the ed of the earch, our et of biaig poit i Etj;I D ijj Φ ν Ψ ν = ;:::; νn. Due to the big umber of olutio, we elected for imulatio biaig oly the N m olutio with the maller Euclidea orm. We mut alo exclude from the error et the quadrat correpodig to the elimiated olutio. Thi prevet very rare ample fallig i thoe remote egmet of Etj;I D ijj, from icreaig the variace of Pe Λ. We ever oticed i our tet the egligible expected uderetimatio for P e. We defer the preetatio of the biaed deity to ubectio III.D. C. Deity Biaig uig AIS Coider ow the product pace D V I, with V beig the N dimeioal oie ample pace ad I the -dimeioal pace of the iitializatio phae error. We adopt the imulatio of the iitializatio error, ice keepig it characteritic i the error oor, would lead here to a uderetimatio of the error probability. We cocluded that ^ffi P i approximately Gauia it parameter beig eaily etimated i a hort imulatio preamble. A for the iitializatio parameter P, we foud it much table, but with a deity imilar to a expoetial. We decided to keep it cotat with it value i the error oor that i P = P exp( ffffi. =) Optimizatio of IS coit ow of biaig i the product pace D V I. For miimizatio of ffis we ue a tochatic earch, becaue we have o iformatio about the error et. The error et i ow E DV I D V I. We ue a parametric AIS techique adapted from that propoed i [5]. We etimate the coditioal mea E f( N;V N ;I) j ( N;V N ;I) E DV IΨ. The error et i coditioed o the tramitted ymbol. Optimizatio mut yield a multiple bia olutio that will cotitute the bia for deity p Λ ( N;V N ;I). We avoid repeatig here the detail i [5]. The propoed etimatio cycle i icreaigly repeated i our cae, a ffv icreae while ffffi i kept cotat. The biaed deity i preeted i the ext ubectio for M>. The major modificatio we have doe to the techique propoed i [5], are the tartig poit for the tochatic optimizatio procedure, ad a bia iolatio trategy i order to prevet the optimizatio to ed o a igle global extremum becaue thi would lead to a poor coverage of E DV I. We ued a tartig poit the optimized biae ν ad o bia at all for V N ad I. Thi horte the time required for tartig the algorithm. Quadrat eparatio i D i eetial to keep the differet bia term eparated durig optimizatio. D. Implemetatio Apect I the error oor, I ijj deped o the reult of etimate ^A k. Our tet howed o differece betwee IS imulatio that ued oly the correct iitializatio I ijj = I jjj =0 ad MC. The correct iitializatio I jjj happe aturally almot all the time, ad it ecod order impact i egligible for all our etimate rage. I fact we coducted tet with all the poible value for I ijj, their a priori probability P I ijj beig etimated recurively, ad the differece were egligible. I coequece we ued alway I jjj i the error oor. For the oiy chael, we ued the ame approach but with I jjj replaced by I N (0; I ) a explaied before with I etimated i a hort preamble due to it depedece from ff ffi ad ff v. We coidered util ow biary igalig (M =). With a M-ary igalig cheme, the error et coditioed o ff i the uio E = M[ i= i6= E ij Thi reder IS biaig uboptimal becaue a bia term iteded ij may reult i a overbiaed term for E jj, with E ij E jj 6= ;. To mitigate thi ource of o-optimality, we itroduce aother level of multiple biaig i our IS imulatio. Thi overcome the poibility of gettig big jump of W ( ) due to o-error ample (wrt E ij ) fallig i E jj whe E ij i targeted. Our biaed deity i the a Gauia mixture of (M ) N m term for appropriate target addreig ad error et coverage. The referred deity i p Λ ( N;V N ;Ijff )= N m X m= MX t= t6= P (B(t; m) ff ) p ( N;V N ;IjB(t; m) ff ) (8) where p ( N;V N ;IjB(t; m) ff ) i the 3N +dimeioal Gauia term with mea B(t; m) ff - the bia vector i D V Iif ff wa tramitted. The P (B(t; m) ff ) are the amplig probabilitie for the bia term. They were made iverely proportioal to the Euclidea orm of the repective bia hift term, but we do ot kow i what extet thi optio approache optimality. Simulatio deity at the error oor, i a mutati mutadi implificatio of (8) ice we are oly geeratig N D. The imulatio coit of a umber of batche with a ytematic geeratio of all the M tramitted ymbol i each batch. For each ymbol we perform a umber of ru, each oe coitig of amplig the bia term, vector N, ad vector [V N ;I] whe applicable. IV. RESULTS The reult preeted i thi ectio were obtaied with a 4- FSK modulatio with f =:6=(ßT ) rad betwee adjacet ymbol; thi correpod to particular value of the modulatio parameter i the igalig cheme coidered i []. The umber of ample per ymbol wa et to N = 0. Simulatio gai, deoted by i defied by the ratio betwee the MC imulatio time, T MC, ad the correpodig IS time,

5 T IS, ( = T MC =T IS ). Simulatio were topped whe empirical preciio reached a value lower tha 0% (ee for example [8]). Fig. 4 repreet imulatio data correpod- ^ P / (rad - ) P ^ o - IS x - MC - Simulatio gai Figure 4: IS ad MC i the error oor. ^P ad veru ff ffi 0 -P ^ o - IS x - MC =0.055 rad =0.050 rad =0.045 rad E b /N 0 (db) Figure 5: ^P veru E b =N 0 for differet value of ff ffi ig to the error oor for value of ffffi ragig from 0: rad =ffffi =0 to 0:03 rad =ffffi. =33:3 Deity biaig wa doe i D accordig to LDT priciple. The left vertical cale repreet the etimate of ymbol error probability, ^P, wherea the right vertical cale repreet imulatio gai. Notice the practical coicidece of ^P value of IS (mark o) with thoe of MC (mark ) i the rage =ffffi =0 to =ffffi =:. Simulatio gai icreae, i thi rage, from =55to = The value of T MC for =ff ffi =: i 3:7 hour uig a PIII@450MHz computer. Extedig compario of imulator for maller phae drift would be ubearable. The IS reult preeted i Fig. 5 were obtaied after adaptive optimizatio a they cocer the receiver performace i a wide rage of operatig coditio (icludig the error oor). Alo repreeted are the value of ^P obtaied with MC for ff ffi =0:05rad ad value of E b =N 0 equal to 9,, 5 ad 3 db; the correpodig gai were 8:, 54:3, 58: ad 45 repectively; for E b =N 0 = 7 db, there i o practical imulatio gai. Oce agai we tre the practical coicidece of etimated value of ^P provided by both imulator. Poit o the curve correpodig to ff ffi =0:045 rad took 0 miute (with the above metioed computer) i the rage of [0; 70] db, while the MC poit i the rage of [7; 50] db would take about 7:5 hour. V. CONCLUDING REMARKS I thi work we combied two IS approache i the aemet of a particular type of ymbol-by-ymbol phae detector. Compario with the MC imulatio how the effectivee of the developed IS algorithm both i term of accuracy ad the time imulatio gai. The receiver we have aalyzed, baed o tochatic oliear filterig implemetatio, were compared i [] with alterative tructure where the etimatio uit are exteded Kalma-Bucy filter (EKBF) applied to the ame problem. With the ame purpoe we are curretly extedig the techique herei deiged to the EKBF baed receiver. VI. REFERENCES [] F. D. Nue ad J. M. N. Leitão, A oliear filterig approach to etimatio ad detectio i mobile commuicatio, IEEE Joural o Selected Area i Commuicatio, vol. 6, o. 9, pp , December 998. [] J. S. Sadowky ad J. A. Bucklew, O Large Deviatio Theory ad aymptotically efficiet Mote Carlo etimatio, IEEE Traactio o Iformatio Theory, vol. 36, o. 3, pp , May 990. [3] J. S. Sadowky, O the optimality ad tability of expoetial twitig i Mote Carlo etimatio, IEEE Traactio o Iformatio Theory, vol. 39, o., pp. 9 8, Jauary 993. [4] J. A. Bucklew, Large Deviatio Techique i Deciio, Simulatio ad Etimatio, New York: Wiley, 990. [5] J. S. Stadler ad S. Roy, Adaptive importace amplig, IEEE Joural o Selected Area i Commuicatio, vol., o. 3, pp , April 993. [6] P. Balaba M. C. Jeruchim ad K. S. Shamuga, Simulatio of Commuicatio Sytem, Pleum, Jauary 994. [7] D. Lu ad K. Yao, Improved Importace Samplig techique for efficiet imulatio of digital commuicatio ytem, IEEE Joural o Selected Area i Commuicatio, vol. 6, o., pp , Jauary 988. [8] Jyu-Cheg Che, D. Lu, J. S. Sadowky ad K. Yao, O Importace Samplig i digital commuicatio - part I: Fudametal, IEEE Joural o Selected Area i Commuicatio, vol., o. 3, pp , April 993.

Comments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing

Comments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing Commet o Dicuio Sheet 18 ad Workheet 18 ( 9.5-9.7) A Itroductio to Hypothei Tetig Dicuio Sheet 18 A Itroductio to Hypothei Tetig We have tudied cofidece iterval for a while ow. Thee are method that allow