A PROPOSED HANGUP FREE AND SELF-NOISE REDUCTION METHOD FOR DIGITAL SYMBOL SYNCHRONISER IN MFSK SYSTEMS

Transcription

1 A PROPOSE HANGUP FREE AN SELF-NOISE REUCTION METHO FOR IGITAL SYMBOL SYNCHRONISER IN MFSK SYSTEMS C.. LEE ad M. ARNELL Iiue of Iegraed Iformaio Syem School of Elecroic ad Elecrical Egieerig, The Uiveriy of Leed, LS 9JT, UK. Abrac-The coribuio of hi paper i cocer digial ymbol ychroier for MFSK yem. Mehod are propoed o achieve hagup free operaio ad o reduce elf-oie. Symbol ychroiaio hagup ca be deeced ad acquiiio ime ca be reduced if all he ample per ymbol from he mached filer oupu are ued. A phae-modulaed-m-ary-frequecy-hifkeyig (PM-MFSK) cheme i iroduced o reduce elf-oie due o radom daa. I he aalyi, he imig error variace of he ymbol ychroier developed are derived. The ychroier are evaluaed via acquiiio ime required ad imig error variace uder addiive whie Gauia oie (AWGN) chael codiio. I. INTROUCTION Mo ymbol ychroier have bee aalyed for M- ary phae-hif-keyig (MPSK) or quadraure ampliude modulaio (QAM) [1]-[4]. The implemeaio of a ymbol ychroier for MFSK i lighly differe from MPSK becaue he former ha M bai fucio, which mea each igalig oe i a bai fucio. Hece, we eed o impleme M mached filer i parallel o ha each ca be mached o a paricular igalig oe. Cohere deecio i ued here ad perfec carrier ychroiaio i aumed. Symbol ychroiaio hagup ad elfoie are he commo problem faced by ymbol ychroier. Sychroiaio hagup i a pheomeo ha occur whe ormalied imig error i cloe o.5 (i.e. he uable equilibrium poi); i will ed o move o a able poi bu, i doig o, i evoluio will be govered by oie ice he eerig force S-curve i mall. A a reul, he occurrece of a hagup give rie o a log acquiiio ime. Boh hermal oie ad elf-oie coribue o imig error variace. The former domiae a mall SNR ad he laer domiae a high SNR. The preece of elfoie i due o he radom daa. Thi paper i divided io ecio a follow: ecio II decribe he operaio of he digial ymbol ychroier; ecio III how how ychroiaio hagup ca be deeced ad correced; ecio IV illurae how elf-oie ca be reduced; he performace of he propoed ychroier i aalyed i ecio V ad evaluaed i ecio VI; fially, i ecio VII, we coclude he paper. II. IGITAL SYMBOL SYNCHRONISER Afer he ymbol ychroier ha obaied he iiial eimae from a kow equece, he deecio proce ca be ared. Coequely, he eimaed equece F ˆ ca be ued o remove daa depedecy. Thi mehod i called deciio-direced () ychroiaio algorihm ad i give by ˆ τ = arg max pr ( c% = cˆ, % τ) (1) τ The log-likelihood fucio i give by [] ( L 1) N 1 1 L( r % τ, c% ) = Re { r( mt) ( mt, % τ) } ( mt, % τ) m= () where L i he obervaio legh i ymbol. The mt τ% ) i he rial igal give by (, ( τ) mt (, % τ)= cˆ ( mt) g mt kt % (3) k k = where ĉ repree eimaed igalig daa equece, g() i a real-valued pule, ad ~ τ i he geeric chael delay. However, he ecod erm i uually dropped becaue he iegral of he erm ha oly a mall variaio wih ~ τ []-[3]. Thu, he approximae loglikelihood fucio i give by ( L 1) N 1 ( τ, c% ) Re { r( mt) ( mt, τ) } L r % % (4) m= Subiuig i (4) from (3), (4) ca be wrie a ( L 1) N 1 ( % τ, c% ) Re { ( ) ˆ k( ) m= k= g( mt kt % τ )} L r r mt c mt (5) Here we make a approximaio ha i valid, epecially whe he obervaio ierval i much loger ha he duraio of he pule g() [1]. I coi of icreaig he ummaio over m o ± while rericig he ummaio over k from o L -1. Hece, (5) ca be approximaed o

2 % (11) % % ( L 1) N 1 L 1 { rmt } ( ˆ mt % τ y kt+ % τ ck) Re ( ) (, ), m= k= (6) where y() i he repoe of r() o he mached filer () ad i give by { } ( + % τ) = Re ( ) ˆ k k( ) ( % τ) ykt rmt c mt gmt kt Thu, m= ( τ c% ) L 1 k = (7) L r %, y( kt + % τ, cˆ ) (8) A ML imig parameer eimaor i a imig parameer eimaor ha maximie (8). The loglikelihood fucio for -MS i defied a L 1 L 1 % k i ( τ k ) ykt ( + % τ, cˆ ) max ykt ( + %, c% ) k i k= k= ck for i = 1,..., M (9) where i c% i he i h igalig oe i he alphabe e a k k h ierval. The ML ymbol ychroier here i implemeed differely from he oe for MPSK [1]- [4] becaue, i MFSK, he geeric daa waveform ~ c i o coa wihi a ymbol period. The baic cocep of he rackig-loop implemeaio i o earch for he τ ha caue he differeial of he approximae log-likelihood fucio o be zero: Λ (% τ) = (1) τ% The rucure of he modified rackig-loop MS (MTMS) i how i Figure 1. I rackig-loop MS (TMS), he ychroiaio epoch i deermie by he poiio of he zero i he differeial buffer. Coequely, followig from (9) ad (1) he ML eimae for he MTMS wih expoeial memory i he ~ τ ha aifie (11) 1 τ L a k = k La 1 i max ( ykt ( + τ, ck )) = L i a ck for i = 1,..., M % where L a i he averagig filer memory. The averagig filer i a ifiie impule repoe (IIR) filer wih feedback gai of (L a -1)/L a. The implemeaio of he averagig filer require he aumpio ha he amplig rae i very cloe o a ieger muliple of ymbol rae. The ierpolaor i ued for fie imig adjume. Accordig o (11), he deciio-direced ui (U) chooe he mached filer oupu ha ha he maximum ampliude. By chooig he maximum ampliude, he U ha idirecly made a ymbol deciio a every amplig ia; hoe ymbol deciio are ored i he ymbol buffer. The imig error deecor i a differeial operaor. The loop filer updae he imig eimae a follow: ( k 1) ˆ( k) e( k) ˆ τ + = τ + γ (1) where e(k) i he imig error igal a he k h ia ad γ i relaed o he oie equivale badwidh B L a follow[1] r(t ) Mached Mached Timig Error eecor Loop Ierpolaor Ierpolaor $τ Corol Figure 1 The rucure of MTMS γ A BT L = ( γ A) (13) U Symbol Buffer Eimaed daa where A i he lope of S-curve whe he imig error i equal o zero. The eimaed imig epoch i produced a he ymbol rae. The ieger par of he eimaed imig epoch i ued o elec oe of he ymbol deciio from ymbol buffer a he eimaed ymbol are obaied a ymbol rae; he fracioal par i ued for fie imig adjume. The differece bewee he ovel ychroier ad he coveioal oe i ha he laer decimae he mached filer oupu o 1 or ample per ymbol ad i doe o have a U. III. HANGUP ETECTION AN CORRECTION Figure how he S-curve for differe umber of ample per ymbol. From Figure, we oice ha differe umber of ample per ymbol will have differe hagup regio. A verical lie draw from he criical poi i Figure ideifie he hagup

3 regio ad o-hagup regio. The criical poi i foud o be a δ 1 = ( N K) (14) T N where δ i he imig error, T i he ymbol period, N i he umber of ample per ymbol ad K i he diace (i erm of umber of ample) o he x-axi (i.e. ime axi) bewee he poi where he differeial i calculaed. Thu, we kow he hagup regio i a S(δ) Criical poi Criical poi Normalied imig error, δ/t Figure S-curve for differe umber of ample/ymbol. δ 1 > ( N K) (15) T N I order o deec a hagup regio, we eed o fid he differece bewee he curre amplig poi ad he poiio of he peak. Thi i illuraed i Figure 3. Noe ha hi could be doe oly if all he N ample were preerved afer he mached filer. Thi require a combiaio of peak earch MS [5]-[6] ad MTMS, ow we called hybrid MS (HMS). The rucure of HMS i how i Figure 4. The peak earch ui imply earche for he poiio of he peak, ad he he imig error deecor will calculae he diace bewee i curre amplig poi ad he peak poiio. If (15) i aified, he a hagup i declared ad he curre amplig poi i immediaely chaged o he poiio of he peak. If hagup i o declared, he he operaio i imilar o MTMS. IV. SELF-NOISE REUCTION Self-oie i caued by he ier-ymbol ierferece (ISI) bewee he raied-coie pule. The elf-oie i he ymbol ychroier icreae whe he rolloff of he raied coie pule decreae. For a rolloff of 1., he mai coribuio of elf-oie i due o he previou ymbol ad he ex ymbol. Thu, by aumig ha he pule duraio greaer ha T i egligible, here are 3 poible cae: i) If all hree ymbol have he ame polariy, he o imig iformaio i available. ii) If oly oe of he ymbol (i.e. eiher he previou or he ex ymbol) ha imilar polariy o he curre ymbol he wrog imig iformaio i obaied. iii) If boh he previou ad he ex ymbol have he oppoie polariy o he curre ymbol, he he imig iformaio i accurae, provided ha he imig error i mall. The opio (iii) i wha we rive o achieve. I MFSK, he daa i coveyed via elecig he frequecy oe. For each MFSK ymbol, we iver he phae of he frequecy oe o ha he phae of ucceive MFSK ymbol alerae bewee ad π. We erm hi phae-modulaed-m-ary-frequecy-hifkeyig (PM-MFSK). Sice he phae modulaio i idepede of he daa, opio (iii) i aified for all poible daa equece. Sice hi ychroier reduce elf-oie, we erm hi elf-oie cacellaio MS (SCMS). Mior modificaio ha o be made o he ymbol ychroier whe PM-MFSK i applied. The modificaio made i how i Figure 5. The phae-modulaed oupu of he mached filer ca be removed by uig a olieariy, uch a a quarer or a abolue fucio, bu hee mehod iroduce more hermal oie io he yem. Thu, a alerae phae demodulaor (AP) i ued iead o remove phae modulaio. Figure 3 Mached filer oupu, y(τ) ( ) y τ τ peak The relaiohip bewee he mached filer oupu ad S-curve. δ T curre amplig poi

4 r(t ) Peak Search Mached Mached Ierpolaor Ierpolaor U iroduce more hermal oie io he yem, i i uavoidable. Coequely, elf-oie cacellaio MTMS (SCMTMS) ad elf-oie cacellaio HMS (SCHMS) were developed by applyig SCMS o MTMS ad HMS repecively. V. PERFORMANCE ANALYSIS Normalied imig error variace i give a [1] 1 = R( m) η( m) (16) T m = r(t ) U Timig Error eecor Loop $τ Corol Figure 4 The rucure of HMS. Mached Mached Alerae Phae emodulaor Symbol eciio To ymbol buffer Ierpolaor Ierpolaor Muliplier SUM Symbol Buffer Eimaed daa To Timig Error eecor Figure 5 The rucure of SCMS. I order o reduce imig error variace i he eady ae, he coribuio of ISI from he previou ymbol mu be he ame a he ex ymbol o ha he curre pule hape remai ymmerical ad he differeial a he opimum poi of a ymmerical pule i zero (i.e. he error igal i zero). Therefore, all he mached filer oupu have o be ummed ice he previou pule ad he fuure pule migh be locaed i differe mached filer brache. Thi mea ha we ow have a o-deciio-direced (N) ML ymbol ychroier. Alhough hi where R (m) = E{(k+m)(k)} i he auocorrelaio fucio of he oie ad η(m) i he covoluio of h(k) wih h(-k), where h i he loop filer repoe o (k), i.e. η ( m) = h( i) h( i m) (17) i = Afer ome maipulaio, i i foud ha [1] η ( m) γ = A ( γa) ( ) m 1 γ A (18) where A i he lope of he S-curve a a imig error equal o zero. A = q ( ) (19) ad q () = g() gmf (- ) i he covoluio of he roo-raied coie pule wih he mached filer repoe. The, ubiuig io (16) yield γ () m = R ( m) (1 γ A) A A T m = ( γ ) Subiuig (13) io () we have BT L = AT m = R ( ) (1 ) m m γ A (1) The compuaio of he imig error variace require a kowledge of he auocorrelaio R N (m) of he oie. Sice he elf-oie (k) ad he hermal oie (k) are ucorrelaed, we have R (m) = R (m) + R (m) () Now coiderig o R (m): he oie from differe mached filer brache i aumed o be ucorrelaed; hu R ( m) = { } E () m= m (3) To evaluae he oie variace, we eed he Fourier raform of g ( ) which i foud o be jπf G (f), where G(f) i he Fourier raform of g(). Thu, he oie variace i give by [1]

5 N { ()} 4 ( ) = E = π f G f df (4) N / i he double-ided power pecral deiy of addiive whie Gauia oie. Noe ha he oie variace for PM-MFSK i M ime larger ha (4) ice he oie from he M mached filer brache i ummed % MN = { } = π M E () 4 f G( f) df (5) Aumig ha he elf-oie i egligible, ad collecig erm from (1), (3) ad (4), he imig error variace i give by NB L = 4 π f G( f) df (6) AT The, ubiuig A from (19), we have NB L = 4 π f G( f) df (7) q () T ad recogiig q () = g() gmf (- ), i ca be ee ha q () ha he Fourier raform 4 π f G( f) ad, hece, q () ca be wrie a 1 BT = L L (3) Hece, by ubiuig (3) io (3), we have 1 1 = (33) 8 E / N πξl which i he well-kow modified Cramer-Rao boud (MCRB) [1]. Hece, MTMS coverge o he MCRB whe elf-oie i egligible. Uig he ame procedure, bu replacig (4) wih (5) we have he imig error variace for SCMS M 1 = 8 L E / N (34) πξ which i M ime larger ha he MCRB. VI. PERFORMANCE EVALUATION The performace of he 4 developed ymbol ychroier MTMS, HMS, SCMTMS ad SCHMS were evaluaed ad compared uig performace meaure uch a acquiiio ime required ad imig error variace. Oly 5 ample per ymbol were ued for a alphabe ize of ad he chael wa AWGN. L wa choe o be 1. A. Acquiiio Time Required ( ) 4 π ( ) q = f G f df (8) Alo he igal eergy i give by [1] E = G( f) df (9) ubiuig (8) ad (9) io (7) produce BT L 1 = 4πξ E / N (3) where ξ i a adimeioal coefficie give by [1] Figure 6 Acquiiio ime required. ξ = T f G( f) df I i alo give ha [1] G( f) df (31) I he imulaio, we le he amplig ar a he wor poible amplig poiio which i whe he ormalied imig error i equal o ±.5. I oher word, amplig a he boudary of he ymbol. Radom daa wa ued ad E b /N = 15 db. The reul i how i Figure 6. MTMS acquire ychroiaio afer approximaely 5 ymbol; HMS oly acquire ychroiaio afer approximaely 75 ymbol; HMS perform wore

6 ha MTMS becaue i i everely affeced by elfoie. SCMTMS require he loge acquiiio ime; a we ca ee from he reul, i achieve a ormalied imig error of approximaely -.35 afer 3 ymbol. Thi how ha MTMS ha a faer acquiiio ime whe he elf-oie i pree. SCHMS ha he be performace; i acquire ychroiaio afer approximaely 1 ymbol. Thi how he propoed ai-hagup algorihm work properly whe he elf-oie i removed. B. Timig Error Variace Figure 7 Timig error variace. The imig error variace how i Figure 7 i ormalied by he ymbol period: ( ) τ ˆ τ = (35) T The HMS ha he wor performace becaue i i everely affeced by elf-oie due o he repeaed ymbol. MTMS coverge o he MCRB whe he hermal oie domiae, bu diverge from he boud a high E b /N due o elf-oie. I i how ha SCHMS o oly ha a fa acquiiio propery, bu i ha a beer imig error variace over SCMTMS a low E b /N. Thi i becaue he imig error eimae i deermied by he joi peak earch algorihm ad imig error deecor which make SCHMS more robu o oie. SCMTMS ha he wor performace a db becaue i i mo ucepible o oie. I i wore ha MTMS becaue i hermal oie variace i larger ha he hermal oie variace i MTMS by a facor of M. Neverhele, he performace of SCMTMS i he ame a SCHMS whe E b /N > 5 db. Sice we imulaed for M=, (34) hould be approximaely 3 db wore ha MCRB. Ideed, from he reul, we ca ee ha he performace of SCHMS ad SCMTMS are approximaely 3 db wore ha MCRB a E b /N > 5 db. Thi demorae ha SCHMS ad SCMTMS do coverge aympoically o a lower boud idicaed by (34) ad i alo how ha (34) i valid. Boh SCHMS ad SCMTMS have beer performace ha MTMS a E b /N > 3.5 db which idicae he domiaio of elf-oie over hermal oie a high E b /N. VII. CONCLUING REMARKS A improved verio of TMS i developed. The advaage of MTMS over TMS iclude a abiliy o impleme fie imig adjume ad lower complexiy, ice oly 1 ample per ymbol i eeded o derive he imig eimae; i work eve whe T/T i irraioal. Neverhele, MTMS i quie imilar o he ymbol ychroier decribed i [1], excep he laer wa o developed for MFSK yem ad i doe o have a U. HMS wa developed o remove ychroiaio hagup, bu he objecive wa o achieved becaue i i everely affeced by elfoie. Neverhele, SCHMS uccefully remove ychroiaio hagup whe he elf-oie i removed. Boh SCMTMS ad SCHMS are almo elf-oie free bu hey are ucepible o hermal oie becaue he ummaio of all mached filer oupu icreae he hermal oie variace by a facor M, where M i he alphabe ize. There i o ymbol ychroier ha i uperior uder all codiio. The be compromie i o fir ue SCHMS o acquire ymbol ychroiaio ice i ha he fae acquiiio ime. Afer ymbol ychroiaio i acquired, we wich eiher o MTMS or SCHMS depedig o chael codiio. VIII. REFERENCES [1] U. Megali, A. N. Adrea, Sychroizaio Techique for igial Receiver, Pleum Pre, [] H. Meyr, M. Moeeclaey, S. A. Fechel, igial Commuicaio Receiver: Sychroizaio, Chael Eimaio, ad Sigal Proceig, Joh Wiley & So, Ic., [3] M. Oerder, erivaio of Garder imig-error deecor from he maximum likelihood priciple, IEEE Tra. o Commu.,Vol 35,pp [4] F.M. Garder, A BPSK/QPSK Timig-Error eecor for Sampled Receiver, IEEE Tra. Commu., vol. COM-34, o. 5, pp , [5] S. M. Bru, Compario of Modulaio erived Sychroiaio ad Coveioal Symbol Sychroiaio Techique i Time iperive Chael, Ph Thei, Uiveriy of Leed, [6] C.. Lee, S.Bru, M. arell, Compario of Variou Type of Modulaio-erived Sychroier uder Low Samplig Rae Codiio, 5 h Ieraioal Sympoium o Commuicaio Theory ad Applicaio, Ambleide, July 1999.