Adaptive Sequence Detection using T-algorithm for Multipath Fading ISI Channels

Transcription

1 1/5 Adapve Sequence Deecon usng T-algorhm for Mulpah Fadng ISI Channels Heung-o ee and Gregory J Poe Elecrcal Engneerng Deparmen, Unversy of Calforna a os Angeles Box os Angeles, CA 995 Emal: poe@csluclaedu Phone: (31) , FAX: (31) Absrac - We develop an adapve, low complexy ree-search recever usng he T-algorhm for mulpah fadng ISI channels Unlke prevous research on sequence based deecon, a symbol spaced channel s no gven a pror raher he recever ulzes he feedforward channel esmaon o derve he mached fler and he symbol-spaced sascs Then, o enhance he effcency of he T-algorhm, he use of a mean-square whenng fler (MS- WF) s proposed We also propose he use of per-survvor processng whch brngs a furher SR advanage and reducon of he average compuaons requred by he T-algorhm recever A subsanal SR benef over a correc decson feedback DFE s acheved a a moderae ncrease n complexy I ITRODUCTIO The recever echnque developed n hs paper s nended o faclae he mplemenaon of a relable, adapve and hghly bandwdh effcen communcaon lnk over me-varyng dspersve channels The recever mus cope wh he unknown channel exhbng Doppler spreadng, frequency-selecve fadng and shadowng In addon, snce he rado specrum s dear, use of a large sgnal se modulaon s hghly desrable We herefore consder sysem wh a large sgnal se (up o 64 QAM), ogeher wh an explc anenna dversy combnng and adapve equalzaon For equalzaon, we nvesgae a low complexy sequence search recever usng he T-algorhm for uncoded use, whch acheves a performance very close o ha of maxmum lkelhood sequence deecon (MSD) A grea deal of research has been underaken o reduce he compuaonal complexy requred o acheve he performance of MSD Research n hs arena ncludes reduced sae sequence esmaon (RSSE), he M-algorhm and he relavely newer T- algorhm [4] Orgnally nroduced by Smmons [4], he T- algorhm has been shown o exhb a superor error-rae versus average-compuaonal-complexy behavor compared o he RSSE and he M-algorhm Smmons has appled he T- algorhm o decode rells coded QAM ransmed over sac ISI channels [5] Oher research exends he T-algorhm o he me-varyng dspersve channel envronmen n [6][7] We develop he opmum dversy combnng fron-end (FE) flers whch provde he symbol-spaced suffcen sascs for he T-algorhm They conss of a fraconally-spaced mached fler (MF) a each dversy branch and a symbol-spaced meansquares whenng fler (MS-WF), boh adapng o he mevaryng channel (See Fg 1 and secon II for deals) Prevous research [6,7] uses a symbol-spaced channel model even for an unknown me-varyng channel However, whenever he channel s unknown, he symbol-spaced channel model s mprecse--or pays sgnfcan amoun of SR penaly n praccal use, snce he MF or WMF canno be denfed In hs paper, he unknown me-varyng channel s esmaed n a feedforward fashon and racked usng he channel esmaon procedure from [1], and he MF and MS-WF are updaed from he channel esmaes For he T-algorhm, we propose o nclude a per-survvor channel rackng procedure The per-survvor processng brngs he addonal benef of lowerng he average complexy of he T-algorhm: In a correc pah, he channel esmae s enhanced; n a wrong pah, he channel esmae quckly degrades, promong he early elmnaon of he pah from he survvor ls A smple overflow handlng roune s suggesed o reduce he sze of he maxmum allowed survvors We show for a sac channel wh a db null n s folded specrum and Raylegh fadng ISI channels ha he proposed recever acheves deecon performance very close o ha of he Verb algorhm (VA), surpassng a correc decson feedback equalzer s performance, a a moderae ncrease n complexy-- less han 1 survvors on average wh 1 maxmum allowed survvors for 4-QAM and less han 5 on average wh maxmum for 64-QAM Ths paper s organzed as follows: The over-sampled dscreeme sysem model s developed n secon II In secon III, we develop he opmal dversy combnng fron-end flers In secon IV, he reduced search T-algorhm s dscussed Secon V dscusses he smulaon resuls and secon VI provdes our concluson II SYSTEM MODE Fg1 defnes he baseband equvalen channel model for an - dversy channel recever We denoe he cascade of he ransm pulse shapng fler g (), he base-band equvalen me-varyng channel c l ( τ; ) and any an-alasng fler a he recever (assumed o be an deal brck wall fler) by h l ( τ; ) We assume g () s an excess bandwdh pulse, and hen he baseband receved sgnal a l -h dversy branch should be fraconally sampled We denoe he samplng nerval as T s s, where s he symbol perod and s We assume he effecve span of h( τ ; ) exends over a h symbol perod, e, he delay spread h( τ ; ) s zero ousde of an nerval [, h T ] The sampled nose s assumed o be addve whe Gaussan wh zero

2 /5 mean and varance For he k -h symbol nerval we have dscree-me samples of x l () whch can be descrbed by x l k, : x l () k s σ n ( + )T s g c 1 ( k) x 1 k, hm 1 ( k) z k w (k) y k ms 1 I h l (( k + s )T T ; kt) + u l (( k + s )T ), for s 1 and,, We now defne he column vecors for he fraconal samples n he k -h epoch as: s xl k, s 1 l xl x k, s l k : h m xl k, h l (( m+ ( s 1) s )T;kT) h, ( k) l (( m+ ( s ) s )T;kT) : and h l ( mt ; kt) u l (( k + ( s 1) s )T ) l u u l (( k + ( s ) s )T ) k : u l ( kt) Thus, a [( ( h + 1) s ) x 1] vecor h l ( k) represens he non-zero poron of he overall channel mpulse response, sampled a he rae of s, e, h l ( k) : ( hl ( k) hl 1 ( k) hl h ( k) ) Then, for he me nerval of neres, ( + h ), he dscree-me sysem equaon s gven by x l H l I + u l ', (1) where x l l l l : ( x + h x + h 1 x ), u l l l l : ( u + h u + h 1 u ), hl ( + h ) hl 1 ( + h ) hl h ( + h ) H l hl ( + h 1) h l h ( + 1 ) h :, hl ( ) hl 1 ( ) hl h ( ) I h I ( h ) :, h s he ( h 1 ) vecor of zeros, and I : ( I 1 I ) s he ransmed daa symbols The s used n place of he ranng segmens for smplcy h In hs paper, we assume connuous ransmsson of frames, where a frame consss of ranng and unknown daa segmens Then he feedforward channel esmaon scheme [1] provdes he esmaes of he me-varyng channel vecors n The feedforward channel esmaon s comprsed of wo modes--he snap-sho channel vecor esmaon durng he ranng segmen and he nerpolaon on a se of channel esmae vecors o capure he channel varaon beween ranng The leas squares channel esmaor (SE) [1] s used n hs paper For deals on feedforward channel esmaon, readers are referred o [1] and references heren In he sequel, we assume he esmaes of channel marces H l n are avalable III DIVERSITY COMBIIG FROT-ED RECEIVER In hs secon, we develop he dversy combnng srucure, c ( k) u 1 () x k, u () h m ( k) Recevers Fg 1 The dversy channels, he dversy combnng mached fler bank and he mean square whenng flers depced n Fg 1 Ths dversy combner s opmum n provdng he symbol-spaced suffcen sascs { } A Opmal dversy combnng for MSE recevers For he ndependen dversy dscree-me receved sequences { x 1 x }, can be shown ha he maxmum lkelhood sequence Î can be found by Î arg max Pr{ x 1 x Ĩ} () x l x l arg mn arg mn { C + M 1 ( Ĩ) + M ( Ĩ) }, (3) where we have defned x l H l : Ĩ, C : x lh x l, M 1 ( Ĩ) Re x lh x l : and Then, M ( Ĩ) : x lh x l usng x l H l Ĩ we have M 1 ( Ĩ) Re Ĩ H H lh x l (4) Re{ Ĩ H z} M ( Ĩ) Ĩ H ( H lh H l )Ĩ Ĩ H ΨĨ, (5) where we have defned z : H lh x l (6) and Ψ : ( H lh H l ) From (6) we noe ha he mulplcaon of H lh and x l represens he fraconally-spaced mached flerng and symbol rae samplng operaons a each dversy branch and ha he summaon mples he dversy combnng, as shown n Fg 1 Thus, z s he se of symbolspaced suffcen sascs for MSD, whch can be descrbed by z ΨI + v, (7) where we have defned v : H lh n l, a vecor wh zero mean and he correlaon marx E{ vv H } σ n Ψ By he use of Cholesky facorzaon, he posve-defne marx Ψ can be facored no he upper rangular band-marx F and s Herman ranspose, Ψ F H F and hus we have Ψ 1 F 1 F H Then z can be rewren as z F H FI + v ow, by applyng F H o z we have: y F H z FI + θ, (8) where he nose erm, θ F H v, s now whened havng he dagonal correlaon marx σ n I Thus, F H s he whenng z k

3 3/5 marx The marx F s an upper rangular marx and hus s causal accordng o our defnon Then, (3) can be shown [] equvalen o Î arg mn { y ỹ } arg mn y k y k k 1 (9) B Fne lengh mean-squares whenng fler The whenng marx F H developed n he prevous secon s opmal, bu no praccal for use wh large block sze For a subopmal, praccal soluon o F H, we propose o use a f - ap mean-square whenng fler (MS-WF) Dealed analyss, as well as comparson wh he whenng fler (WF), can be found n [] Here we wll brefly revew he mehod o oban he f - ap MS-WF The MS-WF s an ancausal fler (realzed wh f 1 delay) ow, ake k for an example o descrbe he mehod o oban he MS-WF We wll use a vecor convenon such ha z( f 1: ) denoes ( z f 1 z ) We denoe he submarx conssng of he las f rows and he las h + f columns of Ψ, defned n (7), by Ψ T Then we have z( f 1: ) Ψ T I( h + f 1 : h ) + v( f 1 : ) (1) Decomposng he frs erm no hree, we have Ψ T I( h + f 1 : h ) Ψ A I( h + f 1 : f 1) + Ψ I( f 1:) + ΨC I ( 1 : h ), (11) where we have defned he frs h columns of Ψ T as Ψ A, he nex f columns of Ψ T as Ψ, and he res as Ψ C ow we defne he ( f f ) marx Ψ ms Ψ + σ n Ξ, where Ξ s he deny marx Then, he MS-WF s obaned from (omng he epoch ndex for smplcy) Ψ ms e f 1 or e f 1F 1 H ms F ms, (1) where Ψ ms F H ms F ms usng he Cholesky facorzaon and e f 1 s a ( f x 1) vecor havng elemens of zeros excep he las elemen beng 1 If Ψ s used nsead of Ψ ms o solve (1) he soluon s he WF The MS-WF s suable for use n he presence of channel esmaon error If he channel has a large n-band null n s folded specrum, he egenvalue spread of marx Ψ s large and he WF would become unsable and enhance he nose and he channel esmaon error For Ψ ms, he smalles egenvalue s resrced o be greaer han or equal o σ n, and hus he MS-WF becomes relavely sabler han he WF ow, mulplyng he MS-WF o we can make he followng observaons: The frs erm approaches zero for a large f, (> h ) The second erm: From (1), we have Ψ σ ( + n I) e f 1 or e f 1F 1 H ms F ms, and hus Ψ e f 1 σ n Therefore, he frs erm n can be wren as Ψ I f 1: ( ) e f 1 σ n w ms I( f 1: ) I σ n I( f 1: ) ( 1 σ nwms, f 1)I + precursor ISI erms for { f 1,, 1} (13) The precursor ISI erms would be zero provded he WF s used The hrd erm w ms Ψ C I( 1 : h ) corresponds o he causal response, whch are he pos-cursor ISI erms corresponds o symbols for { 1,, h } We represen he causal response wh a vecor f, e f Ψ C, (14) for 1,, wh f h ( 1 σ Ths can be nwms, generalzed for wh any epoch 1 ) f( k) k f We now recognze ha { f ( k) } are us a scaled verson of he feedback fler coeffcens of he non-toeplz DFE (T-DFE) [1] We also noe ha (1) s he same equaon used o oban he feedforward fler of he T-DFE The T-DFE s derved under a creron whch mnmzes he mean square error beween predecson and correc symbol, and uses exacly he same se of channel vecors as he MS-WF o derve he feedforward and feedback flers In fac, we can rea he T-DFE recever as a specal verson of he T-algorhm recever ha follows only a sngle pah IV THE REDUCED SEARCH TECHIQUE Fg descrbes he causal symbol-spaced ap fler f( k), whch wll represen he overall channel beween { I k } and { y k } for he purpose of T-algorhm search Ths model dsregards he ancausal erms resulng from he use of fne lengh MS-WF and any esmaon errors n f ( k) These dscrepances would degrade he deecon performance of he complee recever By he use of per-survvor processng, dscussed n secon IV, however, some of he performance penaly can be recovered A The proposed T-algorhm recever Referrng o Fg, he npu/oupu relaonshp s gven by y k h f ( k)i k + η, (15) k where η k s assumed o be whened and he { f ( k) } are as defned n (14) Then, he Eucldean dsance merc (9) for a I k h I k 1 I k f ( k) f f ( k) h 1 ( k) y k η k y k T-algorhm Recever Fg A causal fler (quas mnmum phase) model for he overall response of he ransmer, channels, dversy combnng MFs and he WF for he pos processng of T-algorhm

4 4/5 hypohecal sequence can be compued from Ĩ 1:k J k ( Ĩ 1:k ) J k 1 ( Ĩ 1:k 1 ) + B k, for k,, 1, (16) where B k s he branch merc a k -h symbol epoch B k y k g f ( k)ĩ k y k ỹ k (17) Then, he MSE sequence s deermned from Î arg mn J ( Ĩ) (18) The proposed ree-search T-algorhm uses he Eucldean merc (16) and approxmaes he complee search of (18) The followng seps descrbe he T-algorhm used n hs paper The parameers of mporance are P max, he maxmum number of survvors allowed a an epoch, ζ he hreshold value and D he deph of he ree (Sep-1) Inalzaon: A he zeroh epoch sarng from a sngle pah usng he ranng symbols, se J k ( Ĩ1: h 1) The superscrp denoes he ndex of a survvor runnng from o P max 1 (Sep-) Pah exenson: A he k -h epoch, exend each survvor and oban he cumulave merc J k ( Ĩ1: h 1 + k) J k ( Ĩ1: h + k) + B k, (19) for 1,,, P max M 1 The branch merc B k s defned as B k y k f ( k)ĩ k Ĩ k f h ( k) () 1 A se J mn J k ( Ĩ 1:h 1 + k), updae J mn by a bnary es and he expanded ndex J ( mn ) (Sep-3) Threshold esng and overflow handlng: Frs, coun he number of pahs whose pah merc dfference compared wh J mn s less han ζ and whose sored symbol Ĩ s J ( k D mn ) dfferen from Ĩ k D If he couner reaches P max, lower he hreshold ζ ζ ζ b and repea unl less han P max number of pahs pass Reec all ha fal, release he merged symbol Ĩ k D and concaenae each survvor wh he new symbol (Sep-4) Reurn o Sep- unl he end of he sequence B MS per-survvor channel msmach esmaon We now wan o ake he channel msmach error no consderaon By he use of per-survvor processng usng he leas mean square (MS) algorhm, he channel vecor s refned a each survvor before he pah exenson (Sep-) We now rewre he basc equaon (15) g y k fˆ ( k)i k + η k ε ( k)i (1) + f h + 1 k where we have defned: fˆ ( k) s he esmae obanable from channel esmae, ε ( k) represens he msmach beween he esmae and he acual unknown response, e ε ( k) fˆ ( k) f ( k), for h f + 1,,,, h, () where { f ( k) } represens he unknown, acual response as he resul of he convoluon of he acual channel, he MF and MS- WF; whereas { fˆ ( k) } represens he esmaed response whch s g he convoluon of he channel esmae, he MFs and MS-WF We now defne g ξ( k) y k fˆ ( k)i k g (3) η k + ε ( k)i f h + 1 k I s noeworhy o recognze ha energy of ξ( k) grows wh he sze of he sgnallng se Ths would resul n an ncreased number of survvors for he T-algorhm Thus, elmnaon of he energy s desred o ncrease he deecon SR As seen n (3), he second erm s he convoluon of he ransmed sgnal and he me-varyng msmach vecor ε( k) Snce he prevous symbols are sored n survvors, he causal pars can be esmaed usng he sandard leas mean square (MS) algorhm [], whle he conrbuon of ancausal par s gnored V SIMUATIO RESUTS AD DISCUSSIO In hs secon we sudy he performance of he proposed recever va compuer smulaons Frs, we examne a sac channel ha has a db n-band null n s folded specrum and he parameers of he proposed recevers are uned, such as he feedforward fler lengh, feedback fler lengh of he T-DFE, he hreshold value, he maxmum number of pahs allowed, and he sepsze for he MS channel esmaon error rackng aer, we apply he recever o he fadng ISI channels The smulaon envronmen are manly from [1] We use he -spaced sampled sysem for smulaon, e, s n The square roo rased cosne shapng fler wh 35% roll-off was runcaed a 5 symbol duraon (9 coeffcens) For boh fadng and sac channels, a Mone Carlo mehod wh,-5, ndependen rals was used To evaluae he adapaon on connuously ransmed frames, each ral conssed of 5-16 frames, where a frame s a block of 8 symbols ncludng 11 ranng symbols For he me-varyng channels a each branch, we use he sum of nne-snusodal mehod [1] o generae he ndependen dversy-channel coeffcens, each of whch s connuously vared a a gven fadng rae The ranng symbols are also he same as n [1], bu he magnudes are scaled by ( M 1 + ( M 1) ) for each M -QAM, M 4, 16, and 64 A Sac channel smulaon The example hree-ap sac channel s c ( ) Then, he overall channel s h c g, whch lass 6 symbol perods, h 6 The folded specrum of he channel conans a deep null, whch s abou db down Fg 3 shows he 4-QAM symbol error smulaon resuls on he channel, compared wh he fundamenal mached fler bound (sold lne) The smulaon parameers for all of he sx dfferen recevers are ( f, h ) (6, 6) Frs, compare T- DFE wh he T-algorhm wh ( P max, ζζ, b, D ) ( 1, 4, 1, 3) operang wh perfec knowledge of channel as a benchmark We observe he T-algorhm brngs

5 5/5 1 4 QAM symbol error rae of ch QAM Symbol Error Rae for Raylegh Fadng ISI channel ( ) 1 1 T DFE CDF DFE T alg (1,3) T alg MS (1,3,5) CDF DFE (channel known) T alg (1,4) (channel known) 1 1 T DFE a 1 Hz T alg MS (,45,1) a 1Hz T DFE a 1 Hz T alg MS (,45,1) a 1 Hz Symbol error rae 1 Symbol Error Rae SR per b Fg 3 4 QAM smulaon resuls for he sac channel Average SR per B Fg 5 64-QAM smulaon for fadng ISI channel ( ) abou 15 SR advanage (a ) over he correc decson feedback DFE (CDF-DFE) Ths was acheved by mananng 1 number of survvors on average The same was observed for lower P max 1 (no shown n he fgure) ow, we lower ( P max, ζ) ( 1, 3) for recevers operang wh he leas squares channel esmaor (SE) The T-algorhm recever wh ( P max, ζ) ( 1, 3) shows abou db SR benef over he DFE or 1 db over he CDF-DFE In addon, anoher 1 db SR advanage s obaned usng he T-algorhm equpped wh he MS per-survvor processng (T-alg-MS) wh sepsze B The Raylegh fadng ISI channel The rms delay spread of he Raylegh fadng channel c l ha s used n he smulaon s 357 symbol perods (136 µsec) and E{ c l c lh } dag( 665, 447, 9) Assumng a symbol rae of 4 ksps, fas fadng corresponds o f dm 1 Hz (f dm T 4 or he vehcle speed of 18 km/hr) Sold lnes ndcae he mached fler bounds of he fadng channel whch wll provde us a heorecal benchmark o whch our smulaon resuls can be verfed and analyzed We compare he T-alg-MS recever wh he T-DFE The T-alg-MS parameers suable for each consellaon sze M are emprcally deermned and ndcaed n he fgures Fg 4 s for 4-QAM wh 1 I s evden ha he T-alg-MS recever provdes a sgnfcan SR benef over he T-DFE In parcular, hs s acheved by mananng less han 5 survvors on average when operang n SR regon where 1 or lower SER s acheved Fg 5 dsplays he mos challengng scenaro of he recever We now have dualanenna dversy and 64-QAM We use The Symbol Error Rae 1 4 QAM Symbol Error Rae for Raylegh Fadng ISI channel T DFE a 1 Hz T alg MS (1,3,5) a 1Hz T DFE a 1 Hz T alg MS (1,3,5) a 1 Hz P max Average SR per B Fg 4 4-QAM smulaon for fadng ISI channel ( 1) average number of survvors ncreases o - 5 for SR regons for 1 3 VI COCUSIO We have proposed a low complexy sequence search recever archecure, where he feedforward channel esmaon, opmum dversy combnng fron-end flers and a ree-search T-algorhm wh per-survvor processor are negraed We have shown for sac and fadng ISI channels ha he proposed recever can brng a sgnfcan SR benef compared o he correc decson feedback T-DFE a a moderae ncrease n average complexy Ths recever can be readly exended o nclude decodng of he rells-coded sgnal ransmed over he me-varyng ISI channel Snce a coded sequence has a larger Eucldean dsance, he T-algorhm recever becomes even more effcen Ths s shown n our companon paper [3], where he ree-search T-algorhm s exended o he on decodng of channel nerleaved rells-code ransmed over he fas fadng ISI channels REFERECES [1] Heung-o ee and G J Poe, Fas Adapve Equalzaon/ Dversy Combnng for Tme-Varyng Dspersve Channels, IEEE Trans Commun, vol 46, no9, pp ,Sep 1998 [] Heung-o ee and G J Poe, Adapve sequence deecon: he pre- and he pos-processors for T-algorhm deecon on mulpah fadng ISI channels, IEEE Trans Commun, Unpublshed [3] Heung-o ee and G J Poe, ear-opmal sequence deecon usng T-algorhm of rells coded modulaed sgnal over mulpah fadng ISI channels, Proc of 49h Veh Tech Conf 1999, n press [4] S J Smmons, Breadh-frs rells decodng wh adapve effor, IEEE Trans Commun, vol 38, no 1, pp 3-1, Jan 199 [5] S J Smmons, Alernave rells decodng for coded qam n he presence of ISI, IEEE Trans Commun, vol 4, no /3/4, pp , Feb/Mar/Apr 1994 [6] K D Macell and S J Smmons, Performance of reduced compuaon rells decoders for moble rado wh frequency selecve fadng, ICC 91, vol, pp , Jun 1991 [7] K C Chang and WH am, An adapve reduced-sae channel equalzer wh T-algorhm, Proc of IEEE 44h VTC, vol, pp , Jun 1994

6 1/5 Adapve Sequence Deecon usng T-algorhm for Mulpah Fadng ISI Channels Heung-o ee and Gregory J Poe Elecrcal Engneerng Deparmen, Unversy of Calforna a os Angeles Box os Angeles, CA 995 Emal: poe@csluclaedu Phone: (31) , FAX: (31) Absrac - We develop an adapve, low complexy ree-search recever usng he T-algorhm for mulpah fadng ISI channels Unlke prevous research on sequence based deecon, a symbol spaced channel s no gven a pror raher he recever ulzes he feedforward channel esmaon o derve he mached fler and he symbol-spaced sascs Then, o enhance he effcency of he T-algorhm, he use of a mean-square whenng fler (MS- WF) s proposed We also propose he use of per-survvor processng whch brngs a furher SR advanage and reducon of he average compuaons requred by he T-algorhm recever A subsanal SR benef over a correc decson feedback DFE s acheved a a moderae ncrease n complexy I ITRODUCTIO The recever echnque developed n hs paper s nended o faclae he mplemenaon of a relable, adapve and hghly bandwdh effcen communcaon lnk over me-varyng dspersve channels The recever mus cope wh he unknown channel exhbng Doppler spreadng, frequency-selecve fadng and shadowng In addon, snce he rado specrum s dear, use of a large sgnal se modulaon s hghly desrable We herefore consder sysem wh a large sgnal se (up o 64 QAM), ogeher wh an explc anenna dversy combnng and adapve equalzaon For equalzaon, we nvesgae a low complexy sequence search recever usng he T-algorhm for uncoded use, whch acheves a performance very close o ha of maxmum lkelhood sequence deecon (MSD) A grea deal of research has been underaken o reduce he compuaonal complexy requred o acheve he performance of MSD Research n hs arena ncludes reduced sae sequence esmaon (RSSE), he M-algorhm and he relavely newer T- algorhm [4] Orgnally nroduced by Smmons [4], he T- algorhm has been shown o exhb a superor error-rae versus average-compuaonal-complexy behavor compared o he RSSE and he M-algorhm Smmons has appled he T- algorhm o decode rells coded QAM ransmed over sac ISI channels [5] Oher research exends he T-algorhm o he me-varyng dspersve channel envronmen n [6][7] We develop he opmum dversy combnng fron-end (FE) flers whch provde he symbol-spaced suffcen sascs for he T-algorhm They conss of a fraconally-spaced mached fler (MF) a each dversy branch and a symbol-spaced meansquares whenng fler (MS-WF), boh adapng o he mevaryng channel (See Fg 1 and secon II for deals) Prevous research [6,7] uses a symbol-spaced channel model even for an unknown me-varyng channel However, whenever he channel s unknown, he symbol-spaced channel model s mprecse--or pays sgnfcan amoun of SR penaly n praccal use, snce he MF or WMF canno be denfed In hs paper, he unknown me-varyng channel s esmaed n a feedforward fashon and racked usng he channel esmaon procedure from [1], and he MF and MS-WF are updaed from he channel esmaes For he T-algorhm, we propose o nclude a per-survvor channel rackng procedure The per-survvor processng brngs he addonal benef of lowerng he average complexy of he T-algorhm: In a correc pah, he channel esmae s enhanced; n a wrong pah, he channel esmae quckly degrades, promong he early elmnaon of he pah from he survvor ls A smple overflow handlng roune s suggesed o reduce he sze of he maxmum allowed survvors We show for a sac channel wh a db null n s folded specrum and Raylegh fadng ISI channels ha he proposed recever acheves deecon performance very close o ha of he Verb algorhm (VA), surpassng a correc decson feedback equalzer s performance, a a moderae ncrease n complexy-- less han 1 survvors on average wh 1 maxmum allowed survvors for 4-QAM and less han 5 on average wh maxmum for 64-QAM Ths paper s organzed as follows: The over-sampled dscreeme sysem model s developed n secon II In secon III, we develop he opmal dversy combnng fron-end flers In secon IV, he reduced search T-algorhm s dscussed Secon V dscusses he smulaon resuls and secon VI provdes our concluson II SYSTEM MODE Fg1 defnes he baseband equvalen channel model for an - dversy channel recever We denoe he cascade of he ransm pulse shapng fler g (), he base-band equvalen me-varyng channel c l ( τ; ) and any an-alasng fler a he recever (assumed o be an deal brck wall fler) by h l ( τ; ) We assume g () s an excess bandwdh pulse, and hen he baseband receved sgnal a l -h dversy branch should be fraconally sampled We denoe he samplng nerval as T s s, where s he symbol perod and s We assume he effecve span of h( τ ; ) exends over a h symbol perod, e, he delay spread h( τ ; ) s zero ousde of an nerval [, h T ] The sampled nose s assumed o be addve whe Gaussan wh zero

7 /5 mean and varance For he k -h symbol nerval we have dscree-me samples of x l () whch can be descrbed by x l k, : x l () k s σ n ( + )T s g c 1 ( k) x 1 k, hm 1 ( k) z k w (k) y k ms 1 I h l (( k + s )T T ; kt) + u l (( k + s )T ), for s 1 and,, We now defne he column vecors for he fraconal samples n he k -h epoch as: s xl k, s 1 l xl x k, s l k : h m xl k, h l (( m+ ( s 1) s )T;kT) h, ( k) l (( m+ ( s ) s )T;kT) : and h l ( mt ; kt) u l (( k + ( s 1) s )T ) l u u l (( k + ( s ) s )T ) k : u l ( kt) Thus, a [( ( h + 1) s ) x 1] vecor h l ( k) represens he non-zero poron of he overall channel mpulse response, sampled a he rae of s, e, h l ( k) : ( hl ( k) hl 1 ( k) hl h ( k) ) Then, for he me nerval of neres, ( + h ), he dscree-me sysem equaon s gven by x l H l I + u l ', (1) where x l l l l : ( x + h x + h 1 x ), u l l l l : ( u + h u + h 1 u ), hl ( + h ) hl 1 ( + h ) hl h ( + h ) H l hl ( + h 1) h l h ( + 1 ) h :, hl ( ) hl 1 ( ) hl h ( ) I h I ( h ) :, h s he ( h 1 ) vecor of zeros, and I : ( I 1 I ) s he ransmed daa symbols The s used n place of he ranng segmens for smplcy h In hs paper, we assume connuous ransmsson of frames, where a frame consss of ranng and unknown daa segmens Then he feedforward channel esmaon scheme [1] provdes he esmaes of he me-varyng channel vecors n The feedforward channel esmaon s comprsed of wo modes--he snap-sho channel vecor esmaon durng he ranng segmen and he nerpolaon on a se of channel esmae vecors o capure he channel varaon beween ranng The leas squares channel esmaor (SE) [1] s used n hs paper For deals on feedforward channel esmaon, readers are referred o [1] and references heren In he sequel, we assume he esmaes of channel marces H l n are avalable III DIVERSITY COMBIIG FROT-ED RECEIVER In hs secon, we develop he dversy combnng srucure, c ( k) u 1 () x k, u () h m ( k) Recevers Fg 1 The dversy channels, he dversy combnng mached fler bank and he mean square whenng flers depced n Fg 1 Ths dversy combner s opmum n provdng he symbol-spaced suffcen sascs { } A Opmal dversy combnng for MSE recevers For he ndependen dversy dscree-me receved sequences { x 1 x }, can be shown ha he maxmum lkelhood sequence Î can be found by Î arg max Pr{ x 1 x Ĩ} () x l x l arg mn arg mn { C + M 1 ( Ĩ) + M ( Ĩ) }, (3) where we have defned x l H l : Ĩ, C : x lh x l, M 1 ( Ĩ) Re x lh x l : and Then, M ( Ĩ) : x lh x l usng x l H l Ĩ we have M 1 ( Ĩ) Re Ĩ H H lh x l (4) Re{ Ĩ H z} M ( Ĩ) Ĩ H ( H lh H l )Ĩ Ĩ H ΨĨ, (5) where we have defned z : H lh x l (6) and Ψ : ( H lh H l ) From (6) we noe ha he mulplcaon of H lh and x l represens he fraconally-spaced mached flerng and symbol rae samplng operaons a each dversy branch and ha he summaon mples he dversy combnng, as shown n Fg 1 Thus, z s he se of symbolspaced suffcen sascs for MSD, whch can be descrbed by z ΨI + v, (7) where we have defned v : H lh n l, a vecor wh zero mean and he correlaon marx E{ vv H } σ n Ψ By he use of Cholesky facorzaon, he posve-defne marx Ψ can be facored no he upper rangular band-marx F and s Herman ranspose, Ψ F H F and hus we have Ψ 1 F 1 F H Then z can be rewren as z F H FI + v ow, by applyng F H o z we have: y F H z FI + θ, (8) where he nose erm, θ F H v, s now whened havng he dagonal correlaon marx σ n I Thus, F H s he whenng z k

8 3/5 marx The marx F s an upper rangular marx and hus s causal accordng o our defnon Then, (3) can be shown [] equvalen o Î arg mn { y ỹ } arg mn y k y k k 1 (9) B Fne lengh mean-squares whenng fler The whenng marx F H developed n he prevous secon s opmal, bu no praccal for use wh large block sze For a subopmal, praccal soluon o F H, we propose o use a f - ap mean-square whenng fler (MS-WF) Dealed analyss, as well as comparson wh he whenng fler (WF), can be found n [] Here we wll brefly revew he mehod o oban he f - ap MS-WF The MS-WF s an ancausal fler (realzed wh f 1 delay) ow, ake k for an example o descrbe he mehod o oban he MS-WF We wll use a vecor convenon such ha z( f 1: ) denoes ( z f 1 z ) We denoe he submarx conssng of he las f rows and he las h + f columns of Ψ, defned n (7), by Ψ T Then we have z( f 1: ) Ψ T I( h + f 1 : h ) + v( f 1 : ) (1) Decomposng he frs erm no hree, we have Ψ T I( h + f 1 : h ) Ψ A I( h + f 1 : f 1) + Ψ I( f 1:) + ΨC I ( 1 : h ), (11) where we have defned he frs h columns of Ψ T as Ψ A, he nex f columns of Ψ T as Ψ, and he res as Ψ C ow we defne he ( f f ) marx Ψ ms Ψ + σ n Ξ, where Ξ s he deny marx Then, he MS-WF s obaned from (omng he epoch ndex for smplcy) Ψ ms e f 1 or e f 1F 1 H ms F ms, (1) where Ψ ms F H ms F ms usng he Cholesky facorzaon and e f 1 s a ( f x 1) vecor havng elemens of zeros excep he las elemen beng 1 If Ψ s used nsead of Ψ ms o solve (1) he soluon s he WF The MS-WF s suable for use n he presence of channel esmaon error If he channel has a large n-band null n s folded specrum, he egenvalue spread of marx Ψ s large and he WF would become unsable and enhance he nose and he channel esmaon error For Ψ ms, he smalles egenvalue s resrced o be greaer han or equal o σ n, and hus he MS-WF becomes relavely sabler han he WF ow, mulplyng he MS-WF o we can make he followng observaons: The frs erm approaches zero for a large f, (> h ) The second erm: From (1), we have Ψ σ ( + n I) e f 1 or e f 1F 1 H ms F ms, and hus Ψ e f 1 σ n Therefore, he frs erm n can be wren as Ψ I f 1: ( ) e f 1 σ n w ms I( f 1: ) I σ n I( f 1: ) ( 1 σ nwms, f 1)I + precursor ISI erms for { f 1,, 1} (13) The precursor ISI erms would be zero provded he WF s used The hrd erm w ms Ψ C I( 1 : h ) corresponds o he causal response, whch are he pos-cursor ISI erms corresponds o symbols for { 1,, h } We represen he causal response wh a vecor f, e f Ψ C, (14) for 1,, wh f h ( 1 σ Ths can be nwms, generalzed for wh any epoch 1 ) f( k) k f We now recognze ha { f ( k) } are us a scaled verson of he feedback fler coeffcens of he non-toeplz DFE (T-DFE) [1] We also noe ha (1) s he same equaon used o oban he feedforward fler of he T-DFE The T-DFE s derved under a creron whch mnmzes he mean square error beween predecson and correc symbol, and uses exacly he same se of channel vecors as he MS-WF o derve he feedforward and feedback flers In fac, we can rea he T-DFE recever as a specal verson of he T-algorhm recever ha follows only a sngle pah IV THE REDUCED SEARCH TECHIQUE Fg descrbes he causal symbol-spaced ap fler f( k), whch wll represen he overall channel beween { I k } and { y k } for he purpose of T-algorhm search Ths model dsregards he ancausal erms resulng from he use of fne lengh MS-WF and any esmaon errors n f ( k) These dscrepances would degrade he deecon performance of he complee recever By he use of per-survvor processng, dscussed n secon IV, however, some of he performance penaly can be recovered A The proposed T-algorhm recever Referrng o Fg, he npu/oupu relaonshp s gven by y k h f ( k)i k + η, (15) k where η k s assumed o be whened and he { f ( k) } are as defned n (14) Then, he Eucldean dsance merc (9) for a I k h I k 1 I k f ( k) f f ( k) h 1 ( k) y k η k y k T-algorhm Recever Fg A causal fler (quas mnmum phase) model for he overall response of he ransmer, channels, dversy combnng MFs and he WF for he pos processng of T-algorhm

9 4/5 hypohecal sequence can be compued from Ĩ 1:k J k ( Ĩ 1:k ) J k 1 ( Ĩ 1:k 1 ) + B k, for k,, 1, (16) where B k s he branch merc a k -h symbol epoch B k y k g f ( k)ĩ k y k ỹ k (17) Then, he MSE sequence s deermned from Î arg mn J ( Ĩ) (18) The proposed ree-search T-algorhm uses he Eucldean merc (16) and approxmaes he complee search of (18) The followng seps descrbe he T-algorhm used n hs paper The parameers of mporance are P max, he maxmum number of survvors allowed a an epoch, ζ he hreshold value and D he deph of he ree (Sep-1) Inalzaon: A he zeroh epoch sarng from a sngle pah usng he ranng symbols, se J k ( Ĩ1: h 1) The superscrp denoes he ndex of a survvor runnng from o P max 1 (Sep-) Pah exenson: A he k -h epoch, exend each survvor and oban he cumulave merc J k ( Ĩ1: h 1 + k) J k ( Ĩ1: h + k) + B k, (19) for 1,,, P max M 1 The branch merc B k s defned as B k y k f ( k)ĩ k Ĩ k f h ( k) () 1 A se J mn J k ( Ĩ 1:h 1 + k), updae J mn by a bnary es and he expanded ndex J ( mn ) (Sep-3) Threshold esng and overflow handlng: Frs, coun he number of pahs whose pah merc dfference compared wh J mn s less han ζ and whose sored symbol Ĩ s J ( k D mn ) dfferen from Ĩ k D If he couner reaches P max, lower he hreshold ζ ζ ζ b and repea unl less han P max number of pahs pass Reec all ha fal, release he merged symbol Ĩ k D and concaenae each survvor wh he new symbol (Sep-4) Reurn o Sep- unl he end of he sequence B MS per-survvor channel msmach esmaon We now wan o ake he channel msmach error no consderaon By he use of per-survvor processng usng he leas mean square (MS) algorhm, he channel vecor s refned a each survvor before he pah exenson (Sep-) We now rewre he basc equaon (15) g y k fˆ ( k)i k + η k ε ( k)i (1) + f h + 1 k where we have defned: fˆ ( k) s he esmae obanable from channel esmae, ε ( k) represens he msmach beween he esmae and he acual unknown response, e ε ( k) fˆ ( k) f ( k), for h f + 1,,,, h, () where { f ( k) } represens he unknown, acual response as he resul of he convoluon of he acual channel, he MF and MS- WF; whereas { fˆ ( k) } represens he esmaed response whch s g he convoluon of he channel esmae, he MFs and MS-WF We now defne g ξ( k) y k fˆ ( k)i k g (3) η k + ε ( k)i f h + 1 k I s noeworhy o recognze ha energy of ξ( k) grows wh he sze of he sgnallng se Ths would resul n an ncreased number of survvors for he T-algorhm Thus, elmnaon of he energy s desred o ncrease he deecon SR As seen n (3), he second erm s he convoluon of he ransmed sgnal and he me-varyng msmach vecor ε( k) Snce he prevous symbols are sored n survvors, he causal pars can be esmaed usng he sandard leas mean square (MS) algorhm [], whle he conrbuon of ancausal par s gnored V SIMUATIO RESUTS AD DISCUSSIO In hs secon we sudy he performance of he proposed recever va compuer smulaons Frs, we examne a sac channel ha has a db n-band null n s folded specrum and he parameers of he proposed recevers are uned, such as he feedforward fler lengh, feedback fler lengh of he T-DFE, he hreshold value, he maxmum number of pahs allowed, and he sepsze for he MS channel esmaon error rackng aer, we apply he recever o he fadng ISI channels The smulaon envronmen are manly from [1] We use he -spaced sampled sysem for smulaon, e, s n The square roo rased cosne shapng fler wh 35% roll-off was runcaed a 5 symbol duraon (9 coeffcens) For boh fadng and sac channels, a Mone Carlo mehod wh,-5, ndependen rals was used To evaluae he adapaon on connuously ransmed frames, each ral conssed of 5-16 frames, where a frame s a block of 8 symbols ncludng 11 ranng symbols For he me-varyng channels a each branch, we use he sum of nne-snusodal mehod [1] o generae he ndependen dversy-channel coeffcens, each of whch s connuously vared a a gven fadng rae The ranng symbols are also he same as n [1], bu he magnudes are scaled by ( M 1 + ( M 1) ) for each M -QAM, M 4, 16, and 64 A Sac channel smulaon The example hree-ap sac channel s c ( ) Then, he overall channel s h c g, whch lass 6 symbol perods, h 6 The folded specrum of he channel conans a deep null, whch s abou db down Fg 3 shows he 4-QAM symbol error smulaon resuls on he channel, compared wh he fundamenal mached fler bound (sold lne) The smulaon parameers for all of he sx dfferen recevers are ( f, h ) (6, 6) Frs, compare T- DFE wh he T-algorhm wh ( P max, ζζ, b, D ) ( 1, 4, 1, 3) operang wh perfec knowledge of channel as a benchmark We observe he T-algorhm brngs

10 5/5 1 4 QAM symbol error rae of ch QAM Symbol Error Rae for Raylegh Fadng ISI channel ( ) 1 1 T DFE CDF DFE T alg (1,3) T alg MS (1,3,5) CDF DFE (channel known) T alg (1,4) (channel known) 1 1 T DFE a 1 Hz T alg MS (,45,1) a 1Hz T DFE a 1 Hz T alg MS (,45,1) a 1 Hz Symbol error rae 1 Symbol Error Rae SR per b Fg 3 4 QAM smulaon resuls for he sac channel Average SR per B Fg 5 64-QAM smulaon for fadng ISI channel ( ) abou 15 SR advanage (a ) over he correc decson feedback DFE (CDF-DFE) Ths was acheved by mananng 1 number of survvors on average The same was observed for lower P max 1 (no shown n he fgure) ow, we lower ( P max, ζ) ( 1, 3) for recevers operang wh he leas squares channel esmaor (SE) The T-algorhm recever wh ( P max, ζ) ( 1, 3) shows abou db SR benef over he DFE or 1 db over he CDF-DFE In addon, anoher 1 db SR advanage s obaned usng he T-algorhm equpped wh he MS per-survvor processng (T-alg-MS) wh sepsze B The Raylegh fadng ISI channel The rms delay spread of he Raylegh fadng channel c l ha s used n he smulaon s 357 symbol perods (136 µsec) and E{ c l c lh } dag( 665, 447, 9) Assumng a symbol rae of 4 ksps, fas fadng corresponds o f dm 1 Hz (f dm T 4 or he vehcle speed of 18 km/hr) Sold lnes ndcae he mached fler bounds of he fadng channel whch wll provde us a heorecal benchmark o whch our smulaon resuls can be verfed and analyzed We compare he T-alg-MS recever wh he T-DFE The T-alg-MS parameers suable for each consellaon sze M are emprcally deermned and ndcaed n he fgures Fg 4 s for 4-QAM wh 1 I s evden ha he T-alg-MS recever provdes a sgnfcan SR benef over he T-DFE In parcular, hs s acheved by mananng less han 5 survvors on average when operang n SR regon where 1 or lower SER s acheved Fg 5 dsplays he mos challengng scenaro of he recever We now have dualanenna dversy and 64-QAM We use The Symbol Error Rae 1 4 QAM Symbol Error Rae for Raylegh Fadng ISI channel T DFE a 1 Hz T alg MS (1,3,5) a 1Hz T DFE a 1 Hz T alg MS (1,3,5) a 1 Hz P max Average SR per B Fg 4 4-QAM smulaon for fadng ISI channel ( 1) average number of survvors ncreases o - 5 for SR regons for 1 3 VI COCUSIO We have proposed a low complexy sequence search recever archecure, where he feedforward channel esmaon, opmum dversy combnng fron-end flers and a ree-search T-algorhm wh per-survvor processor are negraed We have shown for sac and fadng ISI channels ha he proposed recever can brng a sgnfcan SR benef compared o he correc decson feedback T-DFE a a moderae ncrease n average complexy Ths recever can be readly exended o nclude decodng of he rells-coded sgnal ransmed over he me-varyng ISI channel Snce a coded sequence has a larger Eucldean dsance, he T-algorhm recever becomes even more effcen Ths s shown n our companon paper [3], where he ree-search T-algorhm s exended o he on decodng of channel nerleaved rells-code ransmed over he fas fadng ISI channels REFERECES [1] Heung-o ee and G J Poe, Fas Adapve Equalzaon/ Dversy Combnng for Tme-Varyng Dspersve Channels, IEEE Trans Commun, vol 46, no9, pp ,Sep 1998 [] Heung-o ee and G J Poe, Adapve sequence deecon: he pre- and he pos-processors for T-algorhm deecon on mulpah fadng ISI channels, IEEE Trans Commun, Unpublshed [3] Heung-o ee and G J Poe, ear-opmal sequence deecon usng T-algorhm of rells coded modulaed sgnal over mulpah fadng ISI channels, Proc of 49h Veh Tech Conf 1999, n press [4] S J Smmons, Breadh-frs rells decodng wh adapve effor, IEEE Trans Commun, vol 38, no 1, pp 3-1, Jan 199 [5] S J Smmons, Alernave rells decodng for coded qam n he presence of ISI, IEEE Trans Commun, vol 4, no /3/4, pp , Feb/Mar/Apr 1994 [6] K D Macell and S J Smmons, Performance of reduced compuaon rells decoders for moble rado wh frequency selecve fadng, ICC 91, vol, pp , Jun 1991 [7] K C Chang and WH am, An adapve reduced-sae channel equalzer wh T-algorhm, Proc of IEEE 44h VTC, vol, pp , Jun 1994