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11 digital coiiiiiiuiiicatic~i~s, dntn sigiials arc transm ittcd tli rough I iucarl y d istortivc analog channcls such RS tclcphone, cablc, ai id wireless ~-adio . In gciicral, thc sill-

glc-uscr systcm modcl is an accnr~tc dcscriptinu of- poi lit-to-poi nt, timc division multi plc ~cccss (Tl>MA), and Li.cqiicncy divisioii iniiltiplc xc- cess (E'lIiMh} sumilmnicxinti spstcins. l 'wo major soiirccs of lincar clianncl distortion in (single-uscr) digird commuuications systciiis arc niii Itipatl I propagation a ud liiiiiccd h;iiid- width. 1,incar channcl distortion Icacls t o iritctsyinbol iiitcrfcrcncc (TSI) at the rcccivcr which, in mrn, may lcad to higl\ m o r ritcs in symlml dctcctimn. l k ~ i i d i 7 ~ 1 - s arc dcsigncd to cornpcnsate fi)r thcsc ch;ilnncl disrorrioiis. One may dircctly dcsigii an equalixr given the rc- ccivcdsignal, or one may Lirstcstiii~arc tlic chm- ne1 iinpulsc i w p o i i s c and rhcn dcsigii ni l cqualizcrbascd on the cstiinnted channel. Tradi- tionally, receivers c w cqunl ixw rely 011 a tram- mittcr assisted training s e s s i m to cxt~icc tlic dcsircd r-etcl-encc signal for clianncl csriiriatiori and cq~i&ation. Such rcccivcrs coiitiiiuc to be highly imporraiir rcscarch subjects bccaase of prx t icd nhstnclcs such as clianicl variiition :ind nonlincariry .

More rcccntly, thcrc h a s bccii niucli intercst i 11 blind (sclf-rcsowring) chiiii~icl csti mat ion and Idiiid cqiidiz;icion whcrc 110 traiiiiiig sc- qticticces arc avai1;lblc or uscd. T u multipoiiit nct- WCII*I<S, whcncvet ii link from rhc scrvcr to OIW oftlic rributary stations is iiitcrruprcd, it is clcarly not fcasible {or dcsirablc) for thc scrvcr to start scnditig a trainiiigscqiiciicc to re-cstablisli n par- ticutai- link. 111 d i g i d comrniiiiicatinns nvcr fad-

ing/i~iul~ipath d~ ;~ i i~ i c I s , ii m t x t is rcqiiircd i d - lowing ;\ teinporsty path intcrruptiou due to sc- V C ~ C L:idiiig. 13uring on-liiic traiismissioii im- p i rinwt i i i c )ii itrwitig, thc t r in ing seq iiciiccs arc olwiviously not supplicd by thc traiisiiiittti-. Con- scq~iciitly, tlic i i i i p o ~ t a i i ~ ~ ofblitid chniiiiel com- pciisatim ~.csc;lrcI~ is also strorigly supported Iy prwtical nccds.

111 this article, wc prcscrit a comprchcii- s i vc si1 111 iii;i ry c i f rcccii t t '~'s(:a IT li dcvc lop inclit 0 1 1 sing IC - I I S C ' I + ch a n IIC 1 cstim 2tri nn ;I ud cq 11 a1 izati 01 I , focusi iig 011 both trail1 ing-bawd arid lilirid appIoachcs. Our ciiiphasis is crn l i i i -

ciir timc-invai-iant clianncls; liiicar. time-varying a s wcll as nonlincar cliniincls :ire oursidc tlic scopc ol'rhis articlc.

System Models In this scction wc first iicscribc clic iiindcls that arc uscd to chnractci izc thc wirclcss aiid iiiobilc cc~mmuiiicatioiis clianncls. 'I'hcn UT iiirn to a brici' discussion of tlic various cqii;ilixer striic- tiiiw that arc iucd to nndn tlic signal distortions cuuscd by tllc cli~l~llicl,

Channel Models The prqxigdtioii of signals through wirclcss channcls (indoors or nutdoors) rcsnlts i t ] the trniismittcd signal iirriving a t thc rcceivcr through multiplc paths. 'llicsc pths arise due to rctlcction, rcfracrion, or diffi.mtioii in t l i ~ chnti- lid. Multip"rli propigition rcsultc i i i s reccivcd signal tlxit is a superposition of several dclaycd and scalctl copics of tlic rraiismitrcd signal giv-

17

ing risc to Sequciicy-sclectivc Gding. b'rcqucncp-sclcctivc fading (dcfined as changes in tlic rccciwd sigiial lcvzl i n tiiiic) is causcd by dcstructivc intcrfcrcncc among mnltiple propagatinii p r h , 't'hc ciivironineiit around thc transmitter aiid thc reccivcr can changc over tiinc, particiilarly in a mobi le setting, Leiding to varin- rimis in chc channcl r e s p " with tirnc. This givcs rise to timc-sclcctivc fading. Also, rlic channcls irlay haw a doiniiinnt path (dircct path in line-of-sight channcls) in aclciitioii to scvcl-a1 sccoiidary paths, or tlicy may bc char- actcriurd as lining; multi plc rci-ancioni" p ~ t h s with no single ciomiiiaiic path.

MLiltipadi ppagnt ion kcads to IS1 at the 1-cccivcr which, in turn, iiiay lcad to high crror rates in syinbul detcctioo. Eqmilixrs arcdcsigiicd to compcnsatc far tlicsc c h a i i i d distortions. Ouc inay dircctly dcsigii an cqildizcr givcu thc rcccivcd signal, or oiic may first c h n a t c the chani~cl iiiipiilsc i'cspoiisc and thcu design an cqiiali7xr based OII tlic csti- niatcd chaiincl. Afm soi i ic procussing (inarchcd filtering, tor imtancc), tlic conti~iiious-tiinc received sigiials arc sain-

at thc baud (symbolj or highcr (fractional) rate bcforc processing tlicm for chaniicl cstimntion and/or equalization. It is t1icrch'ol-c coiivcniciii to w o k with a bnseband-cqui\~n- h i t discrctc-timc cliaiincl model. Considcr a I,aud-i;ltcsatn- plcd systcni. h t s[h] denotc tlic lztli infwiiiation symbol, and lct ~ l , k j clciiotc thc samplctl rccciwcd signal during rlic Rth receivcd symbol. Then tlic two arc relatcd via a timc-varying liiic;ir spstcm rcsponsc as

wIicrchln;W] is tlic chaniicl response at tiiiic n to a unit iii-

put at rime 12. - ,$ and w[nl ~+cprcscnts tlic additive noise (and iiitcrfcrciiccs) at tlic rcccivcr. Modcl (I) 1-cpresciits a tirnc- atid fi-eq~~ciic~;-sclcctivc liiicar chaniicl. A tappcd dclaylinc fiti+i~ctiirc for this rnadclis showii in Pig. 1 . For n slowly (compared to tlic bai id ixtc) timc-varying systcm, oiic o h 1 simplifies (1) to a timc-iiivariaiit systcm as

wlicre hlkl = h [ O ; k ] is tlic tiinc-invariant dianml rcsponse to a unit iiiput at tiinc 0. Modcl (2) rcpresctits a fre- q~ici~cy-scI~ctivc h ica r channel wirh 110 tiinc sckctiviiy. It is thc most commnnly uscd III~CICI for rccciver design.

S~pposc that h m ; k] = h[n]F[ k,O] wlicrc 6[lr,O] is the Krmcclccr dcka located at 0, i.e., 61 h,O] = 1 for 12 = 0 arid S[/z,O] = O for h g 0. 'l'licii wc I ~ v c the time-sclcctive a n d frcqiicucy-nonsckcrive clianncl wliuse outpiit is givcn by

rljt] =h[7z]s[%;l+ IY[R] . (3)

l;inallg, a ti me-iionsclecti vc and ~rcquency-rionsclecti ve clintiricl is rnodelcd as

wlicrc b is a random variablc (or R consrant}. All of clic chatincl rcspoiise functions (1)-(4) riiay be

inodclcd as dcrcrininistic or ~ i i i i d ~ i i i . Also, (1)-(4) result in a singlc-iiipiir single-output (SISE)) complcx dis- crcrc-rim bascbaui~~~cquivalcnt channcl modcl. When the channel of (2) is dctcmiinisric, the outpiir scq~iciice {rl'n]} is discretc-timc stationary. \~liecn tlicrc ia cxccss chaIincl bandwidth [bandwidth > l / 2 x (baiiud rare)], baud rate sampling is hclow tlic Nyquist raw lcadiug to aliasing and dcpciidiiig iip tlic syinbcll timitig phasc, in certain c;iscs, causing dcep spectral notches in thc sampled, aliascrl cliauiicl transfcr fiinctiori 131. Lincar equal- izws dcsigncd 011 t l ic hasis of thc h d - r a t e sainplcd chaiincl rcsponsc NC quite scilsitivrr to symbol timing er- rors. Initially, il l tlic traincd casc, hctio1ial sairiplingwas iuvcsrigatcd to rohustify thc cqidixer pcrformancc ngaiiist timing mors. For liiicar cimc-in\wi:int fit- qiicncy-sclcctivc dctcrmi nistic cliariticls [as i 11 (Z)], whcn sainplcd at higher tliaii tlic baud rare (rypically an intcger niultiplc, p , of baud rate), the sainplcd sigrinl is dis- ci+crc-riinc scalar cyclwtaticmary, and cquivalciitly, it may bc rcprcscnrcd as a cliscrctc-time vcctor srationarg sc- qii e IICC with a 11 11 11 dcr lying single- i i i p lit i m i kip IC - o utpu t (SIMO) ttiodcl wlicrc we stack p consccurivc receivcd satnpks in thc lath symbnl dmxtion to form a p-vector F[n]:

rtl IS), /;I kl is a p vcctnr. For inorc dccils on fading iiiultipath chnniicls, scc [4]

and 151. For rriudeling satiiration nonlincaritics nf power atiiplificrs, notiliiicnr clianiicls of Volterra typc havc also bccn used [ 11 I A disctissioii of basis expansion iiinrlcls for ritiic-varyirig chaiiiicls 11'l;iy bc found in [ 21 ~ I ~ c K , by asuit- nblc sckctinii of die basis fiiirtioiis, a timc-vai-yiiig chamicl can bc 'cti~~~isfi)riiiccl" into a timc-iimiriant cllaiinel.

Equalizer Structures The most comnioti clianiicl cqmdim structut'c is a liiieai: tr"wcrsa1 iilrcr, Giwn rhe baud-ratc samplcd rcccived

ia IEEE SIGNAL PROCESSING MAGAZINE MAY 2000

signal [scc (2)] P Lft], rhc liiicar rransvcrsal cqnalizcr our- put J[H] is an cstirnm ofr[wj, given hy

wlieix {cl:/i]} are the (2N -t- 1) tap wciglitcoefficicnts ofthc equalim+; see ljig. 2. Ar; tioted earlier, linear equalizers dc- sigricd 011 tlic basis nf the baud-rate sampled receivcd signal are quite sensitive tu symbol timing crrors [3]. ’lh crefo r t , fractic 111 a I1 y s p ac cd 1 inc ;i c cqu a I i7~1.s (typic a I ly witli twice the haud-rate sampling: ovcIsampliiig by a factor of two) arc quite widely uscd co riiitigatc scnsiriviry to syinbok timing ci:rors. A fractionally spaccd cqualizcr (FSli) h tlic Iiiicar traiisvcrsal structurc has the output

WIWC wc 1 1 ~ ~ p sa111plCs per syrllboi, ~fcn] ;111d q n j wc p-column vectors [cf, ( 5 ) ] , {Z[rc] } arc tlic (2N t 1) V C C M ~

tap [or p(2.N I- I) scalar tap] wciglit cocificicnts of tlic FSH, and the suycrscript T dciiutcs tlic transpose opcra-

Various crircl-ia aiid cost‘ iiiiictinns cxist to dcsigti the linear cqualizcrs in both harcli and rccursivc (adaptiwj fbrni: wc discuss thcsc later. l’hc linear cqualizcrs c m also bc iiiiplcinented as R Iatticc film [4]. 1.atricc cqii~liz- ers exhibit faster coiivcrgaicc and bctrcr iiumcrical prop- erties [4].

1,incar eqtializcrs do iior pcrforin rvcll wlicii the undcr- lying clianncls havc dccp spcctl-a1 mills iii t1ic piissbaiid. Scvcral nonlilicar equalizcrs have bcen dc\~clopcd ro dcal wirh snch charuicls. ’I’wn cffectivc approaches arc: A l ~ c c i s ~ o r t Fccdbach Eqatnlizer (LIFE): DFE is a iioiiliiicar ecliidizcr that cmploys pcviously dccccrcri syiibols to elirniiiacc thc IS1 duc ro [lie prcvinusly dctcctcd symbols 011 rhc c1irrciit s p l b o l to bc dctcctcd. The ltsc of the ~ I - c - viously dcccctcd symbols rnakcs tlic cqiinlizer niitpitc ii

iioiiliiicar fuiictiori nf‘ tlic data. I3f;li call I>c symbol-spaccd or ti.actioiially s p ” . .A M ~ ~ ~ a z t ~ ~ - L ~ ~ e ~ ~ { ~ ~ o ~ Scperacc Iletector: This esti 111 atcs rhc iiifnrmation sequence to inaxiiniac thc jnint probability d t h e rcceiwd sequence conditioned oti thc informa- tion scqiiciicc.

tion. Note that rllc FSE U L I L ~ I ~ S d:it:l iit ttlc S J W I ~ O ~ filtc.

A detailed discnssbn i i ~ y bc f0ouiid in [4].

algorithm utilizes the avBIablc dara) of thc estimator is particu lady jtnpor ta tit.

We coiisidcr in this scctioii tlirec types ofchaniicl csti- inators based o t i thc framcworli of iilaxii~iiziug tlic likclihood hnctiori. lkferrcd to as thc training-hnserl chaiiiiel estimation, the frrsc typc dcscribcd c m i s i m of the classical techiiiqiics thar cstimatc tlic cliaiiiiel from ;i lciowii training scq~iciicc and its cort.cqnnding ritwmxion. Thc mode of opcratioii is Lctr~iti-l~ef~t.e-tra~-rsr~iit,” which is cffcctivc whcn rhc chatincl does nnt havc significant tirnc variations as iii thc case nf voiccbaud coiimunication overer tclcplioric chiiiiicls. l:nr rapidly varying c l ~ ~ i i i i c l ~ , I i~~vcvcr , such an approach is iwt efftcicnt bccausc rrain- ing has to be pcrforliicd repeatedly, which rcduccs thc available time for traiisinittitig itifimiatinii. Next we de- scribe the approach of blind clianiiel estimation, which iiieaiis that tlic chaiincl estimatiori is perfii-~iicd whilc in- ibrinarian signals are being transmittcd. In ntlm words, tlic goal n f b l i i ~ l chnnncl cstimation is “rraiii-whiIc-tl.ans- mit.” Thc major advantage of thcsc cccliniq~ics is the iiii-

proved 1)anclwidth utilization for tiinc-vnryirig chaiincls. Finally, wc couside~~ the class of tcchniqucs chat f i l l j i i bc- t”cc11 tlic trainiiig-based a n d blind chanticl cstitnation tcchniqucs. Kcferred to as thc scmibliiid chaiuicl estimation, thcsc tccliiiiques aim to cstiimtc the chnnncl using not only thc known data in thc traiisinittcd signa1 nuid its corrcspoiiditig obscrvkon, but alsn the obscrvatioii cor- rcspuiidiiig co C ~ C unkimvii data. ?hc scniiblind chaniid cstirnarion becomcs training-bascd cstiinarioii wlicti m l y tlic obscrvatinii correslirindiiig to thc kiio~vri data is uscd, arid it bccomcs blind-channel estimation whcn tlic obscr- vation is rcsrrictcd to that corresponding to thc iiiiktiowii part. Semiblind channcl cstimitioii i s motivated by the fnct that, in data transmission, thew are always sunit knowti syriibols that sliou Id bc i iicoiymited to iinprovc t IK pcrforrixi iicc .

The Maximum-likeliho od Estlmafor Om of rhc inost popdar paramctcr cstiiiiatioii algorithms is thc maximuii i - l ikcl i l i~~~ ( M I , ) mcrhod. Thc ML estimators can bc dcrived in a systcnxitic way. Pcr- lisps iimrc importaiitly, tlic class nfMT, estimators arc up- tiinal asymptotically [7].

J , t t us cousidcr rhc pvcctor chnniiel modcl given in ( 5 ) where M‘C ~ ~ ~ W ; ~ S S U I I I C that thc cliaiiiiel has a finite iin-

Channel Estimation Uiic oftlic objccrivcs ofrccciwr cicsigii is tn mioimizc the dctcctiou error. In gcncral, thc design nf‘nptimal dctector tequircs the kriowlcdgc ofclicclianncl. Oftcii unknnwii iii practicc, chaiincl pnramcters need to bc estimated, prefer- ably iisirig oiily R liniitcd ainiiunt cif data samplcs. In communicarinn applications, cspccially for packet trails- missions, the efficiency (a I ~ ~ ~ ; I S I T I ‘ C of hmw cffcctivcly a n

:*, 1 . Tapped dehy line model of frequency and time selective chnnnel with finite impulse response. z-’ represents a unit (symbol duration) delay.

MAY 2000 IEEE SIGNAL PROCESSING MAGAZINE 19

q- Nt2] 4 A 2. Structirre of a baud-rate lineor transversal equalizer:

pulsc rcsponse of‘ordcrl. Suppose that we have collcctcd N sninplcs of rhc obscrvationi = [+IN -1],...,?[0]1‘, Wc tlicii liavc ttic folluwing lincar model:

rvhcrc I.,, is B p x p identity matrix, s’ atid I? art: vectors consisting of sninplcs O ~ J I K inpiit scqiicncc s[n] and noisc @[H], w p d v d y , and h is the vector oEthe chatiiicl pa- ramctcn.

Lct 6 bc thc vector ofunkiiown parninctcrs that may include the cIia1i11cl paraineter h and possibly tlic cntirc or part of rhc input vector T. Givcn the probability space that describes joiiitly the toisc vector 3 and possihly tlic input data vector S; wc can then obtain, in principle, the probability density hiictioii (pdf)-assuiiiing it cxisrs-of the ohscrvncioii 1:. As n function oftlie unlriiown pararncter6, thc pdfofthc observation f(T;e) is referred to as thdikcli- hood function. Thc MI, estimator is clefined by ~ I K following optiniiz:ation:

- 0 = arg rpax f(F;0)

O c H

wherc 0 dcfiiies t hc doinaiii of thc optirnixation. While thc ML cstiinatnr is coriceptually sitnplc and it

usually has goad pcifvrmancc wlicii the sainplc sizc is suf- ficiently large, thc iinpkmeIitatioii of MI I estimator is snmctinies computational ly ititctisive. Furthcrirmrc, the optimization o f t l ic liltcliliood function in (9) is often harnpcrccl by the cxistcricc of local maxima. Thercforc, it is ckesirnblc that cffcctivc initializatirm techniqucs arc: used in conjunction with tlic ML csrimation.

We now app ly rlic principle nf triaximining the likclihood fmction to ~ h c t11rcc clianncl cstiination problems: rtic training-bnscd chantid cstiinatim, thc blind chanucl estimation, and the scmiblind channel csrimation.

Twining-Based Channel Fstimatim The training-baxcd chaniicl csrimation assmiles the availability of tlic inpiit vcctm T (as training syiiibnlsj and its corresponding ohservatinii vcctor F . When t l ic noisc samples arc zero incan, white Gaussian, i.c., 6 is a wro mcan, C;aiissian ratidom vector withcovjuhncc o2 1, rhc ML estimator dcfincd in ( 1 a), with 0 = h, is givcn by

where (s) is thc pseudo-iiwcrse of the a(7j dciiiied in ( 8 ) . This is also tlic clxsical lincar least sq~ares estimator which can he implcmcntcd reciirsivcly, and it twnx out ro be thc bcst (in ternis ofl-iavin~mininiuIri mcan squaw crror) among all unbiasctl estiinatnrs and it is the most cfL- cieiir in thc w i s e tliar it achieves thc Cram&-Rao lowcr bouiid . Vari 011 s ad apt ivc imp lcniciita tions can be foil nd io [17].

Blind Channel Estimation Now suppost: that Imth thc i n p t vector F and the clianncl vectnr h arc ii111riiown. The simultaiicous csrimation of r k input vcctor atid the chatiiicl appears to bc ill-posed; how is it possiblc that t l ic chatincl and its input can bc dis- tiiiguishcd using only thc obscwation? l‘hc h y in blind chatiticl csrirr-int ion is thc 11 t ilizat ion of qualitative i nfnr - mation about the chant-icl aiid the input. To this erid, wc considcr hvo diftkrenr typcs of MT, tcchiiiqucs based 011

diffircnt niodels of thc i t i p t szqtmcc I

Sfoch ostic Maximum- Likelihood Estimation Wliilc h e input: vccroo~ i‘ is unknown, it may be tiindclcd as a random vector with a kuown distribution. 111 such a case, rhc likclihond function of the iinkriown paramctcr

=h’ can be nbtaincd by

where f(3) is tlic iiiaigiual pdf of thc input vcctor ;incl j’(i;lS;h) is rlic lilcelihmd h i c t i n n when thc input is known. ASSLII~C, for exatnplc, that the inpm dara symbol r[n] tnkcs, with cqual pmbability, a finite nunibcr of valucs. Cnnscqucncly, the input dara vcctor ? also rakcs Val - iics from rhc signal set {?, ,...,?,>. ’l’hc likelihood function nfrhc channel paratnctcr is then givcti by

where C: is a cuiistanr, and the stnchastic MI, estimator is given by

20 IEEE SIGNAL PROCESSING MAGAZINE MAY 2000

‘l‘hc maxiiiiiratioii nf thc likclihood function defined in (11) is in gencriil difficult hccause J ( F ; O ) is IIOIICOI~VCX.

Tlic cxpcctatioii-i-naxiniizatioii (EM) algoritlirri [ 61, [ 121 can bc applied to transform the complicated optimization to a scq~iciicc of qiiadratic optiinizaticins. Kaleh and ValIct [ l X ] first applied tlic ELM algorithm to the equalization of communicntimi cliaiiriels with inpur scquciicc hwiiig tiitiitc alpliiibet property. By using a hiddcn Markov model (HMM) inodcl, they clcvclopcd a batch (off-line) procedure that iiiclitdes thc so-called fonvard a n d hackrvai-d rccursioiis [ 2 11. Uiifortutiatcly, rhc coni- plcxity of this iilgoi-itlini increases expoiicntially with thc chanticl mcinory.

To rclnx the mcrnory requirements and facilitatc clian- tic1 tracking, r‘o~i-li~i~’’ scqwiitial appro;idics have been proposed iii [30], (341, md [29] for gcncral inpiit atid in [19] fnr inpur with iinitc aipliabct propcrtics iindcr a HMM formulation. Given rlic appropriate regularity cniiditioris [ 2911 and R good iiiitialixatioii giicss, ir can bc shown that these algorithm converge (almost siircly and in t l ic iiican squwc scmc) to the truc chanilel valuc.

Deterministic Maxim urn-likelihood Eslima tion ’Ibc dctci~ninistic ML apprmch assiiincs 110 statistical inndcl for t l ic input scqiicim r[n]. In otlcr words, both tlic chantiel vcctorh nnd tlic input mirce vcctor T kirc p- latiictcrs ro be cstiinatcd. Whcii the iwise is zero-mean Gaussian with covariaiicc D’ I , thc ML cstimates can bc obt.aiiicd bp tlic iioiiliticar Icast sqiims optimization

The joitir ininimimtion ciftliclilcelilionri functinn with IT-

spcct to both the channel and the SC)III-CC paramctcrspaccs is diKiculr. Foi~tnnarcly, thc obsei.vntioii vector E- is linear in both the cliaiiiicl and tlic input parameters individii- ally. I n particular, wc have

whcrc

is tlic so-cnllcd tiltcriog matrix. Wc tlicrcforc h a w a scpa- rablc iinnlimar Icast s q i w c s problan that can bc solvcd scq ucii ti a1 I y

If wc arc ody itircrcstcd in cstiiiiatitig thc chaiincl, thc a b w c minimizatioii can be i+cwritteri as

II

wlicl-e P ( L ) is a projection ti:ansfoi:rii 01: F iiitv, t ~ i c or- tliogonal co~nplcmciit of thc rangc spacc of 7 ( h ) , or tlic rioisc subspacc of clic obscrwrion. Discussinns of a l p - rirhins of rliis typc can bc found in [3 11.

The finite alphabet pmpcrties of the input sequcncc, similar to the HMM for statisticalM1, apprmch, c a n a l s u be incorporated into the deterministic M L mctliuds. TIicsc algorithms, first proposcd by Scshadri [22] atid Gliosh and Wcbcr [ 151, i tcntc bct~vccii csrirnarcs of tlic chaniicl arid rlic inpit . At itcmtioii k, wirh at i inirid gucss of riic c~iannclZ‘~] , rlic algojittini cstiinatcs the input scqucncc s” and thc chatiiiclh‘ rtl) ior rlic iicxt iteration by

where S is tlic (discrete) domain of S. Thc optimization in (21) is a linear least sqiinres prohlcin \~.hcrcas thc upti- iiiizatirrii i i i (20) can be acliicved by using t l ic Vitctbi nl - gclrithin [ 131. The coiivcrgcncc ofsuch approaches is nor guaratitccd h gcncral.

The Method of Moments Although the MI, clianncl cstimatcir usually providcs better perfmniaiice, the cotnputntioii wiiplcxity aiicl thc cxis tc i~e of local optiiiia are thc two major difticulties. ‘I‘he iiiethod of moments, 011 the nthei hand, often has a doscd-form idcntificatinn by cxplnititig tlic ~+cl;ltiniiship brtw.veen tlie cliaiiiiel parameter and rmmients of the obscrvatioii vcctoi: ?.

SeconcZ-C)~d.v Strttz’cicwE Mcthuh: In gciicrd, rhc YCC-

and-ordcr inmiciit of the observation carries ouly thc magnitudc iriforrnatiori of rhc chaiincl. Ir is thcrcfnrc h- sutlicietit for chaonel identificatinu. For SIMn v m o r cliatiiicls, howcvcr, tlic iiiitocorrcliitinti functinn of thc obscrwtion is sufficiciir for rhc idciitikitioti ofthc chauncl impulsc rcspoiisc tip to an iinknown constant [32:1. This obscrvacioii lcd to ;I iiiirnbcr of tcchniqucs ~rndcr bot11 srarisrical aiid dcrcrmitiistic a m i mprioiw of clic in-


put sequciicc [31]. Ry cxploitirig the multichaoiirl as- pccts of the channcl, many of tliesc tcchoiques lead to a cotutraiiied quadratic opti mizatinn

wlicrc Q(?) i s a positivc dcfinite inatrix cotistrncted froin tlicobscwation. Asytiiprotically (eitlicr as thc sample size iiicrcascs ro infinity OS rhc uoisc variaiice approachcs to zero), rlicsc cstiinates corivcrgc to trite channcl pratnc- tcrs.

Hcrc we present a sirnplc yet informativc approach [35] that illustrates thc basic idca. Sqqmsc that wc have only two channels with h i t c impalse respnnscs h, [n] niid k, [raj, rcspcctivcly. I f tlierc is 110 noise, the rcccivcd signals from thc two chaniicls sarislj.

where * is the liiiciircoiivoliitinn. Consequently, wc musr have

Y , [a] * h, [n] =r2 [n] * b, [a]. (24)

Sitice the cotivolution riprratiori is liiicar with respect to the cliaimcl and ri [n] is availablc, (24) is equivalent to solving a hornogc~cncc~~~s liiiear cquarion

It$ =O (25)

rvhcre 11 is a ttiatrix made of observations from the two chatincls. I t can bc slwwii that utidcr c c r t a i n idcnrifiability cnnditioris [ 311, the iiu11 spacc ofR has di- rricnsioii 1, which tiicuis that the channel can bc identi- ficd lip to il cuiistaiit. Wlicii clmc is nnisc, thc cliaiiticl cstitnator can be nbtdiiicd from a cruistraiiicd quadratic optimization

h'=arg mitib'R '.Rk; Ilhl=l (26).

which implies that is thc cigenvectnr corrcsponding to t l ic smallcst cigenvalac nFQ=Rd R.

Alrcriiativcly, one cnii aIso cxploit the subspacc srruc- turcof tlic filtering matrix. For cxaniple, ifit is possible to miistruct a matrix N, fcam data dircctly, such that

NT(h)=O

diic to rlic strtictiire of T($), wc thcn have

OK s i ~ h subspace tccliniquc \vas press"nted in [24]. More recently, tlic problcm ofhlind chaniicl idcntifica-

tion has bccn fmmiilatcd as problcrris o f linear predicrioii

[23 ] , 1141, [66] stid smontliiiig 1331 which liavc sirnplc adaptive impkmeriratioiis [ 361.

H@hei- Ot4der Statisticd (HOS) Methods: Given the tnarticmatical m ~ d e l , thcrc arc two broad classes of appro ac hcs to clxa tiiiel cs tim at io 11, tl ic dist itip~ish ing fea - turc atnong rhcn being thc clioicc of rlic optimization criterion. All nf thc approaches involve (iiiorc or lcss) a lcast-squares crror iiicasurc. Thc error rkfinitioii diffcrs, however, RS follows: A Fittinz Eww: ;Match the modcl-based higher-older ( t yp icaliy four th-or de r) stat is tics to thc cs tiiiia tcci (data-bascd) statistics in a lcast-squarcs scnsc to estiiiiatc the claanncl iriipulsc response, as in [64] and [&I, for es- ample. This approach allows cnnsiricration of-noisy observations. In gciici-al, i t results iii a lionlinear optiiiihation pmblcm. It requires availability of a good initial guess to prcvcnt coiivcrgcnce to a local ininiiiium. Ir yiclds estimates nf thc chaoiicl impillsc respntisc. A Aqwtzon Ewur: I t is based an minimizing 111 "equatimi error" in sonic cqmation which is satisficd idcally. The ap- ptmclies of [h9] and [GX] (amongotlws) fall in this cate- gory, In general, this class cif approaches rcsnlts in R closed-form solurion for the chatiiicl irnpulsc rcsponse sn that il global ~x t r cmi in is always guarantccd provided that tlie chaiiiiel Iciigth (order) is kiio.wn. Tlicsc ap- proaclics iiiay alsa provide good initial guesscs for tlic noillinear fitting error approaches. Quite a fcw of tliesc appmachcs +hi1 if the clianiiel length is iuikiiown.

Further details may bc found in [67] and rcfcrences tlicre i n .

Semiblind Channel Estimation Semib1i;ld chaiinel cstiiiiadon has attracccd coiisiderablc attentinn rccciitly clue to the nccd Eor fixst a d rolmst chaunel cstiiriation and the fact that, for inally packct transmission systciiis, there arc cmbcddcd known symbols that ciiii bc cxploitcd for chatincl csrimatioii and tracking. Wc prcsent here a brief discussio~i abnut the idea aiid rcfcr thc rcadcr to :I rcccntsiirvcy [9] for details.

Seiiiiblind chaniicl estimation assiitiics additional knowlcdgc ofthe input scquciicc. Spcciiicallp, part nftlic input data vccror is lmown. Rntli thc statistical and deterministic ML estirnators rcmain the same cxccpt that the I i Ice1 i 1 io od fU iiccioii iieeds to b e iiiodific d to incu r porate tliclciiowiedgeoftlic input [11], [lo]. Semiblind chatinel cstiiiiaticin may ofhr sigiiificant perfomiaim imp-ow- iiiciir, however, oyer cithcr thc blind or thc training-based methods as dcmoiistiatcd in the evaluarion of Cramir-km lnwcr bound in [ 111.

There are inany gcncralizations nfblind ctlaiinel estimation techiiiqucs to incorporate known symbols. In [ 81, Tsatsanis and Cirpan cxtciided tlie approach of ICalch and Vallet hy rcstricting tlic transition of hiddrn Markov iriodcl. In [20]1, the kuowlcdgc of the kiinwn symbol is uscd to avoid tlic local maxima in tlic maxim- zation nfthc likelihood fiinction. Apopular approach is t o conibinc thc objcctivc fmction uscd to dcrive blind

22 IEEE SIGNAL PROCESSING MAGAZINE MAY ZOO0

cliantiel estimator with tlic Icast squarcs cost i n thc training-bascd chniiiiel csrimatioii. Frw cxat~lplc, a wcighteci liiical- combinarion ofthc cost for blind chaiiiicl estimator and tllar for the training-based cstimatnr c m be uscd [16].

Direct Equalization and Sym bo1 Estimation For the piirposc of comiiiuIiicatint, digital 1-cccivcrs nccd to rccover clinnucl input syiiibols from rcccivcd signals that may suKcr fi.om nnisc and chatincl disrortims. Di- rect channel cclii~lization aiicl synibol cstiiiiarioii arc cnin- rnorily adnptcd in practical systcms. Rccall that dara corrimuiiicatiori input 5r.k I coiiics fruiii il k i i o ~ n coiistella- tiori S that Iins h i r c numbcr. of possible sytnbuls. ' Ih is important iiit?rtnation forins the basis for inaiip direct eqmilization and symbol cstiiiiarion approacchcs to tlic chauiiel cclmlizatioii problcm.

111 this sectioii, we dcscribc sevcral typcs nfapproachcs to the pi*oblciii ofdirccr i u p r signal rcccmcry utirlcr liiicar time-invariant clianiicls. First, wc coiisidcr the classical approach of'adaptiw clianncl equalization basccl on ttaiii- ing. This approach relics on an available scquciicc of training data r h r is trausmirtcd hy t l ic transtnittcr during the senip stage and is ktiowi to tlic i.cccivtr. 'Ilris training apprnacli can be applicd for I'-spaccd c q u a l k ~ m (TSE) as SISO fccd-furwrd filters, fur FS't! as SIMU feed-foiivard filtcis, atid for DYE. Wc thcii nutliiic rhc basic principlc ofblind adaptivc cqdiaatinn bascd an implicit 130s cri- teria. Nut , wc cxplaiii the priiisipk ofsrmc siiiiple algo- 1.i th in s for hli nd symbol cs ti 111 at ion ex1 I r~ i t i iig s CCI ) II d nrdcr statistics. Finah, w e revisit thc mcthod of symbnl csriiiiatim via itcrativc least sqiiarc ci-itcrioii aiid S O I W

variatioiis.

Equalizer Adaptation 8ased on Training Chanticl output (after matclicd filtcr) sarnplcd at baud rm is

- r[rn I = -&qh]J[?Z - k ] + W[?Z].

(29) k=--

Problcms occur whcn thc original aiialng channel docs not satisfy thc Nyq~iist I criwion. Cotiscqiicntly, tmdc- sirablc IS1 is iiitrodiiccd as the channcl output r[a'l dc- pctids on ~iiultiplcsy~mbols { s[n]}. 1.9 is itsualiy causcd hy limited channel bandwidth, multipath, and shaniicl fading. Oiie of t hc simylcw and mosr effect approach to IT- covcritig S[H] from e [ n ] is thc iisc of linciir chaiiriel cqiiualixatinn.

:E'nllnwing clic s u c c c s s f ~ ~ l application of adaptive filccrs by 1,ucIy [37], cqdizatioii piramctcrs arc often irpdatcd tlimugh tlic ininimiiiii nicm squarc error critcria. '~'Iiis rc- quires that a Icnowri cliairncl input scqLicnu bc transmit- ted iriirially. Eqiidizatioii with traiiiiiig is coiiimon to many digital coriirniinicatinii sysrcms siicli as high spccd

tclcphonc inodciii, satcllitc coiiinmnicatiol~ systcins, and digital ccl I tilar systcms .

Thc gcncral striicriirc (,fa chauncl cqualizcr is slimvn in Fig. 3. Ad;iprivc chiinid cqiializcrs begin adaptarion with the assisruncc of a knnwn rraiiiiiig scqucncc trans- mitted dwing thc initial stage hy the transmitrcr. Siiicc tlic inpur signal is Aviiilablc, adaptivc algorithms can bc uscd to adjust. tlic cqu:ilizcr parmictcrs by minimizing a lncall square cr ro~ ' (MSB) Imwccti r ~ i c ctpalizer o u c p ~ t y [ ~ a ] and tlic kmwn chanticl input rvirh a delay s l : a ~ v]. Afrc r tra i 11 i ngl eq uiil i zcr p a f ime tc t's sliou Id bc s u tfici cntl y clasc to tlic dcsircd scttinpr; siicli that much of t l ic IS1 is reiimvcd. As tlic cl~atiiicl iiiput cnii ~ iow bc correctly rc- covcrcd fioiii tlic tqwdizcr output through a meinnrylcss decision dwicc (siiccr), thc sccoiid scagc of real data tmxmissir,n caii Lqin . In the opctxinnal stagc, t h c rcccivcrs typically switch to a decision-dircctcd iiiodc cvhcrc thc cqualizcd signal y [ n ] is sent tn a syiiibol dctcc- tor a n d the dctccrcd syriibnls arc uscd a!: a (pscudn-]training seqmiicc to updatc ec1ii;ilizcr coefficicnrs. 1:ccdi'orward ' I W , FSE, as wcll as 13FE can bc updated. llitring cithcr scssion, the cqLialim. filter parameters can hc dcrcriniiicd using thc well kivxvri rccursive Ic:isr squarc (RLS) o r Icwt iiicnii square (LMS) algorithm.

Maximum-Likelihood Sequence Estimation Chnniicl cqiialization fnllowcd by a r;ymbnl-by-symhrnl cstinintinii sliccr does iiot talcc itico considcratioii thc fact that the cqualizcd noisc is iio Int~gcr white. '~'IILIS, perfor- inancc loss is oftften cricouiicctcd in fced-hrtvard and fccd- hack cqiidizcrs. A inorc cffective bur 11101'~ costly approach is tlic iisc ofVircrbi a lgcir i th for MT, cstima- tinn of thc input sequcncc.

Assume that tlic SISO cha~iucl has finite iiiipiilsc cc- s pm sc

As thc mise i v [r t ] is often whitc Gatissi;in, ML estimarc of the cliauiicl iiipi~t ~ [ n ] bascd 011 a secpciicc of channel output vlnl can bc ohtaiucd iC thc chaiincl i i i i p n l ~ rc- s p i ~ s c i s known or has bccn cstiinateerl via training or blind chaiincl csciinarioii. 'i'hc inpiit scqi~eiicc can bc cs-

rh 3. Feedforword and decision feedbock channel equolizafion filters.


ti m ated ly i m x i ini xi tig tlw 1 i Ire1 i hood fii nct ion or, equivalently, by miuiiiiiziiig

SiiiccSlans only iM sya~l~ols, tlic Vitchi algorithm can bc iiiiplcmciitcd by denoting M ’- btatcs as all possiblc Ltuplcs nf(sln I,sln ~ 1 ] , . , , , 4 ~ ~ - I,+ I ] ) . Tlic trcllis is dc- tcriiiiiwd b y 3 wliilc tlic mctrics ofthe Vitcrbi algorithm clcpeuds on tlac cstimatcd chatinel b{lz].

Thc ML Vitclhi scqiicncc cstimatcir is optimum siiicc ir providcs thc iniiiiniiiiu probability ofsymbol error 1111- der ivliitc tiaussiaii noisc. IC is ;i noiilii~ccurcqiiali~cr, however, and is qiiitcl coiiipIcx iC tIic nu~nbcr ds ta tcs M ‘. is largc. Thc 13FE can bc cr~isidered ;is a suboptimiim sclicmc that iissuincs dl past rkccisioils as corrccct sild only cstimarcs thc iiiost rocciit symbol. To rihnin a mcthod simplcr than MLSE md yet iainrc acciirarc rhan DFE, a rcdnccd state Viterbi algoritliin was proposed bp Lhcl-Hallcn aiid Hccg~rd 1381 that asslimes snmc p s t dccisioiis as corrcct wliilc cstirnnting several most rcccnt symbols. This rcduccd state apprnxh givcs a nicc coni- proinis c bc twcc 11 CO iii p lcxi ty a ~ i d pe rto riiia ticc. It providcs good pcrforiiiance whcn thc channcl impulsc rcspoiise has king but small tails.

SISO 8Ilnd Equalization Based on HOS In many coinmi tiicarioii systcms, traiisni ission of ti-ai n- ing scqi1cnccs is ciclicr inipracticnl or too crxtly. I3lind riddaptivc cIiannc1 sc]uali-/.;ttiun algorithms thar do lint rcly 011 t r i i i ing signals l~avc bccii ilcvclopcd. This property can hc helpful in bl-r,adsasr and multicast systciiis whcre training scqucncc for oiic new nscr uaii bc disruptive to currently conncctcd uscrs.

Gciicrally tlicrc arc two typcs of npproaclics to rhis ~~roblein: Iditid channrl estitnatioii or dircct blind equal- izatioii. I<liud clisnucl cstiinatioii issucs havc bccii dis- cussed prcviously. 1l)ii.ccr blind cq~i~ilization sccks optimum parsiiictcr valnw: for blind cqmilizcr ftlrcrs so that the eyc pattcrn ;It the cqiidlizcr output is opcii to al- low correct slicer dccisiclti. Bccaiisc of’ tlic nonliiicur na- ture of L)f;E, adaptiw bliiid cqualizcrs arc gciici+allp i in 1’1 em eiitcd a s fccd - fo rwci rd .

Tlic kcy to dcsigiiing :I blind cqiializer is to dcsigti ralcs ofcqualizct paratneter xljwtiiieiit. With tlic lack uf trdti- ing S C ~ L I C I I C C , the receivcl- does not havc acccss to tlic dcsircd cqualizci. outpiit s[n] to adopt chc traditional iiiiuiinutn iiiciui sqital-c error criterion. Evidently, bliiid cqualizcr iidaptatioii iieeds to iiiiniiaiixc SOIIIC: spccid, iion-MSK typc cost fiinction which implicitly iiivolvcs higher or-der statistics of thc uhaniicl output sigiial. ‘I’hc dcsign nf thc bliiid cquali~xr thus traiisl;ucs into defining P incan cost fiinctioii E{‘P(y[~11)} wliere Y ( x ) is ;I scalar f~i i ic t ioi i , Thus, tlic stochastic gradieiit dcsccnt miiiimixatioii nlgorithiii is casily dctcrniiucd by t l ic deriv-

atiw fiitictinn ip{x)~’€”(x). I-Iciicc, a blind equalixr can cithcr bc ciciiid by the cost fiinction Y(x) or, ecpiva- lencly, by in dcrivntivc yr(x) tiinction. Idcally, thc function ‘1’t)should bc sclccrcd such that local initiirna oftlic mean cosr corrcspoiid t o a significntit removd of IS1 iu the

Wc ,sLiiiimsrizc scvcral blind adaptation algorithms dc- cq“i’1izcr olltput yI It’[.

sigi id for fccd-forcvard cqunlizcrs.

Decision Directed Argorithm Tlic siinplcst blind equalization algorirliiii is t l ic decision-dircctcd algcxithin witliont rrainhg scquctiw. It miniiiiizes thc iiican sqiixc ci-ror bcrween equdizer out- p i t y [ n ] and chc sliccr o~itlxit r”(n -VI. The pcrformaticc of thc dccisinii-dirccrccd aigorithni depcnds on how closc tlic iiiitial paimiewrs to thcir opti tmm settings. ’I’hc closcr t l icy are, the iiiorc a~ciirilcc tlic sIiccr outpiit is to rhc triic channel iiiput j [ n .-VI. On thc othcr hand, local coiivcrgei~ce is highly likcly if initial parmeter values cmsc significaiit 11 tinbcr of slicer ci-~ors [39] , [40].

Soto AIgarifhm and Some Generalizafions The first truly blind dgorithm was introduced by Sato [41]. ForM-lcwl PAM cIiannel input, it is dcfined by

(33)

Tlic gciicmlixation uses ail odd fiiiictiunyrn (x)whosc scc- orid dcrivative is iionnrgativc Car .Y 2 0.

Stop-and-Co Algorithm Aricitlicr idca callcd rhc ic~t~il)-aiid-gd’ a1gorithm was introduced by Picchi and Priati 1431 to nIlow adaptation “to go’’ only wlicn scvecrid dcrivative functions agrcc iii sign fiw rlic ciirrcnr ou tp~i t y[rl]. Givcn several crircria for bliucl er]ii;~Iization, one c311 cxpcct a inorc accurate dcsccnt direction whcn mort than one of t l ic cxisting algorithms agrcc on sign of tlicil- error funcrions. Wlicri rlic error signs d i t h fbr ii particular output y [ n ] , parainctcr xiaptation is "stopped" to inaintain thcir ciirrciic valucs. A similar idca was cxplniteti i i i 1531.

B ussgang Algorifhm Tlic so-callcd Bussgang nlgorithiiix are derivcd from thc inaxi inum~pos te l~ (MAP) tnrmulatimi 1441, I45 I I De- fine die impiilsc rcspoiisc ofrhc clianiiel-equalizer combi-

24 IEEE SIGNAL PROCESSING MAGAZINE MAY 2000

natiun as ( [ k I = b [ R ] * c [ k ] . Tf Lv has the largcst niagiiitiide, then thc cqualizcr oiitput y [ k ] is

Assuming that thc probability distribution of iioisc is Gaiissian, dic iMAP cstiiiiatc ofs[n -VI

(34)

can bc iiscd as :I rcfc'crencc sigiial fnr LMS equalixl- Lipdatc i n tlic hissgang algorithiii.

Constunf Modulus Algorithm ond Extension Thc best kiiuwri Idind aigoritlmis wesc prcsciircd in 14hJ and [48:1 wirh cust fi~mtioiici

(35)

This class ofGodard a lp i th tns is indcxcd by thc positive integcrq. Using tlic stochastic graclieut dcscent appro'ich, cqual i ze r pi ra in eters ca 11 be adapted acwrdi iigly .

For g = 2, tlic special Godard algorithm 7.1'as devclopcci as thc "coiistaiit iiiodulus algorithm" (CM A) indcpciid- eiltly by Treiclilcr and co-workers [48] using thc philoso- phy of'prqirrty rcstural. For c l ~ a i m l input s i p d that has a constant modulus lsl:v] 1' - I t , , thc CMA cqiialixr {IC- rializcs output samples y [ ~ ] that do not h a w the dcsircd cmstaiit modulus characrcristics. Thc mnduluc error is simply e(n)=Iy[n] ( * -R, , atid thc sqnaring ufthis cri-or yields the constaiit modulus cost fiinccion tlw is the iclcii- tical to thc Godard cost tiitictioii with q =2.

This i~icjd~ilus rutoral colicrpt Iias a particular adni~i- tage in thilt it allows tlic cqiidizcr to be adaptcd i n c h peudent of carricr eccovcry. A carricr i'i-equciicy oftwr of A . caiises a possible phasc 1-otarion of rhc cqu. d I ' 1Zcr

cwtpur. lkcausc thc CMA cast fiitictioii i s itiscnsitive to tlic phase ofy[~z], r l x equalizer p x m c t e r adaptatinn can occ~ir iiidcpciideiitly a i d simulrauicously with thc opcra- tion oftlie carria: rcmvcry system. This propcrty also allows CMA tu bc applicd tn atinlog snodiilatioii sigiials with coiistaiir amplittidc snch as those Lisiiig fi-eqiiciiq or

Thc methods nf Shalvi-Weinstein 1491 gci~eralixc CMA and are explicitly hascd on higlicr ordcr statistics of tlie cqidizcr oiitpiir. Ikf i~ ic thc kurtosis of thc cquializcr flirtput signal y l : ~ ] as

I

~ I U S C lliduliitio~i [48],

SlMO Equdization Symbol Estimation Blind SIMO linear Equalization A y adaprivc blind cq~~~limttinn algoIitlim can bc casily adoptccd fnr liiicar SlMO cqunlixrs [ S O ] . SIMCS blind cqiialization may ofIi3 a convergence advantage given tlic subchatme1 divcrsity 1.5 11. Whilc algorithms snch RS CMA in SISO cqiializatioii may suffer from local corivcr- geiicc [52 ] , CMA aid the supcr-cxpouciirial method [S41 are sliown to cmvcrge to complete IS1 remwal under nciisclcss chaniicls [SO], [SS]. Furthcrmo~-c, thcrc is a ckisc rclatioiisliip bctwecii CMA and tlic nonbliiid mini- nmi~ MSE cqiializcr [71 J, 1721.

Blind Closed-Form Symbol Estimation Jo i iit b I itid cha 11 i i e I c st iiii ;Iti 01 1 and s y mbnl cs ti in at ion mcthods based 011 a dcttriiinistic fi-amcitwrk w r c prc- seiitcd carlier. I t shouId hc iiotcd, howcvcr, that local convcrgmcc and high coinplcxity are tlicir noriblc disad- vantages. A s~rbspacc mcthod was prcscntcd in 1561 tlnt lcnds to closrd-fbrm solutions without high complexity.

Following (8) and (ls), riiultiple snapshrxs of i' c m he cullcctcd as


Viterbi Algorithm for Blind Sequence Estimation The Vitsrbi algorithm can also bc applied for blind se- qwt icc cscimnriori bascd a statistical prcproccssing step [EIS]. Assuming that ’T(h)is fullratik andsjn] is whitc

When tlie chaiincl is noisclcss, 0:” = O and sirigiilar valuc dewinposition yields

R, =U, diag(hi ,..., k:)U:‘

(40) where d = L+ N ,is rank of 7@). The Mnlialaiiobis ortbogorializatinn trmsform can bc iued tu prcproccss t he rcccivcd data vector by

n’:

Thus, wlicn there is 110 noisc

A direct application of thc Vitcrbi algurirhm to q ! k ) can bc w e d to cstirnatc t h e unknown data scqi~cncc { s [ ~ ~ ~ , s [ ~ ~ - l ] , ..., s [ ~ ~ - k - d +I]>. To rcducc thc ilurnbcl- of m t c s , select k = 1 atid cstitiiatc thc unknown iiiput sequcncc via Viterbi algorithm

Rcfcr to 1581 for a i i i ~ r c gciieralized Mahalanobis tixiis- form when iioisc is prcsctit.

Iterative 8 h d Symbol Estimation Thc iterative charmcl arid spmhml cstiniatioii imtliod, as summarized carlicr, also allows direct. clxiiiiicl iiiput estimation. Both the itccrativc lcast .squares with ciiumcration (ILSB) and iterative lcast squarcs with projcction ( I t S l ’ ) cxploit the finite alphabcr iiaturc nf the chaiiiicl input signals, Given that clcmcixs i n Scoinc from S, the task of i i i i -

plcrneiitiiig

(44)’

can be itcrarivcly iniplemeritcd CO iriiprnve tlic cstiiriatc in cach stcp, as in (20) atid (21). ILSP simply rcplaccs thc complex symbol estimation stcp of(20) hy a siuiiplcr projection [54]

(45)

111 [ClOj, a ilccision kcdback nicthod wa,s llrcscntcd. This mctliml utilizcs rlic intcmcdiacc sinciothiiig crmr ot chc lcast sqiiarcs smoothiug (LSS) approach. Assuming past dccisiatis arc correct, dccisioii oftlic lntcst syriibol is based 011 thc closcncss ( i n tci’iiis of;inglc) [xtwccn tlic h i - car prcdictioii (lcastsquare smoothing) error aiid tlic projcction of thc iiiput signal vectcit. O ~ C O ;I piel-ccd ohscrvation s~iI~,ssp;1cc Z, (n).

Clapp and Gcdsill also siicccssfull~~ cxploircd tIic sc- qitential iiiiporraiicc sampling idca for ldiiid. scquciicc cs- tiinatic ) i i 161 1.

Applications of Blind Equalization Cornincrcia II y, blind cqualization h a s h ind IICW applica- tiom in the digital IIDTV systciii and thc digitd cahlc iiiodcni 1701. More rtccntly, proinising rcsiilts have also bccri rcportccl on tlic application of blind cq~idi~,ation in rhc pnpular wirclcss GSM cclhiLar systcin 162 1 nsitig higher-ordcr statistical cicco~ivulnrioii mcthod (621 as wcll as CMA atid sccoiid-order sr<itistical chaii~iel iilcntifi- cntioii [-MI.

Jitcndrtt K. 2kBnait (Fcllow) is a Ptofcssor iii tlie l kpa r t - iticrir of Eicctricd and CoinpLiter Eiigiiiccriiig, Auburn Univcrsity, Auburn, AL. Hc rcccivcd his 1%. 1). dcgrcc from I-bc Uiii vctsi ry of 1 Ilinois, Ulhann-ChampaigI1, i I I

1978 in clcctrical cngiiiccring. V w i i i 1978 tn 1982 he w;1s an Assistant I’rnfessoi- nf Electrical atid Cornpiiter lhgiiiceritig at the Univcrsiry ofIowa, low^ CiLy. Hc was with thc Long lhiigc Rcscarch 1)ivision of tlic Hxxoii I’rodiiction ltcscarch Company, H.ouston, TX, fimm

Jiiiic 1982 to Scpteiiibcr 1989. Hc joincd Aitburn Uiii- vcrsity in Scl)tcniber 1989. His research iiitcrcsts arc in stat is tic a1 s ig I I a 1 p wccssi iig ai id s toc h as ric s ys tc I 11 s a 1 i a I y- sis with appIications to coiiirnLiIiicatioiis. Dr. l‘tigiiait scrvcd :I!: an Associarc Editor ofthc IEEE ~~mz~mtions on S&d I’p.ocessiqq from Dccemher 1.1194 to ikcciiibcr 1907. Kc is currently a mcmber of thc IEEE SignRI l’ru- ccssing for r~mmiiiiical.ioiis ’l’cchiiicnl Committec.

L w g Z’on~ (Membcr) is an Associate Pidicssor in tl-ic School of l k c t k i l Engi~ieering at Cr~riicll llniversity, Itlixcca, NY. Hc rcccivcd his l ’ h . 1 ~ . dcgrcc i i i clcctricnl cii- girieering from rhc Univcwity of Notrc Ihmc, Norrc Damc, IN, in 1990. Aftcr bciiig a Postdoctoral Roscarch Affiliate at rhc Itifonnation Systcnis 1 ,dxmtorp, k in - ford Uiiivasity, Staoford, CA, lie joined chc Dcpartiiiciit mf Elccrrical aiid Cotnputcr Biigiiiccring, Wcst Viiginia Un ivcrs ity , rVI c) I-ga I Itown, and M’M 1.v i tli thc Uii i vcrs itp of C]cjiiticcticut, Storrs. Siiicc tlic fall of 1098, hc has bccii wirh Coriicll Univcrsiiy. His rcsc;~rch inccrcsts iiiclidc statisticd sigiial I)rmcssing, wirclcss coni in11 nication, and systcm thcory. Dr. Tong rcccivcd tlre Young 1nvcsrig;itor Award froin tlic OfTicc of Naval Rcscarcli in I 996 and thc C)utsr,uiding Young Aitthor Award fi-om thc IEEE C h - cuirs arid Systems Socicty. Hc is cLirleiitly a mcmbcr of-

26 [EEE SIGNAL PROCESSING MAGAZINE MAY 2000

2% B i g (Senior Mcmbcr) is ;in Associate l’rofcssor in the lkpnrtment of Electrical and Coinputci- Engiuiccriug ar tlic University of lnwi. He rcccived his PILI). dcgree from thc Scbool of hlcctrical Fhginecring, tijrncll Uni- vcrsity, Itli*icii, NY in August 1.990. From 1990 to 1998, 11c was a faculty ~ncmber in thc 1)cparttnciit of lriectrical Engineering, Auburn University, Auburn, AL, first as an Assistant Prdccssor atid latcr iis a n Associate Profcsso~~. l k . I)ing has hcld visiting positions in t l ic Awl-alianNa- tioiinl University, the Hung I<ong Univcrsity of’ Sciciice atid’l‘cclitiol~~gy, the NASA Lewis Rcscarch Center, a i d the USAE Wright I,aboriwry. Hc was an Associatc Edi- tor af thc XlIFB T”wtimas OF? S&nal I+o*occs~-i~.g. Hc ciir- rcntly sciws as a mcinhcr of the lEEG Signal Pracedng fnr CO iii m U 11 icat ion I( ‘l’cc 1111 ica I Coin in i ttcc. 1-1 is rcsc arch cuvers ;I inimher of research issiics irivvolving statistical signal proccssing that includc coiumonica~i~iis systctn design, sigid detection and classification, as wcll as blind signal scparatioii.


IEEE SIGNAL PROCESSING MAGAZINE MAY 2000

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