LINE ENHANCER METHODS FOR CARRIER TRACKING IN QAM/PSK DATA SIGNALS

Transcription

1 LIE EHACER METHODS FOR CARRIER TRACKIG I QAM/PSK DATA SIGALS Randall Flint (Univrsity of Utah, Salt Lak City, Utah, USA; rkflint@sisna.com); Bhrou Farhang-Boroujny (Univrsity of Utah, Salt Lak City, Utah, USA; farhang@c.utah.du) ABSTRACT In mobil communication systms, th Dopplr shift is a vry common caus of tim-varying changs in th carrirfrquncy offst. This papr discusss two adaptiv lin nhancr algorithms which ar usd to aid in th procss of dtcting and tracking ths tim-varying changs. Spcifically, it dscribs th us of an adaptiv lin nhancr in a quadratur amplitud modulation (QAM), or phas shift kying (PSK), carrir-rcovry systm. It shows that finitimpuls rspons (FIR) adaptiv lin nhancrs ar not wllsuitd for tracking changs in th carrir-frquncy offst. Thn, basd on th LMS algorithm, an infinit-impuls rspons (IIR) adaptiv lin nhancr algorithm is prsntd, which is much mor ffctiv at tracking th carrir frquncy offst. xt, an FIR post-filtring opration is dscribd that is optimal for out-of-band nois rjction, and is usd in conjunction with th IIR adaptiv lin nhancr to furthr improv its lin-nhancmnt capabilitis. Finally, simulation rsults ar prsntd to compar th ffctivnss of th FIR and IIR adaptiv lin nhancr algorithms at tracking changs in th carrir-frquncy offst. 1. ITRODUCTIO In mobil communication systms, th Dopplr shift is a vry common caus of tim-varying changs in th carrirfrquncy offst. If ths changs ar not proprly dtctd, and corrctd, th transmittd data will b lost. This is illustratd by looking at th quation that govrns th si of th carrir frquncy offst du to th Dopplr ffct. It is: Fc Vr Foff = (1) F c s whr F c is th carrir frquncy, F s is th sampl rat, V r is th rlativ vlocity btwn th transmittr and th rcivr, and c is th spd of light. In mobil systms, V r is constantly changing, which rsults in a tim-varying carrir frquncy offst. In ordr to avoid data loss, th rcivr must b abl to track ths tim-varying changs and corrct for thir ffcts. On systm usd for th dtction and corrction of carrir frquncy offsts is shown in Figur 1. This systm, which is targtd for us with QAM or PSK data signals, oprats by synthsiing a complx sinusoid at th ngativ of th frquncy of th carrir frquncy offst, and thn by applying that sinusoid to th input signal to shift its spctrum down to basband. Th lft-most block in this systm, calld th non-linar block, taks th input signal to th M th powr in ordr to convrt th QAM or PSK signal into a complx sinusoid burid in a wid-band nois. Th frquncy of this sinusoid is M tims th carrir frquncy offst, whr th valu of M is dtrmind by th modulation typ. Th nxt two blocks in this systm prpar th signal for th adaptiv lin nhancr by whitning th wid-band nois. Thn, th adaptiv lin nhancr rmovs as much Figur 1: QAM/PSK Carrir Rcovry Systm Procdings of th SDR 08 Tchnical Confrnc and product Exposition, Copyright 008 SDR Forum, Inc. All Rights Rsrvd

2 Figur : Adaptiv lin nhancr block diagram nois as possibl from th input sinusoid, to mak it possibl for th phas lockd loop (PLL) to lock onto th frquncy of th sinusoid. Finally, th PLL and subsqunt blocks synthsi a complx sinusoid, at th ngativ of th carrir frquncy offst, and multiplis th systm's input signal by that sinusoid to shift its spctrum down to basband. A complt xplanation of th opration of this systm can b found in [1]. In summary, th adaptiv lin nhancr must b abl to track changs in th frquncy of a complx sinusoid that is burid in a whit nois, and it must rmov as much nois as possibl from that sinusoid. Figur shows th block diagram of an adaptiv lin nhancr. Th filtr W() is calld th prdiction filtr, which is usd as a band pass filtr in ordr to minimi th nois in th output sinusoid. Th frquncy rspons of th prdiction filtr must constantly b adaptd in ordr to dtrmin th frquncy of th input sinusoid, and to rmov as much nois as possibl. This mans that th optimal frquncy rspons of th prdiction filtr, for maximum nois rjction, is a band pass filtr with a vry narrow passband cntrd at th frquncy of th input sinusoid. This papr will discuss two possibl implmntations of an adaptiv lin nhancr algorithm. Th first algorithm is implmntd using an FIR prdiction filtr. It will b shown that this mthod is lss ffctiv at nois rjction whil tracking changs in th input sinusoid's frquncy. Th scond algorithm is implmntd using an IIR prdiction filtr. It will b shown that this algorithm is much mor ffctiv at tracking ths changs. Thn, a post filtring opration will b discussd that furthr improvs th IIR adaptiv lin nhancr's nois rjction. Finally, th tracking ability of ach adaptiv lin nhancr algorithm will b shown through simulation.. THE FIR ADAPTIVE LIE EHACER It is common practic to implmnt th adaptiv lin nhancr using an FIR prdiction filtr. On advantag of this approach is that th FIR adaptiv lin-nhancmnt algorithm is wll undrstood, asy to implmnt, and its prformanc can b analyd mathmatically. Th main drawback to using this approach is that this algorithm is slow to forgt past input frquncis, as it attmpts to track changs in th input sinusoid's frquncy []. This sction xplains th caus of this drawback, which is rfrrd to as th mmory ffct. In ordr to undrstand th mmory ffct, not that th ability of th adaptiv lin nhancr to rjct nois from th input sinusoid dpnds on th frquncy rspons of th prdiction filtr. Th idal frquncy rspons would pass only th frquncy of th input sinusoid, whil rjcting any othr frquncy contnt in th signal. Th widr th passband of th frquncy rspons, th mor nois is passd through to th output signal. Consquntly, th ffctivnss of th adaptiv lin nhancr at nois rjction, is dtrmind by th numbr of cofficints in th prdiction filtr, or, and th ability of th adaptiv algorithm to dtrmin th optimal filtr cofficints. Th ability of th adaptiv algorithm to dtrmin th optimal filtr cofficints is govrnd by th fact that th prdiction filtr's frquncy rspons will convrg th fastst in th frquncy bands whr th input signal contains th most powr [3]. This is rprsntd mathmatically as th adaptiv algorithm's convrgnc tim constant, which is rlatd to th ignvalus of th input signal s auto-corrlation matrix, according to th quation: 1 τ i = 4µλ i whr µ is th algorithm stp si, and λ i is an individual ignvalu of th auto-corrlation matrix. For a prdiction filtr lngth of cofficints, th adaptiv algorithm will hav convrgnc tim constants, and ignvalus. It can b shown that th magnitud of ach of ths ignvalus is proportional to th powr containd in th input signal in ach of th frquncy bands [3]. In othr words, th input signal's powr-spctral dnsity dtrmins th si of th ignvalus of th input signal's autocorrlation matrix, and th corrsponding convrgnc tim constant in ach of th frquncy bands. In this systm, th powr-spctral dnsity of th input signal has a singl pak at th frquncy of th input sinusoid, and a constant wid-band componnt throughout th rst of th frquncy spctrum. Consquntly, th ignvalus of th input signal's autocorrlation matrix consist of a singl ignvalu at th frquncy of th input sinusoid with a valu of +σ, and -1 rpatd ignvalus with th valu of σ, whr σ is th varianc of th nois in th input signal [3]. From quation, it can b sn that th convrgnc tim-constant is shortst at th frquncy of th input sinusoid, and longr lswhr. In othr words, th rlativ sis of ths convrgnc tim constants rsult in th adaptiv lin nhancr adapting quickly in th frquncy band that contains th input sinusoid's frquncy, but at a slowr rat throughout th rst of th frquncy spctrum. This fact has a significant impact on th tracking bhavior of th FIR adaptiv lin nhancr. Whn th frquncy of th input sinusoid changs, th pak in th ()

3 Figur 4: IIR adaptiv lin nhancr block diagram Figur 3: Th FIR prdiction filtr's frquncy rspons bfor, and aftr, a stp in th input sinusoid's frquncy. powr-spctral dnsity movs with it, and so dos th frquncy band with th largst ignvalu. Th rsult is that th passband of th prdiction filtr is quickly adaptd to includ th nw frquncy of th input sinusoid. Howvr, th portion of th passband at th past frquncy of th input sinusoid is slowly forgottn, du to th smallr amount of powr in th input signal at that frquncy. This rsults in a mmory of th past frquncis of th input sinusoid, and additional nois bing passd through th prdiction filtr. In ordr to illustrat th mmory ffct, Figur 3 shows th frquncy rspons of th adaptiv lin nhancr's prdiction filtr bfor, and aftr, a stp in th input sinusoid's frquncy. Bfor th stp, th passband on th lft is cntrd at th frquncy of th sinusoid. Aftr th stp, this passband disappars slowly, whil th passband on th right, which is at th nw frquncy of th input sinusoid, is quickly dvlopd. Th rsult of this is that unncssary nois is passd through th adaptiv lin nhancr prdiction filtr at th past frquncy of th input sinusoid. 3. IIR ADAPTIVE LIE EHACER A mor ffctiv approach to dsigning an adaptiv lin nhancr is to us an IIR prdiction filtr with th following transfr function: (1-α) W ( ) = 1-α jπϕ jπϕ 1. (3) This transfr function is a singl-pol band pass filtr. Th frquncy rspons of this filtr is controlld by two cofficints, φ and α. Th cofficint φ dtrmins th cntr frquncy of th filtr's passband by rotating th pol location in th -plan. Th cofficint α dtrmins th width of th passband by dtrmining th radius of th pol in th unit circl. Figur 5: IIR φ cost function By adapting ths cofficints in such a way that th cntr frquncy of th passband is placd at th frquncy of th input sinusoid, and th width of th passband is minimid, this filtr is vry ffctiv at rjcting nois from th input sinusoid. Figur 4 shows th block diagram of th adaptiv lin nhancr algorithm usd to simultanously adapt th IIR prdiction filtr's cofficints. Th φ cofficint is adaptd by minimiing th cost function E (, and th α cofficint is adaptd by maximiing th cost function E y α (. Th LMS algorithm is on mthod of adapting ths cofficints in ordr to maximi, or minimi, th valu of th cost function. This is don by approximating th valu of th cost function basd on th currnt filtr cofficints, th input signal, and th rror signal. Thn, basd on th slop of th cost function at that point, th filtr cofficints ar adjustd in th dirction of th slop, which movs th valu of th cost function ithr towards its minimum, or its maximum. A plot of th cost function usd to adapt φ is shown in Figur 5. Clarly, th minimum valu of th cost function corrsponds to th input sinusoid's frquncy. By minimiing th valu of this cost function, th currnt approximation of φ will approach th input sinusoid s

4 Figur 6: IIR α cost function Tabl 1: IIR adaptiv lin nhancr algorithm 3 µ ϕ = µ 0 ( 1 α( ) (4) m( = (1 α ( ) x( + α ( y( i 1) (5) y( jπϕ( = m( (6) ( x( i + 1) y( = (7) ( jπϕ( ( i 1) = [ α( jπm ( ] (8) δϕ( δϕ( i 1) ˆ ( ϕζ = R[ ( ] (9) δϕ( ϕ( i + 1) = ϕ( µ ˆ ζ (10) ϕ ϕ y ( ( ) ( 1) = j i i [ α ( ( i 1)] δα ( δα ( i 1) δ πϕ (11) ˆ 1 + α( ( y( αζ = R[ y( ] + 1 α( δα ( (1) (1 α( ) α( i + 1) = α( 0µ ˆ ζ (13) α α frquncy φ o, and plac th cntr of th prdiction filtr's passband at that frquncy. Th φ cost function is plottd for svral valus of α, in ordr to show that th slop of th cost function is vry dpndnt on α. For α valus nar on, th slop is vry small xcpt nar th frquncy of th input sinusoid. Consquntly, th adaptiv lin nhancr will convrg slowly as for valus of α nar on, and fastr for valus of α nar ro. This has th unfortunat ffct that th filtr Figur 7: Th IIR prdiction filtr's frquncy rspons bfor and aftr a stp in th input sinusoid's frquncy. bandwidth has to b xpandd in ordr to spd convrgnc of th φ cofficint to its optimal valu, and it rsults in two opposing constraints. First, th valu of α must b kpt small in ordr to dcras th φ cofficint's convrgnc tim; and scond, th valu of α must b kpt nar on in ordr to maximi nois rjctd from th input sinusoid by th adaptiv lin nhancr. Consquntly, a cost function must b slctd that will minimi th valu of α, whn φ is far from th frquncy of th input sinusoid, and maximi its valu othrwis. Th cost function E y α ( mts this critria [3][4], whr: 1 + α yα ( = y(. (14) 1 α Figur 6 shows a plot of th α cost function for diffrnt distancs btwn th currnt stimat of φ, and its optimal valu φ o. ot that whn φ is far from φ o, th maximum valu of th cost function is nar 0, and as φ approachs φ o, this maximum movs toward on. Consquntly, by maximiing th valu of this cost function, th valu of α can b adaptd in such a way that th opposing constraints ar mt. Tabl 1 shows a listing of th rsulting adaptiv lin nhancr algorithm. ot that th valu of α must b xplicitly limitd btwn ro and on in ordr to kp th transfr function's pol within th unit circl. Finally, Figur 7 shows th frquncy rspons of th IIR prdiction filtr bfor, and aftr, a stp in th input sinusoid's frquncy. As can b sn, th old frquncy of th input sinusoid is compltly forgottn shortly aftr th stp. In othr words, th IIR adaptiv lin nhancr dos not suffr from th mmory ffct that limits th tracking ability of th FIR adaptiv lin nhancr. Also, th figur shows a tmporary xpansion of th prdiction filtr's passband. This

5 Figur 8: Tim-varying carrir frquncy offst causd by th Dopplr ffct in intr-vhicl communication. is du to th adaptation of α, which is rquird to minimi th convrgnc tim of th φ cofficint. 4. FIR POST-FILTER Aftr th convrgnc of th IIR adaptiv lin nhancr's φ cofficint, th frquncy of th input sinusoid is known by th systm. This allows a band pass post-filtring opration to b applid to th adaptiv lin nhancr's output signal in ordr to achiv furthr nois rjction. This filtr is implmntd as an FIR band pass filtr with th cntr frquncy bing dtrmind by th IIR adaptiv lin nhancr. Th transfr function of this filtr is: jπϕo H ( ) = 1 i= 0 jπϕoi i (15) whr φ is th dsird cntr frquncy dtrmind by th IIR adaptiv lin nhancr. This transfr function rprsnts a filtr mad up of ros spacd qually around th unit circl. It can b shown, using th mthod of Lagrang multiplirs, that this transfr function is th optimal FIR filtr for us in rmoving nois from a sinusoid burid in whit nois [4]. Although th input signal to th post-filtr is a sinusoid burid in nois that has bn colord by th adaptiv lin nhancr, it will b shown that th postfiltr is ffctiv at furthr nois rjction. Finally, by noting that this transfr function is a gomtric sris, an quivalnt transfr function can b drivd that is much mor fficint to implmnt: jπϕo H ( ) = 1 β 1 β jπϕo jπϕo 1 (16) Figur 9: Tracking comparison of th thr adaptiv lin nhancr (ALE) algorithms whr β is a constant that is slightly lss than on. Th purpos for adding β to th transfr function is to kp th pol insid th unit circl, and to nsur stability of th algorithm. 5. SIMULATIO RESULTS A common scnario that ncssitats th tracking of th carrir frquncy offst du to Dopplr shifts is in intrvhicl communications. Assum two vhicls ar communicating wirlssly as thy pass ach othr going opposit dirctions on th frway. Du to th Dopplr ffct, th rlativ vlocity btwn ths vhicls will caus a tim-varying shift in th carrir frquncy offst that must b trackd by th adaptiv lin nhancr. Figur 8 shows an xampl of th Dopplr shift as sn by th carrir rcovry algorithm. Basd on this scnario, simulations hav bn don to compar th ffctivnss of ach of th adaptiv lin nhancr algorithms in tracking changs in th carrir frquncy offst. This is don by comparing a tim avrag of th xcss man-squar rror, during th tracking phas of th simulation, as a rlativ masur of ach algorithm's ffctivnss. Mathmatically, this is dfind as: av ξ xcss = avrag( E ( σ ). (17) Finally, th convrgnc rat of th FIR adaptiv lin nhancr is dtrmind by th signal-to-nois ratio. av Consquntly, ξ xcss is masurd ovr a rang of signal-tonois ratios. Figur 9 shows th rsults of th simulations. As can b av sn, ξ xcss is much highr for th FIR adaptiv lin nhancr du to th mmory ffct. Also, th IIR adaptiv lin nhancr rsults in a significant improvmnt ovr th FIR implmntation, spcially for high signal-to-nois

6 ratios. Finally, th postfiltr adds an additional advantag to that of th IIR adaptiv lin nhancr. 6. COCLUSIO In conclusion, an IIR adaptiv lin nhancr algorithm and a companion postfiltring opration has bn prsntd that rsults in a significant improvmnt in th nois rjction that can b attaind ovr th FIR adaptiv lin nhancr algorithm. This rsults from th IIR adaptiv lin nhancr's lack of a mmory of past input frquncis, and th optimal filtring capabilitis of th postfiltr. Although th xact numbrs in this simulation will not match thos of crtain systms, th ffctiv rsult will b th sam, that of significant nois rjction from th input sinusoid. 10. REFERECES [1] B. Farhang-Boroujny, Signal Procssing Tchniqus for Softwar Radio, Lulu, orth Carolina, 008. [] S. D. Hayward, and C. J. Sprigings, Adaptiv lin nhancmnt via subspac tracking, Signals, Systms and Computrs, Vol., pp , 000. [3] B. Farhang-Boroujny, Adaptiv Filtrs, John Wily & Sons Ltd, Wst Sussx, England, [4] R. S. Flint, Tracking Analysis of an Improvd Infinit Impuls Rspons Adaptiv Lin Enhancr, Mastr's Thsis, Univrsity of Utah, Salt Lak City, Utah, 008.

7