Carrir frquncy stimation ELEC-E54 Signal procssing for communications
Contnts. Basic systm assumptions. Data-aidd DA: Maximum-lilihood ML stimation of carrir frquncy 3. Data-aidd: Practical algorithms 4. on-data-aidd DA stimation algorithms Basd on U Mngali, A.. D Andra, Synchronization Tchniqus for Digital Rcivrs, Plnum Prss,997 Pag
Exampl Considr a systm oprating at GHz. Assum. ppm instability in transmit and rciv oscillators corrsponding to LTE rquirmnts Q: What is th maximum frquncy diffrnc f w can xpct in th two oscillators? Q: Aftr how long tim can w xpct th phas rror to b 36 o? Dopplr shift producs a similar ffct Pag 3
Basic systm assumptions
Basic systm assumptions Frquncy rcovry consists of two tass:. Driving an stimat of th frquncy rror offst f In practic, a rough stimat for th carrir frquncy is nown bforhand almost always. Compnsating for this offst by countr rotating th rcivd signal Assumption: Frq. offst f is much smallr than th symbol rat /T - Timing information can b rcovrd first and is nown for th frquncy stimation algorithm - Phas rotation du to th frquncy offst can b assumd constant during a symbol Pag 5
Basic systm assumptions Typical passband puls amplitud modulation PAM rcivr structur Carrir dmod. Fr osc. Intrpolator control Timing st. Dcimator control x Analog filtr A/D x Int. filt. MF Dc. x Symbol dt&dc Frq. synth. Frq. intgr. Frq. rr.dt. Phas st. Rough corr. Fin corr. Timo I. Laaso and Stfan Wrnr Pag 6
Basic systm assumptions Frquncy stimation approachs Considr first data-aidd DA frquncy rcovry, assuming that a training signal is availabl Last, non-data-aidd DA tchniqus ar brifly discussd Timo I. Laaso and Stfan Wrnr Pag 7
Basic concpts rvisitd
Puls shaping with root raisd cosin wavform bits QAM modulation Transmittr L Root ^cos Root ^cos L Dmodulation Puls shaping is ndd to band limit th transmittd signal wavform h T t in th formulas that follow In simulations of communication systms it is oftn nough to us dirctly th raisd cosin puls.
Tx and Rx signals with root raisd cosin puls shaping.5 Mssag data Tx data Rx data.5 Amplitud -.5 - -.5 5 5 5 3 Tim
Root raisd cosin filtr ormalizd bandwidth is +r, whr r is th roll-off factor Root raisd cosin filtrs form a yquist pair such that thr is no intr-symbol intrfrnc For xampl, 3G WCDMA spcifis roll-off.3 root raisd cosin puls shaping in th transmittr. Max. adjacnt channl laag ratio is -45 db for bas stations and -33 db for mobil stations 9..7
Raisd cosin Convolution of two root raisd cosin pulss o intr-symbol intrfrnc whn sampld at th right tim.4.3.. -. 4 6 8 Tim Raisd cosin puls train in tim domain 9..7
Matchd Filtr Dsign of matchd filtr Maximiz signal powr i.. powr of g t g t* h Minimiz nois powr, i.. powr n t of w t* h t t at t T Givn transmittr puls shap gt of duration T, matchd filtr is givn by h opt t g*t-t for all scalar gains - h opt t is scald, conjugatd, tim-rvrsd, and shiftd vrsion of gt Sampling tim nown gt xt ht yt yt Puls signal wt Matchd filtr t T
DA: ML stimation of carrir frquncy
DA: ML Estimation of carrir frquncy Transmit signal modl: a data symbols to b transmittd h T t transmittd continuous-tim wavform δt Dirac dlta function Timo I. Laaso and Stfan Wrnr Pag 5
DA: ML Estimation Rcivd continuous-tim signal additiv whit Gaussian nois AWG channl: r t x t + w t rt rcivd signal wavform wt additiv nois Th dsird stimat for th rcivd signal is of th form ~ j ft+ x π θ t a h t T τ whr f is to b stimatd, θ is unnown and τ is assumd nown T Pag 6
DA: ML Estimation Considr first th PDF of th discrt-tim signal r assuming AWG: Pag 7 Continuous-tim probability dnsity function PDF: Rplac varianc by nois powr spctral dnsity PSD and sum by intgral: / ~ xp, ; x r f p σ πσ θ r ~ xp x r C σ ~ xp, ; T dt t x t r C f p θ r
DA: ML Estimation Th continuous-tim log-lilihood function LLF is: Λ r; f, θ C T r t ~ x t T T T C r t dt + ~ x t dt R Th first and th scond intgrals ar indpndnt of f and θ > th rlvant part of LLF to b optimizd is dt r t ~ x * t dt Λ r; f, θ R T { r t ~ x * t } dt Timo I. Laaso and Stfan Wrnr Pag 8
DA: ML Estimation Th intgral can b xprssd as Timo I. Laaso and Stfan Wrnr Pag 9 whr z is th intgral sampld at t T + τ and L T T is th obsrvation intrval Λ ~ * R, ; T dt t x t r f θ r R L j z a θ R T ft j j dt T t h t r a τ π θ
DA: ML Estimation By dnoting L a z Z jφ th LLF can b xprssd as Λ r; f, θ R and th corrsponding LF PDF is Z { j φ θ Z } cos φ θ p r; f, θ Z C xp cos φ θ Timo I. Laaso and Stfan Wrnr Pag
DA: ML Estimation... Th phas θ is a nuisanc paramtr and should b liminatd. This can b accomplishd by avraging th PDF ovr [,π]: Pag cos xp, ; ; Z CI d Z C d f p f p θ θ φ π θ θ π π π r r whr I x is th modifid Bssl function of ordr zro: θ π π θ d x I x cos
DA: ML Estimation Bssl function of first ind, ordr zro bssli,x in Matlab Th function is monotonic for positiv valus, and it is only ncssary to maximiz th argumnt of I x 7 6 5 4 3 3 4 5 6 x
DA: ML Estimation... I x is an vn, parabola-li function and monotonic for positiv valus. As th argumnt is positiv in our cas, w can conclud that f max arg max r f Th function to b maximizd is thus with s p. 3 z t G f Z { p ; f } arg max{ Z } L a z jπfξ r ξ ht ξ t dξ f Timo I. Laaso and Stfan Wrnr Pag 3
Exampl of Gf ; fo /; rx xpj**pi*fo*[:-]; xx -.5:.:.5; for ii :lngthxx yii absrx * xp- j**pi*xxii*[:-]'; nd 4
Exampl of Gf Location of th maximum dpnds on th sampling grid -.49:.:.49 9 8 7 6 5 4 3 -.5 -.4 -.3 -. -....3.4.5 Frquncy offst R.W. 5
Exampl of Gf Estimation from a noisy signal. Shiftd max. valu 9 8 7 6 5 4 3 -.5 -.4 -.3 -. -....3.4.5 Frquncy offst R.W. 6
DA: ML Estimation Maximization of Gf: Data symbols must b nown as originally assumd o closd-form solution availabl in gnral Us sarch algorithms instad Problm: svral local maxima > Good initial stimat ndd Timo I. Laaso and Stfan Wrnr Pag 7
DA: Practical algorithms
DA: Practical algorithms Lt us ma simplifying assumptions:. Th Tx and Rx filtrs.g. root raisd cosin filtrs -> raisd cosin filtr form a yquist puls. Data symbols ar from phas shift-ying PSK constllation: a jα π, α m, m,,,..., M M 3. Frquncy offst f is small compard to th symbol rat - Togthr with. this allows th assumption of zro intr-symbol intrfrnc ISI - Typically, uncompnsatd CFOs of th ordr of -% of th symbol rat alrady caus ISI Pag 9
DA: Practical algorithms... Rcivd signal: jπft+ θ r t a iht i Matchd filtring: t it y r t ht t T τ dt jθ a i whr th matchd-filtrd nois is i jπft ht t it τ + n w t ht t T τ dt τ h w t T t T τ dt + n Timo I. Laaso and Stfan Wrnr Pag 3
DA: Practical algorithms... Bcaus f << /T, w can approximat : Timo I. Laaso and Stfan Wrnr Pag 3 so that th MF output bcoms: Data modulation is rmovd by multiplying y with a * T T τ τ τ π π + it t h it t h it f j ft j [ ]. n a y T f j + + + θ τ π [ ] [ ] ' n n a a a y a z T f j T f j + + + + + + θ τ π θ τ π This signal is usd for th dsign of th frquncy stimation algorithms that follow
DA: Practical algorithms ot: Hr T rfrs to symbol rat and {a } is a nown pilot/training squnc In cas unmodulatd pilot ton is availabl a and T rfrs to sampling rat This approach dos not wor aftr FM dmodulation by th discriminator Why? 3
DA: Algorithm Kay 989 Proposd by S. Kay, A fast and accurat singl frquncy stimator, IEEE Trans. on ASSP, Dc. 989. Rarrang Timo I. Laaso and Stfan Wrnr Pag 33 whr For high E s /, th random variabls φ ar approximatly indpndnt, zro-man and Gaussian [ ] [ ] ' T f j T f j n z φ θ τ π θ τ π ρ + + + + + + [ ] θ τ π φ ρ + + + ' T f j j n
DA: Algorithm Kay 989 Approximation at high SR, i.. low nois, ndd to rach th Gaussian assumption in th prvious pag.9.8.7.6 ytan - x yx.5.4.3.....3.4.5.6.7.8.9 x R.W. 34
DA: Algorithm Kay 989... Considr th product of conscutiv sampls th othr conjugatd: z z * ρ ρ j [ πf T + τ + θ + φ ] j[ πf T + τ + θ + φ ] ρ ρ j [ πft + φ φ ] Th argumnt can b considrd as a noisy masurmnt for frquncy: { z z * } πft + φ φ v arg Pag 35
DA: Algorithm Kay 989... Undr th Gaussian assumption, th PDF pv;f can b xprssd in closd form and solvd for th MLE. Th rsult is L fˆ γ arg{ z z * } πt whr γ is a smoothing function: 3 L L γ,,,..., L L L Pag 36
DA: Algorithm Covarianc matrix of vctor v is tri-diagonal Probability dnsity function is givn by Solving R - givs th wights γ 9..7 37
Kay s algorithm applid to Mis calibraton signal @5MHz Uniform wighting simply calculats th man of th angls Kay s wighting is mor stabl w.r.t. th numbr of sampls than th uniform wighting although th diffrncs ar small 9..7 38
DA: Algorithm Fitz 99 Proposd by M. Fitz, Planar filtrd tchniqus for burst mod carrir synchronization, GLOBECOM 9 confrnc, Phonix, Arizona, Dc. 99. Dfin Using L R m z z * m, m L L m z a m y [ πf T + τ + θ ] j + n' yilds R m j πmft + n'' m whr n m is a zro-man nois componnt. Timo I. Laaso and Stfan Wrnr Pag 39
DA: Algorithm Fitz 99 Th frquncy stimat could b solvd from th abov xprssion as fˆ arg{ R m } πmt Howvr, du to th mod π natur of th arg opration, th rsult may b rronous. A bttr stimat is obtaind by avraging. Dfin { R m } πmft m arg Timo I. Laaso and Stfan Wrnr Pag 4
DA: Algorithm Fitz 99 Timo I. Laaso and Stfan Wrnr Pag 4 Undr crtain assumptions < / f MAX T whr f MAX is th max frquncy rror, it holds that which yilds { } m m m mft m R m arg π { } arg + ft m R m π { } + m m R T f arg ˆ π
DA Algorithm comparison Fundamntal diffrnc btwn Kay and Fitz: Kay oprats by first applying th hard arg arctan nonlinarity and thn smooths/wights th rsult Fitz applis smoothing/filtring also bfor nonlinarity Fitz is lss pron to rrors and provids mor accurat rsults in low SR conditions c.f. ML stimation of unnown phas of a sinusoid with nown frquncy: angl of sampl mans instad of sampl man of angls Pag 4
DA Algorithm comparison Prformanc comparison QPSK systm, α.5 xcss bandwidth roll-off factor of root raisd cosin puls, L 3 ormalizd Frq. Error Varianc log scal Fitz Kay CRB 8 E S / db Figur shows that both algorithms attain th Cramr-Rao bound CRB at high SR For low SR, th prformanc of Kay s alg. bras down 8 s Timo I. Laaso and Stfan Wrnr Pag 43
DA: Algorithm 3 Luis & Rgiannini Timo I. Laaso and Stfan Wrnr Pag 44 + m fmt j m m fmt j m m n m R " π π Rcall ow avrag Using '' n m R mft j + π ft j m fmt j ft ft sin sin + π π π π
DA: Algorithm 3 Luis & Rgiannini and th fact that for f /T yilds sin πft sin πft > f arg + R m π T m Accuracy dgrads slowly as dcrass whil stimation rang gts widr f /T Pag 45
Fitz and L&R algorithms applid to Mis signal Fitz Luis&Rgiannini 46
DA Algorithms
DA algorithm Proposd by Chuang & Sollnbrgr 99 Whn th data modulation cannot b liminatd by using training signals or dcision fdbac, othr tchniqus must b usd Assum a small nough frquncy rror f << /T so that th matchd filtr MF output can again b approximatd as y [ πf T + τ + θ ] j a + n. jα π MPSK signal a, α m, m,,,..., M M Us unit-magnitud proprty of MPSK signals to rmov data modulation Pag 48
DA algorithms For vry MPSK symbol it holds that so that y M M [ ] a j [ πmf T + τ + Mθ ] + n'. ow w can us th sam ida as in th DA algorithms: M jπmft [ y y * ] + n'' whr th nois componnt n is again assumd to b zro-man Timo I. Laaso and Stfan Wrnr Pag 49
DA algorithms... Bttr stimats ar obtaind by smoothing: L L M jπmft [ y y * ] + Stting th avragd nois trm to zro and solving for th frquncy givs th stimat L L n'' fˆ ic and simpl! L arg πmt Limitation: f << /MT [ y y * ] M Timo I. Laaso and Stfan Wrnr Pag 5
DA algorithm prformanc Prformanc analysis QPSK systm, xcss bandwidth α.5, L ormalizd Frq. Error Varianc log scal Chuang t al. CRB 5 E S / db 3 Th Cramr-Rao Bound CRB is not clos vn for high SR Usful for coars stimation in bad conditions Timo I. Laaso and Stfan Wrnr Pag 5
Chuang&Sollnbrg with Mis signal Known data uss angl of th autocorrlation with lag QPSK applis yy*- 4 Th diffrnc in th stimatd valu is significant CFO 385 38 375 37 365 36 355 35 Known data C&S assuming QPSK 345 3 4 5 6 7 8 #sampls 9..7 5
Summary W discussd. Basic systm assumptions. DA: ML stimation of carrir frquncy 3. DA: Practical algorithms 4. DA stimation algorithms Timo I. Laaso and Stfan Wrnr Pag 53