5..3 Dciio-Dirctd Pha Trackig [P 6..4] 5.-1 Trackr commoly work o radom data igal (plu oi), o th kow-igal modl do ot apply. W till kow much about th tructur o th igal, though, ad w ca xploit it. Coidr rial tramiio. I Proaki otatio, w rciv j φ rt () = I gt ( T) + zt () t ( ; φ, I) ( z() t i whit oi) o th log-liklihood to b maximizd i (p. 5.1-4) ovr a obrvatio tim o K ymbol i jφ ΛL( φ ) = R r() t () t dt K jφ = R I r( t) g ( t T) dt = 0 MF output y K jφ = R Iy = 0 ad th max i achivd at 1 K φ ML = arg yi. = 0 5.-1
5.-13 I a trackr, w uually work with th gradit o th log-liklihood, ad approach th optimum i vral tp, adjutig th pha i th dirctio o dcraig pha rror (icraig log-liklihood). A poitiv drivativ impli φ =φ φ> 0 drivativ impli φ =φ φ< 0, o rduc proportioal to pha rror φ ar th optimum., o icra φ. Ngativ φ. Th drivativ i approx. o Th gradit ca b xprd a th um o pr-ymbol cotributio K K dλl( φ) jφ jφ = Im y Im I yi = dφ = 0 = 0 v λ ( ) L o W ca mak a corrctio atr vry ymbol uig it trm i th um a a oiy gradit timat: λ ( ) K φ ( ) + oi. Not φ ( ) ad L p φ ( ) ow dpd o ymbol tim. 5.-13
5.-14 o Thi i till a data-aidd (DA) trackr, ic I i aumd kow. To track pha through data gmt, u dciio itad, or a dciio-dirctd (DD) trackr. jφ ( ) Im ( ) Im λ L = y I = v I o Ratioal or DD: th data i a uiac param i pha timatio ( jφ ΛL φ, I) = R yi ad w ca dal with it by joit timatio jφ φ JE = arg max max R y I φ, which u dciio. I Hr i th DSP-bad DD pha trackr. Not icomig igal i aumd to hav a tatic pha φ, ad DVCO output pha φ ( ) tri to match it. Pha rror i φ ( ) =φ φ ( ). W orm a oiy drivativ timat. 5.-14
How do th opratio o Rotatd MF output i vi crat a pha rror timat? ( φ φ ( ) ) jφ ( ) j jφ ( ) = = + ( ) v y I z 5.-15 jφ ( ) = I + z ( ) o A a pictur: So jφ ( ) ( ) = + vi II z I jφ ( ) = E + ζ ( ) + glitch rom dciio rror ζ σ = NE 0 o Ad ( ) ( ) Im jφ = i φ( ) φ( ). Or u arg vi. 5.-15
Hr th ovrall tructur or BPSK. o Form λ = v I = [ v ] ( ) Im Im I (ic I i ral), ad u th L dciio I g R[ v ] = 5.-16 Thi i th trackr rom p. 5.-14, rdraw or BPSK. It ca track lowly varyig tru pha φ. Not th 180 ambiguity du to dciio dirctio: o I φ uddly jumpd by 180, it would b quivalt to t () t (). o Th j + ad I I v I φ z, o which lav th igal part o λ ( ) uchagd. L So 180 dirc ar iviibl. Th trackr ca covrg to a olid timat, but with all th dciio ivrtd. Now look at Dciio Dirctd Trackr or BPSK, i dmo. 5.-16
Th drivativ ad pha rror timat ( ) Im λ L = v I Kpφ ( ) + oi work or othr modulatio, too. jφ ( ) ( ) ( ) Im ( ) λ L = I + z I, ad i o dciio rror, 5.-17 ( ) = E i φ ( ) + ζ ( ) th imagiary part o th oi. i ζi σ = E N 0 o Hr i it or QPSK For QPSK or 4PSK it ha a our-old pha ambiguity (multipl o 90 ). For 8PSK, it ight-old ambiguity. o For 16 QAM, it baically our-old, but thr ar om wakr lock poit, too. S Bigham book o modm dig. Dirt poit hav dirt amplitud, which act th rror timat. 5.-17
How wll do th trackr work? What i th tady tat pha rror variac? How quickly do it pull i? Rpod to chag? Blow, a covtioal liarizd aalyi, or mall φ ( ). o Start by rdrawig th loop, abtractd to pha, itad o complx. 5.-18 o Two o vral choic or PED rom p. 5.-14: U Im vi jφ ( ) Im ( λ ) L = y So K p E I E φ ( ) +ζ ( ) i = ad ξ=ζ i EN ad σ ξ =σ ζ = i 0 U arg vi jφ( ) jφ( ) yi arg yi = arg E So K p = 1 ad ξ =ζ i E ad σ ζ N i 0 1 ξ E E σ = = = γ =φ ( ) +ζi( ) E 5.-18
o Choic o loop iltr... For 1 t ordr loop: H ( z) = K Aalyz thi o. For d ordr loop: H ( z) = K + K z 1 1 z 5.-19 o Firt, th igal compot. Trar uctio rlatig iput pha φ to pha rror φ ( ) i H Φ( z) 1 1 z ( z) = = = Φ( z) K K z 1 ( 1 K K ) z 1+ 1 z p p DC gai chag. H (1) i zro. Good tady tat rror i zro i φ do t Impul rpo 1 KpK h( ) = ( ) ( 1 KpK ), 0 1 K K δ 1 K K p p ha tim cotat ( ) c (i ymbol tim). Calculat it: c 1 1 KpK = o c = l 1 K K ( KpK ) p 5.-19
5.-0 o Nxt, th oi compot. Trar uctio rlatig PED oi ξ ( ) to pha rror φ ( ) i H z K z p ( p ) K z Φ( z) ( z) = = 1 = Ξ( z) K K z 1 1 K K z 1+ 1 z ad th corrpodig uit pul rpo i ( ) K ( ( ) ( 1 ) ) h = δ KpK 1, K 0 Th rultig pha rror atr may tim cotat i φ ( ) = h ( k) ξ( k) k = 0 o th gric tady-tat pha jittr variac i φ () ξ h i ξ k= 0 K p p σ =σ =σ K ( K K ) 5.-0
o Now coidr th PED that u arg vi. With K p = 1 ad 1 ξ σ = γ, w hav: 5.-1 * tim cotat (i ymbol tim) c = 1 K * pha rror variac W that K K σ = 4γ 1 φ γ ( K ) (mall K ). K cotrol variac dirctly ad tim cotat ivrly. Icraig γ dcra jittr variac ivrly. o Nxt, coidr th PED that u Im vi. It ha K p = E ad ξ σ = EN. Thror: 0 * tim cotat c 1 = - i igal gt trogr, loop pd up! E K * pha rror variac EN K 0 NK 0 σ φ = E 4 ( E K ) Now tidy it. W d EK to b dimiol i th ubtractio, o mak K = Kl E, whr th loop gai K l i dimiol. Th = 1 c K ad l φ 14 σ = γ. Similar to th othr PED, but it a uiac havig to adjut K i rpo to igal trgth. 5.-1