Frequency Correction

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1 Chaptr 4 Frquncy Corrction Dariush Divsalar Ovr th yars, much ffort has bn spnt in th sarch for optimum synchronizion schms th ar robust and simpl to implmnt [1,2]. Ths schms wr drivd basd on maximum-liklihood (ML) stimion thory. In many cass, th drivd opn- or closd-loop synchronizrs ar nonlinar. Linar approximion provids a usful tool for th prdiction of synchronizr prformanc. In this smi-tutorial chaptr, w labor on ths schms for frquncy acquisition and tracking. Various low-complxity frquncy stimor schms ar prsntd in this chaptr. Th thory of ML stimion provids th optimum schms for frquncy stimion. Howvr, th drivd ML-basd schm might b too complx for implmntion. On approach is to us thory to driv th bst schm and thn try to rduc th complxity such th th loss in prformanc rmains small. Organizion of this chaptr is as follows: In Sction 4.1, w show th drivion of opn- and closd-loop frquncy stimors whn a pilot (rsidual) carrir is availabl. In Sction 4.2, frquncy stimors ar drivd for known da-moduld signals (da-aidd stimion). In Sction 4.3, non-da-aidd frquncy stimors ar discussd. This rfrs to th frquncy stimors whn th da ar unknown th rcivr. 4.1 Frquncy Corrction for Rsidual Carrir Considr a rsidual-carrir systm whr a carrir (pilot) is availabl for tracking. W considr both additiv whit Gaussian nois (AWGN) and Rayligh fading channls in this sction. 63

2 64 Chaptr Channl Modl Lt r c [k] b th kth rcivd complx sampl of th output of a lowpass filtrd pilot. Th obsrvion vctor r c with componnts r c [k]; k =0, 1,,N 1 can b modld as r c [k] =A j(2π fktsθc) ñ[k] (4 1) whr th r c [k] sampls ar takn vry T s sconds (sampling r of 1/T s ). In th abov quion, ñ[k], k =0, 1,,N 1, ar indpndnt, idntically distributd (iid) zro-man, complx Gaussian random variabls with varianc σ 2 pr dimnsion. Th frquncy offst to b stimd is dnotd by f, and θ c is an unknown initial carrir phas shift th is assumd to b uniformly distributd in th intrval [0, 2π) but constant ovr th N sampls. For an AWGN channl, A = 2P c is constant and rprsnts th amplitud of th pilot sampls. For a Rayligh fading channl, w assum A is a complx Gaussian random variabl, whr A is Rayligh distributd and arg A = tan 1 ( Im(A)/R(A) ) is uniformly distributd in th intrval [0, 2π), whr Im( ) dnots th imaginary opror and R( ) dnots th ral opror Optimum Frquncy Estimion ovr an AWGN Channl W dsir an stim of th frquncy offst f basd on th rcivd obsrvions givn by Eq. (4-1). Th ML stimion approach is to obtain th conditional probability dnsity function (pdf) of th obsrvions, givn th frquncy offst. To do so, first w obtain th following conditional pdf: P ( r c f,θ c )=C 0 (1/2σ2 )Z (4 2) whr C 0 is a constant, and Z = r c [k] A j(2π fktsθc) 2 (4 3) Dfin Thn Z can b rwrittn as Y = r c [k] j(2π fkts) (4 4)

3 Frquncy Corrction 65 Z = r c [k] 2 2AR(Y jθc ) A 2 (4 5) Th first and th last trms in Eq. (4-5) do not dpnd on f and θ c. Dnoting th sum of ths two trms by C 1, thn Z can b writtn as Z = C 1 2A Y cos(θ c arg Y ) (4 6) Using Eq. (4-6), th conditional pdf of Eq. (4-2) can b writtn as [ ] A P ( r c f,θ c )=C 2 xp σ 2 Y cos(θ c arg Y ) (4 7) whr C 2 = C (C1/2σ2). Avraging Eq. (4-7) ovr θ c producs ( ) A Y P ( r c f) =C 2 I 0 σ 2 (4 8) whr I 0 ( ) is th modifid Bssl function of zro ordr and can b rprsntd as I 0 (x) = 1 2π 2π 0 xcos(ψ) dψ (4 9) Sinc I 0 (x) is an vn convx cup function of x, maximizing th right-hand sid of Eq. (4-8) is quivalnt to maximizing Y. Thus, th ML mtric for stiming th frquncy offst can b obtaind by maximizing th following mtric: λ( f) = Y = r c [k] j(2π fkts) (4 10) Optimum Frquncy Estimion ovr a Rayligh Fading Channl W dsir an stim of th frquncy offst f ovr a Rayligh fading channl. Th ML approach is to obtain th conditional pdf of th obsrvions,

4 66 Chaptr 4 givn th frquncy offst. To do so, first w start with th following conditional pdf: P ( r c A, f,θ c )=C 0 (1/2σ2 )Z (4 11) whr C 0 is a constant, and Z and Y ar dfind as in Eqs. (4-3) and (4-4). Sinc A is now a complx random variabl, thn Z can b rwrittn as Z = r c [k] 2 2R(YA jθc ) A 2 (4 12) Th first trms in Eq. (4-12) do not dpnd on A. Avraging th conditional pdf in Eq. (4-11) ovr A, assuming th magnitud of A is Rayligh distributd and its phas is uniformly distributd, w obtain ( ) C4 P ( r c f,θ c )=C 3 xp Y 2 2σ2 (4 13) whr C 3 and C 4 ar constants, and Eq. (4-13) is indpndnt of θ c. Thus, maximizing th right-hand sid of Eq. (4-13) is quivalnt to maximizing Y 2 or quivalntly Y. Thus, th ML mtric for stiming th frquncy offst can b obtaind by maximizing th following mtric: λ( f) = Y = r c [k] j(2π fkts) (4 14) which is idntical to th obtaind for th AWGN channl cas Opn-Loop Frquncy Estimion For an opn-loop stimion, w hav f = argmax λ( f) (4 15) f Howvr, this oprion is quivalnt to obtaining th fast Fourir transform (FFT) of th rcivd squnc, taking its magnitud, and thn finding th maximum valu, as shown in Fig. 4-1.

5 Frquncy Corrction 67 r [k] c FFT Find Max f Fig Opn-loop frquncy stimion, rsidual carrir Closd-Loop Frquncy Estimion Th rror signal for a closd-loop stimor can b obtaind as = λ( f) (4 16) f W can approxim th driviv of λ( f) for small ε as ε) λ( f ε) λ( f) =λ( f f 2ε (4 17) Thn, w can writ th rror signal as (in th following, any positiv constant multiplir in th rror signal rprsntion will b ignord) = Y ( f ε) Y ( f ε) (4 18) whr Y ( f ε) = r c [k] j(2π fkts) j(2πεkts) (4 19) Th rror-signal dtctor for a closd-loop frquncy corrction can b implmntd basd on th abov quions. Th block diagram is shown in Fig. 4-2, whr in th figur α = j2πεts. Now rhr than using th approxim driviv of λ( f), w can tak th actual driviv of λ 2 ( f) = Y 2, which givs th rror signal = Im(Y U) (4 20) whr

6 68 Chaptr 4 Clos Evry N Sampls r [k] c α Dlay T s S j 2π fkt s α Dlay T s Clos Evry N Sampls Fig Approxim rror signal dtctor, rsidual carrir. U = r c [k]k j(2π fkts) (4 21) Not th th rror signal in Eq. (4-20) can also b writtn as = Im(Y U)= Y ju 2 Y ju 2 (4 22) or for a simpl implmntion w can us = Y ju Y ju (4 23) Th block diagram of th rror signal dtctor basd on Eq. (4-23) is shown in Fig Th corrsponding closd-loop frquncy stimor is shown in Fig Th dashd box in this figur and all othr figurs rprsnts th fact th th hard limitr is optional. This mans th th closd-loop stimors can b implmntd ithr with or without such a box Approximion to th Optimum Error Signal Dtctor. Implmntion of th optimum rror signal dtctor is a littl bit complx. To rduc th complxity, w not th

7 Frquncy Corrction 69 Clos Evry N Sampls N 1 Σ x k = 0 k r [k] c j 2π fkt s xk N 1 Σ kx k = 0 k Clos Evry N Sampls Σ j Fig Exact rror signal dtctor, rsidual carrir. r [k] c j 2π fkt s xk N 1 Σ xk k = 0 N 1 Σ kx k = 0 k Clos Evry N Sampls j Σ Numrically Controlld Oscillor (NCO) Loop Filtr Gain δ 1 1 Fig Closd-loop frquncy stimor, rsidual carrir. whr =Im(Y U)= i=0 Im(X 0,iX i1,() ) = C 5 Im(X 0,(N/2)1 X (N/2),) X m,n = (4 24) n r c [k] j(2π fkts) (4 25) k=m Th closd-loop frquncy stimor with th approxim rror signal dtctor givn by Eq. (4-24) is shown in Fig Th paramtrs N w = N/2 (th

8 70 Chaptr 4 r [k] c j 2π fkt s Σ () k=n/2 Upd Dlay NT s 2 Im {} NCO Loop Filtr Gain δ 1 1 Upd Microcontrollr (µc) Fig Low-complxity closd-loop frquncy corrction, rsidual carrir. numbr of sampls to b summd, i.., th window siz) and δ (gain) should b optimizd and updd aftr th initial start to prform both th acquisition and tracking of th offst frquncy Digital Loop Filtr. Th gain δ th was shown in th closd-loop frquncy-tracking systm is usually part of th digital loop filtr. Howvr, hr w spar thm. Thn th digital loop filtr without gain δ can b rprsntd as F (z) =1 b 1 z 1 (4 26) Th corrsponding circuit for th digital loop filtr is shown in Fig Now in addition to th gain δ, th paramtr b also should b optimizd to achiv th bst prformanc Simulion Rsults. Prformanc of th closd-loop frquncy stimor in Fig. 4-5 was obtaind through simulions. First, th acquisition of th closd-loop stimor for a 10-kHz frquncy offst is shown in Fig Nxt th standard dviion of th frquncy rror vrsus th rcivd signal-to-nois rio (SNR) for various initial frquncy offsts was obtaind. Th rsults of th simulion ar shown in Fig. 4-8.

9 Frquncy Error, f f Frquncy Corrction 71 Input Output z 1 b Fig Loop filtr for frquncy-tracking loops SNR = 10.0 db Initial Intgrion Window 32 Sampls Subsqunt Itgrion Window = 32 2 i Frquncy Offst = 10,000 Hz Sampling R = 1 Msps Initial Upd Aftr Sampls Subsqunt Upd = i Initial Dlta = 1024 Hz Subsqunt Dlta = 1024/2 i TIME (ms) Fig Frquncy acquisition prformanc.

10 Standard Dviion of Frquncy Error (Hz) 72 Chaptr f = 15,000 Hz f = 10,000 Hz f = 5,000 Hz f = 100 Hz Initial Window = 32 Sampls Subsqunt Window = 32 2 i Max Window = 256 Initial Upd = 256 Subsqunt Upd = i Initial δ =1024 Hz Subsqunt δ =1024/2 i Hz Min δ = 2 Hz Sampl SNR (db) Fig Standard dviion of frquncy rror. 4.2 Frquncy Corrction for Known Da-Moduld Signals Considr a da-moduld signal with no rsidual (supprssd) carrir. In this sction, w assum prfct knowldg of th symbol timing and da (daaidd systm). Using again th ML stimion, w driv th opn- and closdloop frquncy stimors Channl Modl W start with th rcivd basband analog signal and thn driv th discrt-tim vrsion of th stimors. Lt r(t) b th rcivd complx wavform, and a i b th complx da rprsnting an M-ary phas-shift kying (M-PSK) modulion or a quadrur amplitud modulion (QAM). Lt p(t) b th transmit puls shaping. Thn th rcivd signal can b modld as r(t) = a i p(t it ) j(2π ftθc) ñ(t) (4 27) i=

11 Frquncy Corrction 73 whr T is th da symbol durion and ñ(t) is th complx AWGN with twosidd powr spctral dnsity N 0 W/Hz pr dimnsion. Th conditional pdf of th rcivd obsrvion givn th frquncy offst f and th unknown carrir phas shift θ c can b writtn as p( r f,θ c )=C 6 (1/N0) r(t) i= aip(tit )j(2π ftθc) 2 dt (4 28) whr C 6 is a constant. Not th r(t) i= 2 2 a i p(t it ) j(2π ftθc) = r(t) 2 i= i= a i p(t it ) { R a i r(t)p(t it ) j(2π ftθc)} (4 29) 2 Th first two trms do not dpnd on f and θ c. Thn w hav p( r f,θ c )=C 7 (2/N0)R { i= a i zi( f)jθc } (4 30) whr C 7 is a constant and z i ( f) = (i1)t it Th conditional pdf in Eq. (4-30) also can b writtn as r(t)p(t it ) j(2π ft) dt (4 31) [ ] 2 p( r f,θ c )=C 7 xp Y cos(θ c arg Y ) N 0 (4 32) whr Y = a i z i ( f) (4 33) i= Avraging Eq. (4-32) ovr θ c producs

12 74 Chaptr 4 ( ) 2 P ( r f) =C 8 I 0 Y N 0 (4 34) whr C 8 is a constant. Again, sinc I 0 (x) is an vn convx cup function of x, maximizing th right-hand sid of Eq. (4-34) is quivalnt to maximizing Y or quivalntly Y 2. Thus, th ML mtric for stiming th frquncy offst ovr th N da symbol intrval can b obtaind by maximizing th following mtric: λ( f) = Y = a kz k ( f) (4 35) Opn-Loop Frquncy Estimion For an opn-loop stimion, w hav f = argmax λ( f) (4 36) f but this oprion is quivalnt to multiplying th rcivd signal by j(2π ft), passing it through th mchd filtr () with impuls rspons p(t), and sampling th rsult t =(k 1)T, which producs th squnc of z k s. Nxt, sum th z k s, tak its magnitud, and thn find th maximum valu by varying th frquncy f btwn f max and f max, whr f max is th maximum xpctd frquncy offst. Th block diagram to prform ths oprions is shown in Fig Closd-Loop Frquncy Estimion Th rror signal for closd-loop tracking can b obtaind as = λ( f) (4 37) f W can approxim th driviv of λ( f) for small ε as in Eq. (4-17). Thn w can approxim th rror signal as = Y ( f ε) Y ( f ε) (4 38)

13 Frquncy Corrction 75 r (t) j 2π f t p (t ) Clos t = (k1)t a k Σ () Find Max f Fig Opn-loop frquncy stimion for supprssd carrir, known da. whr Y ( f ε) = a kz k ( f ε) (4 39) Th rror signal dtctor for th closd-loop frquncy corrction is implmntd using th abov quions and is shown in Fig In th figur, DAC dnots digital-to-analog convrtr. Now again, rhr than using th approxim driviv of λ( f), w can tak th driviv of λ 2 ( f) = Y 2 to obtain th rror signal as = Im(Y U) (4 40) and U = a ku k ( f) (4 41) whr u i ( f) = (i1)t it r(t)tp(t it ) j(2π ft) dt (4 42) Thus, u k ( f) is producd by multiplying r(t) by j2π ft and thn passing it through a so-calld driviv mchd filtr (D) also calld a frquncymchd filtr (F) with impuls rspons tp(t), and finally sampling th rsult of this oprion t =(k 1)T. Not th th rror signal in Eq. (4-40) also can b writtn as

14 76 Chaptr 4 r (t) p (t ) Clos t = (k1)t a k Σ () j 2πεt p (t ) j 2πεt Clos t = (k1)t a k Σ () 1 1 j 2π f t Voltag-Controlld Oscillor (VCO) k DAC Loop Filtr Fig Error signal dtctor and closd-loop block diagram for supprssd carrir, known da. = Im(Y U)= Y ju 2 Y ju 2 (4 43) or, simply, w can us = Y ju Y ju (4 44) Th block diagram of th closd-loop frquncy stimor using th rror signal dtctor givn by Eq. (4-40) is shown in Fig Similarly, th block diagram of th closd-loop frquncy stimor using th rror signal dtctor givn by Eq. (4-44) is shown in Fig Th closd-loop frquncy stimor block diagrams shown in this sction contain mixd analog and digital circuits. An all-digital vrsion of th closdloop frquncy stimor in Fig opring on th rcivd sampls r[k] is shown in Fig In th figur, p k rprsnts th discrt-tim vrsion of th puls shaping p(t). W assum th thr ar n sampls pr da symbol durion T. An all-digital vrsion of othr closd-loop stimors can b obtaind similarly.

15 Frquncy Corrction 77 r (t) p (t) Clos t = (k1)t a k Σ ( ) Im() D j 2π f t tp (t ) Clos t = (k1)t a k Σ () VCO DAC Loop Filtr Fig Closd-loop stimor with rror signal dtctor for supprssd carrir, known da, Eq. (4-40). r (t) p (t ) Clos t = (k1)t a k Σ () D tp (t) Clos t = (k1)t a k Σ () j S 1 1 j 2π f t VCO DAC Loop Filtr Fig Closd-loop stimor with rror signal dtctor for supprssd carrir, known da, Eq. (4-44).

16 78 Chaptr 4 r [k] p k Σ i=0 ( ) Clos Evry T = nt s a i Im() D kp k Σ () i=0 1 a i * 1 j 2π fkt NCO Loop Filtr Fig All-digital closd-loop frquncy stimor for supprssd carrir, known da. 4.3 Frquncy Corrction for Moduld Signals with Unknown Da Considr again a da-moduld signal with no rsidual (supprssd) carrir. In this sction, w assum prfct timing but no knowldg of th da (non-da-aidd systm). Again using th ML stimion, w driv th opnand closd-loop frquncy stimors. In Sction 4.2, w obtaind th conditional pdf of th rcivd obsrvion givn th frquncy f and da squnc a. W rp th rsult hr for clarity: ( ) 2 P ( r f,a) =C 8 I 0 Y N 0 (4 45) whr Y = a i z i ( f) (4 46) i= and z i ( f) = (i1)t it r(t)p(t it ) j(2π ft) dt (4 47)

17 Frquncy Corrction 79 Now w hav to avrag Eq. (4-46) ovr a. Unfortunly, implmntion of this avraging is too complx. Instad, first w approxim th I 0 (x) function as ( ) 2 I 0 Y = 1 1 N 0 N0 2 Y 2 (4 48) Now w nd only to avrag Y 2 ovr th da squnc a as E { Y 2} 2 = E a kz k ( f) = i=0 E{a ka i }z k ( f)z i ( f) = C a z k ( f) 2 (4 49) whr C a = E{ a k 2 } and th a k s ar assumd to b zro man and indpndnt. Thus, stiming th frquncy offst ovr th N da symbol intrval can b obtaind by maximizing th following mtric: λ( f) = z k ( f) 2 (4 50) Opn-Loop Frquncy Estimion For opn-loop stimion, w hav f = argmax λ( f) (4 51) f Howvr, this oprion is quivalnt to multiplying th rcivd signal by j(2π ft), passing it through a mchd filtr with impuls rspons p(t), and sampling th rsult t =(k 1)T, which producs th squnc of z k s. Nxt, tak th magnitud squar of ach z k, prform summion, and thn find

18 80 Chaptr 4 th maximum valu by varying th frquncy f btwn f max and f max, whr f max is th maximum xpctd frquncy offst. Th block diagram to prform ths oprions is shown in Fig Closd-Loop Frquncy Estimion Th rror signal for closd-loop tracking can b obtaind as = λ( f) (4 52) f W can approxim th driviv of λ( f) for small ε as in Eq. (4-17). Thn, w can approxim th rror signal as = { z k ( f ε) 2 z k ( f ε) 2 } (4 53) Th rror signal dtctor for th closd-loop frquncy corrction is implmntd using th abov quions, and it is shown in Fig Now again, rhr than using th approxim driviv of λ( f), w can tak th driviv of λ( f) = z k( f) 2 and obtain th rror signal as whr u i ( f) = = (i1)t it Im{z k( f)u k ( f)} (4 54) r(t)tp(t it ) j(2π ft) dt (4 55) r (t) j 2π f t p (t ) Clos t = (k1)t 2 Σ () Find Max f Fig Opn-loop frquncy stimion for supprssd carrir, unknown da.

19 Frquncy Corrction 81 r (t) p (t) Clos t = (k1)t 2 Σ ( ) j 2πεt p (t) j 2πεt Clos t = (k1)t j 2π f t VCO DAC k Loop Filtr Fig Error signal dtctor and closd-loop block diagram for supprssd carrir, unknown da. Not th th rror signal in Eq. (4-54) also can b writtn as = { z k ( f) ju k ( f) 2 z k ( f)ju k ( f) 2 } (4 56) Th block diagram of th closd-loop frquncy stimor using th rror signal dtctor givn by Eq. (4-54) is shown in Fig Similarly, th block diagram of th closd-loop frquncy stimor using th rror signal dtctor givn by Eq. (4-56) is shown in Fig Th closd-loop frquncy stimor block diagrams shown in this sction contain mixd analog and digital circuits. An all-digital vrsion of th closdloop frquncy stimor in Fig opring on th rcivd sampls r[k] is shown in Fig All-digital vrsions of othr closd-loop stimors can b obtaind similarly.

20 82 Chaptr 4 r (t) p (t) D Clos t = (k1)t Σ ( ) Im() tp (t ) Clos t = (k1)t 1 1 j 2π f t VCO DAC Loop Filtr Fig Closd-loop stimor with rror signal dtctor for supprssd carrir, unknown da, Eq. (4-54). r (t) p (t) D tp (t) Clos t = (k1)t Clos t = (k1)t j S 2 2 Σ ( ) 1 1 j 2π f t VCO DAC Loop Filtr Fig Closd-loop stimor with rror signal dtctor for supprssd carrir, unknown da, Eq. (4-56).

21 Frquncy Corrction 83 r [k] p k D Clos Evry T = nt s Σ ( ) i=0 Im( ) j 2π f k T k p k 1 1 NCO Loop Filtr Fig All-digital closd-loop frquncy stimor for supprssd carrir, unknown da. Rfrncs [1] H. Myr and G. Aschid, Synchronizion in Digital Communicions, Nw York: John Wily and Sons Inc., [2] H. Myr, M. Monclay, and S. A. Fchtl, Digital Communicion Rcivrs, Nw York: John Wily and Sons Inc., 1998.

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr