Pendulums and Elliptic Integrals. 1. Introduction. 2. Where Hence Elliptic Integrals? James A. Crawford = (1) dv d dϕ dt dt dt. = mr dt.

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1 Pedulum ad Elliptic Itegral v.doc Pedulum ad Elliptic Itegral Jame A. Crawford. Itroductio May year ago before the advet of the PC o every dektop age, I became faciated with the deig of LC elliptic filter. A part of that edeavor, I alo became itimately acquaited with elliptic itegral. Havig a equal itrigue for umerical preciio, I foud that computig the elliptic itegral with high accuracy wa very difficult if imple itegratio method like Simpo Rule or Gauia quadrature were reorted to. Thu bega my earch for a preciio method of computatio. Some reader will o doubt be familiar with the olutio path ivolved, but to thoe who are ot, I ivite you to read o.. Where Hece Elliptic Itegral? Elliptic itegral how up i may place, electroic elliptic filter for oe. Oe of the ituatio where people ecouter them firt i i coectio with imple pedulum motio. Figure Claical Pedulum m ma of pedulum R legth of pedulum g acceleratio of gravity (e.g., 9.8 m/ ) α tartig agle If we aume that the pedulum arm itelf i both rigid ad of zero ma, it i coveiet to thik about the motio of the pedulum bob i term of motio alog the fixed radiu R where the agle ϕ i a fuctio of time. The tagetial force perpedicular to R that the weight of the bob create i give by F mg ϕ T () i ( ) From Newto Law of motio, thi tagetial force mut be aociated with a tagetial acceleratio which ca be writte a dv d dϕ dt dt dt T FT mat m m R d ϕ mr dt () Proper attetio to ig for the force ivolved reult i the decribig differetial equatio i term of ϕ give a d ϕ g i ( ϕ ) (3) dt R If the agular extet allowed for the pedulum wig are kept mall, we ca approximate i ( ϕ) which lead to the very imple differetial equatio ϕ R α ϕ d ϕ g ϕ (4) dt R If we ow hypotheize that the olutio to thi differetial equatio i give by ϕ () t Ai ( ω t) ad ubtitute ito (4), we quickly ee that thi i ideed the correct olutio with o PE F T W m g A claical pedulum i how i Figure where LC for iductor-capacitor g ω o (5) R Returig ow to the origial oliear differetial equatio (3), thi ca be purued further by multiplyig both ide of the equatio by which create dθ / dt V. 4 Jame A. Crawford

2 Pedulum ad Elliptic Itegral v.doc d d d i o dt dt dt ϕ ϕ ω ( ϕ) ϕ (6) ad itegratig both ide with repect to time reult i dϕ ωo co( ϕ) k dt (7) where k i a cotat of itegratio. Aumig that the pedulum ha a maximal diplacemet of agle ϕ α, the ϕ' ( α ), ad olvig for the derivative ad takig the poitive root lead to d dt ϕ ω co ( ϕ ) co( α ) o Itegratig oe more time produce (8) dϕ o ( ϕ) co( α ) ω t (9) co The time required for ϕ to icreae from to α i α T R d 4 g co co ϕ ( ϕ ) ( α ) () Uig the idetitie co( ϕ) i ( ϕ/) ad co( α) i ( α/) i () lead to T R g with k i ( α /) a i ( ϕ /) k i ( θ ) α dϕ k i / () ( ϕ ). A ew variable ca be defied from which ϕ dϕ co k co( θ) dθ which upo re-arragemet give ( θ) dθ k i ( ϕ /) ϕ k i ( θ ) kco dϕ co () (3) Subtitutig (3) ito () lead fially to T 4 R g π / dθ k i (4) ( θ ) The itegral ivolved i (4) i a elliptic itegral of the firt kid. With k i ( α /), the itegral i very well behaved becaue k i alway < /. I the cae of elliptic filter uage however, k i ofte very cloe to uity thereby makig umerical evaluatio of (4) coiderably more challegig. Aide: Coervatio of eergy may be ued to quickly arrive at the ame tartig poit repreeted by (8). The chage of potetial eergy that occur from agular poitio α to ϕ ca be equated to the icreae i kietic eergy (ice the bob i mometarily motiole at agular poitio α) a mv mgr co( ϕ ) co( α ) (5) Sice the velocity v mut be tagetial to the arc that i cribed by the bob, at ay itat i time v R dϕ / dt. Subtitutig thi ito (5) lead ( ) directly to (8). The elliptic itegral of the firt kid i geerally preeted a (, ) F k x x dθ (6) k i ( θ ) with the complete elliptic itegral of the firt kid give by F(k,π/). It i eay to how that R T π g (7) i3 4 i3i5 6 + k + k + k + i4 i4i6 V. 4 Jame A. Crawford

3 Pedulum ad Elliptic Itegral v.doc 3 Straight forward viual ipectio of (7) eaily how that the erie i low to coverge whe k i reaoably cloe to uity. 3. Accurate Computatio of the Elliptic Itegral of the Firt Kid Gau Traformatio ca be ued to expad the elliptic itegral (6) ito a expaio where ( ϕ, ) ( ) ( φ, ) F k + k F k (8) a b i+ + k k Re curively Compute : a b ai + bi ab i+ i i Upo Covergece : π π F, k a () Thi expaio ca be repeatedly applied ultimately leadig i the limit to lim F( φp, kp) p π. The expaio geerally coverge to or more decimal place accuracy withi oly a few recurio of (8). The other formula that accompay (8) are the followig: k' k k' k + k' φ arci + i ( + k' ) i( φ ) k ( φ ) (9) I the cae where the complete elliptic itegral of the firt kid i to be computed (i.e., ϕ π / ), a differet et of recurive formula [7] ca be ued to compute the deired reult with eve le effort a give by 4. Compario or Liearized Model Reult with Ideal All of the mathematic are greatly implified if the liearized model repreeted by (4) i ued rather tha the complete oliear model. For the liearized cae, the frequecy of the pedulum motio i exactly computable a (5) ad the pedulum motio i preciely iuoidal. For a very large rage of tartig phae, the pedulum motio i very cloely approximated by a iuoid aumig the time period give by (4). I all but the mot rigorou cae, thi i i all likelihood adequately precie. The appreciatio for the liear differetial equatio repreeted by (4) i quickly appreciated over the oliear differetial equatio (3) whe implicit ad or higher-order umerical olutio of the differetial equatio are deired for greater accuracy. The author ha frequetly ued the ecod-order Gear method [5] with good ucce, but thi formulatio i ot poible with the oliear differetial equatio (3). 5. Numerical Solutio of the Differetial Equatio The differetial equatio (3) olutio may be computed umerically i the time domai. 5. Forward Euler Itegratio Alo referred to a Lade Traformatio Although proe to accuracy ad tability iue, the forward Euler method i ofte ued for V. 4 Jame A. Crawford

4 Pedulum ad Elliptic Itegral v.doc 4 olvig differetial equatio becaue it i extremely imple to ue. The forward Euler method i a explicit itegratio method [5-6]. I thi cae, the timederivative i approximated a () ' t ( + ) ( ) t h t () where the time icremet i give by h. Focuig o the tartig differetial equatio (3), it i imple to re-cat thi ecod-order differetial equatio a a pair of firtorder differetial equatio by defiig leadig to h () ϕ () dϕ () t U t t U dt () how. Error propagatio with the forward Euler method i o poor that the amplitude growth i difficult to avoid. Figure Forward Euler Differetial Equatio Solutio Phae, rad T p Pedulum Swig i Time Domai du g i dt R du U dt ( U ) (3) Time, ec Forward Euler Computed Ideal Subtitutig () ito (3) reult i U+ U h U+ U h g i R U ( U ) (4) where the idex repreet the value of the parameter at time t h where h i the cotat time tep ued. Solvig (4) for the parameter value at the ext time tep + produce gh U U + i U R U U + U h ( ) + + (5) The fact that the forward Euler method i a explicit method reult i oly time-idex value beig o the right ide of the equal ide ad the + (future) timeidex value beig o the left-had ide. The et of differece equatio ca be eaily programmed ad i the cae of R meter ad α 3 degree, the reult i a how i Figure. Due to umerical impreciio eve with h 6mec, the computed olutio lowly grow i amplitude rather tha remaiig cotat-evelope a the ideal olutio 5. Backward Euler Itegratio Backward Euler itegratio i a implicit itegratio method ad a uch, it i ot poible to ue thi method ule the differetial equatio i liearized a i (4). Although thi i a hort-cut path that we wih to avoid, thi path will be coidered i order to how the greater tability propertie of the backward Euler method a compared to the forward Euler method. For the backward Euler method we write or i matrix form U U gh U R U U + hu gh U U R + U + U h Solvig thi for the ext-tep tate-variable value, (6) (7) V. 4 Jame A. Crawford

5 Pedulum ad Elliptic Itegral v.doc 5 Phae, rad. U h U gh U + R U + + gh R (8) Thi et of imultaeou differece equatio ca be programmed very eaily alo leadig to the reult how i Figure 3. I the backward Euler cae, the Figure 3 Backward Euler Differetial Equatio Solutio T p Pedulum Swig i Time Domai Time, ec Forward Euler Computed Ideal Liearized Backward Euler umerical impreciio lead to a decay i the evelope magitude, o although thi i clearly a more table ituatio, the extet of the umerical error i about the ame a for the forward Euler method. I the ectio that follow, we will ee that the 4 th order Ruge-Kutta method i dramatically more accurate ad well behaved tha either Euler method coidered thu far. 5.3 Ruge-Kutta Method The derivatio of the Ruge-Kutta method i beyod the cope of thi memoradum, but itereted reader may refer to [4,6]. Reult for the ecod-order ad fourth-order Ruge-Kutta method applied to the ecod-order differetial equatio (3) follow Secod-Order Ruge-Kutta The formula for the ecod-order Ruge-Kutta olutio to the ecod-order differetial equatio are give by (,, ) (,, ) k f t x y j g t x y h h h k f t, x k, y j h h h j g t +, x + k, y + j x+ x + hk y y + hj + (9) I the cotext of the preet et of differetial equatio, () ϕ () dϕ () t U t t U dt du g f dt R du g(...) U dt which lead further to (...) i ( U) g k i ( U ) R j U g h k i U + j R h j U + k U+ U + hj U U + hk + (3) (3) Thee fiite differece equatio are eaily programmed ad the reult for everal differet time tep are how i Figure 4 through Figure 6. A how i thee figure, the reult follow the exact olutio very cloely util the time tep i icreaed too far to mec a how V. 4 Jame A. Crawford

6 Pedulum ad Elliptic Itegral v.doc 6 Phae, rad. i Figure 6 where the oet of ome itability i apparet. Figure 4 d Order Ruge-Kutta with h 3 mec T p d Order Ruge-Kutta Phae, rad. Figure 6 d Order Ruge-Kutta with h mec d Order Ruge-Kutta.5 T p Time, ec d Order Ruge-Kutta Ideal Time, ec d Order Ruge-Kutta Ideal Figure 5 d Order Ruge-Kutta with h 75 mec.6 d Order Ruge-Kutta T p.4. Phae, rad Time, ec d Order Ruge-Kutta Ideal V. 4 Jame A. Crawford

7 Pedulum ad Elliptic Itegral v.doc 7 Figure 7 4th Order Ruge-Kutta with h 3 mec 5.3. Fourth-Order Ruge-Kutta I the cae of the 4 th -order Ruge-Kutta method, the applicable formula are a follow:.6.4. T p 4th Order Ruge-Kutta (,, ) (,, ) k f t x y j g t x y h h h k f t +, x + k, y + j h h h j g t +, x + k, y + j h h h k3 f t, x k, y j h h h j3 g t +, x + k, y + j k f t + h x + hk y + hj (,, ) (,, ) j g t + h x + hk y + hj h x+ x + ( k + k + k 3 + k 4) 6 h y+ y + ( j + j + j3 + j4) (3) 6 Phae, rad. Phae, rad Time, ec 4th Order Ruge-Kutta Ideal Figure 8 4th Order Ruge-Kutta with h 75 mec T p 4th Order Ruge-Kutta Thi et of differece equatio i eaily programmed ad the reult are how for everal time tep i Figure 7 through Figure 9. A how i thee figure, the computed reult match the ideal reult almot exactly eve at the large time tep of mec. Although other techique may be uperior to the Ruge-Kutta method explored here, the implicity of the method combied with the very good preciio make it a highly recommeded method for ue i olvig differetial equatio umerically Time, ec 4th Order Ruge-Kutta Ideal V. 4 Jame A. Crawford

8 Pedulum ad Elliptic Itegral v.doc 8 Phae, rad. Figure 9 4th Order Ruge-Kutta with h mec T p 4th Order Ruge-Kutta Time, ec 4th Order Ruge-Kutta Ideal 6. Coectio with Elliptic Filter Two of the bet treatmet of elliptic filter deig are provided by [,3,8]. Havig bee a log admirer of Sidey Darligto work with elliptic filter, a umber of hi related publicatio are lited here a referece [9-3]. A very iightful ad uifyig view of Butterworth, Chebyhev, ad elliptic filter i provided i [9]. Quotig from [9]: Formula for the critical frequecie ivolved with the deig of Butterworth, Chebyhev, ad elliptic filter are idetical whe expreed i term of appropriate variable. For Butterworth filter, the appropriate variable i imply the frequecy jω. For Chebyhev filter, it i a ew variable defied by a imple traformatio o ω. For elliptic filter, the appropriate variable i determied by a equece of traformatio applied recurively, each imilar to that for the Chebyhev filter. Iterpretatio i term of elliptic fuctio traformatio i a poible but ueceary complicatio. Thi referece provide the mot cocie ad imple method for calculatig the elliptic filter critical frequecie that I am aware of. 7. Referece. I.S. Sokolikoff, R.M Redheffer, Mathematic of Phyic ad Moder Egieerig, 958, McGraw-Hill Book Co.. Atoiou, Digital Filter: Aalyi ad Deig, 979, McGraw-Hill Book Co. 3. R.W. Daiel, Approximatio Method for Electroic Filter Deig, 974, McGraw-Hill Book Co. 4. E. Kreyzig, Advaced Egieerig Mathematic, 3 rd Editio, 97, Joh Wiley & So 5. J.A. Crawford, Frequecy Sytheizer Deig Hadbook, 994, Artech Houe 6. C.F. Gerald, P.O. Wheatley, Applied Numerical Aalyi, 97, Addio-Weley 7. F.R. Ruckdechel, BASIC Scietific Subroutie, Vol. II, 98, BYTE Publicatio 8. P. Amtutz, Elliptic Approximatio ad Elliptic Filter Deig o Small Computer, IEEE Tra. Circuit ad Sytem, December S. Darligto, Simple Algorithm for Elliptic Filter ad Geeralizatio Thereof, IEEE Tra. Circuit ad Sytem, December 978. S. Darligto, Network Sythei Uig Tchebycheff Polyomial Serie, Bell Sytem Techical Joural, July 95. S. Darligto, A Hitory of Network Sythei ad Filter Theory for Circuit Compoed of Reitor, Iductor, ad Capacitor, IEEE Tra. Circuit ad Sytem, Jauary 984. S. Darligto, Aalytical Approximatio to Approximatio i the Chebyhev See, Bell Sytem Techical Joural, Jauary S. Darligto, Filter with Chebyhev Stopbad, Flat Pabad, ad Impule Repoe of Fiite Duratio, IEEE Tra. Circuit ad Sytem, December 978 V. 4 Jame A. Crawford

9 Advaced Phae-Lock Techique Jame A. Crawford 8 Artech Houe 5 page, 48 figure, equatio CD-ROM with all MATLAB cript ISBN-3: ISBN-: X Chapter Brief Decriptio Page Phae-Locked Sytem A High-Level Perpective 6 A expaive, multi-diciplied view of the PLL, it hitory, ad it wide applicatio. Deig Note 44 A compilatio of deig ote ad formula that are developed i detail eparately i the text. Iclude a exhautive lit of cloed-form reult for the claic type- PLL, may of which have ot bee publihed before. 3 Fudametal Limit 38 A detailed dicuio of the may fudametal limit that PLL deiger may have to be attetive to or ele ever achieve their lofty performace objective, e.g., Paley-Wieer Criterio, Poio Sum, Time-Badwidth Product. 4 Noie i PLL-Baed Sytem 66 A exteive look at oie, it ource, ad it modelig i PLL ytem. Iclude pecial attetio to /f oie, ad the creatio of cutom oie ource that exhibit pecific power pectral deitie. 5 Sytem Performace 48 A detailed look at phae oie ad clock-jitter, ad their effect o ytem performace. Attetio give to tramitter, receiver, ad pecific igalig waveform like OFDM, M- QAM, M-PSK. Relatiohip betwee EVM ad image uppreio are preeted for the firt time. The effect of phae oie o chael capacity ad chael cutoff rate are alo developed. 6 Fudametal Cocept for Cotiuou-Time Sytem 7 A thorough examiatio of the claical cotiuou-time PLL up through 4 th -order. The powerful Haggai cotat phae-margi architecture i preeted alog with the type-3 PLL. Peudo-cotiuou PLL ytem (the mot commo PLL type i ue today) are examied rigorouly. Traiet repoe calculatio method, 9 i total, are dicued i detail. 7 Fudametal Cocept for Sampled-Data Cotrol Sytem 3 A thorough dicuio of amplig effect i cotiuou-time ytem i developed i term of the z-traform, ad cloed-form reult give through 4 th -order. 8 Fractioal-N Frequecy Sytheizer 54 A hitoric look at the fractioal-n frequecy ythei method baed o the U.S. patet record i firt preeted, followed by a thorough treatmet of the cocept baed o -Σ method. 9 Ocillator 6 A exhautive look at ocillator fudametal, cofiguratio, ad their ue i PLL ytem. Clock ad Data Recovery Bit ychroizatio ad clock recovery are developed i rigorou term ad compared to the theoretical performace attaiable a dictated by the Cramer-Rao boud. 5

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13 Phae-Locked Sytem A High-Level Perpective 3 are decribed further for the ideal type- PLL i Table -. The feedback divider i ormally preet oly i frequecy ythei applicatio, ad i therefore how a a optioal elemet i thi figure. PLL are mot frequetly dicued i the cotext of cotiuou-time ad Laplace traform. A clear ditictio i made i thi text betwee cotiuou-time ad dicrete-time (i.e., ampled) PLL becaue the aalyi method are, rigorouly peakig, related but differet. A brief itroductio to cotiuou-time PLL i provided i thi ectio with more exteive detail provided i Chapter 6. PLL type ad PLL order are two techical term that are frequetly ued iterchageably eve though they repreet ditictly differet quatitie. PLL type refer to the umber of ideal pole (or itegrator) withi the liear ytem. A voltage-cotrolled ocillator (VCO) i a ideal itegrator of phae, for example. PLL order refer to the order of the characteritic equatio polyomial for the liear ytem (e.g., deomiator portio of (.4)). The loop-order mut alway be greater tha or equal to the loop-type. Type- third- ad fourth-order PLL are dicued i Chapter 6, a well a a type-3 PLL, for example. θ ref Phae Detector K d Loop Filter VCO θ out N Feedback Divider Figure - Baic PLL tructure exhibitig the baic fuctioal igrediet. Table - Baic Cotitutive Elemet for a Type- Secod-Order PLL Block Name Laplace Trafer Fuctio Decriptio Phae Detector K d, V/rad Phae error metric that output a voltage that i proportioal to the phae error exitig betwee it iput θ ref ad the feedback phae θ out /N. Charge-pump phae detector output a curret rather tha a voltage, i which cae K d ha uit of A/rad. Loop Filter + τ Alo called the lead-lag etwork, it cotai oe ideal pole ad oe fiite zero. τ VCO K The voltage-cotrolled ocillator (VCO) i a ideal v itegrator of phae. K v ormally ha uit of rad//v. Feedback Divider /N A digital divider that i repreeted by a cotiuou divider of phae i the cotiuou-time decriptio. The type- ecod-order PLL i arguably the workhore eve for moder PLL deig. Thi PLL i characterized by (i) it atural frequecy ω (rad/) ad (ii) it dampig factor ζ. Thee term are ued exteively throughout the text, icludig the example ued i thi chapter. Thee term are eparately dicued later i Sectio 6.3. ad The role of thee parameter i hapig the timead frequecy-domai behavior of thi PLL i captured i the exteive lit of formula provided i Sectio.. I the cotiuou-time-domai, the type- ecod-order PLL 3 ope-loop gai fuctio i give by 3 See Sectio 6..

14 4 Advaced Phae-Lock Techique ad the key loop parameter are give by G OL ( ) ω ω + τ τ K K (.) d v Nτ (.) ζ ω τ (.3) The time cotat τ ad τ are aociated with the loop filter R ad C value a developed i Chapter 6. The cloed-loop trafer fuctio aociated with thi PLL i give by the claical reult H ζ ω + θout ( ) ω ( ) N θref ( ) + ζω + ω (.4) The trafer fuctio betwee the ytheizer output phae oie ad the VCO elf-oie i give by H () where ( ) ( ) H H (.5) A coveiet frequecy-domai decriptio of the ope-loop gai fuctio i provided i Figure -3. The frequecy break-poit called out i thi figure ad the ext two appear frequetly i PLL work ad are worth committig to memory. The uity-gai radia frequecy i deoted by ω u i thi figure ad i give by ω ω ζ ζ 4 u (.6) A coveiet approximatio for the uity-gai frequecy (.6) i give by ω u ζω. Thi reult i accurate to withi % for ζ.74. The H () trafer fuctio determie how phae oie ource appearig at the PLL iput are coveyed to the PLL output ad a umber of other importat quatitie. Normally, the iput phae oie pectrum i aumed to be pectrally flat reultig i the output pectrum due to the referece oie beig haped etirely by H (). A repreetative plot of H i how i Figure -4. The key frequecie i the figure are the frequecy of maximum gai, the zero db gai frequecy, ad the 3 db gai frequecy which are give repectively by ω π ζ F + Pk 8ζ Hz (.7) F db ω Hz (.8) π ω π 4 F3 db + ζ + ζ + ζ + Hz (.9)

15 Phae-Locked Sytem A High-Level Perpective 5 log ω + 4ω ζ 4 Gai, db (6 db/cm) - db/octave -6 db/octave 4log (.38ζ ) db ω ζ ω ω 3 u 5 Frequecy, rad/ec Figure -3 Ope-loop gai approximatio for claic cotiuou-time type- PLL. 6 H Cloed-Loop Gai 4 F Pk F db F 3dB G Pk Gai, db Aymptotic -6 db/octave - Frequecy, Hz Figure -4 Cloed-loop gai H ( f ) for type- ecod-order PLL 4 from (.4). The amout of gai-peakig that occur at frequecy F pk i give by G Pk 4 8ζ log db 4 8ζ 4ζ + + 8ζ (.) For ituatio where the cloe-i phae oie pectrum i domiated by referece-related phae oie, the amout of gai-peakig ca be directly ued to ifer the loop dampig factor from (.), ad the 4 Book CD:\Ch\u433_figequ.m, ζ.77, ω π Hz.

16 6 Advaced Phae-Lock Techique loop atural frequecy from (.7). Normally, the cloe-i (i.e., radia offet frequecie le tha ω /ζ) phae oie performace of a frequecy ytheizer i etirely domiated by referece-related phae oie ice the VCO phae oie geerally icreae 6 db/octave with decreaig offet frequecy 5 wherea the ope-loop gai fuctio exhibit a db/octave icreae i thi ame frequecy rage. VCO-related phae oie i atteuated by the H () trafer fuctio (.5) at the PLL output for offet frequecie le tha approximately ω. At larger offet frequecie, H () i iufficiet to uppre VCO-related phae oie at the PLL output. Coequetly, the PLL output phae oie pectrum i ormally domiated by the VCO elf-oie phae oie pectrum for the larger frequecy offet. The key frequecy offet ad relevat H () gai are how i Figure -5 ad give i Table -. 5 Cloed-Loop Gai H G F G Hmax -5-3 db F Hmax H Gai, db - F H-3dB F ζ.4-5 F Hz - -5 Frequecy, Hz Figure -5 Cloed-loop gai 6 H ad key frequecie for the claic cotiuou-time type- PLL. F H 3 db F H _ db Table - Key Frequecie Aociated with H () for the Ideal Type- PLL Frequecy, Hz Aociated H Gai, db Cotrait o ] /π 4 GH ( ) _ rad / log ω + ω 4ζ + ω ζ + π 4 4ζ + 4ζ ω π 4ζ F H _ max F ω / π / GH ( ) _ ω log 4ζ ω π ζ 3 H _ max ζ < 4 ( ) G log 4ζ 4ζ ζ < 5 Leeo model i Sectio 9.5.; Haggai ocillator model i Sectio Book CD:\Ch\u435_h.m.

17 Phae-Locked Sytem A High-Level Perpective 9 Aumig that the oie ample have equal variace ad are ucorrelated, R σ I where I i the K K idetity matrix. I order to maximize (.43) with repect to θ, a eceary coditio i that the derivative of (.43) with repect to θ be zero, or equivaletly L rk Aco( ωotk + θ ) θ θ k ( ω θ ) ( ω θ ) rk Aco otk + Ai otk + k (.44) Simplifyig thi reult further ad dicardig the double-frequecy term that appear, the maximumlikelihood etimate for θ i that value that atifie the cotrait ( ω otk θ ) kˆ rk i + (.45) k The top lie idicate that double-frequecy term are to be filtered out ad dicarded. Thi reult i equivalet to the miimum-variace etimator jut derived i (.4). Uder the aumed liear Gauia coditio, the miimum-variace (MV) ad maximumlikelihood (ML) etimator take the ame form whe implemeted with a PLL. Both algorithm eek to reduce ay quadrature error betwee the etimate ad the obervatio data to zero..4.3 PLL a a Maximum A Poteriori (MAP)-Baed Etimator The MAP etimator i ued for the etimatio of radom parameter wherea the maximum-likelihood (ML) form i geerally aociated with the etimatio of determiitic parameter. From Baye rule for a obervatio z, the a poteriori probability deity i give by ( z) ad thi ca be re-writte i the logarithmic form a ( θ ) p( θ ) p( z) p z p θ (.46) ( θ ) ( θ ) + ( θ ) ( ) log e p z log e p z loge p loge p z (.47) Thi log-probability may be maximized by ettig the derivative with repect to θ to zero thereby creatig the eceary coditio that 7 d d { e p( z θ ) e p( θ ) } ˆ log + log θ θ θmap (.48) If the deity p(θ ) i ot kow, the ecod term i (.48) i ormally dicarded (et to zero) which degeerate aturally to the maximum-likelihood form a d d { e p( z θ ) } ˆ log θ θ θml (.49) 7 [5] Sectio 6.., [7] Sectio.4., [8] Sectio 5.4, ad [].

18 3 Advaced Phae-Lock Techique Time of Peak Phae-Error with Frequecy-Step Applied T ta ζ ftep ω ζ ζ Note. See Figure -9 ad Figure -. Time of Peak Phae-Error with Phae-Step Applied T θ tep ( ζ ζ ζ ) ζ ta, ta ω ζ ω ζ See Figure -9 ad Figure -. Time of Peak Frequecy-Error with Phae-Step Applied ζ : θu Tpk ω ζ ζ > : θu + π with u ( ) 3 ζ (.9) (.3) (.3) θ ta 4ζ ζ,3ζ 4ζ (.3) See Figure - ad Figure -. T pk correpod to the firt poit i time where df o /dt. Maximum Frequecy-Error with Phae-Step Applied Ue (.3) i (.8). (.33) Time of Peak Frequecy-Error with Frequecy-Step Applied T pk ω ζ ta ζ ζ % Traiet Frequecy Overhoot for Frequecy-Step Applied Tpk OS% co( T ) ζ ζ ω pk i( ζ ωtpk ) e ζω % ζ T pk ω ζ ta ζ ζ Note. See Figure -3 ad Figure -4. Liear Hold-I Rage with Frequecy-Step Applied (Without Cycle-Slip) ζ ζ Fmax ω exp ta Hz ζ ζ See Figure -5. Liear Settlig Time with Frequecy-Step Applied (Without Cycle-Slip) (Approx.) F TLock log e ec ζω δ F ζ for applied frequecy-tep of F ad reidual δ F remaiig at lock See Figure -6. (.34) (.35) (.36) (.37) (.38) The peak occurrece time i preciely oe-half that give by (.34). See Figure -4 for time of occurrece T pk for peak overhoot/uderhoot with ω π. Amout of overhoot/uderhoot i percet provided i Figure -3.

19 44 Advaced Phae-Lock Techique.3.. Secod-Order Gear Reult for H (z) for Ideal Type- PLL v θ i + Σ - θ e D 4ζ ωt D ζ ω T + _ Σ + T 3 ω + _ Σ + D Σ T 3 + _ Σ + D θ o + 3ζ ωt 4 3 D 4 3 D 3 3 Figure -3 Secod-order Gear redeig of H () (.4). 3ζ 4 + z + z ωt ωt 3 3 GOL ( z) 3 (.5) 4 z + z θo i o D θ θ ( k) a ( k ) + b v( k ) + c ( k ) (.53) a 3ζ + ω T 4ζ a ω T a ζ ω T (.54) b 3 T ω b T ω b T ω (.55) 6 4ζ c + ω T ( ω T ) ( ω T ) ( ω T ) 3 ( ω T ) 4 ζ c ω T c c (.56) 3ζ 3 D + + ωt ωt (.57).3.3 Higher-Order Differetiatio Formula I cae where a preciio firt-order time-derivative f (x + ) mut be computed from a equally paced ample equece, higher-order formula may be helpful. 8 Several of thee are provided here i Table -. The uiform time betwee ample i repreeted by T. 8 Preciio compared i Book CD:\Ch\u48_diff_form.m.

20 5 Advaced Phae-Lock Techique QAM Symbol Error Rate - 64-QAM Symbol Error Rate σ φ o rm -3 σ φ.5 o rm σ φ o rm -4 SER -5-6 Proaki σ φ.5 o rm -7 No Phae Noie E b /N o, db Figure QAM ucoded ymbol error rate with oiy local ocillator. 3 Circled datapoit are from (.87). 3 Book CD:\Ch5\u359_qam_er.m. See Sectio for additioal iformatio. Circled datapoit are baed o Proaki [3] page 8, equatio (4..44), icluded i thi text a (.87).

21 Fudametal Limit 83 A more detailed dicuio of the Cheroff boud ad it applicatio i available i [9]. Key Poit: The Cheroff boud ca be ued to provide a tight upper-boud for the tail-probability of a oe-ided probability deity. It i a much tighter boud tha the Chebychev iequality give i Sectio 3.5. The boud give by (3.43) for the complemetary error fuctio ca be helpful i boudig other performace meaure. 3.7 CRAMER-RAO BOUND The Cramer-Rao boud 6 (CRB) wa firt itroduced i Sectio.4.4, ad frequetly appear i phae- ad frequecy-related etimatio work whe low SNR coditio prevail. Sytem that aymptotically achieve the CRB are called efficiet i etimatio theory termiology. I thi text, the CRB i ued to quatify ytem performace limit pertaiig to importat quatitie uch a phae ad frequecy etimatio, igal amplitude etimatio, bit error rate, etc. The CRB i ued i Chapter to ae the performace of everal ychroizatio algorithm with repect to theory. Owig to the much larger igal SNR ivolved with frequecy ythei, however, the CRB i rarely ued i PLL-related ythei work. The CRB i developed i coiderable detail i the ectio that follow becaue of it geeral importace, ad it widepread applicability to the aalyi of may commuicatio ytem problem. The CR boud provide a lower limit for the error covariace of ay ubiaed etimator of a determiitic parameter θ baed o the probability deity fuctio of the data obervatio. The data obervatio are repreeted here by z k for k,..., N, ad the probability deity of the obervatio i repreeted by p(z, z,..., z N ) p(z). Whe θ repreet a igle parameter ad θ - hat repreet the etimate of the parameter baed o the oberved data z, the CRB i give by three equivalet form a ( θˆ θ ) E ( θˆ θ ) var E log e p( z θ ) θ E log e p θ + p θ ( z) ( z θ ) p ( z) dz (3.46) The firt form of the CR boud i (3.46) ca be derived a follow. Sice θ -hat i a ubiaed (zero-mea) etimator of the determiitic parameter θ, it mut be true that + ( θ ) θ θ p( ) E ˆ z dz (3.47) i which dz dz... dz dzn. Differetiatig (3.47) with repect to θ produce the equality 6 See [] [4].

22 86 Advaced Phae-Lock Techique var var { ωˆ T } o { θˆ o} σ b Q σ bm M σ b M σ Q b M ( ) ( M ) { bˆ o } σ var for all cae (3.6) M Phae kow, amplitude kow or ukow (3.63) Phae ukow, amplitude kow or ukow Frequecy kow, amplitude kow or ukow (3.64) Frequecy ukow, amplitude kow or ukow I the formulatio preeted by (3.55), the igal-to-oie ratio ρ i give by ρ b / (σ ). For the preet example, the CR boud i give by the top equatio i (3.63) ad i a how i Figure 3-9 whe the iitial igal phae θ o i kow a priori. Uually, the carrier phae θ o i ot kow a priori whe etimatig the igal frequecy, however, ad the additioal ukow parameter caue the etimatio error variace to be icreaed, makig the variace aymptotically 4-time larger tha whe the phae i kow a priori. Thi CR variace boud for thi more typical ukow igal phae ituatio i how i Figure 3-. Begiig with (3.57), a maximum-likelihood 7 frequecy etimator ca be formulated a decribed i Appedix 3A. It i iightful to compare thi etimator performace with it repective CR boud. For implicity, the iitial phae θ o i aumed to be radom but kow a priori. The reult for M 8 are how i Figure 3- where the oet of threholdig i apparet for ρ db. Similar reult are how i Figure 3- for M 6 where the threhold oet ha bee improved to about ρ 5 db. Etimator Variace, var ( ω o T ) CR Boud for Frequecy Etimatio Error Variace ρ - db ρ - db ρ db ρ db ρ db Number of Sample Figure 3-9 CR boud 8 for frequecy etimatio error with phae θ o kow a priori (3.63). 7 See Sectio Book CD:\Ch3\u3_crb.m. Amplitude kow or ukow, frequecy ukow, iitial phae kow.

23 Noie i PLL-Baed Sytem 35 would be meaured ad diplayed o a pectrum aalyzer. Havig recogized the carrier ad cotiuou pectrum portio withi (4.65), it i poible to equate 9 ( f ) P ( f ) / θ rad /Hz (4.66) Power Carrier / ( ) f x P θ ( f ν ) o ν o ν o Frequecy Figure 4-7 Reultat two-ided power pectral deity from (4.65), ad the igle-idebad-to-carrier ratio ( f ). Both /( f ) ad P θ ( f ) are two-ided power pectral deitie, beig defied for poitive a well a egative frequecie. The ue of oe-ided veru two-ided power pectral deitie i a frequet poit of cofuio i the literature. Some PSD are formally defied oly a a oe-ided deity. Two-ided power pectral deitie are ued throughout thi text (aide from the formal defiitio for ome quatitie give i Sectio 4.6.) becaue they aturally occur whe the Wieer-Khitchie relatiohip i utilized Phae Noie Spectrum Termiology A miimum amout of tadardized termiology ha bee ued thu far i thi chapter to characterize phae oie quatitie. I thi ectio, everal of the more importat formal defiitio that apply to phae oie are provided. A umber of paper have bee publihed which dicu phae oie characterizatio fudametal [34] [4]. The updated recommedatio of the IEEE are provided i [4] ad thoe of the CCIR i [4]. A collectio of excellet paper i alo available i [43]. I the dicuio that follow, the omial carrier frequecy i deoted by ν o (Hz) ad the frequecy-offet from the carrier i deoted by f (Hz) which i ometime alo referred to a the Fourier frequecy. Oe of the mot prevalet phae oie pectrum meaure ued withi idutry i /( f ) which wa ecoutered i the previou ectio. Thi importat quatity i defied a [44]: f x /( f ): The ormalized frequecy-domai repreetatio of phae fluctuatio. It i the ratio of the power pectral deity i oe phae modulatio idebad, referred to the carrier frequecy o a pectral deity bai, to the total igal power, at a frequecy offet f. The uit 3 for thi quatity are Hz. The frequecy rage for f rage from ν o to. /( f ) i therefore a two-ided pectral deity ad i alo called igle-idebad phae oie. 9 It implicitly aumed that the uit for ( f ), dbc/hz or rad /Hz, ca be iferred from cotext. 3 Alo a rad /Hz.

24 Noie i PLL-Baed Sytem 63 zi pi exp α p (4B.) A miimum of oe filter ectio per frequecy decade i recommeded for reaoable accuracy. A ample reult uig thi method acro four frequecy decade uig 3 ad 5 filter ectio i how i Figure 4B f -α Power Spectral Deity 4 35 α Relative Spectrum Level, db Filter Sectio 5 Filter Sectio Radia Frequecy Figure 4B-3 /f oie creatio uig recurive /f filterig method 4 with white Gauia oie. /f Noie Geeratio Uig Fractioal-Differecig Method Hokig [6] wa the firt to propoe the fractioal differecig method for geeratig /f α oie. A poited out i [3], thi approach reolve may of the problem aociated with other geeratio method. I the cotiuou-time-domai, the geeratio of /f α oie procee ivolve the applicatio of a orealizable filter to a white Gauia oie ource havig α/ for it trafer fuctio. Sice the z-traform equivalet of / i H(z) ( z ), the fractioal digital filter of iteret here i give by Hα ( z) (4B.) α z ( ) / A traightforward power erie expaio of the deomiator ca be ued to expre the filter a a ifiite IIR filter repoe that ue oly iteger-power of z a α α α Hα ( z) z z!... (4B.) i which the geeral recurio formula for the polyomial coefficiet i give by 4 Book CD:\Ch4\u37_recurive_flicker_oie.m.

25 Sytem Performace 85 Strog Iterferer Local Ocillator Spectrum L db Sidebad Noie Deired Chael F Sep Figure 5-9 Strog iterferig chael are heterodyed o top of the deired receive chael by local ocillator idebad oie. 5 Deired Sigal Relative PSD at Baebad F Sep -5 PSD, db db +35 db db Baebad Frequecy, MHz Figure 5- Baebad pectra caued by reciprocal mixig betwee a trog iterferer that i offet 4B Hz higher i frequecy tha the deired igal ad troger tha the deired igal by the db amout how. The firt term i (5.8) BL Floor i attributable to the ultimate blockig performace of the receiver a dicued i Sectio 5.3. The reultat output SNR veru iput SNR i give by SNR out σ + SNRi BL MFX IQ (5.9) It i worthwhile to ote that the iterferig pectra i Figure 5- are ot uiform acro the matched-filter frequecy regio [ B, B]. Multicarrier modulatio like OFDM (ee Sectio 5.6) will potetially be affected differetly tha igle-carrier modulatio uch a QAM (ee Sectio 5.5.3) whe the iterferece pectrum i ot uiform with repect to frequecy. The reult give by (5.9) i how for everal iterferig level veru receiver iput SNR i Figure 5-. Book CD:\Ch5\u357_rx_deee.m. Loretzia pectrum parameter: L o 9 dbc/hz, f c 75 khz, L Floor 6 dbc/hz, B 3.84/ MHz.

26 Sytem Performace of 3 rm phae oie i how i Figure 5B-8. The tail probability i wore tha the exact computatio how i Figure 5-7 but the two reult otherwie match very well o Chael Cutoff Rate R o Bit per Symbol o 6 o 8 o o E b /N o, db Figure 5B-6 Chael cutoff rate, 7 R o, for 6-QAM with tatic phae error a how, from (5B.6). 4 Chael Cutoff Rate R o Icludig Phae Noie Bit per Symbol E b /N o, db Figure 5B-7 R o for 8 6-QAM veru E b/n o for 5 rm phae oie from (5B.8) (to accetuate lo i R o eve at high SNR value). 7 Book CD:\Ch5\u376_rolo.m. 8 Ibid.

27 Fractioal-N Frequecy Sytheizer 37 required, however, becaue the offet curret will itroduce it ow hot-curret oie cotributio, ad the icreaed duty-cycle of the charge-pump activity will alo itroduce additioal oie ad potetially higher referece pur. Sigle-bit -Σ modulator are attractive i thi repect becaue they lead to the miimum-width phae-error ditributio poible. I CP Ideal θ PD Figure 8-7 Charge-pump (i) dead-zoe ad (ii) uequal poitive veru egative error gai. -4 MASH - Output Spectrum with Noliear PD Phae Noie, dbc/hz Frequecy, Hz Figure 8-7 Phae error power pectral deity 48 for the MASH - -Σ modulator how i Figure 8-55 with M, P M/ + 3,, ad % charge-pump gai imbalace. Icreaed oie floor ad dicrete pur are clearly apparet compared to Figure Claical radom procee theory ca be ued to provide everal ueful iight about oliear phae detector operatio. I the cae of uequal poitive-error veru egative-error phae detector gai, the memoryle oliearity ca be modeled a ( ) θ φ + α φ > φ (8.39) pd i i i where α repreet the additioal gai that i preet for poitive phae error. The itataeou phae error due to the modulator iteral quatizatio create a radom phae error equece that ca be repreeted by 48 Book CD:\Ch8\u735_MASH oliear.m.

28 Clock ad Data Recovery 457 the amplig-poit withi each ymbol-period after the datalik igal ha bee fully acquired. I the example reult that follow, the data ource i aumed to be operatig at bit-per-ecod, utilizig quare-root raied-coie pule-hapig with a exce badwidth parameter β.5 at the tramitter. The eye-diagram of the igal at the tramit ed i how i Figure -5. The ideal matched-filter fuctio i the CDR i cloely approximated by a N 3 Butterworth lowpa filter havig a 3 db corer frequecy of.5 Hz like the filter ued i Sectio.4. The reultig eye-diagram at the matched-filter output i how i Figure -6 for E b /N o 5 db. Matched Filter y( t ) y ( t) H MF ( f ) Sample Gate f VCO tah y f No Loop Filter l a k ( t) Sample Gate v( t) A( t) B( t) ( j π f ) H ( f ) MF Figure -4 ML-CDR implemeted with cotiuou-time filter baed o the timig-error metric give by (.)..5 Square-Root Raied-Coie Eye-Diagram.5 Normalized Time, Symbol Figure -5 Eye diagram 5 at the data ource output aumig quare-root raied-coie pule hapig with a exce badwidth parameter β.5. A clear udertadig of the error metric repreeted by v(t) i Figure -4 i vital for udertadig how the CDR operate. The metric i bet decribed by it S-curve behavior veru iput E b /N o a how i Figure -7. Each curve i created by ettig the oie power pectral deity N o for a pecified E b /N o value with E b, ad computig the average of v( kt ym + ε ) for k [, K] a the timig-error ε i wept acro [, T ym ]. The lope of each S-curve ear the zero-error teady-tate trackig value determie the liear gai of the metric that i eeded to compute the cloed-loop badwidth, loop tability margi, ad other importat quatitie. For a give iput SNR, 5 Book CD:\Ch\u44_ml_cdr.m.

29 458 Advaced Phae-Lock Techique the correpodig S-curve ha oly oe timig-error value ε o for which the error metric value i zero ad the S-curve lope ha the correct polarity. A the gai value chage with iput E b /N o, the cloed-loop parameter will alo vary. For large gai variatio, the Haggai loop cocept explored i Sectio 6.7 may prove advatageou..5 Eye Diagram at Matched-Filter Output.5 Amplitude Time Figure -6 Eye diagram 6 at the CDR matched-filter output for E b/n o 5 db correpodig to the data ource how i Figure -5 ad uig a N 3 Butterworth lowpa filter with BT.5 for the approximate matched-filter..4 S-Curve.3.. V -. 9 db 5 db 6 db 3 db db E b /N o 5 db Timig, ymbol Figure -7 S-curve 7 veru E b/n o correpodig to Figure -6 ad ideal ML-CDR how i Figure -4. E b i aumed cotat. A ecod importat characteritic of the timig-error metric i it variace veru iput E b /N o ad tatic timig-error ε. For thi preet example, thi iformatio i how i Figure -8. The variace udertadably decreae a the iput SNR i icreaed, ad a the optimum timealigmet withi each data ymbol i approached. The variace of the recovered data clock σ clk ca be cloely etimated i term of the trackig-poit voltage-error variace from Figure -8 deoted by σ ve (V ), the lope (i.e., gai) of the correpodig S-curve (K te, V/UI) from Figure -7, the ymbol rate F ym ( /T ym ), ad the oe-ided cloed-loop PLL badwidth B L (Hz) a 6 Ibid. 7 Ibid.