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sampling interval which in every case is a lmost ident ica l withthe re s u l t s o f the s imula t ion . T h i s ho lds t rue even fo r the lowS N R . T hus ncri t ica lapp l i ca t ions, he s ecurves ,w h e nm o d i -f i ed t o re f l ec t the co r rec t mes s age bandwid th , cou ld be us ed tode te rmine hem i n i m u msamp ling a te . In rea l i ty , of cours e ,theac tua lva lue o f th ee r ro rva r i anceat hreshold is i n h evic ini ty f 0.25. However , he om puta t iona l i f f i cu l tynapp ly ingm o r eccura tep p r o x i m a t i o n of th earianceequ at io n is not us t i f iablesince all thatw o u l db egained es-sent ia l ly is an ncrease n he vert ica l s lopeof hecurves nFigs. 3-5. T heons i s t ency fhefo rement ionedesul tsp rec ludes the need fo r th i s added d i f f i cu l ty .

C O N C L U S I O N ST heond i t ionsaveeens tabl ishedwhichns ureheva l id i ty o f the , whi te un i fo rm s equence mode l fo r quan t i za t ione r ro r in the ! DPL L . An e f fec t ive S N R is defined which a l lowsq u a n t i z e dy s t e m e r f o r m a n c eo e r e d i c t e dr o m n -quan t i zed e s u l t s .Sampl ing equ i rementshavebeen xpe r i -m e n t a l l y d e te r m i n e d h r o u g h s i m u l a t io n . A m e t h o d h a s b e e ndes c r ibed to p red ic t he s e min im um s ampl ing ra te s us ing hehigh S N R va r iance e la t ions .Sampl ing equ i rements o r hequan t i zed s ys tem may then be de te rm ined us ing the e f fec t ive

REFERENCESS. C. Gupta,Onoptimumdigitalphase-locked oops, IEEETrans. Commun. Technol. (ConcisePapers), ol.COM-16, p.340-344, Apr. 1968.G . PasternackandR. L. Whalin,Analysisand ynthesisofadigital phase-locked loop for FM demodul ation, Bell Sysf. Tech .J . , vol. 47, p p . 2207-2237, Dec. 1968.S. C. Gupta, Status of digital phase-locked loops, in Proc. 3rdHawaii Int. Conf., pp. 255-259, 1970.locked oop or FM demodu lation, in Proc. I n t . Communica-J. Garodnick, J . Greco, and D. L. Schilling, An all digital phase-J . K. Holmes, Performance of a first-order ransition samplingt ionsConf . ,June 1971.digitalhase-lockedoop singandom-walkmodels, IEEEJ . R . Cessna and D. M. Levy, Phase noise and transient times fo rTrans. Commun.,vol. COM-20, pp. 119-131,Apr. 1972.abinaryquantizeddigitalphase-locked oop nwhiteGaussiannoise, IEEE Trans. Commun., vol.COM-20,pp. 94-104,Apr.G. S. Gill and S. C. Gupta,First-orderdiscretephase-locked1972.loop with applications o demodulation ofangle-modulated car-rier, IEEE Trans. Commun. (Concise Papers), vol. COM-20, pp.454-462, June 1972.C. N. Kelly and S. C. Gupta, The digital phase-lock ed loop as anear-optimum FM demodulator, IEEE Trans. Commun. (ConcisePapers), vol. COM-20, pp. 406-411, June 1972.IEkE Trans. Inform. Theory, vol. IT -18, pp. 488-493, July 1972.- Discrete-time demodulation of continuous-time ignals,G . S. Gill and S. C. Gupta, On higher order discrete phase-lockedloops, IEEE Trans. Aerosp.Electron. Syst . , vol. AES-8,pp.615-623, Sept. 1972.C. P. Reddy and S. C. Gupta, Demodulatio n of FM signals by aCon$, Lo s Angeles, Calif.,Oct. 1972.discrete hase-lockedoop, in Proc. I n t . Telecommunicationsanalysis, IEEE Trans. I n d . Electron. onfr.nstrum., vol.IECI-20, pp. 239-25 1 , Nov. 1973.D . R.Polk nd S. C. Gua, Quasi-optimum igital hase-locked oops, IEEE Trans. Commun., vol.COM-21,pp. 75-82 , J y . 1973.-, An pproachothe nalysis f erformance f uasi-opt imu m digital hase-lockedoops, IEEE Trans. Commun.,vol. COM-21, pp. 733-738, June 1973.G.T.Hurstand S. C. Gupta, On th eperformanceof digitalphase-locked loops in the threshold region, to be published.D. G. nyder , Th e Stare VariableApproach to ContinuousEstimation, Res. Mono. 5 1. Cambridge, Mass.: M.I.T. Press, 1969.ofNyquist ampling heory, IRETrans.CircuitTheory, vol.B. Widrow, A study of rough amplitude quantization by meansW . R. Bennett, SPECTRA of quantized signals, Bell Sysr. Tech.CT-3, pp. 266-276, Dec. 1956..A. Papoulis, Probability, Random Variables and Stochastic Pro-cesses. New York: McGraw-Hill, 196 4, p. 133.G. T. Hurst, Sampling, quantizing, and low signal to noise ratioconsiderations in digital phase-locked oops, Ph.D. dissertation,Southern Methodist Univ., Dallas, Tex., Apr. 1973.

- A class of a l l digitalphase ocked oops:Modelingand

J . , VOI . 27, pp. 446-471, July 1948.

IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1974

Exact Dynamics of Automatic Gain ControlJ O H N E . O H L S O N , MEMBER, I E E ~

Abstract-The exact input-output ielationship is derived for a fust-order automatic gain control loop wherein the variable gain is an ex-ponential function of the gain control voltage. The exact solution iscompared to th e linearized solution, an d the condition for validlinearization is given.

I. INTR ODUC TIONAutom at ic ga in con t ro l (AGC) loops a re us ed in v i r tua l ly a llm o d e r n c o m m u n i c a t io n s y s t e m s . T h e w o r k o f O l i v e r [ 11 andV i c t o ran d B r o c k m a n [ 2 ] prov ided a us e fu l heory b o t h o rs ta t ic and small s ignal analyses . In some app l ica t iqns , howeverja m o r e g e n e r a l t h e o r y is requ i red which can p red ic t wha t wi llhapp en whe n la rge va r ia t ions in s igna l l eve l occur . E xam ples o flarge variations ns ignal l evel inc lude s eve re fad ing n u rbanmob i le l inks and l inks to / f rom tum bl ing s a te l l i t e s . Whe n l a rges ignal level varia t ions occur, . the l inear AGC, the ory is no longerus e fu l , and we mus t a t t em pt to s olve the non l inea r p rob lem .Fig. 1 i l l u st r a t es t h e A G C p r o b l e m w h e r e t h e o u t p u t y ( t ) sgiven by

L ( t )= g ( u ) x ( t ) ( 1 )w h e r e x([) s t h e i n p u t , ~ ( t )s the gain control vol tage , g(u) ist he ga in con t ro l cha rac te r i st i c which is a memo ry le s s func t ionof u ( t ) , an d b > 0 is the AGC reference bias .Ses e ra lworke rs [ 3 ] - [ 6 ] as s umed hegaincon t ro lcha rac -teristic to be a inea r func t ion o f w , and have ob ta ined s omeinteres t ing exact resul ts . Plotkin [ 7 ] assumed g(u) t o b e o f t h ef o r m Y O 1 , w h e r e a i s a cons tan t , and ob ta ined a pe r tu rba t ions o lu t ion . However , ne i the r a inea r nor a u- cha rac te r i s t i c i srepresenta t ive of typical a in-con trol led mplif iers .Us ua l lythes e ampl i f i e r s mus t have a ga in dynamic range o f 50-100 dB .F o r p r a c t i c a l m p l e m e n t a t i o n , t h a s b e e n f o u n d h a t a g a i nwhichva r ie sexponen t i a l lywi th u gives thedes i red d y n a m i cr a n g e w i t h a m o d e r a t e r a n g e o f u , and is a lso easy to cha rac te r -iz e $rice a n e x p o n e n t i a l g(u) gives gain in decibels as a inearfunc t io n o f ga in con t ro l vo l t age u . V i c t o r a n d B r o c k m a n [ 2 ]a s s u m e dh iso rmo r g(u ) , an dubs equen t ly lmos t l lmode rn rece ive r des igns have approx ima ted th i s cha rac te r is t i c .T he ana lys i s to fo l low thus app l i e s to a a rge number o f cu r ren tsys tems.

11 . A N A L Y S I SWe shal l assume on the basis of the above that g(u) varies as

w h e r e G > 0 an d a> 0 a re cons tan t s . S ince g(u) is a vol tagegain, the gain in d ecibels is a l inear funct ion of u as discussedabove :g(u ) n dec ibe l s = 2 0 l o g l o g(u )

= G (in ec ibels ) - ( 8 . 6 8 6 ~ ~ ) ~ . 3)We mustn ows pec i fy heAGC oop i l t e r . nm o s tac tua ls ys tems , he oop i l t e r is as imple ow-pass R C filter.H o w -ever, the RC t i m e c o n s t a n t is usual ly so m u c h l o n g e r t h a n t h ec los ed- loop re s pons e t ime tha t the loop f i l t e r can be approx i -ma ted exce l l en t ly by a s im ple in teg ra to r . T h i s we do , and thesys temw eshal lana lyze ss hown nFig . 2. We inc lude he

Paper approved by the Associate Editor for Communication Theory ofthe EEECommunicationsSociety for publicationwithoutoralpre-sentation. Manuscript received January 2 8 , 1973.The author is with he Department of Electrical Engineering, NavalPostgraduate School, Monterey, Calif. 93940.

Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY CALICUT. Downloaded on March 1, 2009 at 09:25 from IEEE Xplore. Restrictions apply.


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CONCISE PAPERS 13

x ( t 1 g ( v ) x ( t i Y ( t )A l N I ( l ) G A I N -Gev - ,Y(l)-a VINPUT 0 ( V I v ' UTPUTA A hV ( l ) +

d l &bK-----

v ( t ) A GCL O W A 07 ; ( I )G A IN -C O N TR O L F I L T E RVOLTAGE BIAS"0

Fig. 1. AGC block diagram. Fig. 2. First-order AGC loop.

q u a n t i t y uo as the initial value of u ( t ) a t t = 0, i.e. , ~ ( 0 )UO.Note also that we incorporate an open- loop gain factor K > 0.We nowwish to f ind y ( t ) a nd u ( t ) in e rms o f x ( t ) . F r o mFig. 2 i t is clear thatd = K [ y - b ] = K (xCe-"' - b )4 )

w h e r ew edropexp l ic i tu se of t in o u rn o t a t i o n ,a n d a d o tdenotes t ime der ivative. From (2) we can see thatg = - & g d , (5 )

a ndbysubs t i tu t ing o r d in 4 )an dbyuse of (2) ,w ec anob ta ing + K a x g ' - K a ! b g = 0. ( 6 )

This is in he orm of Bernoull i ' s quation orwh ich hes o l u ti o n i s k n o w n t o be [ 81

where r = l /Ka!b . S ince a t tenua t ion is the inverse of gain, wesee tha t the vo l tage-con t ro l led a t tenua t ion is a Iinear f u n c t i o nof t h e i n p u t p l u s a d e c a y i n g i n i t i a l c o n d i t i o n . T h e i n t e g r a l i n(7 ) is recognized as a conv olution 12 * x ( t )w h e r e

t < Owhich s a s imple one-pole ow-pass fi l ter whose gain at dc sun i ty . I t i s impor tan t to r ea l ize tha t h ( t ) s no t an ac tua l f i l t e rwh ich i s par t o f the sy s tem, bu t tha t i ts a f ict i t ious or equiva-lent f i l ter which represents a por t ion o f the p rocess ing doneu p o n t h e i n p u t ( t ) .Using (2) an d (8 ) we may r ewr i te (7 ) as

By inver t ing ( 2 ) we f ind that the gain contro l voltage is(10 )

Also , s ince y = x g , we use (9 ) and have

111. TH E STATIONARY CASEThe case of greates t practical in teres t is when the system hasopera ted o r ong ime nddependenceupon he n i t ia l

condit iond isappear s. We may hus d r o p h e r a n s i e n te x p o -nen t ia l t e rms in ( 10) and ( 1 1) and consider the "s tat ionary"case where the ga in con t ro l vo l tage su ( t ) = C + - l o g , [ h * x ( t ) l (12 )1a

where the cons tan t C i s1C = - og , ( G / b ) ( 1 3 )a!

and the ou tpu t i sb x ( t )Y ( t )= ___ (14)h * x ( t ) '

The results in (12 ) and (14 ) a r e the p r inc ipa l c on t r ibu t ion o fth i s work . I t mus t be no ted tha t h * x ( t ) must remain posit iveor (12) and (14) make l i t t le sense. Cer tain ly if x ( t ) > 0 , thenh * x ( t ) > 0 and here i s n o dif f icu lty .However ; i f x ( t ) isnegative of ten enough so t h a t h * x ( t )+ 0 , t h e n h e s y s t e mwill achieve the s tate u = --M f rom which i t cannot recover . I ti s easy to app rec ia te th i s d i f f icu l ty f the loop i s as sum ed to bei n s t e a d y s t a t e d u e t o x ( t ) = Ix l 1, say , and then le t x ( t ) swi tcha b r u p t l y o x ( t ) = - Ix2 1 . I t can hen be shown ha t ~ ( t )+--oo in a f i n i t e t ime, and once there, recovery is not possib le. Innoncoherent receivers , x ( t ) s an enve lope func t ion , and hencealwayshas he ame ign , so there s no prob lem here . I ncoherenteceivers , x ( t ) could ometimes e egative, u trarelywou ld h * x ( t ) benegative,andeven if it w e r e , h efiniteynamicange of ann t e g r a t o rw o u l do ldof in i te max imum f rom wh ich i t cou ld r ecover .An equivalentmodel o rFig . 2 for hes ta t ionarycaseo f(12) and (149 is g iven in Fig. 3w h e n h * x ( t )> 0 . Note tha t i fth is condit ion s v io lated , as far as t h e o u t p u ty ( t ) s coficerned,Fig. 3 can ecoveronce h * x ( t ) again becomesposit ive, al-t h o u g h h e a c t u a l o o p in Fig. 2 cannot recover f rom such acondit ion . Clear ly , u ( t ) will be imagina ry when h * x ( t ) < 0 ,anonphys ica lpossib il i ty . We conc lude hen hatw h e n e v e rFig. 2 is proper ly behaving ( h * x ( t ) > 0), Fig. 3 is an equiva-len t model of the AGC loop .I t is in teres t ing that the h ighly nonlinear AGC loop of Fig. 2has he imp le qu iva len tmodel of Fig. 3. It is indee d e-markab le ha ta inear i l ter ing of t h e n p u t x ( t ) by h ( t ) sc e n t r a l o h e o o po p e r a t i o n . talso is in tu i t ivelypleasingt h a t y ( t ) is p r o p o r t i o n a l o x ( t ) normal izedby an averagedx ( t ) .

Iv . COMPARISON WITH TH E LINEARIZED M O D E LSince we have the exac t dynamics o f the loop , we sha l l nowconsider how good a inear ized analysis can be. Consider hes ta t ic case w i th x ( t ) = x o>0. We clea rly obtai n h * x ( t ) =

x0 > so



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7 4 IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1974.f oa c (G /b )

Fig. 3. Equivalent AGC loop model.

1u ( t ) = c +- og, x0aa nd y ( t ) b. Thus the s ta t ic casegives the gain control voltageas a ogar i thmic unc t ion o f the igna l , nd heou tpu t ss tabil ized exactly to the value of the AGC reference b ias b . Ifthe s igna l now var ies abou tx,, i.e.,

~ ( t )x0 + A ( t ) ,t h e e x a c t u ( t ) a nd y ( t ) re

1- . x. J \ -A ( t )1 +-

( 1 8 )X0

I f h * A ( t )


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CONCISE PAPERS 75

[ 6 ] R. S. Simpson and W. H. Tranter, Baseband AGC inan AM-FMtelemetry system, IEEE Trans. Commun. Tecknol., vol. COM-18,171 S. Plotkin, O n nonlinearAGC, Proc. IEEE (Corresp.),vol. 5 1 ,pp. 59-63, Feb. 1970.181 K . Rektorys, Survey of Applicable Mafhentatics. Cambridge,p. 380, Feb. 1963.191 J . Omura and T. Kailath, Some useful probability distributions,Mass.: M.I.T. Press, 1969, pp. 746, 834.StanfordElectron.Lab., tanford,Calif.,Tech.Rep. 050-6,SU-SEL-65-079, Sept. 1965, pp. 47-49.

Digital Com panding TechniquesC. J. K I K K E R T , MEMBER, IEEE

Abstuact-This paper deals with he requirements for the design ofdigital companding techniques in either delta or pulse-code modulation.

Both delta and pulse-code modulation convert analogue signals ntobinary signals and in both the se systems the dy namic rang e s normallysmall. By th e use of companding, the dynamic range can be extended.Since both delta and pulse-code modulation are digital methods, theyare well suited to theuse of digital companding techniques.

Thebinary ransmittedsignalnormallycontainsameasure of thesystem erformance.By bservingertain atternsnhis inarysignal and using the occurrence or nonoccurrence of these patterns tochange thegainof the modulator and demodulator, syllabic compandingcan be obtained. The selection of the binary pattern and t he rate ofchange of gain of the modulator and demodulator, determines both thepoint at which the companding operates and the attack and decay times.

The ratio of th e largest to th e smallest value of the gain determinesthedynamic ange. By th e use of digitalcircuitry, hegaincanbecontrolled with sufficient accuracy over a large dynamic range.

The paper deals with the principl es involved n selecting th e binarypatterns to control the gain of the modulator and as examples a deltamodulation system and a pulse-code modulation system with compand-ing ratios of60 dB are discussed.

I . INTRODUCTIONCode modu la t ion i s a g roup o f m odula t ion me thod s where

th eana log ue npu t i s s ampledan dw h e r ea teach ampl ingins tan t a code word rep re s en t ing the inpu t i s gene ra ted . PCM,APCM, AM and their variants are exam ples of code m odula t ionsys tems.S inceode odu la t ionpprox ima te sh enpu tignal ,dis tort ional leduant iza t ion i s to r tion i s in t rodu ced . Atypical plot of the esul t ings ignal- to-quant iza tion-noise a t io(SNR ) ve rs us inpu t s ignal i s s hown in F ig . l ( a ) . I t c an be s eentha t a h igh SNR i s on ly ob ta ined fo r a na r row range of i n p u ts ignals . Com pand ing is used to increas e hed y n a m i c a n g eover which a high SNR occurs . There are wo bas ic ypes ofc o m p a n d i n g : )n s t a n ta n e o u s o m p a n d i n g n d 2 ) syl labicc o m p a n d i n g .Ins tan taneouso m p a n d i n gaseenpp l i edy a n yw o r ke r s t o b o t h PC M [ 1 ] -[ 3 I and de l t a modula t ion [41- [ 8 I .Ins tan taneous compan d ing a l t e r s the s hape o f the SNR curveandhe eakS N Rmay ven e ighe rhan o rhe n -c o m p a n d e d s y s t e m .Sy l l ab ic compand ing i s simi la r to th e ac t io n of a n a u t o m a t i cv o l u m econ t ro l n ha t he t ep s ize s changed lowlyands hou ld dea l lyb epropor t ioned o heave ragepower o f theinput ignal .Sy l l ab iccompand inghason lybeenapp l i ed od e l t am o d u l a t i o nbu t , a s i s hown n h i spape r , tcanb eappl ied to PCM as wel l .

of the IEEE Communications Society for publication after presentationPaper approved by the Associate Editor for Communication Theoryat the 1972 Electronic Instrumentation Conference, Hobart, Australia.Manuscript received July 21, 1972; revised January 25, 1973. The author is with he James Cook University of North Queensland,Townsville, Australia.

0 T5 companding

0,>overload

Y

0,E

0,>overload

Y

- 2 o v \ I0 20 LO 60 80Relative inputndB.Fig. 1 . Comparisonbetween heperformance of companded and u n -companded code modulators.

Syllabic ompanding anbedivided nto wo ypes [ 9 1 :1 )c o m p a n d i n gwi th ncomple tecon t ro la nd 2) compandingwi th comple te con t ro l . Com pand ing wi th ncomp le te con t ro ls t re tches a region of t he SNR curve in F ig . ] (a ) hor i zon ta l lywhi le compand ing w i th comple te con t ro l s t re tches one po in to n h eS N R c u r v ehor izon ta l ly t o g ive F ig . (b ) . tc anb es een tha t com pand ing wi th com ple te con t ro l g ives a f l a t SNRo v e r h een t i rec o m p a n d i n g a n g ea ndconse quen t ly gives abe t t e r pe r fo rmance than compand ing wi th incomple te con t ro l .I t is poss ible to apply sy l labic com panding to a code modula -t ion s ys tem which has ns tan taneous compand ing a s we l l , SOt ha t doub le compand ing can be ob ta ined .T he re ave eenmany y l l ab ic ompand ing chemes o rde l t a modula t ion , us ing ana logue t echn iques [IO]-[ 1 4 1 . D u etoh a r d w a r e i m i t a t i o n so n l y o m p a n d i n gw i t h n c o m p l e t ec o n t r o l h a s b e e n o b t a i n e d .I n o r d e r t o o b t a i n c o m p a n d i n g w i t h c o m p l e t e c o n t r o l , e i t h e ra s epa ra te igna l on ta in ing he n fo rma t ion e la t ed o hei n p u tp o w e rm u s tb es en t [ 1 5 1 , [ 161 ord ig i t a l e chn iquesmust be used [ 171 [20]. T he a s t two papers use a one b itm e m o r y , c o r r e s p o n d i n g t o a c o n t r o l w o r d f 2 b .T h i sp a p e rp re s en t san ewa p p r o a c h o h edes ignof hecompand ing s t ra t egy which enab le s oneo s e lec t con t ro l words ,o r m e m o r y , of any l eng th , the reby a l lowing one to s e lec t thera t io o f the a t t ack and decay t ime cons tan t s o f the compand-ing .heigi ta lyl labicompand ingel taodu la t ion(DSCDM) hardware discussed in this paper has a c o m p a n d i n gra t iow h i c h is a b o u t 20 dBmore hanhasprevious lybeenposs ible .T he inpu t pow er to a code modula to r can be norma l ized bydividingt b yhe qua re fhe t ep i ze .T he e s u l t ingnorma l ized npu tpower is averyusefulparameter or hedesign of th ec o m p a n d i n gs ince t is ameasure of overload.The c o r r e sp o n d i n g e r m n ampl i tude modu la t ion i s modula -t i o nd e p t h .F o rac o d em o d u l a t o rw i t h o u tc o m p a n d i n g h es tep s izes f ixed nd henorma l ized npu tpower i s thusd i r e c t l y p r o p o r t i o n a l t o t h e i n p u t p o w e r .

11 . DIGIT ALCOMPANDINGPRINCIPLESFig. 2 shows blockdiagram of a c o d em o d u l a t o rw i t hc o m p a n d i n g .T h ec o d eg e n e r a t o rg e n e r a t e sd i f f e r e n tc o n t r o lword s un t i l he co r rec t one i s ob ta ined , which is then rans-m i t t e d . A meas ure o f the norma l ized inpu t power i s de tec tedand h i s i s us ed to con t ro l he s t ep s i ze s to re which n u rncon t ro l s he mul t ip l ie r . P rov ided no rans mis s ion e r ro rs haveoccur red , the s t ep s i ze a t the t rans mi t t e r and rece ive r wi ll bet h e s a m e .In o r de r to de r ive a meas ure of t he norma l ized inpu t power ,one mus t s e lec t one o r more b ina ry pa t t e rns o r con t ro l words ,the e la t iveoccur renceofwhich varies with he norma l izedinpu t power . T he bes t con t ro l can be ach ieved if the func t io nof re la t ive occurrence of t he con t ro l word v e rs us norma l izedinpu t p ower i s a monoton ic inc reas ing o r dec reas ing func t ion .Fig. 3 s hows a yp ica lg raph o f the re l a t ive occur rence o f asui table ontrolword e r s us o rm a l izednpu t ower .T he


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