a variable rate transmission for rayleigh

8/12/2019 A Variable Rate Transmission for Rayleigh

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-20, NO. 1, FEBRUARY 1972 15

Variable-Rate Transmission for RayleighFading ChannelsJ A M E S K.

Abstract-Amodulation system is proposed hat continuouslyadjusts its data rate in response to signal strength variations n afading channel. The optimumvariation of dataratewithchannelconditions is determined, and includes the effects of feedbackchannel time delay, the interval between rate changes, and restric-tion of the number of allowable rates. Application of these resultsto a fullduplex situation allows determination of the optimumfraction of the data stream tobe devoted to service information.Comparison of this scheme with diversity transmission on the basisof error probability and bandwidth utilization reveals a reduction onthe order of 14 dB in transmitter power for a typical duplex link.

I. INTRODUCTIONSCATT ER comm unicat ion channel provides thereceiverwithamultitu de of reflected an d de-layed s ignal components , and as the relat ive de-lays of these componen ts dr i f t they tend to reinforce orcancel each other random ly. The result s fading of thesignal, a problem which has received considerable atten-tion since the earl y 1950s.

T he s im ples t wayof coping with fading is just. to makethe t ransm it ter power arge enough, or the bi t ra te lowenough, that he r rorprob abi l i ty is a t is factory orsomespecified fractio n of the ime . Rat her obv ious n-efficiencies ar e involved n this me thod .A more common technique is known as divers i ty t rans-mission, n which the signal is transmitted over severalchannelswith tatisticallyndependent ading,nhehope that not a l l recept ions will be badly faded. Oaf t h evar ious receiver s t ructures which can be used, he mosteffective is maximal at iopredetect ioncombining [ l ] ,in which the gain and phase a.re tracked on each diver-s i ty hannel so th at he s ignals an be brought n tophase coincidence, weighted optimally, and summed be-fore detect ion. Although divers i ty t ransmiss ion can pro-videaconsiderable mprovement nperformance, i t isat he expense of equipme ntcos t n hecase of spacediversit.y, or increasedbandw idth n he ase of fre-quency divers i ty .If afeedback ink s available , he nteresting possi-bility arises of allowing the receiver to monitor the chan-nel condi t ions and reques t compen satory changes in cer-

mittee of the EEE CommunicationsSociety for ublicationPaper approvedby theDat a Communication SystemsCom-without ral resentation.Manuscript received Jun e 3, 1971;revised July 23, 1971. This workwas supported by he NationalResearch Council of Canada,.Theuthors with theDivision of Systems ngineering,Carleton University, Ottawa, Ont., Canada.

C A V E R S

tain param eters of the transmitted signal. Haye s r3] hasconsidered a Rayleigh fading channel in w hich ,he ampli-tud e of the ransmittedsignal sundercontrol of thereceiver through use of the feedbac k channe l. For noise-less delayless eedback onditions he determi ned hatfunct ion elat ingpulseampli tude o he ns tantaneouschannel gain which minimized he error probability fora specified average ransmitter powe r. The metho d wasdemonstrated o be remarka bly effective incombatingth e effects of fading .Another nvestigationwasbegunby Clowes [2] , whoalso considered a Rayleigh fading channel and noiselessdelayless feedback, but t reated var iable durat ion ratherthan var iable ampli tude pulses . Although no g eneral re-sul ts were repor ted, he tudy ndicatedapotent ial lysuccessful scheme.

Bello andCowan lo]analyzed heperformance ofon/off transm ission, whic h can be considered as a specialcase of a two-rate t ransmiss ion sys tem. They calculatedthedegradationcausedbydelayandestimationerror,but because of their complicated system model, optimiza-tion of thesys temparam ete r shad o be approximate.Never the les s, hey found ha t he wo- ra te sys tem wasas l i t t le as 4 .34 dB away f rom the equivalen t nonfadingchannelunder dealconditions,andwasalwaysbetterthan fixed-rate transmission.In this pap er, we present the description and analysisof a scheme for varying the t ransmit ted data rate opt i -mally in response to changes in the channel gain hy thesimple xpedient of chang ing hepulsedurations. n-cluded in the an alysis a re the effects of feedback channeldela y and of time- and amplitude-discrete feedback forsituationsnvolving oisyeedback hannel . I t i sshown th at th e best performance, obtainable with noise-less delayless eedback, is typic ally 5 0 d B b e t t e r h a nth at of an quivalent ixed-rate ondivers i ty ys tem.

The egradat ion esul t ing rom ach of delay, ime-discreteeedba ck, nd uantizatio n of the allowablerates , is examined ndividual ly .Final ly , hepaper ex-amines a duplex ink or two-waysystem n which thefeedback channel carries messages as well as the service(ratecontrol) nformation.Hereone wishes to use thesmalles t service bi t ra te to the greates t . ef fect . A graphi-calmethod of optimizin g he wo-w ay ystem is pre-sented, and used t o show t h at even with significant feed-back de lay , the adap t ive r a te con t ro l s chem e can ach ievean impressive 14-17-dB reduction in required transm itter


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16 IEEE T R l N S A C T I O N SO NC O M M U N I C A T I O N S ,F E B R U A R Y 197Matched FiltersandEnvelope Detectors

Delay ta t )

R t )

Fig. 1 . Model of variable rate system

power rom n quivalen tmaximal a t iopredetectioncombineddiversi tysystem.11. SYSTEMMODEL

Descrip t ion of t he Var iab l e Ra te Sys temTh e communication ystem smodeledas nFig. 1.The sourcebuffer,which s ssumed to be infini te oavoid difficulties with overflow, su pplie s hemodula tor

with message bits a t a ime-var ying rate of R ( t i bit /swhich:is under control of the receiver. It will be under-s tood t ha t t h i s ra t ewil l not be changed during the trans-miss ion of a bit ; indeed, R t ) an be changed oniy peri-odically,andmust beheld constantover nterv als ofdurat ion A , which is much onger than he dur at ion ofindividual bi ts.The modula tor in turn emi ts the bandpass s ignal ml it )cos act, where k is 1 or 0, for each b i t . The two low-passn~odula t ingw avefo rms m k t ) areorthogonal inusoidsof durat ion T ( t ) = l / R ( t ) , as northogonal PSK orFSK. Ino rde r oma in t a in heo r thogona i i t y , nd oensureadiscretecarriercomponent , he inusoidsareswi tched cohere nt ly ; th i s impl ies tha t T t ) s an integermultiple of the half- perio d of the m k ( t ) . owever, i f thefrequency of m k ( t ) s much greater than R ( t ) hen fo ranaly t ica l purposes T ) can be treated as a continuousvariab le .TheRayle igh ad ingchannel is assum ed o be lowand flat , that is, the signal is received as

a ( t ) m d t ) os w,t + e@ + n ( t >where the Rayle igh d is t r ibu ted gain( t )and the uni formlydistributedphase O t) a rebo thcons t an tover he pulsed u r a t i o n , a n d n ( t ) is white Gaussian noise of single-endedpower densi ty N o . The fad ing bandwidth ( the bandwidthof ~ 2 ) ) will b e denoted by v an d the second m om ent ofthe channel ga in az will be deno ted by B.

In order to de termine the ident i ty of th e t ran sm i t teddig i ts , the receiver employs f i l te rs matched to m, t) a n dmo t),w hichare fol lowed by envelopedetec torsand adevicewhich elects the arg erou tpu t .The e su l t i ngbit error probabi l i ty for incoherent de tec t ions well know n:

P, (a , R ) = exp (- P a 2 / N o R ) 1)where P i s he ransm i t ter power. Th e receiver l sot racks the t ime-vary ing gain a ( t ) and phase 8 ) ei therwithdecision-directedmethods orwi th a narrow-band

fi l te ro x t rac the i scre te arr ier omponent .Thereference thus produced is assumed to be clean, or noisefree , which can be the case for da ta ra tes much greaterthan he fading bandwidth . On th e b asis of these mea -surements of the hannel ondit ionshe eceiver anrequest a chang e n data ra te a t n tervals of A secondhy forming an ins tan taneous funct ion a ) . Th e val ue op ( a ) is the rate request , a request for a possible changein t ransmission ra te , and i s sen t back to the t ransmi t terUpon ecept ionby he ransmi t ter , h i s eedback a terequest becomes the new d ata ra te R ( t + T . T he op timization of the ystem s oncerne dwith elec tion ofthe ra te funct ion p ( a ) .In rder or he ystem o pera te roperly , o tht ransm i t ter and receiver must know the curren t da ta ra texactly.This mpliese i ther tha t he feedbackchannelis noise-free or, more rea li s tica lly , tha t the ra te requestis ampl i tudequant ized o N nonzero evels, as well at ime quant ized by A , then protected by coding from feedback hannel oise.The uanti t ies N and A , whichdetermine the service bi t rate, wil l be considered as pa-rameters of the feed back channel . The unavoidable rotrippropagat iondelay T between t.he t im e he rate request i s fo r lnula ted and the time it take s effect can alsobe a t t r ibu ted o he eedback han nel .Thus we candescribe the act ion of the feedback channel by he pa-rameters N , thenum ber of nonzero rate s A ( t h e a t ep a ) = 0 will always beallowed n ,hedeterminationof the ate unction, houghnotnecessari lyused) , het ime separat ion between he periodic rate requests, andthe round-t r ip de lay T.Mathem at ical Formulat ion of t he Prob lem

If a ( t ) s ergodic, the n in a very. long tiine inter val Ia will be between a and CY + d a ) f o r Ip,(a)cFCv secondswhere p a ( @ ) s the probabi l i ty densi ty funct ion (pdf) ofa . D ur ing h i s ime p ( a ) l p , ( a ) d a bits willbe t ransmi t ted ,and hus heaverage a te ,or o ta lnumber obi t s t ransmi t ied during I div ided by I , is given by

R,, =lrn.~>P~ .> ia.For a specified R,, we wish to minimize he averageerrorprobabil i ty,which scalculatedbysumming heerror probab il i t ies for each bit in a very long string anddiv id ing the resu l t by the numbe r of b i ts in the s t r ing .


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C A V E R S : V A R I A B L E R . 4T E T R A N S M I S S I ON 17W c first consider the case where the rate change spac-ing A is negligible compared with he fading ime cori-s t an t v - ~ .Waveform values a t t ime t will he denoted bythesubscr ipt 0, andwaveformvalues a t t im e ( t + T)by the subscr ipt 7. Then thc contr ibut ion to the sum ofthe error probabilities when a is between CY and C Y + & )is given by

h ) P ( 4 da 1- AP, P a ) ) P a , l a o P 1 . 1 dP.In the p reced ing equa t ion P o @ , ) is the er ror p robabi l i tyas a funct ion of th e da ta ra te p an d the channel gain P a tthe ime he ate eque s t akes ef fect, 7 seconds ater.T he func t ion Pa,,ao(Pla)s the pdf of a( t + 7) onditionedon a ( t ) . It fol lows that he average er ror probabi l i ty isgiven by

F e = 1 P ( a P c b , (.(>)P..(4 d f f1 nwhere thepredictederrorprobabi l i ty P , isdefined as

Fading StatisticsBefore we consider th e sol uti on of 5 ) a n d (6), let usexamineheorm of the redicted rror robabilityP , ( x , 1.). According to (2) a n d 3) we need second-orders tat is t ics for the hannelgain a ( t ) . Fortunately , heseare avai lable for he Rayleigh fading channel . It is wellknown [ ] ha then-phase nd uadra tu re hanne l

gains u ( t ) a n d v ( t ) , respectively, can be characterized asnarrow-bandGaussianproce sses of power B/2, a n d tcan be shown hat if the f requency scat ter ing funct ion[.5] iseven, hen u t) a n d v ( t ) are ndependent .The e-ceiver, s ince it tra cks a ( t ) a n d e t ) , the polar coordinaterepresen tation of the wo processes, can alculate hequadrature ains as u t) = a ( t ) cos O ( t ) a n d v t ) =a( i ) s in O ( t ) . Th e con ditio nal pdf of a(t + 7) can be ob-tainedby inear ly i l ter ing u t) a n d v ( t ) separately toproduce the condi t ional means p,, a n d p with some fixederror ar iance.Con version of theseGaussian andomvariables u t + 7) n d v(t + 7) back to polar coordinatesgives a Rician pdf for a( t + 7):

abilitiesover he PA bits ransmit ted in the nterval[ t + T, + T -t. A ] , we obtainwhere i = a( t + 7 + i / p ) a n d pa, l , ,o ( f l i [u )s the pd f ofP i conditionedon a ( t ) . E quat ion 3) is just a sum ofterms s imilar to (2).A convenientnormalizationcanbeperformed if wedefine r p/RaV n d x A Pa2/NnR,,. The pdf of x isthen exponent ial

1pdP> = 6 exp ( - x / b ) , x 2 0 4)where the verage nergy-to-noiseatio er it b APB/N,R,,. T he ns tan taneous r ro rprobability of (1)has the form P L ( x , ) = 3 exp (-x/21.).The variable rate problem now consists of f inding therate unct ion ~ ( x ) hichminimizes the verage r rorprobabi l i ty

p e = 1- P)Pe P, )pz P> P ( 5 )while maintaining the average rate

p = l mP>pz P>P = 1 6)subject, of course, to ~ p ) 0.

BR h ) = = xp - y I r l )for two reasons. First, this form has a first-order Butter-worth power spectrum and it is known [ 6 ] that for th isspectrumkno wled ge of theval ue of thewaveform att ime t alone is sufficient for the prediction of the value a ttinw ( t + 7). his simplifies ( 7 ) considerably sinc,e 7 nnow dependsonly on a ( t ) , ndnot 8 t ) . Second,andmoremportant,ecentleasure men ts of th e time-frequencycorrelat ion unct ion on a n H F l inkbetweenHawaii ndNcw .Jersey [ 7 ] show an utocorre lationfunction R, y) which an he closely appro xima ted byexp(-vjyl) . For his autocorrela t ion funct ion we have

m = k a ( t ) , u2 = (B/2)(1 A )wherc k = e x p ( -vT). Subst i tu t ion nto 7 ) gives heconditional pdf

which, as noted before, is independcnt of O ( t ) Thc re-ceiver needs to track only the gain ci ( t )Final ly , we subs t i tu te (8) in to 2 ) andnormalize oobtain the p redicted er ror probabi l i ty for A = 0:P e ( x , ) = r 2- exp2r + b(1 k ) 2r + b(1 k 2 )


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18If A 0, then rom 3) we write hepredictederrorprobabi l i ty as

IEEETRANSACTIONS O N COMMUNI CATIONS , FEBRUARY 1972channel is show n in F ig. 2.T he top th ree cu rves a re fo rm ax im al a t io 'p rede tec t ionc o m b i n e d M R P D C )diver-s i ty t ransmiss ion with no divers i ty (s ingle t ransmiss ion) ,dua ldivers i ty , ndquadrup ledivers i ty , as calculatedfrom Bel lo 18, appendix 111. Th e curve represent ing theper fo rm anceof headap t ive a tecon t ro l ys temwi thperfect feedback is the lowest on the graph. In fac t , noscheme which employ s incoherent detection of orthog onalRayleigh faded pulses can at ta in a lower error proba bil-i ty han dap t ive a te on t ro lw i thper fec t eedback ,since theerrorprobabi l i ty P , = exp ( - b / 2 ) ispre-cisely that of a cons tant channel with the same value ofenergy- to-noise rat io per b i t b . T he po ten t ia l s av ing in -t roduced by adap t ive r a te con t ro l s ve ry a rge , on heorder of a 20-dB reduction in power requirement from adua l d iver s i ty MR . PDC sys tem .3andwid th E xpans ion C ons t ra in t

T he pena l ty pa id fo r th i s im provem ent i s a bandwidthwhich luc tua tesabou t heaverageva lue R,,, a n d att imes, ecause no constra int as been place d nhemagnitudeof he ate unct ion + x), can become in-finite. It is easonable oaskwhatdegradat ion esul tsf rom an upper bound ro on r (x), ince i t i s the maximumbandwidthwhichdetermines he pacing of frequencymult ip lexedsubchannels .

The opt im um rate is now propor t ional to x, as before,b u t s a t u r a t e s a t To that is ,r (x ) = x 5 UT

-io/: uro < x .where the cons ta n t u s selected to sat is fy he averageratecons t r a in t ( 6 ) . The resul t ingaverageerrorprob-ability of (5) is shown on Fig. 2 for a maximum band-widthexpansion ro of 2 . T h ecurve o r r,[)= 4 snotshownbecause i t i s a lmost ndis t inguishable rom hem i n i m u mat ta inab leerrorprobabi l i tycurve .A l thoughthe cons traint ro can be seen to hav e l i t t l eeffect on errorperform ance, it should be included in the calcula tion oftheopt imum ate unct ion ince hem axim umband-widthoccupancy is an mpo rtant p a r a m e t e r ns y s t e mdesign.

It should be noted that if theM R P D C i v e r si t ysystem s whose performance s shown n Fig . 2 are con-sidered as f requency divers i ty sys tems, then the var iablera te cu rves fo r ro = 2 a n d ro = 4 r e d i r ec t ly com parab let o h e M R P D C c ur ve s o r I, = 2 a n d L = 4, espec-t ively . With the same values of bandwidth and energy-to-noise at ioperbit,adapt ive atecontrolprovides avery argepower eduction, on the ordero f 18 d B f o rro = L = 2.Effect of Feedback Delay and Rate Change Per iod

T he f eedbackdelay T and he atechangeper iod Ah a v e a similareffect, in that hey ncrease sys tem e-sponse t im e t o a change in z ( t ) . However , A, r a t h e r t h a nbeing fixed inadvance,can be selected to compromise

r R n v A r* 2 2r + b(1 ki 2r + b(1 k:)2- exp k i2 x -) (10)

where kj = exp [ - V ( T + j / ( r R R , , ) ) ] .111. SYST E M ERFORMANCEIT H OPT IMUMRAT EFUNCTIONNow th at we have obtained an express ion for the pre-

dicted error probability, we ca n procee d to calcula te theopt imum a te unct ion, heonewhichsatisfies (5) a n d6), and to determ ine the reduc tion in erroi- proba bilityresult ing f rom ts use. In th is sect ion, the minim um at-ta inable er ror proba bi l i ty will be der ived, then the effectof each of the feedbac k param eters T, A , a n d N will beexamined separately .Minimum Attain ,able Error Probabil i tyT he m in im um a t t a inab le e r ro r p robab i l i ty is achievedwhen eedback ond i t ions r e dea l , ha t is, VT 0,v A 0, a n d N = a n this case, he predicted er rorprobabi l i ty P , ( x , r ) of 9) a n d (10) reduces to he in-s tan taneous e r ro r p robab i l i ty P s ( x , ) = 3 exp ( - x / 2 r ) .The problem of dete rmining the cont inu ous r ( x ) whichminimizes (5) while at is fyinghe ons traint (6) is atypical problem in ,var iat ional calculus , made even s implerby the lack of der ivat ives . We adjoin the two equat ionsw i t h a Lagrangemultiplier X anddif ferent iate he n-teg rand wi th r espec t to r t o o b t a i n

+ 1+ Z / 2 i ) exp ( - x / 2 r ) = h 11)as the im pl ic i t equa t ion fo r the so lu t ion r x ) .Since theleft-handside of (11) isa unctiononly of x / r and sequa ted oacons tan t , hesolut ion s r = k x , with Isom eonstant .Usinghe pdf 4) in 6) gives theopt imum rate funct ion as r ( x ) = x/ b , which holds th eerror probabi l i ty cons tant a t P , = 9 exp --b/2).

Since the ransm it ted power sconstant,use of th erate funct ion r z ) = x / b al lows the ransmit ted energy ,per it P b/R , , , x to become extremelyargewhen x issmall . Never theless , the average energy per b i t is f in i te ,T h e energy r ansm i t t ed n a long ime nterval I isj u s t P I . T h e o t a lnu mb er of bits ransferred n I isgiven by

. as he fol lowing alculat ion hows.

I o Lrn avr(a)p ) da = IR,,,so that he average energy , he rat io of the wo s us tP / R a v as one would expec t. The energy is f inite becausethose bits with very large energy, simply because of theirlong durat ion, make u p only a t iny f ract ion of th e to ta lbi t s t ream.T he e r ro r p robab i l i ty fo r the var iab le r a te sys tem andfor some other systems operating on a Rayleigh fading


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CAVERS : VARIABLE RAT13 TR.ANSMISSION 19r-.

ra te contro lfeedback

2 5 IO 50 100Energy to noise rat io b = P W N o Rav __

Fig. 2 . Comparison of errorperformance of diversi. ty ransmis-sion to hat of variable ate ransmission with andwithout abandwidth constraint.between the requirements of low errorprobabi l i tyandlow service bit. rate . Th e effect of these two param eterscan be observed when other conditions are ideal, th a t is ,To = co a n d N = M . T he im pl ic i t qua t ion o rheopt inlum rate funct ion T Z ) c an b e o b t a i n d i n a m a nn e rs imilar to that for the casewhen al l feedback parametersare deal . Adjoining (5) a n d 6) with a Lagrange m ult i -p l ier and dif ferent iat ing he ntegrand with respect o ryields

r -$+ P e ( Z , r ) x = 0Pas the im pl ic i t equa t ion fo r the op t im um T z). T h e p re -dictederrorprobabi l i ty p , ( ~ ,.) isgiven bye i ther (9)o r (lo), depending on whether or no t A = 0.Fig. 3 disp lays the effect of the feedba ck delay T a n dtheratechangeper iod A (expressed as fractions of thefad ing t im e cons tan t V-') on the performance of a n a dap -t ive rate control sys tem us ing the rate funct ion given by12). T heerrorprobabi l i ty waseva lua tednum er ica l lyusing thediscrete ate unct ion,developed in thenextsection, with 16 nonzero rates. It is evident th at even asmall increase in the system response time can seriously

degrade heperform ance of thevar iable ate ys tem.For tuna te ly , m any sca t t e r channe l s have a VT product onthe orde r of 0.01, so th at th e system s still reasonBblycontrollahle.Effec t of R a t e Qumt ixa t ion

It was observed in Section I1 th at th ere will he a finitenunlber of t ransmiss ion rates in order hat he rate re-quest be protected from feedback channel noise. Here wecons ider he ef fect on er ror probabi l i ty of the resul t ingquantization of the rate function. It should be noted th atal though heerrorprobabi l i tymust ncreasewithde-

rLLt

\ \v2 5 IO 50 100Energy to noise ratio b= P W N o Rav

ig. 3 . Errorperformance of variable ate ystem withdelayand with time-discrete rate requests.

creasing N the required feedback channel capaci ty de-creases.T hequant ized ate unct ion Fig . 4 ) is a s taircasefunctionwith N nonzero atesand 1V switching hres-holds. The lowest rate rl has beenset. to zero n orderto al low the t ransm it ter to shut down, or cease informa-tion transmission, when channel conditions are very bad.We wish tose lec t heset of rate s { T ; } and hresholds{ x i } to minimize the average er ror probabi l i ty

while sat isfying he average rate cons traintN + 1

,? = s < rip ) dP = 1= 1 zi - -1

where x,,= 0 a n d zNil= m .Onceagain we canadjoin he woequat ionswithaL,agrange multiplier A to form the augm ented cost. func-tion,which epends on the 21V var iables { T ~ } ; = ~ , a n d{x;};=,,. Differentiation of this cost unctionwith e-spect o each variable produces he necessary and suffi-cient [9] conditions foram in im um :

?. , . l [P , P , * + d XI = d P , P , T i ) X I 14)where each ?*i must sat is fy 13) and each x i must sat is fy14). T hepredictederrorprobabi l i ty P , P , T ) hasbeenused so t h a t 13) a n d 14) can erve as t h e basis fo rcalculating the combined effect of the p aram eters N , T, A,a n d ro.T he a tesand hresh olds of theo p t i m u mquan t izedra te unc t ion anbeob ta ined rom 13) a n d 14) b yiterat ion on a s ingle parameter , say t , , thereby avoiding

a search over the entire 2N-dimensional space T ; } X {x i .


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20 IEEE T R A N S A C T I O N S O NC O M M U N I C A T I O N S , F E B R U A R Y 197

X I X 2 x3Instantaneous energy o noise ratio XFig. 4 . Quantized rate function.

The terat ionproceedsbyselect ing a value or tl a n dsolving l ternately (14) an d (13) to btain he therra tes and hresholds . If X, a n d y N + , so obtained sat isfy(13) , then the i te ra t ion i s com plete . Otherwise the valueof t , i s a d justed and the process repeated . The value ofX isad justedunt i la a te unct ionsat i sfy ing (13) an d(14) alsoatisfies theverageateonst ra in t = 1 .If we sub sti tu te he pdf of x (4) and hep red i c t ederror robabil i ty 10) andperform he n tegra t ion n(13), hen we can write he condit ions (13) a n d 14) s

@ xi, ; , X) = r i , X)xi,X> = +(Ti, xi, >

where@ x, r , X) = X exp -x/b) r R e AA q i = lc exp ( - P / b )

u b(1 i 2 ) I 2rki2 xu

in which q = 2r + b and uj q b k j 2 .If there is delay-free eedback 7 = 0) and he a terequests are fed back continuously in time A = 0) , henwe canohserve he solated effect of allowingonlyfini tenumber of ra tes .F ig. 5.illustra tes he effect onsystem erform ance of this uantizat ion of the atefunction for selected values of N , the num ber of nonzerora tes . It can be seen th a t even n termi t ten t ( N = 1)t ransmission provides considerable improvement over thefixed ra te ystem .With ncreasing N theperformanceimproves; unti l a t N = 16 here is only a 0.2-dBdeg-radat ion from hat ob ta inable wi th he cont inuous ra tefunction.

IV. O PTIMIZA TIO NF A T W O - W A Y A R IA B LER A T E S Y S T E MAt th is po in t we have the ab i l i ty to ca lcu la te the errorprobabil i ty P , as a function of th e energy-to-noise rat ioper bi t b for any choice of the param eters ro , N , A, a n d r.I n a duplex l ink, however, we ar e inte res ted not only inachieving a low error probabi l i ty but in using as smal l afract ion as possible of the bi t stream for serv ice inform a-t ion. In this sect ion , graphical method of invest igatingthe radeoffsamong heseparameterswhich affect th e

-

2 5 IO 50 100Energy to nolse mtio b- P W N o Rav

Fig. 5 . Error performance of variable rate systemwithatesquantized to N nonzero values.3000

2000

1000900800700600500400

2 300v,2 200

Wa;W

I&5 100aW 90

80706050

I I I I ICurves f r o m t o p t bot tom , have:q = l N.1)q.2 N.3)q=3 N.7)

FRACTIONAL RATE CHANGE SPACING 2/AFig. 6. Duplex variable rate system.

bandw id thand he message apacity of the eedb ackchannel will be presented.Letusconsidera i tuat ion inwhich tat ion A andsta t ion R t ransmit oeachotheroverRayle igh ad ing


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C h V E R S : V A R I A B L ER A T R TRANSMISSIONchannels, so t h a t A s forward channel can act as B s feed-back channel , and vice versa (Fig . 6 ) . E v e r y A secondsthe message bi t t ream is n ter ruptedand he e rv icehi tsrateeques t) renser tcd a th e u r r e n ta t e .Service information reduces the message data rate , so theproblem is to determ ine the optimu m amou nt. of serviceinformation on each channe l , hat is, to p ick N a n d Awhich yield the mo st efficient system.

W eassum e ha tchanne l AB andchannel BA h a v eidentica l s tatistics, so tha t they ca r ry the sam e am o untof service inform ation. W e fu r ther as sum e tha t they a res tat ist ical ly ndependent s ince, even if th ey occupy thesame scatter volume, the center frequencies will usuallybe ufficient ly fa r offset, th at he Gauss ianquadra tu recomponents of the ade willbe uncorrelated.Thus wecanconsider the chann els separately .If PC M transmiss ion of the rate reques t is employed,th at is , he request s digitized with no redundancy re-moval, hen he ervice nformationbit ate s R, =M log, ( N = l ) / A , where each service bit is repeated Mt imes for protect ion. This e lementary coding s me antto ensure that service nformation is received correctly.W e define q = log, ( N + 1) .It i s easi ly shown tha t the avera ge b i t r a te R,, usedearlier is the su m of the averagemessagebit ateandthe service bit rate, that. is ,

R,, = R , -k R. = R,, -+ M q / A . 15)In he design of a commu nication ystem we mightreasonablyexpec t to beable o choose the number ofnonzero rates N , the rate change per iod A , the m ax im umbandwidth expans ion yo, a n d t h epower P . T he param ete r smore ikely to bespecified in advance a re he bit error

probability p e , he message rate R,, and he eedbackdelay T . I n such i rcumstances a graphdisplayingallcombinat ions of the eedbackparam ete r s b , uA, and. Nwhich give a specified error probability p , fo r a specifieddelay U T , and bandwidth expans ion yo, is l ikely to proveuseful.Fig. 7is such a graph or p e = UT = 0.01,a n d 70 = 2. The funct ion displayed will be denoted byAs a n example of th e use of Fig. 7 , suppose tha t at roposcat ter sys tem must be des igned o handle a datar a t e R , at an average bit error proba bility of lo-'. T h eround- tr ippropagat iondelay is 0.01 ading ime con-s tan t sa n d h em a x i m u m bandwidth expans ion ro is 2.We w ish to select q a n d A so th at th e required power-to-noiseatio P B / N o is minimized. Fro m he definition

b = P B / N , R , , and f rom (15)

b ( q , A ) .

b ( q ,) R , + M q / A ) = P B / N , , (1G)where b ( q , A) is thefunctionshown on Fig. 7.T o ca l -cu la te he op t im um q a n d A , we first pick q = 1, thenfind the A whichminimizes the eft-hand side of (16).Next , heprocedure s epeated or q = 2, q = 3 , a n dso on to q =: 4 afte r which th e effect on error probabilityof increasing q is negligible. Th e q , A combination whichgives t he lowest value for P B / N o is optimum.

21AB message EA serv ice

Channel EA

TransmitterEA message A B servlce

Fig. . Tradeoffs amqng feedback system parameters with constantP,=10-6, v s = O . O l , r 0 = 2 .

A nalter nati ve use of Fig . 7 occurs n thes i tua t ionwh ere, inste ad of minimizing P B / N o for a given R,,,,wemaximize R,, for a given P B / N o . In hi s case, we re-arrange (16) and f ind the q , A combination which maxi-mizes

A th ird way in which Fig . 7 can be used s the situ a-t ion n which, as well as a cons traint on the maximumbandwidth expans ion, there is a cons traint on the maxi-m u mbandwidth TOR;,,.. This impliesa pecified Ray. fP B / N o is fixed (t hu s fixing b ) hen we wish to maximizeR,,, = R,, Vq/A. We simply calculate he value of band hen, on Fig . 7 , calculate q / A foreach of the fou rvalues of q a t t h a t . b . T h e q , A combination hus foundwhichminimizes q / A isptimum . lternatively , ifP B / N , , is not fixed (so nei ther s b ) , a n d R , = M q / A( and hus R,,,) is fixed,we canminimize the equiredvalu e of P B / N o by f inding the q , A A = Mq/R ,) pointon Fig. 7which has the minim um value of b ( q , A ) .

As a numerical example, suppose a modulation systemis to be designed for a 300-mile roposcatter ink whichhas a fading bandwidth of 0.5 H z , a typical value, whichgives u = 3.14 r ad / s .Ea ch of several requencymulti-plexed subchannels is to h a v e a message data rate R,,, =100 kbi t /s , an er ror probabi l i ty E , = lo-', a n d a maxi-m umbandwidthexpans ion 7.0 = 2 . W h t c ho ic e of Na n d A gives theminimum equired eceiverpower-to-noise ra t io P B I N , ?Firs t we pick M = 3 so tha t a double error is requiredfor a mis take in a service bi t . Th e bi t er ror probabi l i tyis lready so th a t we can eglect the possibility ofincorrectreception of the rate request. Now since VT =0.01,Fig. 7 summarizes henecessary nformation. W ewill try t o m inimize P B / l v o = b (9, A ) ( i o 5 t 3 q / A ) . Onthe q = 1 curve, tr ial and error gives a minimum valuefor P B / N , , of 9 .1 X lo6, a t t a i n e da t A = 0.75 XSimilar ly , on the q = 2 curve hem in im umvalue ofP B / N , , is6.5 X lo , also a t A = 0.75 X Continuingin his ashion, we ind them in im umvalue of P R / N l lis 6.03 x loo,a t t a ined a t q = 4 a n d = 1.25 xLetuscompare his resultwith herequired P B / N ofor anM R P D Cdiver s i ty ys temwi th n qu iva len tbandwidth expans ion; th at is , a dual- f requency divers i tysystem. The perform ance of this system is il lustrat.ed on


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22Fig. 2, where i t can be seen tha t the requi red va lue of bis 2800. The req uired pow er-to-noise rat io P B / N o is then2.8 X 10. Thus a t he s a me da t a r a t e a nd e r ro r p rob -abil i ty, and for only 9.6 percent . more bandwidth (causedby he se rvice inform at ion) , hevar iable a te schemeeffects the emarkable avingof16.7dB n equi redt ransm i t te r power .NO W, as a second example , e t us suppose ha t band-width cons t ra in t s are very t ight and we can allow onlya 1-percent ncrease in bandwidth for he service nfor-ma t i on . Thus R, = 0.01 X R,,, = 1 kbi t / s , and A = 3qX Examining ig .wi thhese q , A pairs, weeet ha t he mi n imumvalue or b is 105, andoc c urs a tq.= 1, A = 3 X Then since R,, = R,,,+ R, = 101kbi t / s , we f ind tha t PB,/N, = 1.06 X l o7 Thi s i sstilla savin g of 14.2 dB from a dua l - f r e que nc y MR P DCdivers i ty sys tem.

V. CONCLUSIONSWe have seen th at when feedback conditions are per-fect , adaptive rate control can completely el irninate he

effect of fad ing, hereb y savin g 10-50 d B in energy-to-noise rat io per bi t from a fixed rate nondiversi ty systemfor ypic,alvalues of err or pro bab il i ty. n view of thefac t tha t t roposca t te r sys tems a re charac te r ized hy veryhigh tr ansm it ter pow er, typical ly 10-50 kW, th i s savingis most a t t ract ive. The disadvantage of t .he variable ratescheme i s th a t i t expands he bandwidth; f i rs t, becausei t is a varia ble rate scheme, and second, because serviceinformat ionmus t he t ransmi t ted .How ever , h i s s aminordrawback,sinceallowing a maximum bandwidthexpansion of only 2 results in a mere 0.9-dB degradationfrom the infini te bandwidth case.T he c loses t nonadapt ive competi tor i s M R P D C diver-si ty ransmission, which requires both he gain and hephase to be tracked on eachransmission.Diversitytransmissionheremeans requencydiversity.Spacedi-versi ty would require everalphysically eparatedan-tennas , a considerable expense. Time diversi ty not onlyrequires buffer storagea t . he ransm i t te r (as does ratecontrol),but or heses lowly adingchannels equiress torage of ve ry many ana log va lues a t t he rece iver . Weare eftwi th requencydiversi ty,whichalso ncreasesthe Ijandwidth. A comparison of a variable rate systemt o a n MR P D C sys t e m, wi t h t he s a me t yp i c a l va l ue s ofe r ro rprobabi l i ty , da ta a te ,an d banc1widt.h ha s showntha t dapt ive a te ont ro l an chieve n mpress ive14-17-dB savingover M R P DC , t s c lo ses tcompetitor.Ye t a fur the r advantage a r i ses f rom the fac t tha t , inorder to be s ta t i s ti ca l ly independent , the separa te d ive r-si tyransmissionsmust bewidelyeparated inre-quency, a t l eas t grea te r than the corre la t ion bandw idth

IEEE TRANSACTIONS O N COMMUNICATIONS,F E B R U A R Y 197of thechannel. Since a typical roposcatter ink woulhave a totalbandwidth of abo ut1 0M H za n d a correlat ionbandwidth of a bou t4MH z , he re i sobviousdifficulty inusingmore handual-frequencydiversi ty.Channels with larger correlat ion bandwidths present evmoreproblems incenot l l ubchannels anuse re-quencydiversi ty.Adaptive atecontrol does not suffefrom his disadvantage, since he bandw idth required inot n disjoint regions of the spe ctru m.It would appear tha t the adapt ive ra te cont ro l schemdesp ite its com plexity , is an excellent, metho d of improving commun icat ion system performance o n Rayleigh fading channels.

AC KNOW LEDGMENTT h e a u t h o r is indebted to his research supervisor, DR . TV. Donaldso n, of the Universi ty of B ri t ish Colum biaforhisadvice and encouragement.during hecourse othis work.

REFERENCESD. G.Brennan, Linear iversity ombiningechniques,Proc. IRE, vol. 47, pp. 1075-1102, June 1959.G. J. Clowes, Variable rate data transmission for a Rayleigfading channel, Commun. I,ab., Defence R es. TelecommunEstablishm ent, Ottawa, Ont., Canada, Tech. Memo. 22, Feb1969.J. F. Hayes, Adaptive eedback ommunications. IEEEH. N. Shaver, B. C. Tuppe r, and J. B. Lomax, EvaluationTrccns.Commt~n. echnol. , vol. COM-16, pp. 29-34, Feb. 196of aGaussian HF channelmodel, IEEE Trans .CommtcnR. S Kenncdy, Fading Dispers ive Communicat ion ChannelTechnol. , vol. COM-15, pp. 79-88, Feb. 1967.New York: Wiley, 1969.A . Papoulis, Probabilzty, Random VariablesandStochasticF. David. A . G . Franco, H. Sherman, and 1,. V. ShucavagProcesses. New Pork : McGraw-Hill, 1965.IEEE Trans. Commtm. Technol., vol. COM-17, pp. 245-256Correlationmeasurements n n HF transmissionink,Apr. 1969.P. A. Bello, Selection of multichanne l digital data systemsvol. COM-17, pp. 138-161, Apr. 1969.for ,roposcatter channels, IEEE Trans . Commun. TechnolJ. K. Cavers, Adaptive rate control for ime-varying communica,tionchannelswith a feedback ink. Ph. D. dissertation, Dep. Elec.Eng..Univ.BritishColumbia.Vancouver,B. C., Canada, Aug. 1970.P. A . Bello and W . M. Cowan, Theoreti cal study of on/oftransmission over Gaussian multiplicative circuits, n ProcIRE 8 thNo t .Communica t i onsSymp . , Utica, N. Y., Oc1962.

James K. Cavers wasborn nPort AliceB. C . , Canada,onSeptember 19,1944. Hreceived the B.A.Sc. degree in engineerinphysicsn 1966 and hePh.D. degree ielectrical engineering in 1970, both from thUniversity of British Columbia, Vancouver.I n 1970 he joined the Systems EngineerinDivision of CarletonUniversity,Ottawa,Ont., Canada. His current research interestsare nadaptive echniques for signaling itime-varyingispersivehannelsndnhuman-fact,or evaluation of video communication links.

a variable rate transmission for rayleigh

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