Adaptive Feedback Communications

7/30/2019 Adaptive Feedback Communications

1/6

IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOQY,OL. C O M - ~ ~ ,O. 1, FEBRUARY 1968 29

eight detector and converterombinations with112ARSs.In either arrangement, the equipment quantities can bevaned tomeet individualconditions.

The average savings in ime for a seven-digit call insetting up a connection for a subscriber with a TONE-DIAL telephone as compared to a rotary dial subscriber is21/2 econds. This is attributable to the fastereying time,of course, coupled with the controlled outpulsing by theconverter. Greater time savings are possible by increasingthe outpulsing speedwhere the associated switching systemis capable of responding to higher speeds. It has also beennoted that these subscribers make less dialing errors thando rotary dial subscribers. These advantages added o thegreater ease in dialing provide an attractive featureor thetelephone subscriber.

REFERENCESR. N. Battista, C. G. Morrison, and D. H. Nash, Signalingsystem and receiver for touch-tone calling, ZEEE Trans. Communi-cation and Electronics, vol. 82, pp. 9-17, March 1963.

cation of touch-tone calling in the Bell system, ZEEE Trans. Com-L2 1 M. L. Benson, F. L. Crutchfield, and H. F. Hopkins, Appli-munication and Electronics, vol. 82, pp. 1-5, March 1963.Bell Lab. Rec., vol. 39, pp. 201-204, June 1961.L3] C. G. Morrison, Centra l office receiver for touch-tone calling,

touch-tone dialing, Bell Lab. Rec., vol. 39, pp. 437441, December[dl N. Lazo and A. S. Martins, Central office modifications for1961.R. V. Burns and J. M. Hartz, Touch-calling provides high-speed control signaling, Automat. Elec. Tech. J . , vol. 8, pp. 266-273,October 1963.

L. Kelly Armstrong was born inNoblesville, Ind.,on January 12,1920. HeattendedButler Univer-sity,ndianapolis, Ind., nd, fol-lowing service in the U.S. Army,he attendedurdue University,Lafayette, Ind., where he receivedthe B.S.E.E. degree in 1949.

He joined Kellogg Switchboardand Supply Co., Chicago, Ill., wherehe designed circuits for manualtelephone switchboards rom 1949to 1952. Ele then joined Stromberg-Carlson Gorp., Rochester, N.Y., and

has served as Circuit Designer, Project Engineer, and EngineeringSupervisor in the design of dial telephone, automatic toll, and com-mon control systems. He holds three patents on design of telephoneequipment.

Adaptive Feedback CommunicationsJEREMIAH F. HAYES, ME MB E R, IEEE

REFERENCE :Hayes, J. F. :ADAPTIVE FEEDBACK COMMUNI-CATIONS, Purdue University, Lafayette, Ind. Recd 7/30/67.Paper 67TP977-COM, approved by theEEE CommunicationTheory Committee for publication without oral presentation. IEEETRANS. ON COMMUNICATION TECHNOLOGY, 16-1, February1968, pp. 29-34.ABSTRACT: A scheme for ransmitting binary signals hrough aRayleigh fading multipath medium s analyzed. Thescheme en-visions an adaptive receiver and a feedback channel. The feedbackchannel conveys information to the ransmitter on the forwardchannel s tate learned at the receiver. The transmitt er uses this in-formation to modify transmission. Results that show the improve-ment in performance of the system under the assumptionsf perfectreceiver learning and noise-free feedback are presented. In this-way,upper bounds on system performance are established.

INTRODUCTIONT E PROBLEM under study is the design of binarysignals for transmission througha Rayleigh adingmultipath medium. The approach to the problem is, inessence, a fusion of two concepts tha t have appeared in

N00014-67-A-226-0004 Mod. AB.1 This research was supported by Joint Services under Contract

recent communication theory iterature,adaptivityandfeedback.

In the system being considered, an adaptive receiverlearns the present state of the channel. This information,presumably gleaned from past receptions, is employed inthedetection process. The informationon the channelstate is also sent to the transmi tter via aeedback channel.The role of this feedback nformation is to modify thetransmitted signal in such a manner that the probabilityof error is minimized, subject to power constraints on thetransmitted signal.

The approach in this paper is to evaluate the perfor-mance of this adaptive eedback system under deal condi-tions. Accordingly, analysis is carried out based on theassumptions that receiver learning is perfect and the feed-back channel is delayless and noise free. The results indi-cate thecircumstances under which the adaptive eedbacktechnique is most promising. The results alsoshow theforms of transmitted signal and receiver tha t performmost effectively.

This approach is similar to th at of previous work tha thas been concentrated on adaptivity and feedback sepa-


2/6

30 IEEE T R A N S A C T I O N S O N C O M M U N I C A T I O N T E C H N O L O G Y , F E B R U A R Y 1968

MESSAGESOURCE

*I F A D I N Gxr(t,i)- -ECEIVERULTIPATHRANSMITTER MESSAGESINKM E D I U Mt L I 4 t ) ,F O R W A R D C H A N N E LFEEDBACK CHANNEL

Fig. 1. Adaptive feedback system.

rately. The effect of receiver adaptiv ity has been studiedby assuming perfect learning of the channel state in one-way systems.[11~[21he potential of feedback in combatingadditive noise in the forward channel has been demon-strated by he work of T ~ r i n , [ ~ l ~ [ ~ lchalkwijk,[51~[61 ndIiailath.[51 In this paper the feedbackhannel was assumedto be delayless and noise free.

S Y S T E M M O D E LThe system under studys depicted in Fig. 1. In charac-

terizing the random multipathorward channel, the Turinmodel[] is followed. If the input to the orward channel ism(t)cos mot, its output is5 a,m(t - j ) cos [wo( t - 7,)+e,] +n( t> (1)where a,, e,, and 7, are the random gain, phase shift, anddelay, respectively, in the jth multipath. n ( t ) is additivewhiteGaussian noise with single ended power densityspectrum N ,. Theadditive noise s independent of themultipath channel components a j , j , and T ~ ,= 1, 2, . . .,n. Each multipath is assumed to be resolvable, i.e., the ef-fective duration of the transmitted signal is less than thedifferences in the delays of the separate paths. Finally,the multipa th channel is assumed to be constant over sev-eral successive transmissions. Clearly, if any effectivelearning is to take place, this condition is necessary.

At any instant of time, the state of the multipath ischaracterized by the vectorsa = (a l , . . .,an ) ,0 = (01, . . .,e) and 7 =(rl, . ., rn). These vectors are learned at thereceiver. In this analysis, two levels of receiver learningare examined in urn. In he coherentcase the receiverlearns the vectors a, 0, and 7 , whereas in the incoherentcase the receiver learns the vectors and 7 . It is convenientto define the state vectors s1 = (al , . . ., a,, el , . . ., O n,T ~ , . ., 7,) for the coherent case and s2 = (UI, . . ., a,,T ~ , . ., rn) for the incoherent case. In the sequel many ofthe steps of the analysis are the same for both cases. Wedenote the coherent case by the subscript i = 1 and theincoherent case by i = 2 .

The receiver employs these learned vectors slor s2 toassist in the detect ion process. The channel state vectorsare also fed back to the transmitter.2 Depending on theoutput of the message source, the ransmitted signal iseither x+( t , si)cos mot or x-(t , si) os mot, i = 1, 2. Notethat the transmit ted signal is a function of the channel

j=1

state vector. The sum of the squares of the multipath gains is suf-2 As will be seen, it is not necessary to feed back the entirechannelficient.

state vector (coherent case) or sz (incoherent case). Inthe sequel we shall find the form of x*(t, si )which, ineach case, minimizes the average probability of error sub-ject to aconstrainton he average transmitted power.Since the message rat e is constant, a constraint on heaverage transmitted power s equivalent to a constrainton the average energy per message.

AVERAGERANSMITTED ENERGYAN D PROBABILITYF ERROR

For both the coherent and the incoherent case let usassume that the outputs f the message source are equallyprobable and thatE(&) = x+Z(t, si)& = x - y t , s,)dt, i =1, 2. (a )LT LT

X@,) =ST+(t, s,)z-(t, Si)&, i =1, 2. (3)The cross correlation is denoted by0

If for agivenchannel state he ransmit ted signal isz+(t, si) os mot, the transmi tted energy is

1E&) =2 E(s,) , i = 1, 2. (4)The ransmitted energypermessageaveraged over hechannel states is

wherep(si) s the multidimensional density functionof thechannel state.

The outputof the forward channel isn

d t ) = C a j 4 - ,, si ) os [ w d t - TJ +e,] +n(t),j = 1i =1, 2. (6)

Since the multipa ths n the forward channel are resolvable,we have from ( 2 ) and (6) the received signal energy for agiven channel state

The cross correlation between the possible received signalsfor different messages is

X@. )X&,) = -2- aj2, i =1, 2 .2 . j= l


3/6

HAYE S: ADAPTIVE FEEDBACK COMMUNICATIONS 31

Now in the coherent case, the signal component of g ( t )[see (6) ] is completely known. Thus, the etection problemis simply tha t of detecting known signals whose energyand cross correlation are given by (7) and (S), respectively,(i = 1 in this case). The probability of error for a givenchannel state is easily shown to be

PE(S1) = I - e r fdXR(s l ) I - X(SJ 1 (9)where

Averaging over all channel sta tes, the average probabilityof error is

n

where f(sl) is the 3-n variate distribution of the channelstates. From (10) it is clear tha t whatever the transmittedenergy is, the probability of error willbe minimized ifX(sl) = -1 for all sl, .e., antipodal transmitted signals.From thispoint on it will be assumed that the ransmittedsignals are antipodal in the coherent case.

In the ncoherent case it is assumed that the transmi ttedsignals are orthogonal, i.e., h(s2)=0 for all sz. hus, from(8) the received signals areorthogonal. The probabilityof error for this case has been worked ou t by Pri~e .[ *l ~[ ~lFor a given channel state, the probability of error is

12E ( S Z ) - exp [-R(s JX/ ~]

where

Again the probability of error is computedover the channel states

I n

(11)

by averaging

pE2 =- J, exp [ - R( sJ x/ ~I ~( sJ ~s ~ (12)where f(s2)s the 2 n variate density function of the chan-nel states.

SIGNALDESIGNIn both cases considered, coherentndncoherent

reception, we want to find transmitted signals that mini-mize the averageprobability of error [see (10) or (12)]subject to a constraint on theaverage transmitted energy

ETt =f E(s , ) f ( s , )d s ,5 M, i = 1, 2 (13)where M is some prescribed energy level.on the signal-to-noise ratio. Thus

It is convenient to work with an equivalent constraint

:*R(s,)f(s,)ds, I - M , i= 1, 2.MNOR 1 & - - -

We also have the obvious constraint

E ( s t )- - R(s,)2 0, i = 1, 2.NOA peak power constraint is not considered on the trans-mitted signal, although the optimization technique can beextended to include this constraint.The derivation of the optimum signal is very similar inboth the coherent and the incoherent case. The coherentcase is considered first. We form

where p is a Lagrange multiplier. If R(s1)is such that 1 -erfd2XR(sl) + pR(sl) is minimized for each channelstate sl,hen L1 is minimized. (Note that X, the sum of thesquares of t he channel gains, is specified by the channelstate.) Thus L1 is minimized by minimizing

I,(R) =1 - e r f d Z R +pR (17)for any X 2 0. Differentiating successively with respectto R we have

d21x(R)- X 2 exp [-2RX]dR2 - 2/x

From (19) it is seen that d21x(R)/dR2 2 0, and Ix(R) isa convex function of R. Thus, Ix(R) is minimum for

dIx(R*)dR*-- - - exp [-2R*X]

or rearranging terms the implicit equation for R* is

R*(X) =X exp [-2R*(X)X]27rp2Note that (15) is satisfied since

The Lagrange multiplier p was chosen such that (14) issatisfied with equality holding. It is not difficult to showthat such a p can always be found given any M.

We claim that of all transmitted signals whose averageenergy is equal to M , those specified by (20) minimizePE1 .Suppose we have another candidate hose normalizedenergy is given by M as we have seen

f Js [l - rf 42XR*(sl) +pR*(sl)]f(sl)dsl6. f s, [ l - erf d 2 X R ( s l )+pR(sl)]f(sl)dsl.


4/6

32 IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOGY, FEBRUARY 1968

But R*(sl ) nd R'(sl) are such that

and therefore

I [ l - rf d2XRr(sl)]f(sl)dsl .It is easy to see that the optimum signal just comes fromthe class such that (14) is satisfied withequality. If asignal has energy less than M , hen the probability f errorcan be decreased simply by increasing R in (9).

The analysis in the ase of incoherent reception proceedsin much the same way. Again the Lagrangian is formed

LZ= Js ds2f(s2) {exp [-R(sdXI +pR(s2) . (21)1Clearly, Lz is minimized if exp [-R(sz)X]+pR(s2) sminimized for each channel tate. We form

J,(R) =exp [ - R X ] +p R .1Differentiating successively with respect t o R we have

d2Jx(R)dR2 = X 2 esp [ -RX] .

From ( 23) it is seen th at I&) is a convex function of R- for all X . From (22) it is clear that I x (R) attainsa

global minimum at

But forX 5 p, R 5 0 and (15) is violated. However, sinceJx (R) is a convex function of R for any channel statewithin the region R 2 0, I, is minimized by R =0. Thuswe have the solution

As in the former case, p is chosen so that (14) is satisfiedwith equality. It is not difficult to show that such a valueof p can always be found for any given M . Following thesame line of reasoning as in the coherent case, one canshow that PE 2 s minimized by R*(X)given by (25).

EVALUATIONF SOLUTIONThe solution o thesignal design problem in the case of

coherent reception given by (19) has several interestingproperties. First of all, we note that its a function onlyofX , the sums of the squares of the channel gains. For agiven channel state, the feedback channel is required tosupply only ne number o the transmitter rather than the3-n numbers tha t specify the channel state .

It is not difficult to prove th at R ( X )as given by (19) hasthe following properties.

1) R(X) s continuous and differentiable.2 ) R(0) =0.3) R(X) attains a maximum of [ 2 p d Z exp ( 1 / 2 ) ] - 1 .4) R(X) is unimodal with a maximum at X = p 4 35 ) For large X , R ( X ) log X/4X .These properties indicate how the ptimum signal

achieves its economy of operation. The transmitted nergyis the highest in a middlerange of total channel gain.When the channel gain is low, little energy is wasted inattempting to compensate. As the channel gain increasesbeyond X = p d ~ T 2xp (1 /2 ) the transmitted energy de-creases as log X / X while the energy received increases aslog x.

Note that in deriving the optimum transmitted signal,knowledge of the probabili ty distr ibution of t he channelstates was not required. It is only in he computation of p ,the Lagrange multiplier, that his knowledge is used.Assuming th at he channel gains are ndependentandRayleigh distributed with density

exp ('/2>.

The distribution of

is chi-square with n - 1 degrees of freedom. The densityfunction of the random variableX is

f x ( 4 =zn--l exp [-z /k2]r(n)k2" , 0 I < m. (27)The average probabilityof error is

The average transmitted energy is

In Fig. 2 P E s plotted as a function of 2k2Rlwith n, thenumber of forward channel multipaths , as a parameter.For comparison purposes PE 1 s also plotted as a functionof the normalized signal-to-noise ratio 2k2R1, or the casewhere there is receiver learning of the channel state butno feedback. This corresponds to one of the cases examinedby Lindsey and Hancock.['l Curves for the no feedbackcase are computedusing the same numerical echniqueused to compute the curves for the feedback case.It is seen th at for the lower values of probability oferror, the improvement in performance can be significant.For example, for n =4 the feedback system requires 4 dBless average power to achieve a probability of error of


5/6

HAYES:ADAPTIVE EEDBACKCOMMUNlCATIONS 33

Fig. 2. Coherent case comparison curves.

!

FEEL

t10-7 -

1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 I I I I I I I I I I I I I I I I I I I-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 IO 12 14 16 182kK-b d B

Fig. 3. Coherent case cumnlative effect of adaptivity and feedback.

We point out that this mprovement is due to feed-back alone since the comparison is madewithsystemshaving the same level of receiver learning. In Fig. 3 thecumulative effect of receiver learning and eedback isillustrated. For the case n = 4, probability of error isplotted as a function f 2k2Rfo r the following three cases:

1) no eedback, receiver learning of r alone, and or-thogonal transmitted signals;3

FEEDBACKNO FEEDBACK---

n=NUMBER of MULTIPATHS -

lo- t \J I I I I l I l l l l l l l l l I I I I J I I I I I I I I I I I I I I I I I16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 IO 12 14 16 18 20 22

2k E2, B

-

Fig. 4. Incoherent case comparison curves.

--------

-FEEDBACK----NO FEEDBACK

n = 4 t \t \10-7 - \

1 1 1 1 1 1 , 1 1 1 1 , 1 1 15 -14 -12 -10 -8 -6 -4 -2 0 2 4 . 6 8 IO 12 14 16 18 20 2:1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1

2k2R d B2

Fig. 5. Incoherent case cumnlative effectof adaptivi ty and eedback.

2 ) no feedback, receiver learning of a, 8, and r,and anti-podal transmitted signals;

3) feedback, receiver learning of a, 0, and r, and anti-podal transmitted signals.

Note that if phase information is available at the re-ceiver, no ambiguity will result if antipodal signals aretransmitted. Such is not he case when the phase isunknown at the receiver. Comparing curves 1 and 2 it isseen that by learning a and 8, an advantage of approxi-This curve was taken from Hancock and Lindsev.[1lLI mately 6 dB results. Half of this improvement isdueo


6/6

34 IEEE TRANSACTIONS ON C OM M U N I C A TI ON TECHNOLOGY, FEBRUARY 1968

differences in thecorrelation coefficient. Comparing curves3 and 3 it is seen that the mprovement due to eedback isat least 4 dB for reasonable values of probability of error.

The results for the case of incoherent reception aresimilar to those of coherent reception.Note tha t hetransmit ted signal energy is a function onlyof X , he sumsof the squares of the channel gains. Averaging over X it isfound tha t the average probability of error is

1-[Xn-l exp (-X)]r 4 dX

In the course of analysis it was pointed out that thecoherent case is simply the familiar problem of coherentdetection of known binary signals. If a feedback link isavailable, it can yield improvement in performance overand above that already discussed, simply by applying thesignal design worked out by T ~ r i n . [ ~ l ~ [ ~ ] I nhis applica-tion, the feedback channel would carry Turins uncer-tainty feedback signal as well as nformationon thechannel state. For the case of uncerta,inty feedback, thedeterioration in performance caused by feedback channelnoise and delay js known.[lO1,[lll

exp ( - X ) REFERENCES(30) f l l J. C. Hancock and W . C. Lindsey, Optimumperformance ofself-adaptive systems operating through aRayleigh-fading medium,December 1963.and hat he avera,getransmitted normalized energy is IEEE -Trans. Communication Systems, vol. CS-11, PP. 443453,

R2 =L m l o g X / ~ P )xp ( - X ) X n - 2 *r ( n ) k 2n- x. (31)In Fig. 4 P E 2 s plotted as a function of 2 k 2R zwith n,

the number of multipaths, as a parameter. For omparisonPE 2s also plotted as a functionf 2 k2R2 or a systemwhosereceiver learns a and c but which has no feedback link.Again for reasonably low values of probability of error,feedback provides a significant improvement in perfor-mance. For example, for a probability of error of 10-4 thereis a 4 dB improvement in performance. The cumulat iveeffect of learning in the incoherent case is shown in Fig. 5where the probability of error is plotted as a function of2 k2R2 atio for the following cases:1) no feedback, receiver learning of c alone, and orthog-onal transmitted signals;

3) no feedback, receiver learning of a and e, and orthog-onal signals;3) feedback, receiver learning of a and e, and orthogonalsignals. ,

A comparison of curves 1 and 2 confirms the result ofHancockand Lindsey[] that learning the gain in eachmultipath does not significantly improve performance.A comparison of curves 2 and 3 shows th at if a feedbackchannel is available, learning of the multipath gains cansignificantly improve performance.

CONCLUSIONAs we have seen, for low enough probability of error,

the adaptive eedback technique can improveperformancesignificantly over systems employing the adaptive featurealone. This improvement is predicated on the assumptionof noise-free, delayless feedback. Ultimately the validityof this assumption is intimately related to the time con-stant of the mul tipath component of t he forward channel.It is recalled, that for a given channel state, only the sumof the squares of t he channel gains was fed back. If thisnumber varies slowly compared to the ra tef transmittinginformation, then he capacity of the feedback channelcan be far less than that of the forward channel. Further ,if the time constant is large compared to the delay in thefeedback channel, then the transmittersknowledge of thecurrent channel sta te is up-to-date.

amplitude and phase adaptivity in the detection of signals subjec tR. Esposito, D. Middleton, and J. A. Mullen, Advantages ofto slow Rayleigh fading, IEE E Trans. nformation Theory, vol.IT-11, pp. 473481, October 196.5.with feedback, IE EE Trans. Information Theory, vol. IT-11, pp.L3] G. L. Turin, Signal design for sequential det.ection systems401-408, July 1965.I4 ]-, Comparison of sequential and nonsequential detectionsystemswith uncertainty feedback, IEEE Trans.nformationTheory, vol. IT-12, pp. 5-8, Janua ry 1966.

f51 J. P. M. Schalkwijk and T. Kailath, A coding scheme forstraint, IE EE Trans. nformation Theory, vol. IT-12, pp. 172-additive noise channels with feedback-part I : no bandwidth con-183, April 1966.

f61 J,. P. M. Schalkwijk, Coding scheme for addit.ive noise chan-nels wlth feedback-part 11: band-limited signals, IEEE Trans.Information Theory, vol. IT-12, pp. 183-189, April 1966.[?I G. L. Turin,Communicationhrough noisy random-multipath channels, 1966 IRE Conv. Rec., pt. 4, pp. 154-166.ception of binary signals, IR E Trans. Information Theory,ol. IT-8,[a ] R. Price, Error probabilities for adaptive multichannel re-pp. 305-316, September 1962.of binary signals, Lincoln Labs., M.I.T., Tech. Rept. 258, JulyIs] -, Error probabilities for adaptive multichannel reception1962.

[ l o ] J. F. Hayes,Studies of sequentialdetectionsystems withuncertainty feedback, Ph.D. dissertation, University of California,Berkeley, Calif., June 1966.tainty feedback, IE EE Trans. Information Theory, vol. IT-14, pp.63-7 1, January 1968.

[111- Studies of sequentialdetectionsystemswith uncer-

Jeremiah F. Hayes (S65-N161)wasborn in New York, N. Y., on July 8,1934. He received the B.E.E. degreefrom Manhattan College, NewYork, N.Y., in 1956. He receivedtheM.S. degree in mathematics fromNew York University, New York, in,1961, and the Ph .D. degree in elec-trical engineering from the Uni-versity of California; Berkeley,Calif., in 1966.

From 1956 to 1960 he was a Rilem-ber of the Technical Staffat the ellTelephone Laboratories, Inc., Mur-

ray Hill,N. J . He worked at he Columpia University Elec-t.ronics ResearchLaboratories, New York, N. Y., from 1960 to1962. Presentlyhe is an Assistant Professor of Engineering a tPurd ue University, Lafayette, Ind. During the summer of 1967 hewas employed by the Jet ropulsion Laboratory, California Institu teof Technology, Pasadena, Calif.

Dr. Hayes is a member of Et a Ka pp a Nu, Sigma Xi, and theAmerica11 Association of University Professors.

Adaptive Feedback Communications

Documents

Transcript of Adaptive Feedback Communications