Contains reports on one research project.National Aeronautics and Space Administration (Grant NGL 22-009-013

Dolinar, S. J., Jr.

DSpace@MIT

COMMUNICATION SCIENCESANDENGINEERINGVII. PROCESSING AND TRANSMISSION OF INFORMATIONAcademic and Research StaffProf. P. Elias Prof. R. S. Kennedy Prof. C. E. ShannonProf. R. G. Gallager Dr. E. V. HoverstenGraduate StudentsDonna Jean Brown J. G. Himes N. PippengerV. Chan R. J. Metzger L. RapisardaS. J. Dolinar, Jr. A. Z. Muszynsky S. R. RobinsonR. A. Flower N-S. Myung M. A. Sirbu, Jr.L. M. Hawthorne F. Nourani V. VilnrotterA. AN OPTIMUM RECEIVER FOR THE BINARY COHERENTSTATE QUANTUM CHANNELNational Aeronautics and Space Administration (Grant NGL 22-009-013)S. J. Dolinar, Jr.Recently an (ideally) implementable receiver for the detection of binary coherent-state signals has been discoveredl whose performance is exponentially optimum, dif-fering from that of the optimum quantum receiver by at most a factor of two in the errorpro- bability. By suitably generalizing the structure of that receiver, we have been ableto obtain one that achieves quantum optimal performance precisely for this particulardetection problem.Our problem is to decide, with minimum probability of error, between two possiblemessages m and ml, with a priori probabilities Tro and rl, respectively, when thereceived field, conditioned on m. being sent, is a linearly polarized narrow-band planewave corresponding to the quantum coherent state sj), j = 0, 1. The minimum errorprobability for this problem is well known,P(E) = I1 - 1 - 4 -ro 1  K s s l 2 (1)where ( s o sl) is the inner product of so) with s]). For coherent states, the mag-nitude squared of this inner product may be simply expressed in terms of the complexenvelopes So(t), S (t) of the classical received fields corresponding to the states I so )s).(s Isl) 12 = exp - f kX(t) dt), (2)where we have defined (t) to be the photon-counting rate for direct detection of a planeQPR No. 111 115(VII. PROCESSING AND TRANSMISSION OF INFORMATION)wave of complex envelope So(t) - SI(t), and [0, T] is the signaling interval.X(t) = C IS(t) - Sl(t) 2 , (3)where C is a constant related to the aperture area, the impedance of space, and theenergy of photons at the carrier frequency.The near-optimum receiver processes the received field by adding to it a fixed lin-early polarized local reference plane wave 1(t), equal to the negative of one of the sig-nals So(t) or Sl(t), and direct-detecting the result with a photon counter. We generalizethis concept by allowing the choice of the local reference field added at the receiver todepend causally on the actual output of the photon counter, via a feedback arrangement.This generalization immediately introduces complexities in the analysis of the photoncounter output, since it is no longer a Poisson process conditioned on the message sent.It remains at least a regular point process, and so it is characterized by the setof times t 1 < t 2 < ... at which counts occur. Thus, the class of receivers that we areconsidering is identifiable with the set of all possible feedback functions, {((t: t), t =(tl ... tn): 0 <t < . tn < t < T, n= 0, 1, 2,.. . . Here, 1(t: t) represents the feed-back field at time t when the vector of counts observed prior to t is t. The structureof such a receiver is illustrated in Fig. VII-1.BEAM PHOTONCOMBINER DETECTOR EVOLVINGEVENT TIMES OPTIMUM DECIDE=(t 1 , t2 t .,n) DECISION RULE (t) ( IDE, O<tl<t2<....< tn< t BASED ON tRECEIVED FIELD: , AT TIME tLINEARLY POLARIZEDPLANE WAVE OFCOMPLEX AMPLITUDE50(t) OR Sl(t)LINEARLY POLARIZEDPLANE WAVE OFCOMPLEX AMPLITUDELct: t)rGENERATEFig. VII-1. Receiver structure considered here.When a particular feedback function f( ) is specified, the statistics of the photoncounter output may be determined from the following observation. Conditioned on themessage m. being sent and also on the entire history of event times t of the photoncount process up to time t, the probability that a single count will occur in the smallinterval (t, t+ A) is Xj(t:t)A, within o(A), and the probability of multiple counts in thisinterval is o(A), whereQPR No. 111~116(VII. PROCESSING AND TRANSMISSION OF INFORMATION)\.(t:t) = C S (t)+ f(t: t) . (4)This incremental analysis may be easily extended to yield the probability density p (t: T)for the set of event times t within the entire observation interval [0, T], conditionedon m..n n+l 1 1pj(t: T) = I J(t: t.1 )  1 exp - (a-:t ) do- , (5)i= 1 tiwheret = (t 1  ... tn) to 0ti 1 = (t 1  ... . ) tn+ 1 - T.The optimum decision rule is obvious from (5), and in principle the probability of errormay be obtained by integrating the expressions in (5) over the appropriate decisionregions in t, for any feedback function f(). As expected, explicit evaluation of per-formance is possible only for certain special cases, one of which is the optimum feed-Aback function f( . ).In this report, we shall describe, rather than derive, the optimum feedback f( ),and then calculate performance and verify that it is quantum-optimal. We first definea deterministic weighting function f (t), which depends on S ( ), (S1(), and the ratioy T Irl o  Without loss of generality, we assume -y > 1.f (t) = em(t) - 1 + )2 e2m(t) - 4 em(t), (6)where m(t) = k(o-) d- = C f So(o)-S1() 2 do-. Next, we define two weighted com-binations of the signals So(t), S 1 (t).S ( t ) f (t) - So(t)f (t) = (7a)f (t) - 1S o() f (t) - S ( t )So(t) = (7b)f (t) - 1The optimum feedback function C( . ) depends very trivially on the observations t:QPR No. 111 117(VII. PROCESSING AND TRANSMISSION OF INFORMATION)n evenn oddIf we use this feedback function in (4) and (5) and make use of the fact that f ( ')satisfies the integral equationX([f () + 1t Yk(o-) do-,o Y0 < t < t < T,it is straightforward, albeit tedious, to obtain an expression for the likelihood ratioA(t: T) = Pl(t: T)/p (t: T).n(t: T) = [yA(t: T) = [f (T)](-1 t = (t .... , t ).-- nSince f (T) > 1, the decision regions for minimum probability of error are very simplydefined:D = {t = (t ... , t ):yA(t: T) > I} = {t= (t .... t ): n even}1 - 1' n 1. nDo = {t= (t .. , t ):yA(t: T)< 1} = {t= (t,o ( . ' n =''"The probability of error is given byP(E) = fD0 Tr1 p1 (t: T) dt + fD Iropo(t: T) dto 1., t n): n odd}.(Ila)(llb)= D yA(t: T) To o(t: T)odt + fD 1D1 'yA(t: T) rlpl (t: T) dtD 1of (T)cTr op (t: T) dt + fD00-11 1 (t:T)f ( T)To o (t: T) dt + fDI1 Trlp(t:T) dt].The quantity in brackets is the probability of correct detection, or 1 - P(E). Solvingfor P(E), we obtainQPR No. 111A ( t ), t  ( t  ... tn)(t:t) = if0(t), t =(tl, , t n),f (t)In -(10)1f (T)C ID(12)118(VII. PROCESSING AND TRANSMISSION OF INFORMATION)1P(E) =1 + f (T)S1 -m(T)21- 1 - em(T1(1+ )21 - 1 - 4  r em(T)]. (13)Recalling the definition of m(T), we see that this is identical to the quantum-optimalresults in (1) and (2).Let us briefly discuss some aspects of the behavior of our optimum receiver(Fig. VII-2). The feedback field alternates between fl(t) and fo(t) with each count. Atthe same time, In yA(t:t) changes sign at each count, but its magnitude is adeterministically increasing function of time, In f (t). Thus the occurrence ofeach successive count constitutes a more and more conclusive negative test of thehypothesis last considered more probable. As time goes on, the optimum feedbackSWITCHESBEAM PHOTON BACK AND FORTHCOMBINER DETECTOR WITH EACH A (t) = mC COUNT I DECIDECOUNTERRECEIVED FIELD: (t) mLINEARLY POLARIZEDPLANE WAVE OFCOMPLEX AMPLITUDES0(t) OR S l ( tLINEARLY POLARIZEDPLANE WAVE OFCOMPLEX AMPLITUDE0(t) OR I ( 0GENERATE, -- - lo(t)GENERATE 1Fig. VII-2. Optimum receiver for known So(t), Sl (t).alternates less frequently between I(t) and fo(t) and with increasing probability tendstoward nulling whichever signal was actually sent. In fact, if lim m(t) = cc, there is,t- ocwith probability one, a final count even in the case of an infinite signaling interval(T = c).We remark that in the case of equal a priori probabilities (y = 1), Eqs. 4, 7,QPR No. 111 119(VII. PROCESSING AND TRANSMISSION OF INFORMATION)and 8 provide only a formal description of the optimum processing, since the requiredfeedback field, and hence the photon counting rate, are infinite at t = 0. It is possibleto define rigorously the random process which represents the photodetector output forthis optimum processor, but it is not a regular point process on [0, T] and so the opti-mum receiver technically does not belong to the class we are considering. We point out,however, that we may approach optimum performance as closely as desired with boundedfeedback fields by appropriately redefining k(t:t) for t in some small interval[0, E), 0 < E < T, and letting E 0. One redefinition that has this convergence propertyis-S ( t ) ,  t E [0, c)f (t: t) = (14)f(t: t), tE [E, T]From a practical point of view, the foregoing result might seem insignificant, sincean exponentially optimum receiver for this particular detection problem is already avail-able. The attainment of precisely optimum performance, however, warrants an investi-gation into the possible connection between the type of feedback receiver considered hereand quantum-optimal measurements for more general problems for which exponen-tially optimum receivers have not yet been found. Further study is being conducted onthis question.References1. R. S. Kennedy, "A Near-Optimum Receiver for the Binary Coherent State QuantumChannel," Quarterly Progress Report No. 108, Research Laboratory of Electronics,M.I.T., January 15, 1973, pp. 219-225.2. C. W. Helstrom, "Detection Theory and Quantum Mechanics," Inform. Contr. 10,254-291 (1968).QPR No. 111 120

Processing and Transmission of Information

http://dspace.mit.edu/bitstream/1721.1/56414/1/RLE_QPR_111_VII.pdf

Processing and Transmission of Information

Abstract

Similar works

Full text

Available Versions

DSpace@MIT