488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 2, MARCH 1998The Poisson Multiple-Access ChannelAmos Lapidoth, Member, IEEE, and Shlomo Shamai (Shitz), Fellow, IEEEAbstract—The Poisson multiple-access channel (MAC) modelsmany-to-one optical communication through an optical fiber orin free space. For this model we compute the capacity regionfor the two-user case as a function of the allowed peak power.Focusing on the maximum throughput we generalize our resultsto the case where the users are subjected to an additionalaverage-power constraint and to the many-users case. We showthat contrary to the Gaussian MAC, in the Poisson MAC themaximum throughput is bounded in the number of users. Wequantify the loss that is incurred when Time-Division MultipleAccess (TDMA) is employed and show that while in the two-usercase and in the absence of dark current the penalty is rathermild, the penalty can be quite severe in the many-users case inthe presence of large dark current. We introduce a generalizedTDMA technique that mitigates this loss to a large extent.Index Terms—Capacity region, infrared, multiple-access chan-nels, multiuser, optical CDMA, optical TDMA, Poisson.I. INTRODUCTIONTHE Poisson channel attracts much interest as it serves asthe standard model for optical communications [1]–[3].Its conceptual simplicity and the advent of many uncodedand coded communications techniques [1]–[4] have propelledan extensive information-theoretic study of communicationover this channel in an effort to identify and quantify theultimate limits and the ultimate potential of this channel.The overwhelming majority of these papers [4]–[12] treat thesingle-user channel only. In this model, which is depicted inFig. 1(a), the channel outputis a doubly sto-chastic Poisson process with instantaneous rate,whereis the channel input, and is a constant.The outputcorresponds to the number of counts registeredby the direct detection device (usually a p-i-n diode) in theinterval; the input is proportional to the squaredmagnitude of the optical field impinging on the detector at timeintegrated over its active surface; and the constant standsfor “dark current” and accounts for spontaneous emissions dueto background radiation.Manuscript received December 15, 1996; revised August 20, 1997. Thework of A. Lapidoth was supported in part by the U.S. Army under GrantDAAH04-95-1-0103. The work of S. Shamai was supported by the BroadbandTelecommunications R&D Consortium administered by the Chief Scientist ofthe Israeli Ministry of Industry and Trade. The material in this paper waspresented in part at the IEEE Information Theory Workshop, June 9–13, 1996,Haifa, Israel.A. Lapidoth is with the Department of Electrical Engineering and ComputerScience, Massachusetts Institute of Technology, Cambridge, MA 02139-4307USA.S. Shamai is with the Department of Electrical Engineering, Technion–IsraelInstitute of Technology, Haifa 32000, Israel.Publisher Item Identifier S 0018-9448(98)01632-0.The input signal is often peak- and average-powerlimited [5]–[8] so that(1.1)wherestands for the peak power and denotes the allowedaverage power. Heredenotes the expectation operator, andsubscripts, if attached, denote the random variables over whichthe expectation is taken. The timestands for the transmissionduration and is usually assumed to approach infinity. Thecapacityin nats per second under these constraints is givenby [5]–[7](1.2a)where(1.2b)and where(1.2c)The capacity of the single-user Poisson channel is maximizedin the absence of dark current () and when the average-power constraints are relaxed. In this case, the capacity isgiven by. Thus(1.3)To achieve capacity, input signals of infinite bandwidth arerequired, and the capacity is typically reduced if the input issubjected to bandwidth-like constraints [10]–[12].The Poisson single-user channel is one of the few channelsfor which, in addition to the channel capacity, the reliabilityfunction at all rates below capacity is also known [5]. In fact,in the absence of dark current and under capacity-reducingaverage-power constraints, the reliability function is evenknown in the presence of a noiseless feedback link from thereceiver to the transmitter [13].In recent years optical multiuser communication systemswere introduced and intensively investigated [14], [15]. Avariety of multiple-access techniques such as Wavelength-Division Multiplexing (WDM), Time-Division Multiple Ac-cess (TDMA), and Code-Division Multiple Access (CDMA)are commonly considered [14], [15]. While these accessingmethods have natural counterparts in the radio channel, the0018–9448/98$10.00  1998 IEEELAPIDOTH AND SHAMAI: THE POISSON MULTIPLE-ACCESS CHANNEL 489(a)(b)Fig. 1. Schematic diagram of the single- and multiple-access Poisson channel. (a) The single-user channel.y(t)is a conditional Poisson process withinstantaneous ratex(t)+0. (b) The multiple-access Poisson channel.y(t)is the observed Poisson process combined of the Poisson processesfyk(t)g,which correspond to the individual rates of the independent usersfxk(t)g;k=1;2;111;K.D(t)is the dark-current Poisson process with rate0.Poisson channel is unique in that the channel input must benonnegative.Multiuser optical channels with a variety of single-user andmultiuser detection methods were studied [16]; optical CDMAwas particularly studied in [17]–[29] and in references therein.The constraints of having nonnegative inputs fundamentallyimpacts the design of good spreading sequences [26]–[28].In fact, TDMA can be viewed as a special case of syn-chronous CDMA where the disjoint time slots of the differentusers are determined by properly selecting the spreadingsequences. Most of the reported studies examine uncoded,possibly spread, communication systems; but see [29]–[32],where coding is addressed in the context of multiuser opticalcommunication and in particular in combination with CDMA-based methods.The model for the Poisson multiple-access channel (MAC)that we study is shown in Fig. 1(b). The input of theth userdetermines the rate of the corresponding doublystochastic Poisson processwhile the overall observationis also a doubly stochastic Poisson process with instantaneousrateHere is a homogeneous Poisson process of rate (thedark current), anddesignates the number of users. Thischannel model is equivalent to having an inputto the single-user Poisson channel. Clearly, this multi-user channel model accounts for any possible