IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 2007 2349 The Poisson Fading Channel Kaushik Chakraborty, Member, IEEE, and Prakash Narayan, Fellow, IEEE Abstract—In this first paper of a two-part series, a single-user single-input single-output (SISO) shot-noise-limited Poisson channel is considered over which an information signal is trans- mitted by modulating the intensity of an optical beam, and individual photon arrivals are counted at the photodetector re- ceiver. The transmitted signal, which is chosen to satisfy peak and average constraints, undergoes multiplicative fading, which occurs over coherence time intervals of fixed duration. The fade coeffi- cient (channel state) remains constant in each coherence interval, and varies across successive such intervals in an independent and identically distributed (i.i.d.) fashion. A single-letter characteriza- tion of the capacity of this channel is obtained when the receiver is provided with perfect channel state information (CSI) while the transmitter CSI can be imperfect. The asymptotic behavior of channel capacity in the low and high peak-signal-to-shot-noise ratio (SNR) regimes is studied. Index Terms—Channel capacity, channel state information, co- herence time, free-space optical communication, lognormal fading, Poisson fading channel, single input single output. I. INTRODUCTION T HE development of the technology of free-space optical communication has long been motivated by deep space ap- plications, which, in fact, have served as an important impetus for much of the literature on communication in the Poisson regime ([3], [16], [19], [23], [25], [28], [29], among others). Additionally, free space optics has emerged recently as an at- tractive technology for several applications, e.g., metro network extensions, last mile connectivity, fiber backup, radio-fre- quency (RF)-wireless backhaul, and enterprise connectivity [38]. The many benefits of wireless optical systems include rapid deployment time, high security, inexpensive components, seamless wireless extension of the optical fiber backbone, im- munity from RF interference, and lack of licensing regulations. Consequently, free-space optical communication has received Manuscript received January 23, 2006; revised December 19, 2006. This work was supported by the Army Research Office under ODDR&E MURI01 Program Grant DAAD19-01-1-0465 to the Center for Communicating Net- worked Control Systems (through Boston University), and by the National Science Foundation under Grant ECS0636613. The material in this paper was presented in part at the 42nd Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, October 2004, and at the IEEE Inter- national Symposium on Information Theory, Adelaide, Australia, September 2005. K. Chakraborty was with the Department of Electrical and Computer Engi- neering and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA. He is now with the California Institute for Telecommu- nications and Information Technology, University of California at San Diego, La Jolla, CA 92037 USA (e-mail: kchakrab@ucsd.edu). P. Narayan is with the Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail: prakash@eng.umd.edu). Communicated by Y. Steinberg, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2007.899559 much attention in recent years [14], [15], [17], [22], [39], [41], [42]. In free-space optical communication links, atmospheric tur- bulence can cause random variations in the refractive index of air at optical wavelengths which, in turn, result in random fluc- tuations in both the intensity and phase of a propagating op- tical signal [18], [36]. Such fluctuations, which in practice can routinely exceed 10 dB, lead to an increase in link error proba- bility thereby degrading communication performance [42]. The fluctuations in the intensity of the transmitted optical signal, termed “fading,” can be modeled in terms of an ergodic log- normal process with a correlation time or “coherence time” in- terval of the order of 1–10 ms [34]. For the systems under con- sideration, data rates typically can be of the order of gigabits per second. Therefore, the free-space optical channel is a slowly varying fading channel with occasional deep fades that can af- fect millions of consecutive bits [15]. A general approach that is often followed to achieve higher rates of reliable communication over fading channels is to use estimates of the channel fade (also referred to as path gain or channel state) at the transmitter and the receiver. For RF com- munication, a comprehensive review can be found in [31]. In op- tical fading channels, instantaneous realizations of the channel state can be estimated at the receiver; at typical data rates, more than bits are transmitted during each coherence time, a small fraction of which can be used by the receiver to form good esti- mates of the channel fade. 1 Then, depending on the availability of a feedback link and the amount of acceptable delay, the trans- mitter can be provided with complete or partial knowledge of the channel state, which can be used for adaptive power control, thereby achieving higher throughputs (cf., e.g., [5], [31] in the context of RF communication). Another approach for combating the detrimental effects of fading entails the use of spatial diversity in the form of mul- tiple transmit and receive elements. In RF communication, the use of multiple transmit and receive antennas has been shown to significantly improve communication throughput in the pres- ence of channel fading; for a survey of recent results, see [11]. In the context of free-space optical communication, in several experimental studies [2], [19], [21], multiple laser beams were shown to improve communication performance. Attempts have since been made to characterize analytically the benefits of mul- tiple-input multiple-output (MIMO) communication over op- tical fading channels [14], [15], [22], [39]. The study of the Shannon capacity of the direct detection optical channel without fading, known popularly as the Poisson channel, has a rich history, beginning with [9]; see [37] for 1 Clearly, the degree of accuracy of the receiver’s estimated knowledge of the channel state (and the fraction of transmitted bits used for this purpose) will affect receiver performance. This issue is beyond the scope of the present work. For a related study in the context of RF communication, see, for instance, [27]. 0018-9448/$25.00 © 2007 IEEE