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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 2003 1385
Optical Intensity-Modulated Direct DetectionChannels: Signal Space and Lattice CodesSteve Hranilovic, Student Member, IEEE,and Frank R. Kschischang, Senior Member, IEEE
AbstractTraditional approaches to constructing constellationsfor electrical channels cannot be applied directly to the optical in-tensity channel. This work presents a structured signal space modelfor optical intensity channels where the nonnegativity and averageamplitude constraints are represented geometrically. Lattice codessatisfying channel constraints are defined and coding and shapinggain relative to a baseline are computed. An effective signal spacedimension is defined to represent the precise impact of coding andshaping on bandwidth. Average optical power minimizing shapingregions are derived in some special cases. Example lattice codes areconstructed and their performance on an idealized point-to-pointwireless optical link is computed. Bandwidth-efficient schemes areshown to have promise for high data-rate applications, but requiregreater average optical power.
Index TermsFree-space optical communications, lattice codes,optical intensity modulation, signal space, wireless infraredchannel.
T HE free-space, direct detection, optical intensity-modu-lated channel offers the modem designer interesting newchallenges. Most practical wireless optical channels use light-emitting diodes as transmitters and photodiodes as detectors,as shown in Fig. 1. These devices modulate and detect solelythe intensity of the carrier, not its phase, which implies that alltransmitted signal intensities are nonnegative. Furthermore, bi-ological safety considerations constrain the average radiated op-tical power, thereby constraining the average signal amplitude.Both multipath distortion in signal propagation and the limitedresponse times of the optoelectronics create sharp constraintson the channel bandwidth. Conventional signalspace modelsand coded modulation techniques for electrical channels cannotbe applied directly to this channel, since they do not take thesignal amplitude constraints into consideration. Conventionaltransmission techniques optimized for broad-band optical chan-nels (e.g., optical fibers) are not generally bandwidth efficient.
Historically, optical intensity channels have been modeled asPoisson counting channels. In the absence of background noise,the capacity of such channels is infinite , , and-arypulse-position modulation (PPM) can achieve arbitrarily small
Manuscript received September 26, 2001; revised February 24, 2003. Thiswork was supported in part by the Natural Sciences and Engineering ResearchCouncil of Canada. The material in this paper was presented in part at the 20thBiennial Symposium on Communications, Kingston, ON, Canada, May 2000and at the 2001 IEEE International Symposium on Information Theory, Wash-ington, DC, June 2001.
The authors are with the Edward S. Rogers Sr. Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada(e-mail: email@example.com; firstname.lastname@example.org).
Communicated by R. Urbanke, Associate Editor for Coding Techniques.Digital Object Identifier 10.1109/TIT.2003.811928
Fig. 1. A simplified block diagram of an optical intensity direct detectioncommunications system.
probability of error for any transmission rate , . Under apeak optical power constraint, the capacity is finite and achievedwith two-level modulation schemes , . These schemes donot consider bandwidth efficiency; indeed, schemes based onphoton counting in discrete intervals require an exponential in-crease in bandwidth as a function of the rate to achieve reli-able communication . In the more practical case of pulse-am-plitude modulated (PAM) signals confined to discrete time in-tervals of length and with a given peak and average opticalpower, Shamai showed that the capacity-achieving input dis-tribution is discrete with a finite number of levels increasingwith . For high-bandwidth cases the binary leveltechniques found earlier are capacity achieving, however, lowerbandwidth schemes require a larger number of levels.
There has been much work in the design of signal sets foruse in optical intensity channels under a variety of optimalitycriteria; see, e.g., . The most prominent modulationformats for wireless optical links are binary level PPM andonoff keying (OOK). For example, low-cost point-to-pointwireless infrared infrared data association (IrDA) modemsutilize 4-PPM modulation . Spectrally efficient variationshave been considered , , but these two-level schemesoffer relatively limited improvement in bandwidth efficiency.The results of Shamai  show that nonbinary modulation isrequired to achieve capacity in bandwidth-limited channels.Subcarrier modulation  has been suggested as one possiblemultilevel scheme to achieve high bandwidth efficiency. Shiuand Kahn developed lattice codes for free-space optical inten-sity channels by constructing higher dimensional modulationschemes from a series of one-dimensional constituent OOKconstellations .
In this paper, we present a signalspace model for theoptical intensity channel using time-disjoint symbols, anddefine coding and shaping gain measures that are relevant inthis framework. Unlike previous work, no assumptions aremade about the underlying pulse shape, only that independent,time-disjoint symbols are employed. Previous work then ap-pears as a special case of this work. A more accurate bandwidthmeasure is adopted which allows for the effect of shaping on
0018-9448/03$17.00 2003 IEEE
1386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 2003
the bandwidth of the scheme to be represented as an effectivedimension parameter. We suggest techniques for achievingcoding and shaping gain using multidimensional lattices, andwe compare the bandwidth efficiency of conventional PPMschemes with a new raised-quadrature amplitude modulation(QAM) scheme, and suggest that such schemes may be advan-tageous for high-rate short-distance optical communication.
The remainder of this paper is organized as follows. Section IIgives background on the wireless optical channel and presentsthe communications model used in the remainder of the work.Section III presents a signal space model suited to the inten-sity-modulated channel and presents theorems regarding the setof signals satisfying the channel constraints. Lattice codes forthe optical intensity channel are defined in Section IV and thegain is presented versus a PAM baseline. New expressions forthe shaping and coding gain of these schemes are presented. Op-timal shaping regions that minimize the average optical powerare derived subject to certain conditions. The peak transmittedoptical amplitude is taken as a design constraint and representedin the signal space. Section V presents example schemes andcompares them versus a baseline. Design guidelines are givenfor the design of modulation schemes for this channel. Finally,in Section VI, the results of this work are summarized and gen-eral conclusions drawn.
II. COMMUNICATIONS SYSTEM MODEL
A. Channel Model
The optical intensityof a source is defined as the opticalpower emitted per solid angle. Wireless optical links transmitinformation by modulating the instantaneous optical intensity
in response to an input electrical current signal . Thisconversion can be modeled as , where is the op-tical gain of the device in units of W/(Am ). The photodiodedetector is said to performdirect-detectionof the incident op-tical intensity signal since it produces an output electrical pho-tocurrent proportional to the received optical intensity. Thechannel response from to in Fig. 1 is well approxi-mated as
where denotes convolution andis the detector sensitivity inunits of A m /W, is the noise process, and is thechannel response , . Substitutinginto (1) gives
where the product is unitless. Without loss of generality, set, to simplify analysis. In this manner, the free-space
optical channel is represented by a baseband electrical model.Throughout the remainder of this paper, unless explicitly stated,all signals are electrical.
The additive noise arises due to the high-intensity shotnoise created as a result of ambient illumination. By the centrallimit theorem, this high-intensity shot noise is closely approx-imated as being Gaussian distributed. The noise is modeled as
additive, signal-independent, white, Gaussian with zero meanand variance .
The channel response has low-pass frequency responsewhich introduces intersymbol interference. The low-passchannel response arises in two ways: i) front-end photodiodecapacitance and ii) multipath distortion. The use of large pho-todiodes with high capacitances in free-space optical channelslimits the bandwidth of the link. The achievable bandwidthin inexpensive systems, such as the point-to-point IrDA fastinfrared (IR) standard  is on the order of 1012 MHz whichis approximately three orders of magnitude smaller than inwired fiber-optic systems. Multipath distortion gives rise to alinear, low-pass response which limits the bandwidth of someexperimental links to approximately 1050 MHz depending onroom layout and link configuration , .
In this work, it is assumed that if a signaling scheme is essen-tially band limited to the frequency range hertz,in the sense defined in Section IV-D, that the channel is nondis-torting. Consequently, the received electrical signal can bewritten as
The physical characteristics of the optical intensity channelimpose constraints on the amplitude of which can equiva-lently be viewed as constraints on . Since the physical quan-tity modulated is