Optical intensity-modulated direct detection channels ... nbsp;· Optical Intensity-Modulated Direct Detection Channels: Signal ... OPTICAL INTENSITY-MODULATED DIRECT DETECTION CHANNELS ... The average optical power calculation

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<ul><li><p>IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 2003 1385</p><p>Optical Intensity-Modulated Direct DetectionChannels: Signal Space and Lattice CodesSteve Hranilovic, Student Member, IEEE,and Frank R. Kschischang, Senior Member, IEEE</p><p>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.</p><p>Index TermsFree-space optical communications, lattice codes,optical intensity modulation, signal space, wireless infraredchannel.</p><p>I. INTRODUCTION</p><p>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.</p><p>Historically, optical intensity channels have been modeled asPoisson counting channels. In the absence of background noise,the capacity of such channels is infinite [1], [2], and-arypulse-position modulation (PPM) can achieve arbitrarily small</p><p>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.</p><p>The authors are with the Edward S. Rogers Sr. Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada(e-mail: steve@comm.utoronto.ca; frank@comm.utoronto.ca).</p><p>Communicated by R. Urbanke, Associate Editor for Coding Techniques.Digital Object Identifier 10.1109/TIT.2003.811928</p><p>Fig. 1. A simplified block diagram of an optical intensity direct detectioncommunications system.</p><p>probability of error for any transmission rate [2], [3]. Under apeak optical power constraint, the capacity is finite and achievedwith two-level modulation schemes [4], [5]. 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 [3]. 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 [6]. For high-bandwidth cases the binary leveltechniques found earlier are capacity achieving, however, lowerbandwidth schemes require a larger number of levels.</p><p>There has been much work in the design of signal sets foruse in optical intensity channels under a variety of optimalitycriteria; see, e.g., [7][11]. 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 [12]. Spectrally efficient variationshave been considered [13], [14], but these two-level schemesoffer relatively limited improvement in bandwidth efficiency.The results of Shamai [6] show that nonbinary modulation isrequired to achieve capacity in bandwidth-limited channels.Subcarrier modulation [15] 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 [16].</p><p>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</p><p>0018-9448/03$17.00 2003 IEEE</p></li><li><p>1386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 2003</p><p>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.</p><p>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.</p><p>II. COMMUNICATIONS SYSTEM MODEL</p><p>A. Channel Model</p><p>The optical intensityof a source is defined as the opticalpower emitted per solid angle. Wireless optical links transmitinformation by modulating the instantaneous optical intensity</p><p>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</p><p>(1)</p><p>where denotes convolution andis the detector sensitivity inunits of A m /W, is the noise process, and is thechannel response [14], [18][20]. Substitutinginto (1) gives</p><p>where the product is unitless. Without loss of generality, set, to simplify analysis. In this manner, the free-space</p><p>optical channel is represented by a baseband electrical model.Throughout the remainder of this paper, unless explicitly stated,all signals are electrical.</p><p>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</p><p>additive, signal-independent, white, Gaussian with zero meanand variance [14].</p><p>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 [12] 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 [14], [18].</p><p>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</p><p>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 a normalized power, this constrains all trans-mitted amplitudes to be nonnegative</p><p>(2)</p><p>The optical power transmitted is also limited by the biolog-ical impact that this radiation has on eye safety and thermalskin damage. Although limits are placed on both the averageand peak optical power transmitted, in the case of most prac-tical modulated optical sources, it is the average optical powerconstraint that dominates [17]. As a result, the average ampli-tude (i.e., average optical power)</p><p>(3)</p><p>must be bounded. This is in marked contrast to conventionalelectrical channels in which the energy transmitted depends onthe squared amplitude of transmitted signal.</p><p>Note that this channel model applies not only to free-spaceoptical channels but also to fiber-optic links with negligible dis-persion and signal independent, additive, white, Gaussian noise.</p><p>B. Time-Disjoint Signaling</p><p>Let be a finite index set and let: be a set of optical intensity signals satisfying</p><p>for for some positive symbolperiod . In the case where such time-disjoint symbols are sentindependently, the optical intensity signal can be formed as</p><p>(4)</p></li><li><p>HRANILOVIC AND KSCHISCHANG: OPTICAL INTENSITY-MODULATED DIRECT DETECTION CHANNELS 1387</p><p>where is an independent and identically distributed (i.i.d.)process over . Since the symbols do not overlap in time, (2) isequivalent to</p><p>(5)</p><p>The average optical power calculation in (3) can also simplifiedin the time-disjoint case as</p><p>which, by the strong law of large numbers, gives</p><p>(6)</p><p>with probability one, where is the probability of trans-mitting . Thus, the average optical powerof a schemeis the expected value of the average amplitude of .</p><p>III. SIGNAL SPACE OFOPTICAL INTENSITY SIGNALS</p><p>This section presents a signal space model which, unlike con-ventional models, represents the nonnegativity constraint andthe average optical power cost of schemes directly. The proper-ties of the signal space are then explored and related to the setof transmittable points and to the peak optical power of signalpoints.</p><p>A. Signal Space Model</p><p>Let be a finite index set, ,and let : be a set of real or-thonormal functions time-limited to such that</p><p>. Each is represented by the vectorwith respect to the basis set</p><p>and the signal constellation is defined as : .The nonnegativity constraint in (5) implies that the average</p><p>amplitude value of the signals transmitted is nonnegative. It ispossible to set the function</p><p>(7)</p><p>where</p><p>otherwise</p><p>as a basis function for every intensity modulation scheme. Notethat by assigning , the upper bound on the dimensionalityof the signal space is increased by a single dimension to be</p><p>. Due to the orthogonality of the other basis functions</p><p>(8)</p><p>The basis function contains the average amplitude of eachsymbol, and, as a result, represents the average optical power ofeach symbol. In this manner, the average optical power require-ment is represented in a single dimension. The average optical</p><p>power of an intensity signaling set can then be computed from(6) as</p><p>(9)</p><p>where is defined as</p><p>The term can be interpreted as the component ofwhichdepends solely on the constellation geometry.</p><p>Note that unlike the case of the electrical channel where theenergy cost of a scheme is completely contained in the geometryof the constellation, the average optical power of an intensitysignaling scheme depends on the symbol period as well. Thisis due to the fact that in (7) is set to have unitelectricalenergy since detection is done in the electrical domain. As aresult, the average amplitude and hence average optical powermust depend on .</p><p>B. Admissible Region</p><p>Not all linear combinations of the elements ofsatisfy thenonnegativity constraint (5). Define theadmissible region ofan optical intensity-modulation scheme as the set of all pointssatisfying the nonnegativity criterion. In terms of the signalspace</p><p>(10)</p><p>where for , : is defined as</p><p>The set is closed, contains the origin, and is convex. This claimcan be justified since for any and any ,</p><p>since it describes a nonnegative signal.It is instructive to characterize in terms of its cross section</p><p>for a given value or, equivalently, in terms of points of equalaverage optical power. Define the set</p><p>as the set of all signal points with a fixed average optical powerof . Each forms an equivalence class orshellof trans-mittable symbols. This is analogous to spherical shells of equalenergy in the conventional case [21]. As a result, the admissibleregion can be written in terms of this partition as</p><p>(11)</p><p>The set of signals represented in eachcan further be analyzedin the absence of their common component by defining theprojectionmap : that mapsto . The important properties of are sum-marized in Theorem 1.</p><p>Theorem 1: Let denote the admissible region of points de-fined in (10).</p><p>1) For , .2) .3) is closed, convex and bounded.</p></li><li><p>1388 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 2003</p><p>4) Let denote the set of boundary points of. Then</p><p>i) andii) : .</p><p>5) is the convex hull of a generalized-cone with vertexat the origin, opening about the -axis and limited to</p><p>.Proof:</p><p>Property 1. The set is a set of pulses with averageoptical power . Therefore, . Similarly,</p><p>which implies that .Property 2. Follows directly from (11) an...</p></li></ul>