SHORT-RANGE line-of-sight optical links have the potential

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER Optical Repetition MIMO Transmission With Multipulse PPM Stephen G. Wilson, Senior Member, IEEE, Maïté Brandt-Pearce, Senior Member, IEEE, Qianling Cao, and Michael Baedke Abstract We study the use of multiple laser transmitters combined with multiple photodetectors for atmospheric, line-of-sight optical communication, and focus upon the use of multiple-pulseposition-modulation as a power-efficient transmission format, with signal repetition across the laser array. Ideal (photon counting) photodetectors are assumed, with and without background radiation. The resulting multiple-input/multiple-output channel has the potential for combating fading effects on turbulent optical channels, for which both log-normal and Rayleigh-fading models are treated. Our focus is upon symbol error probability for uncoded transmission, and on capacity for coded transmission. Full spatial diversity is obtained naturally in this application. Index Terms Multiple-input/multiple-output (MIMO) systems, optical communications, pulse-position-modulation (PPM). I. INTRODUCTION SHORT-RANGE line-of-sight optical links have the potential for providing high-bandwidth access to a larger wired network, for linking intranets within corporate campuses, or even for links between aircraft. Such optical transmission, often called free-space optics (FSOs), is attractive for several reasons, notably the license-free nature of the spectrum, the highly directive nature of the radiation (providing spatial isolation), relatively low infrastructure costs, and rapid redeployment. FSO technology is summarized in recent articles by Willebrand et al. [1] and Acampora [2]. Two primary challenges are attached to free-space optical communication. First, the narrow beamwidth (on the order of a few milliradians typically) implies the need for careful pointing, and building vibration or sway can induce signal strength fading in the link. Second, is the need to combat link fading due to scattering and scintillation. Even in clear sky conditions, links may experience fading due to inhomogeneities of the index of refraction in the optical beam; one may view these pockets of inhomogeneity as refraction zones that distort the phase front of the optical field, leading to interference patterns in space at the detector s location. Surveys of optical propagation effects Manuscript received January 1, 2004; revised January 23, 2005 and March 30, This work was supported in part by the National Science Foundation under Grant CCR This work has been presented in part at ISIT 04, Chicago, IL, July 2004, and at the Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November S. G. Wilson, M. Brandt-Pearce, and Q. Cao are with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA USA ( sgw@virginia.edu; mb-p@virginia. edu; qc4d@virginia.edu). M. Baedke is with Dominion Generation, Mineral, VA USA ( michael_l_baedke@dom.com). Digital Object Identifier /JSAC are found in [3] [6]. It is apparently not uncommon to incorporate 15 db of link margin into the design merely to provide protection against fading. To address both challenges, we consider the use of (noncoherent) optical arrays, analogous to the use of antenna array technology for microwave systems, as a means of combatting fading. Specifically, we envision separate lasers, assumed to be intensity-modulated only, together with photodetectors, assumed to be ideal noncoherent (direct detection) receivers. Due to laser incoherence across the array, their powers add at the receiver array. The sources and detectors are physically situated so that the fading experienced between source-detector pairs is statistically independent and, thus, diversity benefits can accrue from the multiple-input/multiple-output (MIMO) channel. Obviously, the assumption of independence may not be valid dependent upon the spacing of the devices, and on the nature of the fading. For example, a cloud or fog bank that fills most of the link will obviously induce large fades on all source/detector pairs. Alternative means of operation in such environments must be considered. Research in optical free-space communication dates to the 1960s, [5], [7], [8], where it was shown that standard pulse-position-modulation (PPM) is an average-energy efficient strategy as the number of slots increases, and this also mitigates against background radiation. Work on coded PPM includes that of [9] [12]. Multipulse modulation has been studied in [13] [16], some combining modulation with error control codes to improve performance. Recent work related to this paper on optical MIMO with direct detection is found in [17], where the focus is upon receiver diversity and binary PPM. Channel capacity for optical MIMO channels with only peak and average power constraints has been studied in [18]. The authors have considered optical MIMO transmission for standard PPM in [19] and [20]. II. SYSTEM MODEL In this section, we formulate the assumptions for the transmission, channel character, and detection, prior to developing performance results. A. Multipulse Modulation We assume laser sources, all pointed toward a distant array of photodetectors, are pulse-position-modulated by an information source. The laser beamwidths are narrow, but sufficiently wide to uniformly illuminate the entire photodetector array. The path pairs may experience fading, as further detailed below, and we designate as the amplitude of the path gain (field strength multiplier) from source to detector /$ IEEE

2 1902 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER The aggregate optical field is detected by each photodetector, assuming an ideal photon-counting model with a quantum efficiency. It is known from earlier work on single-sender, single-detector optical links [7], [8], [10], that classic -ary PPM represents an energy-efficient transmission format if average energy per symbol (or bit) is the efficiency criterion. However, the required peak power from the lasers must become arbitrarily large as increases. Also, as becomes large for a fixed bit rate, the spectral efficiency diminishes (though normally not a major issue in optical transmission), the required switching speed for the electronics increases, and the receiver synchronization becomes more difficult. This motivates study of multipulse modulation, with notation as follows. To modulate, we divide a symbol interval into time slots, and turn the lasers on for of a possible time slots to communicate a digital symbol. Of course, represents conventional PPM. In general, there are possible codewords; a minor practical difficulty is that the number of codewords is not a power of 2 except when and. When the laser pulse patterns are viewed as binary codewords, the encoding is a constant-weight code with minimum distance 2, [21]. For a given bit rate the symbol interval is, and this interval is subdivided into slots, each of length. We assume that during the designated slots, the laser power, measured at the receiver after all link losses, is watts. Thus, represents the peak power. In the multilaser case, we will assume that is the power presented by all transmitters to a single photodetector in the absence of fading, regardless of number of transmitters. This parallels the standard assumption for the microwave MIMO setting. In this paper, we assume repetition coding across lasers, that is, each of the lasers transmits the same -pulse pattern on each symbol. While this constraint restricts the permissible bit rate, relative to an unconstrained set of patterns, the receiver processing is simple, performance analysis is more direct, and as we shall see, performance is remarkably good on the MIMO channel. In particular, full transmit and receiver diversity is achieved without further coding. B. Channel Modeling Several models exist for the aggregate amplitude distribution of the channel gains. 1 Most prominent among the models is the log-normal model, justified both by analysis in light turbulence and by empirical studies [3], [4]. As the clear-air turbulence increases, the log-normal model begins to deviate from experimental observation. Multiple empirically derived probability density functions have been proposed for the moderate turbulent environment (e.g., [22] and [23]). These expressions are difficult to manipulate due to their parametric nature. In the limit of strong turbulence, a negative exponential distribution for the irradiance holds (both theoretically and experimentally, [22], [24]), which reduces to Rayleigh statistics of the optical 1 The distribution of the phase is not important, since in the noncoherent detector the phase of the field is not important. Fig. 1. Probability density functions for fading variable A. field amplitude. In our analysis, we have considered the two extremes, the log-normal and the Rayleigh, with the expectation that the two models bound the range of performances observed in application. In the log-normal case, the amplitude of the random path gain is, where is normal with mean and variance. Thus, the logarithm of the field amplitude scale factor is is also log- normally distributed. The optical intensity normally distributed in this case. The pdf for is (1) So that the mean path intensity is unity, i.e.,, it can be shown that. For the log-normal distribution, the scintillation index, defined as is a measure of the degree of fading. This index can be related to the parameter by. Typical values appearing in the literature for S.I. are in the range Rayleigh fading emerges from a scattering model that views the composite field as produced by a large number of nondominating scatterers, each contributing random optical phase upon arrival at the detector. The central limit theorem then gives a complex Gaussian field, whose amplitude is Rayleigh where we have normalized so that. In this case, the random intensity is a one-sided exponential random variable. The scintillation index for the Rayleigh situation is 1, though the distribution is quite different from the log-normal case, especially, in the small-amplitude tail, as shown in Fig. 1. Beyond the marginal distribution, we assume that the spatial coherence distance of the field at the detector is large relative to the optical aperture of one detector (on the order of one centimeter diameter), but that the field is independent among (2) (3)

3 WILSON et al.: OPTICAL REPETITION MIMO TRANSMISSION WITH MULTIPULSE PPM 1903 detectors. The spatial coherence distance in the turbulent atmosphere is modeled to be approximately, where is the path length and is the wavelength [4]. This typically produces correlation distances in the range of 5 to 50 cm. Thus, for small detectors spaced on the order of one meter, we can model the received field as spatially constant (though random) over any one detector aperture, while the path gains from any one source to various detectors are independent. Finally, we assume the channel fading process is flat across the optical frequency band and slow relative to the symbol duration. The latter is justified by the high signaling speeds contemplated in FSO systems and the relatively slow dynamics of the channel turbulence, while the flat fading follows from the large coherence bandwidth relative to the laser linewidth and modulation rate. C. Optical Detection The photodetector analysis is based on a semi-classical treatment of photodetection, where the incident field is viewed as a wave, and this wave produces a modulated Poisson point process of photoelectrons that contribute to the detector current at the output. The rate of the Poisson process is proportional to the total power incident on the photodetector. The relevant theory is summarized here, with further details found in [25] and [26]. We neglect additive thermal noise of receiver electronics, which is an idealization but one that can be approached with avalanche photodiodes (APDs) [27]. Any dark currents in the detectors are assumed to be handled by injecting an equivalent level of background radiation. Extension of our transmission method to the case of thermal-noise-limited reception is found in [28]. Each of the photodetectors forms an integral of its photodetected current over each of slots, or equivalently counts photoelectrons. The probability mass function for the number of counts in an on interval is related to the optical intensity by the Poisson distribution [25] (4) where is the mean number of signal photoelectrons produced at detector, and is the mean number of photoelectrons produced per slot due to background radiation. Similarly, in an off interval, the count variable is Poisson with parameter. 2 The signal mean parameter is given by where is the detector s quantum efficiency, assumed to be 0.5 here, is Planck s constant 10, and is the optical center frequency, here taken to be 10, corresponding to a 1.55 m wavelength in the infrared region. is the power provided at one detector by the entire transmitter array, and is the duration of a slot. 2 We assume the number of spatial modes in the background radiation field is large, leading to the Poisson statistics [25]. (5) The background mean count parameter is with denoting the background radiation power incident on each photodetector. This power is due to diffuse scattered light whose spectral radiance is a function of wavelength, and is proportional to receiver field-of-view, receiver area, and optical bandwidth. In the results section, we use dbj, which corresponds to a field-of-view of about 5 milliradians, an aperture diameter of 2.5 cm, an optical bandwidth of 1% of the center frequency, a data rate of 100 Mb/s, and the estimated diffuse background radiance at 1.55 m taken from [6]. The Poisson variates are independent, conditioned on the pulse pattern and channel gains, from slot-to-slot and across detectors. We designate the collection of counts by matrix. By writing the likelihood function, taking the logarithm, and eliminating common bias terms, we find that the maximum-likelihood detector finds the set of columns with largest weighted column sum, i.e.,, and decides in favor of the corresponding symbol. Actually, in three of the four cases considered below, equal-gain combining is optimal, namely for the nonfading (equal-gain) scenario or for the cases of no background radiation. Even in the situation of fading and background radiation, we show that equal-gain combining is essentially optimum. Thus, in this optical MIMO setting, one does not need to know the individual channel gains, a practical benefit. In the remainder of this paper, we treat two related problems. First, we study the uncoded symbol error probability for various MIMO fading scenarios, with and without background radiation. Then, we turn to a study of channel capacity for the quasi-static MIMO optical channel under the multipulse PPM constraint. III. ERROR PROBABILITY ANALYSIS We consider two cases: without or with background radiation, and for each, nonfading or fading links. We discuss a general theory, and illustrate with specific results for the most interesting cases. The situation without background radiation is the easiest and is treated first. A. Case I: No Background Radiation With no loss of generality, we assume the symbol with the first slots illuminated is sent. An error will occur if some other subset of columns has a larger column sum, but this cannot occur in the case of no background radiation. Instead, likelihood ties represents the only mechanism for decision error. This occurs when one or more of the on slots register zero counts at all detector outputs. Specifically, suppose slots among the first slots in each detector register zero counts, i.e., of the on slots produce a column of zeros in the array, where nonzero counts are expected. Then, a likelihood tie occurs among candidates and tie-breaking errs with probability (6) (7)

4 1904 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER 2005 By the Poisson property and independence we have that the probability of exactly of columns registering zero counts is of columns (8) where, and. Putting this all together, we have that the symbol error probability, conditioned on some set of channel gain variables, is We may expand the last term using binomial expansion, i.e.,, then combine terms to get a finite series expansion for conditional error probability (9) Fig. 2. Symbol error probability for nonfading case M = N = 1 no background radiation. P represents either P or P here. (10) where again. 1) Nonfading Case, No Background Radiation: The previous expression leads to the error probability on nonfading (equal-gain) links by simply setting, giving (11) Notice the dependence on vanishes in the nonfading case, whereas increasing helps due to larger effective aperture. 3 Equation (11) can be plotted for any pair as a function of the optical energy parameter. To cast this in terms of peak (alternatively, average) power and a common information rate, we note that so (12) (13) This reveals the energy efficiencies as a function of and : the multiplier on is, while the multiplier attached to peak-power normalization is. Thus, multipulse transmission becomes more efficient under peak power comparisons when, with efficiency maximized when. Single-pulse designs are most efficient under an average energy comparison. The peak-power constraint is usually the most relevant for semiconductor lasers. The symbol error probability is shown in Fig. 2 for the nonfading case versus and. Both classic PPM and multipulse cases are shown. We have emphasized and for illustration only, and show the binary PPM case for comparison. In the nonfading case, conversion to other values of and is straightforward, as described above. Notice that, as expected, the multipulse case ( and ) exhibits a superior energy efficiency when comparison on a peak-power constraint is made. On the other hand, if average N. 3 In [17], the assumption is to fix the total receiver aperture, independent of energy per bit is the criterion, classic PPM is much superior. 2) Fading, No Background Radiation: To extend the analysis above to the case of link fading, we assume the fading is fixed over a symbol duration and simply average the (conditional) symbol error probability (10), with respect to the joint fading distribution of the variates. It is emphasized that this produces the symbol error probability averaged over fades. Formally, we find by (14) where the integral is interpreted as an -dimensional integral. Since the variables are assumed independent, the above averaging leads to where (15) (16) is a function of the received energy per slot and the fading distribution. In the case of Rayleigh fading, this averaging may be done analytically, and produces (17) Log-normal fading can at least be handled numerically in evaluating (15). Study of the Rayleigh-fading expression as a function of reveals that is an inverse- -power function of signal energy in the large signal regime, which we take as definition of achieving full diversity. It is worth noting that attainment

5 WILSON et al.: OPTICAL REPETITION MIMO TRANSMISSION WITH MULTIPULSE PPM 1905 Fig. 3. Symbol error probability for Rayleigh and log-normal (S:I: = 1) fading, no background radiation, Q = 8, w = 4, M 2 f1; 2; 4g, N = 1, and peak-power definition. Fig. 4. Symbol error probability for optimal combining (dashed-dot line) and equal-gain combining (solid line), Rayleigh and log-normal (S:I: =1)fading, and background radiation, Q =8, w =4, M =1, and P T = 0170 dbj. of full transmit diversity is obtained without resort to exotic space time constructions here, in contrast to the microwave counterpart. This is essentially the result of intensity modulation and noncoherent detection here. As in the nonfading case, the symbol error probability expressions can be plotted versus either or. Fig. 3 presents the performance for both log-normal and Rayleigh cases versus peak power for,,, and. Rayleigh-fading performance shows linear asymptotes revealing the diversity order 1, 2, or 4 is attained. Similar conclusions pertain to average power as were made for the nonfading case. It may also be noted from the expressions for that the interchange of and is not symmetric, due to the power division by at the transmitter [note the factor in (17)] Thus, though they have the same diversity order, and (1,2) cases are 3 db different in favor of the latter. Actually, if we fix the total receive aperture as in [17], then and are interchangeable. B. Case II: Background Radiation For the case of background radiation, the evaluation of error probability is more complicated. Here, some incorrect symbol can have higher likelihood (incorrect set of largest weighted column sums) in addition to likelihood ties. One can formally sum conditional probabilities over the error region and correctly handle ties, but we have resorted to Monte Carlo simulation using importance sampling instead; see, e.g., [29]. We tested optimal weighting and equal-gain combining, the latter suboptimal in the case of fading and background radiation, and only when. In cases where it is an issue, Fig. 4 shows that the loss of the equal-gain combiner is very small, and this probably represents a logical practical choice for receiver processing. To perform the simulation, without loss of generality, we assume the message with the first slots illuminated is sent. For a given background and signal power, a normal Monte Carlo simulation would generate Poisson variates having the proper mean count values, then perform the detection and count errors. This procedure takes a very long simulation time if low error probabilities are sought. The method of importance sampling instead produces variates that are much noisier, i.e., producing a much higher frequency of errors than one expects, then weighting the error counts appropriately to obtain an unbiased estimate of. If the sampling probability density is correctly chosen, the variance of the estimate can be greatly reduced relative to that of the Monte Carlo procedure with the same number of trials. In our procedure, we bias the fading distribution to a one-sided exponential for the amplitude variable, which has the effect of decreasing the mean signal counts. Fig. 5 shows simulated results for for a nonfading case, with 10 joules, fixed. Notice the shape of the curves is similar to those of Fig. 2 (without background), except the new results are steeper essentially, some minimal level of signal power is required to overcome the background noise, and once this level is exceeded, performance improves sharply. It is interesting to compare the signal level required to obtain 10, say, with and without the given background power. With,, these values are approximately 167 and 173 dbj, respectively, in terms of peak power. In the regime of strong background power and low error probability, a bounding argument shows that a 6 db increase in background power is offset by a 3 db increase in optical power. Following an argument provided by a reviewer, we first let and, respectively, represent the on-slot and off-slot mean counts, after combining. A union bound on symbol error probability will be dominated by error events with symbol Hamming distance 2, of which there are occurrences. Thus distance error (18) The latter two-signal probability can be bounded, using the appropriate Poisson distributions, by a Chernoff bound distance error (19)

6 1906 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER 2005 Fig. 5. Symbol error probability for nonfading case with background radiation P T = 0170 dbj. where. Expanding the exponent with a Taylor series in for small gives (20) This reveals that a quadrupling of background power can be offset by a doubling of signal power. The same argument also applies in the fading case by conditioning upon the fading state. The same simulation procedure handles the case of fading with background radiation. A sample result is shown in Fig. 6, for both Rayleigh and log-normal fading. Background power remains fixed at dbj. Again, full diversity is observed. IV. CHANNEL CAPACITY Our analysis now shifts to the information-theoretic capability of multipulse PPM on optical MIMO channels, looking to eventual overlay of coding on multiple-pulse-position-modulation (MPPM) MIMO transmission. We study the distribution of the random capacity variable, and analyze ergodic capacity (appropriate for coding over long-time spans) and upon outage probabilities, the quantity of interest for coding over quasi-static channels. The discussion divides into two cases as before. A. Case I: No Background Radiation We begin by considering the mutual information between, the random array of counts for the slots, and the input symbol (21) Due to channel symmetry, mutual information will be maximized with an equiprobable input distribution, so only the second term needs further discussion. For any message, slots will be on slots, and will be off. Without background radiation, is nonzero only when all of the detectors generate zero photoelectrons for Fig. 6. Symbol error probability for Rayleigh and log-normal (S:I: = 1:0) fading, Q =8, w =4, N =1, background energy = 0170 dbj. some of the on slots, in which case the detector must choose from equally probable codewords. The resulting capacity is of columns For each of the on slots, the variable of columns of columns (22) where photoelectrons generated by detector slots. Letting and thus variable is a Poisson random (23) is the mean number of during one of the, as in (8) gives of columns (24) (25) The last term can be expanded using the binomial formula, which after simplification gives (26) Observe that the capacity in (21) depends on the channel matrix through, and is thus a random variable. One may then study the so-called ergodic (or mean) capacity, relevant to coding applications with very large time span, or the capacity

7 WILSON et al.: OPTICAL REPETITION MIMO TRANSMISSION WITH MULTIPULSE PPM 1907 distribution pursuant to outage capacity plots appropriate for quasi-static channels. The ergodic capacity is the expected value of the above (random) capacity (27) 1) Nonfading Case: In the nonfading case, and capacity can be expressed as (28) As with symbol error probability, in the no-fading case, there is no dependence of mean capacity on. However, there is an advantage to increasing the number of detectors (the aforementioned gain in aperture size). Fig. 7 depicts the mean capacity for the nonfading case with,, with conversion to other values straightforward. 2) Fading Case: To find the ergodic capacity with channel fading, we multiply the conditional capacity expression by the pdf for and integrate Fig. 7. Ergodic capacity for nonfading and log-normal (S:I: = 1) fading cases, Q =8, w =4, and no background radiation. (29) Given the assumed independence of the fading variables, it is straightforward to show that (30) In the Rayleigh case, the last integral can be computed easily, giving (31) Fig. 7 shows the ergodic capacity for log-normal fading for different MIMO configurations, clearly showing the benefits of additional lasers and/or photodetectors. Also, the loss relative to the nonfading case is minor when, even with the strong turbulence assumed. In delay-constrained applications, ergodic (or average) capacity is less relevant, and the outage probability is of more Fig. 8. Outage probability for log-normal fading (S:I: =1), Q =8, w =4, and coding rate R =log =2. interest. This is the probability that a random channel s mutual information falls below a certain fixed rate target, defining an outage in the information-theoretic sense. To study this, we presume that the information rate is half the maximum allowed by the modulation. By forming an ensemble of random channels according to a desired fading distribution, we may compute the probability that the mutual information falls below the adopted rate. These are shown in Figs. 8 and 9 for log-normal and Rayleigh models, respectively. Note the outage probability curves have linear asymptotes in the Rayleigh case, illustrating the diversity order. Clearly, it is advantageous to have a steeper outage probability characteristic provided by MIMO designs, so that an appropriate fixed-rate design is reliable.

8 1908 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER 2005 Fig. 9. Outage probability for Rayleigh fading, Q =8, w =4, (M; N) = (1; 1), (2,2), and (4,1), coding rate R =log =2. Fig. 10. Capacity for nonfading channel, Q =8, w =4, M =2, N =2, and varying P T. B. Case II: Background Radiation With background radiation present, the problem is further complicated. Here, uncertainty exists for any realization of. Again, the channel is symmetric, so we can make all codeword probabilities equal to achieve capacity. Here, we use an alternative definition for mutual information To generate the random variables, it is necessary to first look at the probability distribution for given.if we define to be the set of on slots in codeword, and to be the complement of that set signifying the off slots, we have that the conditional probability mass function is (32) By symmetry each term in the outer sum is the same, and we can condition upon any of the codewords, say (36) (33) The marginal probability for is which is simply the conditional expectation (34) Therefore, to evaluate the capacity of the channel with background radiation, we can perform a Monte Carlo simulation for and use (35) where the random variables are conditioned by choice of sending energy in the first slots, and is the number of independent trials. (37) Therefore, see (38) shown at the bottom of the page. To eliminate the product over, we can include it as a summation in the exponents. Each can be written in product form, which allows it to be pulled outside the outer summation in the numerator (since it is independent of ), as shown (38)

9 WILSON et al.: OPTICAL REPETITION MIMO TRANSMISSION WITH MULTIPULSE PPM 1909 (39) in (39) at the top of the page. We can define a new parameter and cancel like terms, to give (40) From this, it becomes easy to devise a programming strategy for simulating the terms in (35) for each trial. Fig. 10 presents results for nonfading channel capacity for a (2,2) MIMO system with background power varying as a parameter. We observe the capacity growth versus signal power is sharper when background radiation is present, paralleling the results for error probability. The penalty for a given level of background power is difficult to state, but one can notice that in the region of strong background power, every 10 db increase in background power implies only a 5 db increase in signal power to retain the same capacity. V. CONCLUSION AND EXTENSIONS We have analyzed an optical MIMO system employing multipulse PPM across laser sources, together with ideal photon counting reception from multiple small apertures. Both nonfading and log-normal and Rayleigh-fading models have been treated, assuming independent fading on source/detector pairs. The analysis shows the beneficial effects from a diversity standpoint of multiple sources and detectors, and transmit diversity is achieved here without additional special coding. Further, multipulse transmission exhibits clear superiority over classic PPM when peak laser power is constrained. Use of maximizes the energy and spectral efficiencies in this case. A particularly attractive choice appears to be,, providing 70 patterns per symbol interval. Many extensions to this work can be made. The most arguable of our assumptions is the lack of any electronic noise. As in fiber systems, avalanche photodetectors provide one way to make the optical (shot) noise exceed the electronic/thermal noise. This comes at the expense of an excess noise penalty, however, that will degrade performance from the ideal photon-counting analysis here [30]. Analysis of APD receivers with thermal additive noise is currently under study. More general space time patterns could be considered, in order to increase the throughput even further. We have concluded in other research [31], however, that attainment of full transmit diversity is precluded when one moves out of the repetition coding regime. Channel coding overlays with block, e.g., Reed Solomon, or trellis codes are certainly possible, but will not further increase the diversity order in quasi-static channels. Coding gain is possible however. Study of MPPM MIMO capacity helps in choice of optimal coding rates. ACKNOWLEDGMENT The authors wish to acknowledge helpful comments of the reviewers, in particular the suggestion of the Chernoff bounding procedure for assessing impact of background radiation. REFERENCES [1] H. Willebrand and B. Ghuman, Fiber optics without the fiber, IEEE Spectrum, vol. 38, no. 8, pp , Aug [2] A. Acampora, Last mile by laser, Sci. Amer., vol. 287, pp , Jun. 17, [3] V. Tatarski, Wave Propagation in a Turbulent Medium. New York: Mc- Graw-Hill, [4] E. J. W. Strohbehn, Laser Beam Propagations in the Atmosphere. New York: Springer-Verlag, [5] E. Hoversten, R. Harger, and S. Halme, Communication theory for the turbulent atmosphere, Proc. IEEE, vol. 58, pp , Oct [6] S. Karp, R. Gagliardi, S. Moran, and L. Stotts, Optical Channels. New York: Plenum, [7] R. Gagliardi and S. Karp, M-ary Poisson detection and optical communications, IEEE Trans. Commun. Technol., vol. COM-17, pp , Apr [8] J. Pierce, Optical channels: Practical limits with photon counting, IEEE Trans. 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Baedke, Optical MIMO transmission with multipulse PPM, M.S. thesis, Univ. Virginia, Charlottesville, VA, [29] R. Srinivasan, Importance Sampling: Applications in Communications and Detection. New York: Springer-Verlag, [30] F. Davidson and X. Sun, Gaussian approximation versus nearly exact performance analysis of optical communication systems with PPM signaling and APD receivers, IEEE Trans. Commun., vol. 36, no. 11, pp , Nov [31] M. Brandt-Pearce, S. Wilson, Q. Cao, and M. Baedke, Code design for optical MIMO systems over fading channels, in Proc. Conf. Record 38th Asilomar Conf. Signals, Syst., Comput., vol. 1, Nov. 7 10, 2004, pp Stephen G. Wilson (S 65 M 68 SM 99) received the B.S. degree from Iowa State University, Ames, the M.S. degree from the University of Michigan, Ann Arbor, and the Ph.D. degree from University of Washington, Seattle, all in electrical engineering. He is currently a Professor of Electrical and Computer Engineering, University of Virginia, Charlottesville. Prior to joining the University of Virginia faculty, he was a Staff Engineer for The Boeing Company, Seattle, WA, engaged in system studies for deep-space communication, satellite air-traffic-control systems, and military spread spectrum modem development. He is the author of the graduate-level text Digital Modulation and Coding. He has authored over 100 technical articles on communication system design and signal processing systems. He also acts as consultant to several industrial organizations in the area of communication system design and analysis, and digital signal processing. His research interests are in applications of information theory and coding to modern communication systems, specifically, digital modulation and coding techniques for satellite channels, wireless networks, spread-spectrum technology, and space time coding for multipath channels. Prof. Wilson has been recognized for Outstanding Teaching with Distinguished Professor Awards from the University of Virginia Alumni Association, the State Council on Higher Education, VA, and the ASEE-Southeastern Section. He is presently Area Editor for Coding Theory and Applications of the IEEE TRANSACTIONS ON COMMUNICATIONS. Maïté Brandt-Pearce (S 91 M 83 SM 99) received the B.S.E.E., M.E.E, and Ph.D. degrees in electrical engineering from Rice University, Houston, TX, in 1985, 1989, and 1993, respectively. She worked with Lockheed in support of the NASA Johnson Space Center from 1985 until In 1993, she joined the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, where she is currently an Associate Professor. Her fundamental interests lie in deciphering stochastic signals with multiple simultaneously received components from different sources. This interest has found applications in a variety of research projects including spread-spectrum multiple-access schemes, multiuser demodulation and detection, study of nonlinear effects on fiber-optic multiuser/multichannel communications, free-space optical multiuser communications, and radar signal processing and tracking of multiple targets. Dr. Brandt-Pearce is a member of Tau Beta Pi, and Eta Kappa Nu. She is the recipient of a National Science Foundation (NSF) CAREER Award, an NSF RIA, and an ORAU Junior Faculty Enhancement Award. She is currently an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS. Qianling Cao received the B.S. degree in engineering from the Xi an University, Xi an, China, in 1992 and the M.S. degree in engineering from the University of Virginia, Charlottesville, in She is currently working towards the Ph.D. degree at the University of Virginia, as a Research Assistant in electrical engineering. From 1999 to 2001, she was a Research Engineer in the Department of Electrical and Computer Engineering, National University of Singapore. From 1992 to 1998, she worked as a Research and Development Engineer at the Fourth Research Institute of Ministry, Post and Telecommunication of China. Her major research interests include wireless communication and optical communication, the latter of which is the focus of her Ph.D. research. Michael Baedke was born in Dunkirk, NY, in He received the B.A. degree in German from the University of Richmond, Richmond, VA, in 1998, the B.S. degree in electrical engineering from Virginia Commonwealth University, Richmond, VA, in 2003, and the M.S. degree in electrical engineering from the University of Virginia, Charlottesville, in 2004, while researching the performance and channel capacity of MIMO free-space optical communication links. Since then, he has been employed as an Electrical Systems Engineer at Dominion Generation, Mineral, VA.