Performance Analysis of Multipulse PPM on MIMO Free-Space Optical Channels

Transcription

1 Performance Analysis of Multipulse PPM on MIMO Free-Space Optical Channels A Thesis Presented to the Faculty of the School of Engineering and Applied Science University of Virginia In Partial Fulfillment of the Requirements for the Degree Master of Science Electrical Engineering by Michael L. Baedke August 2004

2 Approval Sheet This thesis is submitted in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering Author This thesis has been read and approved by the examining Committee: Dissertation advisor Accepted for the School of Engineering and Applied Science: Dean, School of Engineering and Applied Science August 2004

3 Acknowledgements Over the course of the past few years, there have been many people who helped to make this point in my education possible. In particular, I would like to thank my advisor, Professor Wilson, whose dedication to learning and contagious passion for the field has been an inspiration to me. Thank you for taking a personal interest in sharing your knowledge and providing me with the best education imaginable. I would also like to thank my wife, Alison for her unfailing love and support. Thank you for putting your dreams on hold this past year so I could follow mine, and thank you for always believing in me and encouraging me to be my best. Thanks also goes to my parents for always offering their financial and emotional support over the past few years. Being able to concentrate fully on our educational goals would have been impossible without a safety net to rely on whenever we needed it. Finally, I would like to thank Dr. Guess and Dr. Brandt-Pearce, who also happen to be on my committee, for their time and assistance this past year. You each made time for me whenever I needed your help, and I ve really enjoyed getting to know both of you. i

4 Abstract Free-space-optics (FSO has emerged as a technology that has the potential to bridge the last-mile gap that separates homes and businesses from high speed access to the Internet. In FSO systems, information is transmitted between two points by modulating a light source, much like with traditional fiber optic communication. However, FSO is a wireless technology in that it operates via line-of-sight, transmitting the data through the air, potentially over distances on the order of 1 km. Its main advantages over other competing technologies are that it can provide high speed access into the Gbps range, it operates in an unregulated frequency domain, and unlike fiber optic systems or other wired services, it avoids the need for trenching, which is much slower and more costly. The main challenge to FSO systems is the atmosphere itself. The systems must be designed such that the potentially harsh atmospheric effects of turbulence and aerosol scattering can be mitigated. Traditional solutions include increasing the link margin (supplying excess power to overcome potentially deep fades, and keeping the link distances small. However, these solutions are limited: Link margins become prohibitively difficult (often impossible to maintain during deep fades, and keeping link distances small is often impractical in real-world implementations. In this thesis, we study a method to combat link fading based on the multiple-input, multipleoutput (MIMO approach that has seen much success in the RF domain. By using multiple transmitters and receivers spaced sufficiently far from one another, we are able to create multiple, uncorrelated paths over which to send the data. The probability that all of the paths are simultaneously faded is much lower than when only relying on a single path from transmitter to receiver, as with a single-input, single-output (SISO system. For our analysis of the MIMO FSO system, we explore multiple pulse position modulation ii

5 iii (MPPM. This is a modulation technique where the duration of a signal is divided into Q slots, and the laser array is pulsed simultaneously during w of them, creating ( Q w possible patterns. Traditional PPM is a special case of MPPM, where w = 1. We show that MPPM is superior to PPM with respect to bandwidth efficiency (and maximized when w = Q, and exhibits supe- 2 rior symbol error performance when the system is peak-power-limited. PPM exhibits superior symbol error performance when the system is average-power-limited. In this thesis, we develop the maximum likelihood detectors for the system operating in the perfect photon counting (Poisson and thermal-noise-limited (Gaussian regimes. We demonstrate that for non-faded channels, having multiple receivers improves symbol error performance due to the increase in receiver aperture size. We also demonstrate that for faded channels, performance gains are seen for increases in the number of transmitters and receivers. Full transmitter and receiver diversity is observable by analyzing the Rayleigh fading case. We also analyze the information-theoretic channel capacity of the Poisson regime by looking at the ergodic (average capacity and the outage probability (probability of the instantaneous capacity dropping below some set threshold. We see from these results that the maximum capacity is achieved at lower power levels for MIMO and SIMO systems over non-faded channels, and for MIMO, SIMO, and MISO systems over faded channels. The outage probability curves are steeper and show the beneficial effect of transmitter and/or receiver diversity. Full diversity is once again observable by looking at the outage probability curves in the Rayleigh fading case. We conclude that MIMO system design is a technique that improves MPPM FSO system performance under various fading cases, and that full transmitter and receiver diversity is achievable in this system.

6 Contents 1 Introduction What is free-space optics? The history of free-space optical communication Uses of free-space optics Military communication Satellite and deep-space communication The last-mile solution System overview Information Source and channel encoders Multiple pulse position modulator The transmitter The channel Turbulence Aerosol scattering Fading models The receiver p-i-n photodiodes Avalanche photodiodes Bandwidth and noise considerations in p-i-n and APD receivers Bandwidth Shot noise Background noise iv

7 v Thermal noise Excess APD noise MPPM demodulator Source decoder, channel decoder, and retrieved information MIMO applied to FSO systems using MPPM - background, system model and definitions Research on MIMO and MPPM FSO communications systems MIMO in wireless systems Application of MIMO concepts to free-space optics Multiple pulse position modulation (MPPM Error probability - Gaussian vs. Poisson Power comparisons between PPM and MPPM System model MPPM signaling Transmitter array Receiver array Channel Detector and observable Poisson Regime Gaussian Regime Maximum likelihood (ML detection ML Detection for the poisson regime Case 1: ML detection with no background and no fading Case 2: ML detection with no background and fading Case 3: ML detection with background radiation and no fading Case 4: Background radiation and fading General ML detection in the Gaussian regime Thermal noise dominates over shot noise No fading present Fading present

8 vi Shot noise dominates over thermal noise Fading present No fading present Shot and thermal noise are not dominant Fading Present No fading present Error analysis of MIMO FSO system using MPPM Error analysis in the Poisson regime No background radiation No background, no fading No background, Rayleigh fading No background, log-normal fading Error probability in the presence of background radiation Background radiation, no fading Background radiation, Rayleigh fading Background radiation, log-normal fading Error probability in the Gaussian regime No fading Rayleigh fading Log-normal fading Capacity of the MIMO FSO system using MPPM in the Poisson regime No background radiation No background, no fading No background, Rayleigh fading No background, log-normal fading Background radiation and no fading Background radiation, no fading Background radiation, Rayleigh fading Background radiation, log-normal fading

9 7 Conclusions 88 vii

10 List of Figures 2.1 FSO system block diagram Output light power vs. input drive current for all three most common light sources [1] Probability density functions for Rayleigh and log-normal distributions A typical FSO transceiver [2] Simplified detector circuit employing a p-i-n photodiode Simplified model of an APD and integrator The MIMO concept Relative energy efficiencies vs. Q for w {1, Q/2 } Norton equivalent noise model Poisson and Gaussian p.d.f. s with equal means and variances of λ = Poisson and Gaussian p.d.f. s with equal means and variances of λ = Symbol error probability vs. average power with no fading and no background radiation, and Q = Symbol error probability vs. peak power with no fading and no background radiation, and Q = Symbol error probability vs. average power with Rayleigh fading, no background radiation, and Q = Symbol error probability vs. peak power with Rayleigh fading, no background radiation, and Q = Symbol error probability vs. average power with log-normal fading (S.I. = 1.0, no background radiation, and Q = viii

11 ix 5.6 Symbol error probability vs. peak power with log-normal fading (S.I. = 1.0, no background radiation, and Q = An example of a definite error An example of an indefinite error (where the receiver chooses incorrectly from the 3 possible modulator symbols Symbol error probability vs. average power with no fading, P b T b = 170 dbj, and Q = 8 dashed and solid lines overlap Symbol error probability vs. peak power with no fading, P b T b = 170 dbj, and Q = 8 dashed and solid lines overlap Symbol error probability vs. average power with Rayleigh fading, P b T b = 170 dbj, and Q = Symbol error probability vs. peak power with Rayleigh fading, P b T b = 170 dbj, and Q = Symbol error probability vs. average power with Log-normal fading, S.I. = 1.0, P b T b = 170 dbj, and Q = Symbol error probability vs. peak power with Log-normal fading, S.I. = 1.0, P b T b = 170 dbj, and Q = Symbol error probability vs. average power with no fading, Q = 8, R = 100 Ω, T 0 = 290 K, P b T b = 170 dbj, and R b = 100 Mbps Symbol error probability vs. peak power with no fading, Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Symbol error probability vs. average power with Rayleigh fading, Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Symbol error probability vs. peak power with Rayleigh fading, Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Symbol error probability vs. signal power with Rayleigh fading using equal gain combining (EGC or optimal gain combining (OGC. Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Symbol error probability vs. average power with log-normal fading, S.I. = 1.0, Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps

12 x 5.21 Symbol error probability vs. peak power with log-normal fading, S.I. = 1.0, Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Symbol error probability vs. signal power with log-normal fading using equal gain combining (EGC or optimal gain combining (OGC. Q = 8, R = 100 Ω, T 0 = 290 K, P b = 90 dbw, and R b = 100 Mbps Ergodic capacity vs. average power with no fading, no background radiation, and Q = Ergodic capacity vs. peak power with no fading, no background radiation, and Q = % outage probability vs. average power with no fading, no background radiation, and Q = % outage probability vs. peak power with Rayleigh fading, no background radiation, and Q = Ergodic capacity vs. average power with Rayleigh fading, no background radiation, and Q = Ergodic capacity vs. peak power with Rayleigh fading, no background radiation, and Q = % outage probability vs. average power with Rayleigh fading, no background radiation, and Q = % outage probability vs. peak power with Rayleigh fading, no background radiation, and Q = Ergodic capacity vs. average power with log-normal fading, no background radiation, Q = 8, and S.I. = Ergodic capacity vs. peak power with log-normal fading, no background radiation, Q = 8, and S.I. = % outage probability vs. average power with log-normal fading, no background radiation, and Q = % outage probability vs. peak power with log-normal fading, no background radiation, and Q = Ergodic capacity vs. average power with no fading, Q = 8, and P b T = 170 dbj

13 xi 6.14 Ergodic capacity vs. peak power with no fading, Q = 8, and P b T = 170 dbj % outage probability vs. average power with no fading, Q = 8, and P b T = 170 dbj % outage probability vs. peak power with no fading, Q = 8, and P b T = 170 dbj Ergodic capacity vs. average power with Rayleigh fading, Q = 8, and P b T = 170 dbj Ergodic capacity vs. peak power with Rayleigh fading, Q = 8, and P b T = 170 dbj % outage probability vs. average power with Rayleigh fading, Q = 8, and P b T = 170 dbj % outage probability vs. peak power with Rayleigh fading, Q = 8, and P b T = 170 dbj Ergodic capacity vs. average power with log-normal fading, Q = 8, and P b T = 170 dbj Ergodic capacity vs. peak power with log-normal fading, Q = 8, and P b T = 170 dbj % outage probability vs. average power with log-normal fading, Q = 8, and P b T = 170 dbj % outage probability vs. peak power with log-normal fading, Q = 8, and P b T = 170 dbj

14 List of Symbols xii a Substitution for a nm in integration. A M N matrix representing the path gains from each receiver to each transmitter. a nm One realization of the individual path gain from transmitter m to receiver n; this quantity is squared to find power gain. b, g, i, j, k, l Generic variables used to index summations and for other tasks. C Information theoretic capacity of a channel, measured in bits per channel use. C d d 0 E s f F (M G Junction capacitance of a photodetector. Correlation distance, measured in meters. Energy per modulator symbol contributed by the entire transmit array to one receiver element. Frequency of transmitted signal. For a 1550 nm laser diode, this would be approximately Hz. Excess noise factor in an avalanche photodiode. Average gain of an avalanche photodiode. h Planck s constant, approximately equal to H(X Entropy of X, measured in bits. L Link length, measured in meters. m Denotes a single transmitter. m {1, 2,..., M} M Number of transmitters in the system. n Denotes a single receiver. n {1, 2,..., N} N Number of receivers in the system. P Peak on power observed at the receiver. P ave P b P def On power observed at the receiver, averaged over the duration of the modulator symbol. Background power observed at the receiver. Probability of a definite error; occurs when one or more off slots have a higher count rate than on slots.

15 xiii P indef P peak Q Q i on Q i off R b S.I. T T b T s w ˆX Z Probability of an indefinite error; occurs when one or more off slots have a count rate equal to one or more on slots, with no off slots having a count rate greater than any on slot. Also used to denote peak on power at the receiver. Number of slots per MPPM symbol. The set of w on slots for a MPPM symbol. The set of Q w off slots for a MPPM symbol. Bit rate of system. Scintillation index. A measure of strength of fading in the log-normal fading model. Duration of one slot of a pulse position modulation symbol, measured in seconds. Duration of a single bit. Inverse is the bit rate. Duration of the entire pulse position modulation symbol, measured in seconds. Number of on slots per MPPM symbol. The estimate of the receiver; the modulator symbol chosen by the receiver as the most probable symbol received. A N Q matrix representing the received observations by all N detectors over all Q slots. Z nq Observable at output of integrator at receiver n at slot q. β Probability of a deep fade detrimentally affecting a link path from transmitter m to detector n. ζ n η λ λ on λ on,n Defined as (λ on,n + λ off /λ off Quantum efficiency of detector; the average number of electron-hole pairs generated per incident photon Wavelength of the transmitted signal; we have used 1550 nm for our analysis. Poisson parameter denoting average number of photoelectrons observed during an on slot, summed over all N photodetectors. Poisson parameter denoting average number of photoelectrons observed during an on slot at a single detector n.

16 xiv λ off Poisson parameter denoting average number of photoelectrons observed during an off slot at a single detector n. µ on Used for the Gaussian regime, as the average number of photoelectrons observed during an on slot, summed over all N photodetectors. µ on,n Used for the Gaussian regime, as the average number of photoelectrons observed during an on slot at detector n. µ off Used for the Gaussian regime, as the average number of photoelectrons observed during an off slot at detector n. µ 2 X Denotes the mean of the log-normal fading variable. σ 2 σon 2 σoff 2 σx 2 τ 0 Used for the Gaussian regime, when thermal noise dominates, as the variance of the number of photoelectrons observed during an on or off slot. Used for the Gaussian regime, when thermal noise is not dominant, as the variance of the number of photoelectrons observed during an on slot. Used for the Gaussian regime, when thermal noise is not dominant, as the variance of the number of photoelectrons observed during an off slot. Denotes the variance of the log-normal fading variable. Correlation time, measured in seconds. φ pdf (k, α Probability that a Poisson random variable with parameter α equals k. φ cdf (k, α Probability that a Poisson random variable with parameter α is less than or equal to k.

17 Chapter 1 Introduction 1.1 What is free-space optics? Free-space optics is a method of communication that involves using a light source, usually a laser, to transmit information through space or the atmosphere to a receiver. It is similar to traditional fiber-optics communication, in that it uses light to communicate information, but the difference is the medium through which the information travels. Fiber-optics, as the name implies, uses a fiber to carry the light wave from transmitter to receiver. Free-space optics, however, relies on a line-of-sight approach. As a result, the medium could be a near-vacuum, as it is for satellite-to-satellite communication, or it could be the atmosphere, which includes atmospheric and other natural obstructions that come into the path of the light. 1.2 The history of free-space optical communication Although free-space optical (FSO communication as we know it originated in the 1970s [3], the history of optical communication through free-space using light really began when signal fires were used to send messages across long distances. Paul Revere s lanterns, as well as manually operated lanterns on ships are examples of early FSO communication systems [4]. The first major technological advancement in FSO communication, that moved us past the ages of signal fires, happened in 1880, when Alexander Graham Bell invented the photophone. The device could send intensity-modulated sunlight over a distance of a few hundred feet. How- 1

18 2 ever, there were no major advancements in FSO communication for nearly a century, until the laser was invented in the 1960s [4]. Within a few months of the invention of the laser, there was interest in using the technology to communicate through the atmosphere. Bell Labs engineers brought an early ruby laser to the top of a microwave tower at Murray Hill, NJ, and pointed it at a large screen 40 km away. A colleague watched the screen for signs of the red pulses coming through, but not many pulses made it. Although the results were poor, the engineers were able to make a spot as large as a dining table glow like a fireplace at a distance of 2-6 km [5]. FSO communication, which also sometimes goes by the names optical wireless communication, lasercom (patented by McDonnell Douglas in the 1980 s, wireless optical communication, and fiber-free optical communication, really started to take off in the 1980s. Funding was increased in the United States and Europe as governments tried to plan for next generation communication technologies for air-to-air, satellite-to-submarine, air-to-satellite, and satellite-to-satellite links [6]. There was also an interest for non-military communication, as well, but after the first few trials of FSO communication, interest began to decline. One reason is that communicating through optical fibers seemed so far superior to communicating through the atmosphere [5]. A lack of reliable components was also always an issue with early optical systems. Almost every single component had to be developed, including laser sources, detectors, high-speed electronics, and high accuracy pointing components to name a few. Because of these technical difficulties, as well as financial and political issues as well, one-by-one, almost all early FSO communications programs met with an early demise [6]. However, in the past few years, mainly due to a need to solve the last-mile problem, an increased interest has been seen in non-military uses of FSO communication [5]. In fact, Acampora cites a study that predicts that the FSO communications industry could grow from an annual 120 million dollars in 2000 to more than 2 billion dollars annually by 2006 [3].

19 3 1.3 Uses of free-space optics Military communication Free-space optics is an attractive method of communication for military applications. The reason is security. Using traditional radio frequency (RF communications methods, eavesdropping on a conversation is much easier than with free-space optics, since the RF waves are transmitted over a large area. This makes it possible to receive the signal while in the vicinity of the system, although it is still necessary to demodulate and decode it. A FSO link, on the other hand, has a very narrow beam divergence [4], typically milliradians, so the only way to intercept the signal is to be in the path of transmission Satellite and deep-space communication Free-space optics has several advantages that make it well suited to satellite communication. First, it can provide high data-rate communication links between satellites at geosynchronous distances and beyond. Second, it has several attributes that are superior to traditional RF communication methods. Traditionally, satellite communication has been accomplished using microwaves, but these systems are bulky and expensive [4]. Free-space optics has the advantage of being much smaller and far more inexpensive, which is an enormous asset to any space vehicle. For this reason, NASA is developing a deep-space optical communication transceiver in its X2000 program, also known as the Advanced Deep-Space Systems Development Program [7]. Early in the X2000 program, NASA plans to support tens of kilobits per second of data from the Mars range [8]. It also plans on building the first of two 10 m class ground receiving telescopes by 2008 [7]. The International Space Station (ISS Engineering Research and Technology Development program is sponsoring the development of a high data rate FSO transmitter from the low-earthorbit range (on board the ISS [7], that is projected to be able to support a data rate of 2.5 Gbps. FSO communications appears to be the technology that will meet the needs of future space ventures, including near-earth, solar-system, and interstellar missions [8].

20 The last-mile solution Probably one of the most compelling and timely uses of free-space optics is to provide a solution to the last-mile problem (or first-mile problem, [9] depending on your perspective. The multibillion-dollar optical fiber backbone that was built to provide high-speed broadband access to offices and homes has come up less than one mile short for 9 of 10 US businesses with 100 or fewer workers. As a result, only 2-5% of the fiber network is actually being used today [3]. Most businesses and homes are currently connected to the fiber backbone using traditional copper wires, which do not possess the gigabit-per-second capacity required to carry bandwidth-intensive applications [3]. Laying fiber optic cable to each home and business that needs broadband access would be the ideal solution to this problem, but it is slow and expensive. The process can take 6-12 months, and can cost anywhere from $100,000 to $500,000 per mile[3], with up to 85% of the cost due to trenching and installation [2]. Free-space optics, on the other hand, can be up and running in a few days, and costs 1/3 to 1/10 of the cost of a fiber installation [3]. Trenching also causes traffic jams, displaces trees, and can destroy historical areas. For these reasons, Washington D.C. is considering a moratorium on fiber trenching [2]. FSO communication is thought by many experts to have the best chance at succeeding over other fiber-free technologies (like DSL, microwave radio, etc. at bridging the last-mile gap [3]. In addition to the cost and speed benefits, it has a greater potential because it operates in an unlicensed band (which is an enormous cost benefit as well, it s scalable (unlike RF networks [9], and it can be set up in a mesh configuration to carry full duplex gigabit-persecond communications around a city town or region [3]. One challenge that faces FSO communications systems is building sway. Because of the very narrow beamwidths possible from the lasers, very small changes in the position of either the transmitter or receiver can cause the laser beam to miss its target. The two possible solutions to this problem are to increase the beam divergence at the receiver (which also reduces the power density or, for especially tall buildings or narrow beamwidths, to use an active tracking system; mirrors continually adjust to keep the beam centered on the target. The biggest challenge facing free-space optics in terrestrial applications, which is also a major focus of this thesis, is overcoming limitations caused by the atmosphere. Bad weather, especially thick fog, can severely attenuate the signal before it reaches the receiver [3]. In fact,

21 5 weather is the reason links in non-desert regions are often kept to m to ensure carrierclass availability (99.999% availability [2]. In this thesis we analyze an approach called spatial diversity which attempts to overcome these atmospheric difficulties. This approach employs multiple transmitters and/or receivers simultaneously sending and/or receiving the information. The idea is to keep the transmitters and receivers sufficiently far from one another (which is a surprisingly small distance, as we will see, such that all of the individual paths from transmitters to receivers would have to be simultaneously faded (a much lower probability event in order to degrade system performance.

22 Chapter 2 System overview In this section, we will describe the free-space optical link from beginning to end. The physical devices, channel, and noise and other disturbances will be considered, however in this section we are most concerned with the hardware used to construct such a system, and a better understanding of the physical phenomena that affect both the hardware and the channel to make them non-ideal. An overall block diagram depicting the FSO system is shown below in Figure 2.1. Figure 2.1: FSO system block diagram. 6

23 7 2.1 Information For the purpose of this paper, we are going to assume that the information to be transmitted is already in binary format. This could be any kind of data including, but not limited to, Internet and intranet traffic, multimedia applications (including streaming audio and video, and file transfers or data exchanges of any kind. 2.2 Source and channel encoders From an information-theoretic point of view, the raw information described in Section 2.1 contains natural redundancies that can be removed to make the system more efficient. This is known as compression or source coding. The channel encoder then adds intelligent redundancies back into the stream of data that make the system more resistant to errors. More detail on these topics can be found in texts by Wilson [10] and Cover and Thomas [11]. Upper limits on the performance of channel coding will be investigated in Chapter Multiple pulse position modulator For this system, we are focusing on multiple pulse position modulation (MPPM, which is a intensity modulation technique. This is, of course, not the only possible way to build this system. Since a laser is, in effect, an optical oscillator [5], any modulation that is possible with RF communication is also possible with optical communication (including coherent modulation/demodulation techniques. The drawback, however, comes from the fact that coherent techniques like phase or frequency modulation are far more complex and expensive to build [12]. The analysis and a detailed description of MPPM will be completed in Section Until then, it suffices to say that the modulator can take a certain number of bits (from the channel encoder, and map them to a single MPPM symbol that will be sent across the channel by the transmitter.

24 8 2.4 The transmitter In Figure 2.1, the transmitter and modulator are depicted as being separate entities, but there are actually different ways to construct this. It is possible for the transmitter to be constantly on, and then be modulated as it is passed on to the channel, or the laser can be directly modulated in one step. Here we consider the transmitter to be the light source, which simply has the task of sending light over the channel. There are three different types of light sources that are commonly used in free-space optics: Light Emitting Diode (LED: LED s can produce light in the nm band, they are cheap, and they can produce radiation with low current drive levels. However, they have limited output powers (1-10 mw, there is more frequency spreading than the other light sources, and the light tends to be incoherent and unfocused [1]. Laser: Lasers have power outputs of W, but are much bulkier than LED s. The laser is an optical cavity filled with light amplification material and mirrored facets at each end. When the cavity lases, an initiated optical field crosses back and forth in a self-sustaining reaction. A small aperture in one of the mirrored facets allows some of the energy to escape as radiated light. In the linear range of operation (see Figure 2.2, lasers are unstable, so they are usually operated as continuous-wave devices at peak power [1]. Laser Diode: Like LED s, laser diodes are semiconductor junction devices [1], but they operate more like lasers with reflecting etched substrates which act like small reflectors (like the reflectors in the laser. Laser diodes are small, rugged, and very power-efficient. They require more drive current than LED s, but also generate more power. A laser diode produces about a hundred milliwatts of useable optical power [4] with a more focused beam than with LED s [13]. All of the three light sources have the same output power characteristics, shown in Figure 2.2. From this, one can see a distinct linear region of operation, where an increase in input current would result in a proportional increase in output light power. The wavelength chosen for FSO systems usually falls near one of two wavelengths, 850 nm or 1550 nm. The shorter of the two wavelengths is cheaper and is favored for shorter distances.

25 9 Figure 2.2: Output light power vs. input drive current for all three most common light sources [1]. The 1550 nm light source is favored for longer distances since it has an allowed power that is two orders of magnitude higher than at 850 nm [2]. These power limits are determined by the American National Standards Institute (ANSI Z136.1 Safety Standard [9]. The reason for the higher allowed power is that laser-tissue interaction is very dependent on wavelength. The cornea and lens are transparent to visible wavelengths (such as 850 nm so the power can reach the retina at the back of the eye. At 1550 nm retinal absorption is much lower, since the power is absorbed mostly by the lens and cornea before it can reach the retina. The power at 1550 nm is not unlimited, however, since it can still cause photo-keratitis and cataracts at higher levels [14]. The 1550 nm wavelength is also preferred since more photons per watt of power arrive for longer wavelengths, and therefore more photocurrent is produced per watt of incident power for equal efficiency devices[4]. For the remainder of this thesis, we will assume the laser to be an ideal, infinite bandwidth light source. For the simulations in Chapters 5, 6, and 5.2 we will make use of the 1550 nm wavelength. 2.5 The channel In a terrestrial free-space optical link, the channel is simply the atmosphere plus any other disturbances through which the optical signal will pass. This is a very important component of

26 10 our system, since the channel is often the limiting factor for how long the link can be. The atmospheric channel is uncontrolled in that the designers have no way of preventing obstructions and other disturbances from coming between the transmitter and receiver. The engineer will attempt put the system in a location where it is unlikely for obstructions to occur, but it is always possible for a bird, for example, to temporarily pass through the beam. However, in a packet-switched network, short duration interruptions are easily handled by retransmitting the data [9]. A more serious threat is the atmosphere itself. Zhu and Kahn classify atmospheric effects on the FSO channel into two categories, atmospheric turbulence and aerosol scattering [12]. These are discussed further below Turbulence Atmospheric turbulence is also known as scintillation. Even on a clear day, there are continual variations in the intensity of the light at the receiver due to inhomogeneities in the temperature and pressure of the atmosphere. The Kolmogorov turbulence model is often used to describe atmospheric turbulence [12, 15] and predicts that changes in the air temperature as small as 1 degree Kelvin can cause refractive index changes as large as several parts per million [15]. These pockets of air with different refractive indices, or eddies, act like time-varying prisms [2] whose size ranges from a few millimeters to a few meters [12], and whose time scale is related to wind speed [16] among other things. These eddies cause the light to diffract along the path to the receiver in a time-varying manner, affecting the intensity of the light. This phenomenon is visible to the naked eye by watching the stars twinkle at night, or by watching the horizon shimmer on a hot day [2]. The effect of scintillation on a FSO communications link can be a wandering beam when the eddies are bigger than the beam diameter and move the beam completely off target [2], fluctuating power at the receiver [9], and changes in the phase of the received light wave [12]. For weak turbulence, the intensity of the received signal is a random variable best approximated by a log-normal distribution [13, 16]. This model is described further in Section To describe turbulence-induced fading, we do so using parameters in the spatial and temporal domains. The first useful parameter is the correlation length, which we call d 0. This is simply the distance for which the intensity of a light wave at two points in the atmosphere is

27 11 essentially uncorrelated. This distance can be approximated by d 0 λl, where λ is the wavelength of the transmitted wave, and L is the length of the FSO link [13]. This approximation is valid for most FSO communication systems using visible or infrared lasers for link lengths ranging from a few hundred meters to a few kilometers [12], and is approximately 1-10 cm for most terrestrial links [13]. The importance of correlation distance will become evident as we talk about spatial diversity as a method of mitigating the effect of turbulence on FSO links. The second useful parameter is the correlation time, which we call τ 0. When observing a single point in the atmosphere at two different times, τ 0 represents the amount of time between observations for which the atmospheric parameters are uncorrelated. The time scale for scintillation is about the time it takes a volume of air the size of the beam to move across the path, and is therefore related to wind speed [16]. Typical values for terrestrial links are 1-10 ms [13]. Correlation time is important to our discussion in order to justify spatial diversity as a method of mitigating block fading. At the transmission rates desirable for a FSO system (2.5 Gbps for example, a deep fade that could last 1-10 ms could potentially affect 2.5 to 25 megabits of data. The normal approach to counteract block fades is to interleave the data before coding, but this is an unattractive solution due to the enormous size of the interleaver that would be necessary to be effective [15]. Spatial diversity, which will be discussed in Section is a method that avoids the need for such large interleavers Aerosol scattering The most detrimental atmospheric phenomenon that affects FSO links is fog, which is classified as aerosol scattering. According to Acampora, susceptibility to fog has slowed the commercial development of free-space optics, since it so severely limits the range of a FSO link [3]. The exact amount of signal attenuation caused by fog varies with its density. Acampora states that the link might lose 90% of its power for every 50 meters in moderately dense fog [3]. This translates into a loss of 200 db/km. Other sources give ranges in attenuation from 16 db/km in light fog [9] to 300 db/km in dense fog [17]. There are various ways to combat link fade due to fog. One such method is to simply increase the power, also known as increasing the link margin. The link margin is simply extra transmit power that is in excess of what is normally needed to communicate. The only problem

28 12 with increasing the link margin is that power levels are limited for any system, both because of eye safety as well as practical limitations in the system itself. For moderately dense fog, increasing the link power by a large amount, 60 db (a factor of one million for example, would still only allow for an extra 300 m in link length. Fading can also be mitigated by making the link length as small as possible. Longer links can be accommodated by arranging the transmitters and receivers in a mesh-topology. In an urban setting, the mesh could jump from building-to-building or house-to-house, so that the signal propagates only over shorter distances and has multiple paths to reach any point in the network [3]. The wavelength of the link also affects the link s susceptibility to fog. Future FSO systems will most likely take advantage of the long wavelength infrared range (LWIR spectrum (8µm < λ < 14µm, also known as the night-vision spectrum. LWIR systems are called all-weather systems because they are times less sensitive to fog, rain, smog, and other atmospheric disturbances. These wavelengths are also far less dangerous to eye safety so allowable power levels are higher than those for the µm range. [18]. Point-to-point microwave radio is an alternative to free-space optics that is immune to fog. However, this technology requires spectrum licensing, which is a major disadvantage when compared to FSO systems [3]. Effectively overcoming challenges imposed by foggy weather for any particular FSO link would most likely involve a combination of the aforementioned solutions. Using spatial diversity to combat atmospheric effects, which is the focus of this thesis, can be incorporated into almost any well-designed system that also uses link margin, a mesh topology, and LWIR lasers. The effect of also using spatial diversity is to introduce yet one more weapon in the arsenal of the communications engineer to combat link fading Fading models There are two widely used models for fading. One is the log-normal distribution, the other is the Rayleigh distribution. In all of the fading cases, we keep the expected path gain E[A 2 ] equal to one, in order to make fair comparisons between them and the non-fading case. The log-normal distribution is very often used in the literature to describe atmospheric turbulence as experienced in FSO systems. In the log-normal distribution, the amplitude of the

29 path gain is a random variable A where A = e X and X is normal with mean µ X and variance σx 2. By definition, the logarithm of A follows a normal distribution. The optical intensity I = A 2 is also log-normally distributed. The p.d.f. for A is f A (a = 1 (2πσ 2 X 1/2 a exp( (log e a µ X 2 /2σ 2 X, a > 0 (2.1 In order to keep the mean path intensity unity, i.e. E[A 2 ] = 1, it can be shown that µ X = σx 2. For the log-normal distribution, we make use of a parameter called the scintillation index, defined as 13 S.I. = E[A4 ] E 2 [A 2 ] 1 (2.2 This quantity is proportional to the degree of fading, as seen in Figure 2.3, and can be related to the variance σ 2 X by S.I. = e4σ2 X 1. Typical values appearing in the literature for S.I. are in the range They Rayleigh distribution is used less often in the literature than log-normal fading to analyze FSO systems, but has some nice properties that make it an attractive model to use. First of all, the Rayleigh fading case exhibits deeper fading than log-normal fading because of the higher concentration of low-amplitude path amplitudes (see Figure 2.3. Second, with Rayleigh fading, the diversity order of the MIMO system becomes apparent when analyzing the slopes of the curves for symbol error probability. In Rayleigh fading, the amplitude of the path gain follows a Rayleigh distribution. The wavelength of the light is modeled to be large compared with the size of the scatterer, such that the composite field is produced by a large number of non-dominating 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: f A (a = 2ae a2, a > 0 (2.3 where we have normalized so that E[A 2 ] = 1. The random intensity I = A 2 is a one-sided exponential random variable, whose density function is heavily concentrated at low (deeply faded values. 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 Figure 2.3 [19].

30 Log normal fading, S.I. = 0.4 Log normal fading, S.I. = 0.6 Log normal fading, S.I. = 1.0 Rayleigh fading f A (a a Figure 2.3: Probability density functions for Rayleigh and log-normal distributions. We will make use of both of these fading models in Chapters 5 and 6 when we discuss the effect of fading on error probability and channel capacity. 2.6 The receiver Once the transmitted signal passes through the atmosphere, it must be collected and measured by the receiver. As we mentioned before, both coherent and noncoherent detection schemes are possible, but for complexity and cost reasons noncoherent (or direct detection is preferred. For this thesis, we will only consider noncoherent systems. In noncoherent optical signal detection, the detectors rely on the photoelectric effect incident photons are absorbed by the detector and free-carriers are generated and can be measured. This is a probabilistic phenomenon, since it is possible for a photon to pass through the photodetector without generating any free-carriers. However, in a well-designed photodetector, the probability of an incident photon causing a free-carrier is high [4]. There are two models that we will use in our analysis of the system. In the ideal photon counting model, we assume no thermal noise is present, and the system is capable of counting current blips that occur as each photoelectron is produced. Integrating the photocurrent over

31 15 a certain period of time (called a slot is equivalent to counting the current blips. In the Gaussian model, we assume that zero mean, additive white Gaussian noise (AWGN is added to the generated photocurrent. We still integrate over a slot, and the integration process should, on average, remove the noise power from the observable. Figure 2.4 shows what a FSO transceiver (receiver and transmitter in one unit might look like, showing other components also present in many FSO systems. Figure 2.4: A typical FSO transceiver [2]. According to Alexander, photodetectors fall into one of four categories: photomultipliers, photoconductors, photodiodes, and avalanche photodiodes [4]. There are numerous configurations and variations in these four categories, so we will concentrate on the two most popular detectors in optical receivers for communication, p-i-n photodiodes and avalanche photodiodes p-i-n photodiodes A p-i-n photodiode is made up of a p-type, and an n-type layer of semiconductor, separated by an intrinsic layer (hence the name p-i-n photodiode. The p-type layer is made to be very thin, so incident photons can pass directly through to the intrinsic region where they can generate electron-hole pairs. Any pairs that are generated are quickly swept into the p- and n-type layers where they contribute to the photocurrent.

32 16 The p-i-n photodiode has a quantum efficiency associated with it that depends on the reflectivity of the p-type layer, the absorption length of the intrinsic region, and the length of the depletion region [4]. The quantum efficiency is often denoted by η and is a measure of the average number of electron-hole pairs generated per incident photon. In a practical p-i-n photodiode, η ranges from 0.3 to 0.95 [20]. A simplified model of the p-i-n photodiode with its biasing voltage and integrator is shown in Figure 2.5 below. Figure 2.5: Simplified detector circuit employing a p-i-n photodiode Avalanche photodiodes An avalanche photodiode (APD is constructed similarly to a p-i-n photodiode. In some models, there is a second p-type layer between the intrinsic layer and the n-type layer (the layers are p-i-p-n. Incident photons still generate electron-hole pairs, but now there is an avalanche effect each free electron and/or hole has the potential to create more free electrons and/or holes as it traverses the gain region (the extra p-type layer and part of the n-type layer. Each newly created electron or hole can then repeat the process until all carriers have exited the gain region. This avalanche process creates multiple carriers for every incident photon. This increase in the number of carriers is known as the APD gain. It is the ratio of observable photocurrent at the APD terminals to the internal photocurrent before multiplication [4], and is a random variable with mean G. A simplified model of an APD is shown in Figure 2.6.

33 17 Figure 2.6: Simplified model of an APD and integrator Bandwidth and noise considerations in p-i-n and APD receivers Bandwidth An ideal photodetector would be noiseless and have an infinite bandwidth. An actual photodetector, however, has neither of these attributes due to the physical qualities of the photodetector itself and the accompanying electronics that take part in the detection process. A photodetector has a junction capacitance C d associated with it that is proportional to the aperture size. Increasing the capacitance of the detector also increases the time-constant which lowers the bandwidth. Therefore there is a tradeoff between increasing the field-of-view (FOV for a detector and the detector s bandwidth. The time-constant, and consequently the bandwidth is also dependent on the load resistor (R in Figures 2.5 and 2.6. Kedar and Arnon give an estimate for the data rate based on these parameters [21] R b 1 2πRC d (2.4 which is simply the inverse of the time constant converted to frequency in Hz. Clearly, increasing the capacitance (by increasing the aperture size or increasing the resistance decreases the data rate Shot noise There are also many sources of noise that must be considered when doing analysis on photodetection circuits. The first source we consider is optical shot noise, which occurs because

34 18 of the randomness of the creation of photoelectrons. We adopt the semi-classical view of photodetection, in which light arrives as a wave, and produces a stream of photoelectrons from the detector. The number of photoelectrons produced during a slot time can be described by a Poisson random variable with a mean and variance λ. This variance in the generation of photoelectrons can be seen as noise, which can affect the probability of error Background noise In a terrestrial free-space optical system, there is also a very good chance that background noise will enter the receiver along with the signal. Sources of background noise include the sun and artificial lighting, and can enter the receiver s aperture directly or by reflecting off of other surfaces. One way of minimizing background noise is by blocking other sources of radiation so that light can enter the receiver only from approximately the direction of the transmitter. Another method for minimizing background noise is to use a frequency selective filter in front of the receiver to only allow a narrow band around the center frequency of the laser. The potential drawback to this method is that it reduces the strength of the received signal Thermal noise Thermal noise is also called Johnson noise, and is a result of thermally induced random fluctuations in the charge carriers in a resistive element. Thermal noise is technically present in any semiconductor where thermally induced charge carriers can be present, which even includes the photodetector itself, but it is only significant in the load resistor (see Figures 2.5 and 2.6 whose resistance is higher than the other sources. Both the p-i-n photodiode and APD are affected by thermal noise, but it is more detrimental for the p-i-n photodiode. The APD has internal amplification that can be seen as a low-noise amplifier, whereas the p-i-n photodiode relies completely on the circuitry for amplification Excess APD noise The APD has the advantage of internal multiplication to raise the overall SNR, but it does so in spite of the excess noise factor F (M, caused by the random nature of the gain mechanism. F (M depends on the semiconductor material, the average gain of the APD, the ratio of the ionization coefficients for electrons and holes, and is largest in devices where both holes and

35 electrons produce ionizing collisions [4]. The SNR after the avalanche process for the APD is reduced by multiplying the SNR before amplification by F (M MPPM demodulator The MPPM demodulator has the task of taking the electronic signals delivered by the receiver and deciding which of the MPPM symbols was sent. This is a non-trivial task, and the maximum likelihood decision metric is investigated further in Chapter Source decoder, channel decoder, and retrieved information Once the MPPM demodulator has decided which MPPM symbol was sent, the source and channel decoders perform the inverse operations of the source and channel encoders. If the symbol was chosen correctly, the retrieved information matches the input information to the system. If the symbol was chosen incorrectly, the error may be detected and possibly even corrected (depending on the error and the coding scheme. It is also possible that the error could remain undetected and the retrieved information would not exactly match input information, i.e. bit errors would occur. These aspects are studied in detail in Section 5.

36 Chapter 3 MIMO applied to FSO systems using MPPM - background, system model and definitions In this chapter we look at the application of multiple-input, multiple-output (MIMO techniques to the FSO system using multiple pulse position modulation (MPPM. 3.1 Research on MIMO and MPPM FSO communications systems MIMO in wireless systems Multiple Input Multiple Output (MIMO systems have recently emerged as one of the most significant breakthroughs in modern communications. The idea behind MIMO systems can be explained quite simply. At both the transmitter and receiver end, the system employs multiple antennas. The effect of MIMO approaches is that the signals at the transmitter and receiver can be combined such that the bit error rate or the data rate (in bits/sec is improved. In the wireless RF domain this is done at no extra cost of spectrum only added hardware and complexity [22]. The MIMO concept is depicted in Figure 3.1. The advantage from the MIMO setup can be utilized through two different concepts, spatial multiplexing and spatial diversity. 20

37 21 Figure 3.1: The MIMO concept. In spatial multiplexing, such as with the BLAST technique, the incoming high-rate data is decomposed into M independent data streams, and sent to all M antennas to be transmitted simultaneously over the channel. The receiver array, having learned the mixing channel matrix through training sequences, can identify each of the individual data streams and recombine them to retrieve the original message. The result is that the spectral efficiency improves; the transmitter array can send at a new data rate M times faster than with a single antenna [22]. The second beneficial concept is spatial diversity. In MIMO systems, the M N path gains from each transmitter to each receiver can be described in a M N matrix form (see Figure 3.1. If the antennas are situated far enough apart, the paths can be considered decorrelated and the effects of random fading caused by multipath or other phenomena can be mitigated. The improvement of a MIMO system is directly related to the number of decorrelated antenna elements, also known as the diversity order, whose maximum is MN [22]. There has been much research into the performance advantages of MIMO systems. The consensus is that MIMO design can increase the capacity and decrease the bit error rate over a single input, single output (SISO, MISO, or SIMO system with a given power and bandwidth. The results of this research have focused heavily on RF systems, and the reader is directed to [11] or [22] (among many possibilites for more detail.

38 Application of MIMO concepts to free-space optics Free-space optical communications systems can also benefit from spatial diversity, as has been shown in [15, 23 25]. Although a well-designed FSO link will not suffer from traditional multipath effects (except for diffuse FSO systems, atmospheric fading (the most serious problem facing FSO systems can be mitigated through MIMO design. The key is to place the lasers and photodetectors sufficiently far from one another to ensure with a high degree of probability that each of the MN path gains are independent. The distance that each laser or photodetector should be placed away from one another can be calculated from the correlation distance d 0 (see Section This is the distance for which two points in the atmosphere are uncorrelated and can be approximated by d 0 λl, where λ is the wavelength of the transmitted wave, and L is the length of the FSO link [13]. As an example, if the length of the FSO link is 1 km and the wavelength is 1550 nm, the correlation distance would be approximately 4 cm. Therefore, keeping the lasers and photodetectors separated by at least 4 cm would ensure with a high degree of probability that the MN path gains are independent. This small of a separation distance is perhaps surprising but it is small enough for MIMO to be considered for terrestrial FSO links, where the laser and photodetector arrays would be placed on rooftops or even behind windows. The result of keeping the M N path gains independent is seen by considering the probability that a path gain a nm is sufficiently small, such that the signal falls behind the background level. If we call the probability of this event β, then the probability of a deep fade detrimentally affecting all of the paths for one realization of the path gain matrix A is β MN. Therefore, the system has the potential to achieve the diversity order MN Multiple pulse position modulation (MPPM The capacity and performance of PPM and MPPM has been studied in detail in [26 33]. We will first investigate the attributes of PPM signaling, and then transition to MPPM. In pulse position modulation (PPM, an observable time period (the duration of a modulator symbol, T is divided up into slots each having a duration of T s. A symbol is represented by sending a pulse in only one of the time slots. If there are Q slots in a symbol, then there are consequently Q possible symbols, each representing up to log 2 Q bits of information. PPM is a

39 23 Q-ary orthogonal signaling scheme. To increase the throughput of a PPM system, it is necessary to increase Q, which decreases the pulsewidth [28]. This is an attractive solution for a number of reasons. First, decreasing the width of the slot also decreases the number of background photons that will be received [29], since we are assuming the background count rate remains the same. Second, the probability of symbol error for noncoherent detection of M-ary orthogonal signals decreases for an increasing number of symbols for a fixed energy per bit, E b [10]. Decreasing the slot width, however, has its limitations. Namely, an increase in the required bandwidth, implying more thermal noise. For high data rate applications, MPPM is a more attractive alternative [34]. Multipulse transmission works with the same concept as PPM, but instead of having only one on slot, there are w of Q slots that can be on for each modulator symbol, giving ( Q w possible symbols. The bandwidth efficiency, defined as the number of bits that can be transmitted per is superior in MPPM, and as we will see later, the MPPM system has an improved performance when the system is peak-power-limited. This was studied in detail by Atkin and Fung in [33]. In their analysis, they compared different schemes with similar bandwidths, and found that MPPM can outperform standard PPM in coded and uncoded systems. Our analysis differs from theirs in a few ways. We allow all ( Q w symbols to remain in the set, whereas they would limit the symbol set size to a power of two, which is a logical limitation to place on the modulation scheme. The result is that their error analysis was limited to an upper bound on error probability. By allowing all ( Q w symbols to remain in the set, we preserve symmetry in the problem, and can often obtain closed form expressions for error probability. We also consider MIMO techniques overlayed with MPPM, whereas one of their focuses was on Reed-Solomon coding of a MPPM system Error probability - Gaussian vs. Poisson In optical detection the Gaussian approximation is often used to analyze systems that employ APD s as detectors [35, 36]. The properties of APD s and the accuracy of the Gaussian approximation were established in the early 1970 s in work by McIntyre, Conradi, and Webb [37 39], and the approximation is used to take many factors specific to APD s into consideration. However, we are concerned with detection using p-i-n photodiodes, and details of the operation of APD s is beyond the scope of this thesis.

40 24 Instead, when we speak of the Gaussian approximation, we are either interested in approximating the Poisson point process at the output of the photodetector when the signal and background power levels are both large, or when additive white Gaussian noise (thermal noise is introduced by the receiver. 3.2 Power comparisons between PPM and MPPM Care must be taken to fairly compare system performance as w varies. The reason for this is that increasing w with a constant peak signaling power (the on power during a slot increases the average power consumption for the system. Also, increasing w while keeping the symbol duration T s constant would increase the bit rate. To address the first concern, we will need to make a distinction between peak-power-limited systems, and average-power-limited systems. In a peak-power-limited system, the signal power in an on slot is limited to P peak, regardless of how many on slots there are. This means the total received optical energy per symbol is equal to the peak power multiplied by the duration of the on slots. E s = P peak T w (3.1 In an average-power-limited system, the average power in all Q of the slots (both on and off observed over the duration of the symbol must remain constant, and the optical energy per symbol is equal to E s = P ave T Q (3.2 Therefore, combining (3.1 and (3.2, we can state that the relationship between average power and peak power is P ave = P peak w Q P ave is easily recognized as P peak times the duty cycle of a symbol. (3.3 To consider the second concern, we observe that the bit rate of the system is related to w. The bit time T b multiplied by the number of bits per modulator symbol is equal to the symbol time, T s = T b log 2 ( Q w (3.4

41 25 and the slot time is therefore T = T ( b log Q 2 w (3.5 Q We can address both of these concerns by defining a general optical energy parameter P T with which to compare systems (used in plots in Chapters 5 and 6, where P is the signaling power and T is the slot duration. Peak and average power are handled by the following conversions: ( T b log Q ( 2 T w b log Q 2 w P T = P ave or P T = P peak (3.6 w Q For a better understanding of the effect of average or peak power limitations on system performance, we can define a relative energy efficiency for equal asymptotic performance, based on (3.6 as the multiplier on P T b : ρ ave = log ( Q 2 w w bits/slot (3.7 ρ peak = log ( Q 2 w bits/slot (3.8 Q which shows that for a given (Q, w pair, multipulse is more efficient than single pulse in the peak-power-limited system. In fact, for the peak-power-limited system, the relative energy efficiency is at a maximum when w = Q/2. The efficiencies as a function of Q are shown below in Figure 3.2. It is important to be careful in interpreting this plot. The first thing that the plots reveal is that for a given P T b, the average-power-limited system will always outperform the peak-powerlimited system. The interpretation of this goes back to (3.3. Notice that if we are given P, and interpret it as average power, the corresponding peak would be Q/w times greater than if P were interpreted as peak power. Also interesting to note is that for the peak-power-limited system, the MPPM system has a superior relative energy efficiency, and in an average-power-limited system, standard PPM is superior. It is also interesting to note from this plot that the efficiency is unbounded for the averagepower-limited system where w = 1 (unlike all of the other curves. From a logical standpoint, this makes sense; as Q increases, the efficiency increases at a rate of log 2 Q. However, what

42 (Q,w = (8,1 ρ ave, w=1 ρ ave, w= Q/2 ρ peak, w= Q/2 ρ peak, w=1 Relative Energy Efficiency (Q,w = (8, Q Figure 3.2: Relative energy efficiencies vs. Q for w {1, Q/2 }. is not seen from this plot is that for the average power to remain constant as Q becomes large, the peak power must also become large. Another way to think of this is that all of the power is concentrated in a single slot the on slot is constantly becoming narrower as Q grows, which squeezes the peak power (as well as the bandwidth up toward some large value. Because of power and bandwidth limitations in any real system, the advantages of this phenomenon become impractical to implement for large values of Q. The zig-zag trajectories are due to the fact that w = Q/2 only changes at even values of Q. Another important advantage of the MPPM system is its spectral efficiency. We can define this to be ψ, which is measured in bps per Hertz. If we state that the bandwidth is inversely proportional to the slot duration T, ψ is calculated as ψ = R b 1/T ( T b log Q 2 w = R b Q ( log Q 2 w = R b R b Q = log ( Q 2 w Q (3.9 (3.10 (3.11 (3.12

43 27 where R b is the bit rate of the system. Since ( Q w is maximized when w = Q/2, we can state that the spectral efficiency of MPPM is also maximized for w = Q/2. As Q becomes large, the spectral efficiency of MPPM approaches one for the w = Q/2 system, and approaches zero for the w = 1 system. This can also be seen in Figure 3.2, since the spectral efficiency ψ is equivalent to the relative energy efficiency for the peak-power-limited case, ρ peak. Although bandwidth conservation is not of particular interest in FSO communications, since it operates on unregulated spectrum, the spectral efficiency is more of a measure of the necessary electronics speed to operate at a given bit rate. Alternatively, the spectral efficiency shows us that for a given bandwidth (or minimum slot duration allowed by the electronics, the MPPM system will allow the system designer to transmit at a higher bit rate. This is one of the most attractive features of MPPM for high data rate communications. 3.3 System model MPPM signaling We send binary information using multipulse pulse-position modulation, where each modulator symbol is of duration T s seconds and is comprised of Q slots, each T seconds long. The symbols are created by turning the laser on for w of the Q slots. This results in ( Q w different symbols which we assume are equiprobable. Each symbol can represent log 2 ( Q w bits of information. For simplification of the analysis, we ignore the fact that ( Q w may not be a power of two. We will often revert to the notation that an on slot is denoted by a one and an off slot by a zero. For example, a Q = 8, w = 4 symbol, where the first w slots are on can be denoted by [ ]. We also make use of the notation that Q i on denotes the set of on slots for modulator symbol i, and Q i off is the set of off slots. In our example above, for q Q i on, q = {0, 1, 2, 3} and for q Q i off, q = {4, 5, 6, 7}.

44 Transmitter array For our system, the transmitters are modeled to have infinite bandwidth, so it is assumed possible to have the laser on at some constant power for the duration of a slot. Each laser is also completely off (full extinction for the entire duration of any off slot. The number of lasers is denoted by M. In order to compare systems differing in the number of lasers, we divide the transmitter power at each laser by M so the power delivered by the entire transmitter array is constant as we vary M. We also assume that the lasers are noncoherent, without any special precautions. The wavelength we have chosen in simulations for the system is 1550 nm, which corresponds to a frequency of Hz Receiver array Each of the N receivers is assumed to be perfect-photon-counting devices, also with infinite bandwidth. The arrival of the signal and background photons at the receiver is modeled as a Poisson process, where the number of photons arriving at detector n during a single slot time has a mean and variance of λ on,n for on slots and λ off for off slots. We also assume perfect synchronization, so signal photons are only received during on slots, and off slots can only contain background photons Channel The channel was described in detail in Section 2.5. We assume that the M transmitters and N receivers are placed sufficiently far from one another such that each of the individual paths from transmitter to receiver is independent. The intensity gain along each path from transmitter m to receiver n is denoted by a 2 nm, and it is a random variable following the distributions described in Section We denote a single realization of all of the amplitude path gains as a M N matrix, A. The signal received at detector n is a composite of the signals received from all of the M transmitters simultaneously. Therefore, at each detector n, the signal power received is proportional to m a2 nm, assuming all transmitters send with the same power.

45 Detector and observable Poisson Regime We first focus on the Poisson regime, which is applicable when the background count rate is low and the variance in the observable at detector n for slot q is Z nq due to thermal noise in the amplifier is small. Each of the N receivers is modeled according to Figure 2.5, and consists of a photodetector and integrator. The photodetector is analyzed using the semi-classical view of photodetection. The optical wave is received, and the output is a flow of photoelectrons that obey Poisson statistics over any slot interval. The observable at detector n and slot q is depicted as Z nq, and has a mean on slot count λ on,n, and a mean off slot count λ o ff. These are given by and λ on,n = ηp T hfm λ off = ηp bt hf m a 2 nm + ηp bt hf (3.13, n = 1,..., N (3.14 respectively, where P b is the power received due to background radiation, and η is an efficiency factor for the detector, defined as the ratio of generated photoelectrons to incident photons Gaussian Regime For this case, it is convenient to model the noise using a Norton equivalent circuit, as in Figure 3.3. Figure 3.3: Norton equivalent noise model.

46 In the figure, I is the mean current, equal to the average number of photoelectrons created by the photodetector times the charge associated with each photoelectron, q, giving ( I = ( ηp hfm ηp b hf m a2 nm + ηp b hf q, signal present; q, no signal present. 30 (3.15 The voltage across the (noiseless resistor R is equal to IR. Since the mean values of the optical shot noise i s (t and thermal noise i t (t are equal to zero, and the mean of an on or off slot is assumed to be constant over the slot time T, the output of the integrator Z has a mean equal to E[Z] = ( ( ηp T hfm ηp bt hf m a2 nm + ηp bt hf Rq, Rq, signal present; no signal present. (3.16 The generation of photoelectrons is a Poisson point process, so the variance in the photoelectron count during an on or off slot is equal to (3.13 and (3.14. The voltage at the input to the integrator caused by this optical shot noise current is this value multiplied by a constant qr. Since a random variable X with a variance of σ 2 x times a constant k has a new variance of k 2 σ 2 x, we can state that the variance of the output due to optical shot noise is equal to V ar[z] shot = ( ( ηp T hfm ηp bt hf m a2 nm + ηp bt hf R 2 q 2, R 2 q 2, signal present; no signal present. (3.17 The variance in the thermal noise current i t (t can be described by its spectral noise density S t (f = 2kT 0 /R, where T 0 is the absolute temperature of the resistor R [40]. Multiplying this current by the resistance again means multiplying its variance by R 2. Taking the integral over T seconds gives V ar[z] thermal = 2kT 0 T R (3.18 Since the thermal and shot noise processes are independent, we can add them together to get the variance at the output of the integrator. Putting everything together, we have that the output of the integrator is a random variable Z described by E[Z] = ( ( ηp T hfm ηp bt hf m a2 nm + ηp bt hf Rq, Rq, signal present; no signal present. (3.19

47 31 and V ar[z] = ( ( ηp T hfm ηp bt hf m a2 nm + ηp bt hf R 2 q 2 + 2kT 0 T R, R 2 q 2 + 2kT 0 T R, signal present; no signal present. (3.20 At this point, we may choose to use (3.19 and (3.20 in a Gaussian approximation to model Z. However, we must be careful to determine when this approximation is valid. The thermal noise is modeled as a Gaussian random variable. Therefore, if its variance is much larger than the optical shot noise variance, the first term may be neglected in (3.20, and Z may be accurately approximated by a Gaussian random variable. Also, if the level of background power P b is high enough, the p.d.f. for the count of photoelectrons, which is Poisson in nature, becomes increasingly Gaussian in shape (see Figure 3.4. When this is added to the Gaussian noise, the resulting p.d.f. may also be modeled as Gaussian. This fact is also true of high signal power P, but high background power P b is a sufficient condition for the Gaussian approximation to hold, and necessary when thermal noise is low. Figure 3.4: Poisson and Gaussian p.d.f. s with equal means and variances of λ = 200. In the case where the background power P b and thermal noise power levels are both low, the Gaussian approximation ceases to yield accurate results. The Poisson distribution would

48 32 be poorly approximated by a Gaussian distribution (see Figure 3.5. As a result, we must let the model for the distribution of the off slots remain as Poisson. Thus, the convolution (due to adding independent random variables of the Gaussian and Poisson p.d.f. s would no longer have a Gaussian shape. Instead, for low mean count rates, the convolution of the two would create a p.d.f. having many non-overlapping Gaussian shapes, each centered and weighted at the different integer values given by the Poisson p.d.f., which would be poorly approximated by a Gaussian distribution. Figure 3.5: Poisson and Gaussian p.d.f. s with equal means and variances of λ = 2.

49 Chapter 4 Maximum likelihood (ML detection In this section, the general ML detection rules are derived for MIMO FSO detection in both the Poisson and Gaussian regimes. 4.1 ML Detection for the poisson regime Let Z nq be the photoelectron counts for detector n and slot q, and Z = {Z nq, n = 1,..., N, q = 0,..., Q 1} represent the set of received observations. If we send one of ( Q w binary patterns represented by X i, then the ML decision is ˆX = arg max X i f(z X i (4.1 Using the definitions of λ on,n and λ off from Section 3.3.5, and recognizing that Z nq at each of the N detectors and Q slots is independent, the conditional distribution of the N Q random matrix Z can be written as a N Q-fold product over all of the individual elements in Z. These elements, which are represented as Z nq, are conditioned on whether they are in an on slot, or an off slot. ˆX = arg max X i n q Q (i on exp( λ on,n (λ on,n Z nq Z nq! q Q (i off exp( λ off (λ off Z nq Z nq! (4.2 Since the Z nq! terms in the denominator are invariant to X i, they can be removed without affecting the outcome of the ML decision. Thus 33

50 34 ˆX = arg max X i n q Q (i on exp( λ on,n (λ on,n Z nq q Q (i off exp( λ off (λ off Z nq (4.3 If we assume that the number of on slots for all modulator symbols is equal to w, the exponential terms are equal to exp ( wλ on,n and exp ( (Q wλ b for all X i and can also be eliminated, leaving ˆX = arg max X i n q Q (i on (λ on,n Z nq q Q (i off (λ off Z nq (4.4 Next, we take the logarithm of the entire quantity to find the log-likelihood function. ˆX = arg max X i which can be rewritten as slots. ˆX = arg max X i = arg max X i = arg max X i n ln((λ on,n Z nq + n q Q (i on q Q (i on Z nq ln(λ on,n + n q Q (i off q Q (i off ln((λ off Znq (4.5 Z nq ln(λ off (4.6 Z nq ln(λ off Z nq ln(λ off (4.7 Z nq ln(λ on,n + q Q (i all q on q Q (i on ( λon,n Z nq ln (4.8 n q Q (i on λ off Therefore, the ML detector would make a decision based on a weighted sum over the on What will be interesting to note in the following subsections is that of the four cases we will consider in the Poisson regime, the optimal detector simplifies to finding the w largest column sums with equal gain combining for three of them. The exception to this rule is the case where background radiation and fading are both present, and the optimal detector searches for a weighted sum over the on and off slots Case 1: ML detection with no background and no fading For this case, we start with (4.6, and express λ on,n and λ off explicitly:

51 35 ˆX = arg max X i = arg max X i [ Z nq ln(λ on,n + n q Q (i on [ n q Q (i on ( ηp T Z nq ln hfm q Q (i off m Z nq ln(λ off a 2 nm + ηp bt hf q Q (i off ] + (4.9 ( ] ηpb T Z nq ln (4.10 hf For the cases where no background radiation is present, P b = 0 and Z nq = 0 for all off slots. Since (4.10 can be restated as lim x log(x = 0 (4.11 x 0 + ˆX = arg max X i n q Q (i on Z nq ln ( ηp T hfm m a 2 nm (4.12 ln In the no fading case, all of the fading variables A are equal to one, meaning that the term is constant for all X i leading to ( ηp T hfm m a2 nm ˆX = arg max X i = arg max X i n q Q (i on q Q (i on If we define n Z nq = S q, which we will refer to as a column sum, Z nq (4.13 Z nq (4.14 n ˆX = arg max X i q Q (i on S q (4.15 which shows that the optimal detector simply searches for the w largest column sums. In the case where ties occur and there are more than w column sums that could provide an optimal solution, the detector must resort to making a random choice among all of the equally likely codewords.

52 Case 2: ML detection with no background and fading With no background and fading, we know that any slot that has a non-zero count must be an on slot, and any slot that has a zero count may be an off slot. Intuitively, we can easily state that if w slots have non-zero counts at one or more detectors, there is no ambiguity in the transmitted symbol and the system will err with zero probability. Therefore, no outside information is required to make the maximum likelihood decision. Any scheme involving weighting each column sum differently based on the current realization of A cannot assist in the outcome of the decision, so the optimal detector simply searches for the w largest (or simply non-zero column sums Case 3: ML detection with background radiation and no fading In this case, we again start with ˆX = arg max X i [ Z nq ln n q Q (i on ( ηp T hfm a 2 nm + ηp bt + hf m ( ] ηp b T Z nq ln hf q Q (i off (4.16 Here, the m a2 nm term is equal to M, and bias terms can be eliminated, yielding ˆX = arg max X i [ Z nq ln (P + P b + n q Q (i on q Q (i off Z nq ln P b ] (4.17 Here, we can make the observation that logarithms are monotonic increasing functions, and P and P b are both positive numbers or zero. Therefore we can state that ln(p + P b ln P b for all P and P b. If we make a substitution and let τ = ln(p b and τ + = ln(p + P b (so that 0 is the difference between the two logarithmic terms, we can rewrite (4.17 as ˆX = arg max X i = arg max X i [ (τ + Z nq + τ n [ n q Q (i on q Q (i on Z nq + τ all q q Q (i off Z nq ] Z nq ] (4.18 (4.19

53 Since the τ all q Z nq term is independent of X i, it can be eliminated. is also a scale factor which is independent of X i, leaving the following ML decision rule: 37 sums. ˆX = arg max X i = arg max X i = arg max X i n q Q (i on q Q (i on q Q (i on Z nq (4.20 Z nq (4.21 n S q (4.22 This once again shows that the optimal detector simply searches for the w largest column Case 4: Background radiation and fading When background radiation and fading are both present, we again start with ˆX = arg max X i [ Z nq ln n q Q (i on ( ηp T hfm a 2 nm + ηp bt + hf m ( ] ηp b T Z nq ln hf q Q (i off (4.23 Eliminating bias terms leaves ˆX = arg max X i [ ( Z nq ln n q Q (i on P M a 2 nm + P b + m q Q (i off Z nq ln P b ] (4.24 If we again employ the same method applied to the previous case, we can state that ln [(P/M m a2 nm + P b ] ln P b for all P, P b, M, and a nm. Therefore, we can make the following substitutions: Let τ = ln P b and τ + δ n = ln [(P/M m a2 nm + P b ], such that δ n 0 is the difference between the two logarithmic terms, and is dependent on n through the fading distribution A. We can then rewrite the ML decision rule as

54 38 ˆX = arg max X i = arg max X i = arg max X i [ (τ + δ n Z nq + τ n n q Q (i on [ (τ + δ n Z nq + τ [ n δ n q Q (i on q Q (i on Z nq + τ all q q Q (i off q Q (i off Z nq ] Z nq ] Z nq ] (4.25 (4.26 (4.27 The last summation term is independent of X i and can be eliminated, which leaves ˆX = arg max X i [ n δ n q Q (i on Z nq ] (4.28 where δ n = ln [(P/M m a2 nm + P b ] ln P b. This shows that the optimal detector will perform a weighted sum of the on slots. A reasonable approximation to this detector is to perform equal gain combining and search for the w largest column sums, treating this case like the other 3 cases. This is only slightly suboptimal, as was shown in [19], and is the method that was chosen for the analysis in Chapters 5 and General ML detection in the Gaussian regime Assuming the necessary conditions are met for the Gaussian approximation to be justified (see Section , we can start to develop the general ML detector in the Gaussian regime. For now, we will not assume that thermal noise or shot noise is dominant, and we will assume the background and signal power levels are such that the Gaussian approximation is justified. This will allow us to develop a more general ML detector. More specific cases will be considered in the following subsections. Based on the analysis in Section we can state that the observable Z is a Gaussian random variable with parameters µ on,n = ( ηp T hfm m a 2 nm + ηp bt hf Rq (4.29

55 ( ηpb T µ off = hf 39 Rq (4.30 σ 2 on,n = (µ on,n Rq + 2kT 0 T R (4.31 σ 2 off = (µ off Rq + 2kT 0 T R (4.32 Similar to the development from the Poisson regime, we can claim that N Q elements in Z are independent, and therefore the distribution of Z is simply a N Q-fold product over all detectors and slots of the (Gaussian distribution of each Z nq, which are conditioned on q Q on or q Q off : ˆX = arg max X i N n=1 q Q (i On ( 1 e (Znq µon,n 2 2σon,n 2 2πσ 2 on,n q Q (i Off 1 2πσoff 2 Eliminating the 1/ 2π scale factors and taking the logarithm gives e (Znq µ off 2 2σ off 2 (4.33 ˆX = arg max X i = arg max X i [ N n=1 q Q (i On [ N n=1 q Q (i On ( (Z nq µ on,n 2 2σ 2 on,n q Q (i Off ln(σ on,n + ( (Z nq µ off 2 ln(σ off ] (4.34 2σ 2 off ( ( Z 2 nq + 2µ on,n Z nq µ 2 on,n ln(σ on,n + q Q (i Off 2σ 2 on,n ( ( Z 2 nq + 2µ off Z nq µ 2 off ln(σ off ] (4.35 2σ 2 off The µ 2 on,n, µ 2 off, ln(σ on,n, and ln(σ off terms are constant for all X i, so

56 40 ˆX = arg max X i = arg max X i = arg max X i = arg max X i N n=1 q Q (i On [ N n=1 q Q (i On [ N n=1 q Q (i On [ N n=1 q Q (i On ( 2µon,n Z nq Znq 2 + 2σ 2 on,n q Q (i Off ( 2µon,n Z nq Znq 2 + 2σ 2 on,n ( 2µon,n Z nq Z 2 nq 2σ 2 on,n q Q (i All q Q (i On 2µ offz nq Z 2 nq 2σ 2 off { (µon,nσ 2 off µ off σ 2 on,n Znq + ( 2µoff Z nq Z 2 nq 2σ 2 off ( 2µoff Z nq Z 2 nq 2σ 2 off ( 2µoff Z nq Z 2 nq 2σ 2 off ] ( σ 2 on,n σ 2 off 2 (4.36 ] }] Znq 2 We will use this as the starting point for the different cases in the Gaussian regime. (4.37 (4.38 ( Thermal noise dominates over shot noise When the thermal noise is the dominant noise in the system, we can neglect the variance due to the shot noise, and state that the slot count at each receiver, Z nq follows a Gaussian distribution with mean and variance of ( ηp T µ on,n = a 2 nm + ηp bt Rq, signal present (4.40 hfm hf m ( ηpb T µ off = Rq, no signal present (4.41 hf σ 2 = 2kT 0 T R, either case (4.42 Therefore, to determine the ML decision in the Gaussian regime, we start with (4.39, but let the variance terms be dominated by 2kT 0 T R. This results in ˆX = arg max X i N (µ on,n µ off Z nq (4.43 n=1

57 After making the appropriate substitutions, eliminating the scale factors gives the ML detector as No fading present ˆX = arg max X i N n=1 q Q (i On ( m a 2 nm Z nq (4.44 When no fading is present (and regardless of the presence of background radiation, m a2 nm = M. This becomes a scale factor that can be eliminated, giving ˆX = arg max X i = arg max X i = arg max X i n q Q (i on q Q (i on q Q (i on Z nq (4.45 Z nq (4.46 n S q (4.47 which shows that the optimal detector searches for the largest column sums. This is an identical result to the Poisson regime when no fading was present, where the optimal detector also searched for the w largest column sums Fading present With fading present (and regardless of the presence of background radiation, (4.44 is irreducible and gives the optimal detector, which searches for the largest weighted column sums, dependent on the fading distribution A Shot noise dominates over thermal noise For this case, we will start with (4.39, and simply plug in the mean and variance expressions given by µ on,n = ( ηp T hfm m a 2 nm + ηp bt hf Rq (4.48

58 ( ηpb T µ off = hf 42 Rq (4.49 σ 2 on,n = (µ on,n Rq (4.50 σ 2 off = (µ off Rq ( Fading present After plugging in the appropriate means and variances into (4.39 and eliminating the scale factors, the ML detector is ˆX = arg max X i N n=1 q Q (i On ( m a 2 nm Z 2 nq (4.52 This differs from the Poisson regime, where the optimal detector looks for weighted column sums. Here, the optimal detector will square the observable at each detector before multiplying by the weighting coefficient and summing over detectors and on slots No fading present With no fading present, m a2 nm = M, and just as when thermal noise was dominant, the optimal detector becomes ˆX = arg max X i = arg max X i n q Q (i on q Q (i on Z 2 nq (4.53 Znq 2 (4.54 This also differs from the Poisson regime since the optimal detector will square the observable at each detector, and then look for the w largest column sums (as opposed to simply looking for the w largest column sums in the Poisson regime. n Shot and thermal noise are not dominant In this case, we use the means and variances defined earlier as

59 ( ηp T µ on,n = a 2 nm + ηp bt Rq (4.55 hfm hf m ( ηpb T µ off = Rq (4.56 hf σ 2 on,n = (µ on,n Rq + 2kT 0 T R (4.57 σ 2 off = (µ off Rq + 2kT 0 T R ( Fading Present Plugging these values into (4.39 and eliminating scale factors terms gives us ˆX = arg max X i { { anm} 2 (2kT 0 T Z nq + n q Q (i on m ( q 2 Z 2 nq} (4.59 This result is similar to other fading results in the Gaussian and Poisson regimes, since it shows the optimal detector to be a weighted sum based on the fading distribution, A. In most instances, one would expect (q/2z 2 nq to negligible, in which case 2kT 0 T becomes a scale factor. In these instances, the optimal detector is identical to the optimal detector for the other fading cases in the Gaussian regime No fading present When no fading is present, (4.60 becomes ˆX = arg max X i n q Q (i on { ( q (2kT 0 T Z nq + Z 2 nq} 2 (4.60 Again, when (q/2z 2 nq is negligible, 2kT 0 T becomes a scale factor and the optimal detector is simply searching for the w largest column sums.

60 Chapter 5 Error analysis of MIMO FSO system using MPPM In this chapter, we will perform symbol error probability analysis of the MIMO MPPM FSO system in both the Poisson and Gaussian regimes. For simplicity of the analysis, we will assume equal gain combining for all cases. This is optimal in three of the four cases under consideration, and was shown to be a very good approximation for the Poisson regime in the case where background radiation and atmospheric fading are both present [19]. We also show this to be a very good approximation in the Gaussian regime in Section Error analysis in the Poisson regime For the Poisson regime, we consider a number of possible scenarios that are combinations of the following: no fading, Rayleigh fading, log-normal fading, background radiation, and no background radiation No background radiation For the case of no background radiation, the w column sums with the largest counts will be the w on slots in the transmitted symbol X j unless i > 0 of them are zero (due to quantum effects in the photodetector. With i of the column sums equaling zero, the detector may have partial information (or no 44

61 information if i = w, with which it can make a guess at the transmitted symbol. The receiver must decide among ( Q w+i i equally probable symbols. This quantity can be interpreted by noting that this is the number of ways the receiver can place the i non-received signal pulses in Q w + i vacant slots. The detector errs with probability 45 t(q, w, i = ( Q w+i i ( Q w+i i 1 (5.1 As described in Section 3.3.5, for on slots, we can model the photoelectron slot count Z nq at each of the N receivers as a Poisson process with a Poisson parameter of λ on,n and λ off for on and off slots. With equal gain combining, the sum over all receivers for each on slot is also a Poisson random variable (conditional on A with a parameter of λ on = ηp T hfm N n=1 m=1 M a 2 nm (5.2 If we define S q = N n=1 Z nq, an on slot will have zero photoelectrons with probability P (S q = 0 = p = e λon (5.3 Treating each column count as an independent random variable, we get P [i of w columns = 0] = w i=1 ( w p i (1 p w i (5.4 i Therefore, we can state for the no background case, that the error probability conditioned on the fading variables A is P s A = w ( w t(q, w, i p i (1 p w i (5.5 i i=1 We can take the (1 p w i term and rewrite it using the binomial expansion: This gives the probability expression w i ( w i (1 p w i = ( 1 l p l (5.6 l l=0 P s A = w ( w t(q, w, i i i=1 p i w i l=0 ( w i ( 1 l p l (5.7 l

62 46 which is easily rewritten P s A = w w i ( Q w+i ( 1 l i 1 i=1 l=0 ( Q w+i i ( w i ( w i l p (i+l (5.8 This is the probability of symbol error, conditioned on no background radiation and the fading variables A. In each of the next three subsections we will use different distributions of A, and show how the system performs No background, no fading In the non-fading case, a 2 nm = 1 for all n and m. Plugging this value into (5.2, (5.3, and (5.8 gives λ on = ηp T N hf (5.9 P (S q = 0 = p = e ηp T N hf (5.10 and respectively. P s no fading = w w i ( Q w+i ( 1 l i 1 i=1 l=0 ( Q w+i i ( w i ( w i l e ηp T N(i+l hf (5.11 Observe that the average number of photoelectron counts and the probability of error are independent of the number of lasers M in the non-fading case. This is because the M path gains from each laser to each detector are unity, and we kept the total transmit power constant by dividing the transmit power at each laser by M. However, performance does depend on the number of receivers because the total aperture size increases by a factor of N. For consistency, we could have removed that dependency on receivers by also decreasing the total aperture size by N (as done by Shin and Chan in [15], but in a realistic setting increasing the number of receivers will also increase the aperture size, so our choice is justified. As can be seen by Figures 5.1 and 5.2, no change in performance is observable for varying M. However, a 6 db gain in performance is seen in the cases where N is equal to four. This can be attributed to the exponent of p being increased by a factor of four.

63 47 It is also important to note that multipulse PPM outperforms standard PPM only in a peakpower-limited system. This makes intuitive sense, since average-power-limited systems would require that the total signal ( on power be distributed among the w on slots in the averagepower-limited system, as w increases, each on slot has a lower peak power, which increases the probability that a zero is observed at the detector during an on slot M=1,N=1,w=1 M=4,N=1,w=1 M=1,N=4,w=1 M=4,N=4,w=1 M=1,N=1,w=4 M=4,N=1,w=4 M=1,N=4,w=4 M=4,N=4,w= P s PT b,dbj Figure 5.1: Symbol error probability vs. average power with no fading and no background radiation, and Q = No background, Rayleigh fading For the Rayleigh fading case, we refer back to Section 2.5.3, which discusses different fading models. (2.3 gives the p.d.f. for Rayleigh fading, and is repeated here for convenience: f A (a nm = 2a nm e a2 nm, anm > 0 (5.12 We assume that each realization of a channel fading gain a nm follows this distribution, and therefore the expected error probability can be found by averaging (5.8 with respect to the Rayleigh distribution for all MN channel paths.

64 M=1,N=1,w=1 M=4,N=1,w=1 M=1,N=4,w=1 M=4,N=4,w=1 M=1,N=1,w=4 M=4,N=1,w=4 M=1,N=4,w=4 M=4,N=4,w= P s PT,dBJ b Figure 5.2: Symbol error probability vs. peak power with no fading and no background radiation, and Q = 8. P s Rayleigh fading = = N M n=1 m=1 w w i i=1 l=0 0 w w i i=1 ( Q w+i i ( Q w+i i M l=0 ( Q w+i i ( Q w+i i 1 ( w i ( w i l ( 1 l 2a nm e [ηp T a2 nm (i+l]/hfmq da nm (5.13 ( ( 1 w w i ( 1 l i l N m=1 n=1 0 2a nm e [ηp T a2 nm(i+l]/hfmq da nm (5.14 This is a NM-fold integration, but each term is identical so it can be rewritten P s Rayleigh fading = w w i i=1 l=0 ( Q w+i i ( Q w+i i ( ( 1 w w i ( 1 l i l [ MN 2ae da] [ηp T a2 (i+l]/hfmq (5.15 and applying the solution to this integral gives the final closed-form result 0 P s Rayleigh fading = w w i i=1 l=0 ( Q w+i i ( Q w+i i 1 ( w i ( [ w i ( 1 l l ηp T (i+l hfmq ] MN (5.16

65 49 This is plotted in Figures 5.3 and 5.4. Since the error probability drops by a factor of 10 MN for every 10 db increase in signal power, we claim that the system achieves a diversity equal to MN. This shows a performance advantage to increasing the number of lasers as well as the number of receivers. However, comparing the M = 1, N = 4 to the M = 4, N = 1 curves for constant w shows that for a given diversity, increasing the number of receivers has more of a beneficial effect than increasing the number of transmitters, due to the increase in overall receiver aperture size P s M=1 N=1, W=1 M=4 N=1, W=1 M=1 N=4, W=1 M=4 N=4, W=1 M=1 N=1, W=4 M=4 N=1, W=4 M=1 N=4, W=4 M=4 N=4, W= PT, dbj b Figure 5.3: Symbol error probability vs. average power with Rayleigh fading, no background radiation, and Q = No background, log-normal fading With log-normal fading, the channel path gains a nm follow the distribution in (2.1, repeated here for convenience f A (a nm = 1 (2πσ 2 X 1/2 a nm exp( (log e a nm µ X 2 /2σ 2 X, a nm > 0 (5.17 where µ X = σ 2 X and S.I. = E[A4 ]/E 2 [A 2 ] 1 = e 4σ2 X 1 [0.4, 1.0]. The symbol error probability could be found by averaging over the log-normal distribution:

66 P s M=1 N=1, W=1 M=4 N=1, W=1 M=1 N=4, W=1 M=4 N=4, W=1 M=1 N=1, W=4 M=4 N=1, W=4 M=1 N=4, W=4 M=4 N=4, W= PT, dbj b Figure 5.4: Symbol error probability vs. peak power with Rayleigh fading, no background radiation, and Q = 8. P s Log normal fading = = N N n=1 m=1 0 w w i i=1 l=0 ( Q w+i i ( Q w+i i 1 ( w i ( w i l ( 1 l e [ηp T (i+l]/hfmq 1 e ( (ln a nm µ X 2/2σ2 X da (2πσX 2 nm (5.18 1/2 a nm w w i ( Q w+i ( ( i 1 w w i ( Q w+i ( 1 l i l l=0 i [ ] MN e [ηp T (i+l]/hfmq 1 (2πσX 2 1/2 a e( (ln a µ X2 /2σ2 X da (5.19 i=1 0 However, unlike the Rayleigh fading case, the resulting integral is not easily solved in closed form. Instead, we can choose from a number of methods that still give reasonable accuracy, including numerical integration, importance sampling, and Monte Carlo simulation. The method chosen for this thesis is numerical integration. The log-normal density is sampled for the values 0 to 10.0 in step sizes of 10 5, and using a S.I. of 1.0. The resulting plots can be seen in Figures 5.5 and 5.6. Clearly, the log-normal fading case causes a degradation in system performance compared to the non-fading case, although not as severe as with Rayleigh fading. Most notable are the

67 M=1,N=1,w=1 M=4,N=1,w=1 M=1,N=4,w=1 M=4,N=4,w=1 M=1,N=1,w=4 M=4,N=1,w=4 M=1,N=4,w=4 M=4,N=4,w= P s PT,dBJ b Figure 5.5: Symbol error probability vs. average power with log-normal fading (S.I. = 1.0, no background radiation, and Q = M=1,N=1,w=1 M=4,N=1,w=1 M=1,N=4,w=1 M=4,N=4,w=1 M=1,N=1,w=4 M=4,N=1,w=4 M=1,N=4,w=4 M=4,N=4,w= P s PT b,dbj Figure 5.6: Symbol error probability vs. peak power with log-normal fading (S.I. = 1.0, no background radiation, and Q = 8.

68 52 effects of transmitter and receiver diversity. Just as in the Rayleigh fading case, a considerable performance gain is also achievable by increasing only the number of lasers. However, the increased aperture size resulting from adding receivers is also visible, showing that receiver diversity is even more effective than transmitter diversity Error probability in the presence of background radiation The analysis of error probability for the case of background radiation present is more complicated than for no background radiation. In the no background case, errors only occurred when an on slot was received as a zero at all N receivers. In the background case, the off slots have the potential to have an equal or higher count than any of the on slots, in which case the receiver could pick the wrong modulator symbol. For this reason, a closed-form solution, even for the non-fading case is an intractable problem. For the non-fading case, an infinite-summation expression is possible. For both fading cases, however, some sort of Monte Carlo simulation is necessary Background radiation, no fading To obtain the infinite-summation solution we first classify the possible symbol error types into two categories. The first are errors caused when one or more of the off slots has a higher count than one or more of the on slots. In this case, the receiver will certainly make an error. We will refer to this as a definite error and use the symbol P def. This scenario is depicted in Figure 5.7. We will start by analyzing definite errors. When the receiver receives a vector Z representing the Q slots summed over all N photodetectors, w of them were originally sent as on slots, and Q w as off slots. Of the on slots, there will be i {1,..., w} of them that will have the lowest slot count u. If j {1,..., Q w} of the off slots have a value (labeled v in Figure 5.7 which is greater than u, a definite error has occurred. Letting {S on,l, l = 1,..., w} denote the column sums for on slots, and {S off,l, l = 1,..., Q w} denote the column sums for off slots, we can state this for all possible values of u as

69 53 Figure 5.7: An example of a definite error. P def = = u=0 u=0 [ P ( {l : S on,l = u} = i ( {l : S on,l > u} = w i (5.20 ] ( {l : S off,l > u} = j ( {l : S off,l u} = Q w j w i=1 (5.21 ( w P (S on = u i P (S on > u w i i Q w ( Q w P (S off > u j P (S off u Q w j (5.22 j j=1 For simplicity, we will make the following symbolic substitutions regarding Poisson random variables with parameter α: φ pdf (k, α = P (slot count = k α = αk e α, k = 0, 1, 2... (5.23 k! φ cdf (k, α = P (slot count k α = k α b e α, k = 0, 1, 2... (5.24 b! (being careful to note that φ cdf and φ pdf are both equal to zero for k < 0. If we let λ on = ηp T n m a2 nm/hfm + NηP b T/hf and λ off = NηP b T/hf, b=0