PERFORMANCE ANALYSIS OF COMMUNICATION SYSTEMS OVER MIMO FREE SPACE OPTICAL CHANNELS

Transcription

1 PERFORMANCE ANALYSIS OF COMMUNICATION SYSTEMS OVER MIMO FREE SPACE OPTICAL CHANNELS By Qianling Cao SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT UNIVERSITY OF VIRGINIA CHARLOTTESVILLE, VIRGINIA JAN 2005 c Copyright by Qianling Cao, 2005

2 UNIVERSITY OF VIRGINIA DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled Performance Analysis of Communication Systems over MIMO Free space Optical Channels by Qianling Cao in partial fulfillment of the requirements for the degree of Master of Science. Dated: Jan 2005 Supervisor: Maïté Brandt-Pearce Readers: Stephen G. Wilson Yibin Zheng ii

3 UNIVERSITY OF VIRGINIA Date: Jan 2005 Author: Qianling Cao Title: Performance Analysis of Communication Systems over MIMO Free space Optical Channels Department: Electrical and Computer Engineering Degree: M.Sc. Convocation: Jan Year: 2005 Permission is herewith granted to University of Virginia to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED. iii

4 To Xiao Qun and my parents. iv

5 Table of Contents Table of Contents List of Figures Acknowledgements Abstract v vii ix x 1 Introduction 1 2 System Model Q-ary Pulse Position Modulation Free Space Channel Model Transmitter and Receiver Link Budget Performance Analysis Case I: No Channel Fading, No Background Radiation Error Probability Analysis Peak Power Constraint and Average Power Constraint Modulation Efficiency Case II: Fading Channel, No Background Radiation Case III: No Channel Fading, with Background Radiation Case IV: Fading Channel, with Background Radiation Conclusion and Summary Conclusion Future work v

6 Bibliography 44 vi

7 List of Figures 2.1 System model of free space optical communications System block diagram of free space optical communications Probability density functions of channel gain A under Rayleigh and log-normal distribution Optical detection model of free-space communication system System error probability for non-fading, no background radiation case System error probability for fading, no background radiation case with M = 1 and N = Symbol error probability for Rayleigh and log-normal fading, no background radiation, Q = 8, w = 4, M (1, 2, 4), N = 1, and peak power definition Symbol error probability for log-normal fading (S.I = 1.0), no background radiation, Q = 2, w = 1, M (1, 2, 4), N (1, 2), and peak power definition Symbol error probability in different background radiation levels without channel fading, for binary PPM and 8-ary PPM with w = 1, M = 1, N = Symbol error probability under background radiation without channel fading, for binary PPM and 8-ary PPM with w (1, 4), M = 1, N = 1, P b T b = 170 dbj vii

8 3.7 Simulation result of symbol error probability for optimal combining and equal-gain combining, Rayleigh and log-normal fading, and background radiation, M = 1, N = 4, P b T b = 170 dbj Simulation of symbol error probability for Rayleigh and log-normal (S.I.=1.0) fading, Q = 8, w = 4, background energy = -170 dbj viii

9 Acknowledgements I would like to thank Professor Maïté Brandt-Pearce, my supervisor, for her many suggestions and constant support during this research. I am also thankful to Professor Stephen G. Wilson for his guidance through the years of my study in this university. Dr. Bo Xu expressed his interest in my work and supplied me with the preprints of some of his recent work, which gave me a better perspective on my own results. Michael Baedke shared with me his knowledge and provided many useful references and friendly encouragement. Of course, I am grateful to my husband for his patience and love. Without him this work would never have come into existence (literally). I also give thanks to my parents, their support is my confidence in my study. Finally, The author gratefully acknowledges the support of the National Science Foundation for funding the research under the project Space-Time Coding for Optical MIMO Channels. Qianling Cao, January, 2005 ix

10 Abstract The global telecommunications network has seen massive expansion over the last few years. Optical communication through clear atmosphere provides a means for high data rate communication over relatively short distances (e.g. 2km). However, the turbulence in the atmosphere leads to fades of varying depths, some of which may lead to heavy loss of data. Free space optical communication system using multi-input multi-output with QPPM is described in this thesis. We use spatial diversity at both the transmitter and receiver as a means to mitigate the channel fading. Using direct detection receivers and QPPM modulation, we derive the symbol error probability of MIMO systems that use ML detection and equal gain combining with and without background radiation. We demonstrate that for faded channels, performance gains are seen as the number of transmitters and receivers increases. Full transmitter and receiver diversity is obtained and observed by analyzing the Rayleigh fading case. We also show that multipulse QPPM is superior to PPM with respect to bandwidth efficiency, and exhibits superior symbol error performance when the system is peakpower-limited. Hence, we conclude that MIMO systems can be used effectively as a technique for atmospheric optical channels. x

11 Chapter 1 Introduction The global telecommunications network has seen dramatic expansion over the last decade, catalyzed by the telecommunications deregulation of First came the tremendous growth of long-haul, wide-area networks (WANs), then followed by a more recent emphasis on metropolitan area networks (MANs). Meanwhile, local area networks (LANs) and gigabit Ethernet ports are being deployed with a comparable growth rate. In order for this tremendous capacity to be exploited, and for the users to be able to utilize the broad array of services becoming available, network designers must provide some flexible and cost-effective means for the users to access the telecommunications network. As a consequence, there is a strong need for a high-bandwidth bridge (the last mile ) between the LANs and MANs or WANs. Free-space optical communication systems represent one of the most promising approaches for addressing the emerging broadband access market and its last mile bottleneck. Since light travels through air less expensive than through fibre and it is more convenient to build a line-of-sight link without fiber, it provides a natural, reliable approach for broadband access. Mention optical communications and most people think of fiber optics. But light 1

12 2 travels through air in a simpler way. So it is hardly a surprise that smart entrepreneurs and technologists are borrowing many of the devices and techniques developed for fiber-optic systems and applying them to what some call fiber-free optical communications. Free-space optical systems, which establish communication links by transmitting laser beams directly through the atmosphere, have been engineered to provide robust performance that is highly competitive with other access approaches, offering high capacity, excellent availability, fiber-like bandwidth, low operational cost per bit per second, and rapid deployment (for example, 2 hours). Available systems offer capacities in the range of 100 Mbps to 10 Gbps. These systems are compatible with a wide range of applications and markets and they are sufficiently flexible as to be easily implemented using a variety of different architectures. As we know, modern free-space optical communication originated in the 1970 s [11]. It really started to develop in the 1980 s. But as communications through optical fibers boomed, the interest in free-space optical communication began to decline. However, as the requirement for solving the last-mile problem arise, the market for free-space optical communications began to grow. Free-space optical interconnects can provide high bandwidth with no physical contact, but are hampered by signal fading effects due to particulate scattering in the line-of-sight path caused by atmospheric turbulence. In particular the atmospheric turbulence causes fluctuations in both the intensity and the phase of the received light signal, producing additional space losses as well as possible beam distortion. Even in clear weather, channels may suffer fading due to inhomogeneities in the index of refraction of the optical path. Related research results can be found in several references [2], [3] and [11]. Incorporating large link margins to combat the optical

13 3 propagation effects is not efficient. Furthermore, the narrow optical beam-width also implies critical pointing from the transmitter to the receiver, especially when we establish the free-space optical system between two high buildings since the sway can make it impossible to keep both transmitter and receiver pointing towards each other without any active tracking. To address these challenges, especially the intensity fluctuations, we consider the use of optical arrays instead of a single transmitter laser and a single photodetector to reduce fading effect. This creates a multiple-input multiple-output channel. Specifically, we use M separate lasers, assumed to be intensity modulated (IM), together with N photodetectors, assumed to be ideal direct detection (DD) receivers. The sources and detectors are situated separately enough so that we can assume the fading experienced between every laser-photodetector pair is statistically independent. Thus diversity benefits can accrue from the multiple-input multiple-output (MIMO) channel and the pointing issue can also be improved. However, the assumption of independence is based on the fading character and it may not merely depend on the spacing of the devices. For example, in foggy weather, all the laser-detector pair links will induce large fades. Related work on MIMO optical communication can be found in [17] by Haas et al by analyzing pairwise code error probability. Shin et al [18] also treat the problem of receiver diversity. The multi-source, multi-detector configuration also helps the pointing issue. Horizontal roof-top transmit/receiver arrays will experience fading instead of total pointing loss in the presence of substantial sway. MIMO processing has been used with great success to combat fading in RF wireless communication systems [44][45]. The free-space optical intensity modulation,

14 4 direct detection communication system that we propose has several aspects different from traditional RF wireless technology. First, the MIMO channel input symbols are non-negative real intensities. Second, the channel gains are real and non-negative. This is unlike the RF wireless system where both input symbols and channel gains are typically described as complex numbers. Third, the noise mechanisms are quite different - in this thesis, we analyze the circumstance that the signal-dependent shot noise in optical communications limits the system performance. On the other hand, RF wireless communications are often thermal-noise-limited. Our study focuses on the performance analysis of optical MIMO systems in various fading environments with and without background radiation. For this system, we are focusing on multiple pulse position modulation (MPPM), which is an intensity modulation technique. Research in pulse position modulation (PPM) in optical free-space communication can be dated as early as 1960 s, [1] and [14]. These works showed that PPM becomes more average-energy efficient as the number of time slots per symbol increases. Further work on coded PPM includes [32] and [33] in the 1980 s and some recent work [12]. In the 1990 s, MPPM has been studied and combined with traditional block codes to improve performance [19]. Some recent studies related to MIMO channels with direct detection are found in [18]. In this thesis, we analyze the system performance by an approach called spatial diversity which attempts to overcome the atmospheric difficulties. The rest of the thesis is organized as follows. A theoretical system model constructed based on the MIMO channel is described in Chapter 2 including a detailed link budget. In Chapter 3 we analyze the performance of the system and also offer simulation results. Chapter 4 then summarizes the results and also provides a brief

15 discussion of the advantages and the limitations of the proposed method. 5

16 Chapter 2 System Model A free-space optical communication system is composed of three basic parts: a transmitter, the propagation channel and a receiver. A typical simple diagram illustrating the system is displayed in Figure 2.1. In our system, M lasers, intensity-modulated by input symbols, all point toward a distant array of N photodetectors. Every laser beamwidth is sufficiently wide to illuminate the entire photodetector array. The M N laser-photodetector path pairs may experience fading and the amplitude of the path gain from laser m to detector n is designated as a nm. Figure 2.2 shows the block diagram of the proposed free space optical MIMO system. In the transmitter, binary data bits are converted into a stream of pulses corresponding to QPPM symbol described below, and sent to the M lasers. All lasers send the same symbol towards every photodetector (repetition coding). Every photodetector counts the photoelectrons it receives in every QPPM symbol slot. The received symbol of the nth photodetector, is a vector of Q photoelectron counts {Z nq, q = 1,, Q} and it is passed to the processor and finally decoded to binary data bits. 6

17 meters Laser Array Photodetector Array Figure 2.1: System model of free space optical communications. bits in MPPM Modulation x 11 x X 21 x 12 x x 1 Q Laser 1 x 1 Q Laser 2 Turbulent Atmosphere Medium PD1 PD2 z z z z z1 Q z2 Q Processor output bits out PDN z N1 z N2... z NQ x M1 x M2 Laser M... x MQ Figure 2.2: System block diagram of free space optical communications.

18 8 The wavelength chosen for free space optical systems usually falls near one of two wavelengths, 0.85µm or 1.55µm. The shorter of the two wavelengths is cheaper and is favored for shorter distances. The 1.55µm light source is favored for longer distances since it has an allowed power that is two orders of magnitude higher than at 0.85µm [16]. The reason for the higher allowed power is that laser-tissue interaction is very dependent on wavelength. The eye hazard at 1.55µm is much lower than at 0.85µm. 2.1 Q-ary Pulse Position Modulation Pulse-position modulation, or PPM, is a powerful and widely used technique for transmitting information over an optical direct-detection channel [14]. PPM is a modulation technique that uses pulses that are of uniform amplitude and width but displaced in time by an amount depending on the data to be transmitted. It is also sometimes known as pulse-phase modulation. It has the advantage of requiring constant transmitter power since the pulses are of constant amplitude and duration. PPM also has the advantage of good noise immunity since all the receiver needs to do is detect the presence of a pulse at the correct time; the duration and amplitude of the pulse are not important [13]. Q-ary Pulse Position Modulation, or QPPM, is an energy-efficient and well developed modulation method. At the transmitter, the encoder maps blocks of L consecutive binary data bits into a single PPM channel symbol by placing a laser pulse into one of several time slots. In this method, every symbol interval of duration T s is subdivided into Q slots, each of duration T Q = T s /Q. If each bit is T b seconds in duration, then L bits take T s = L T b seconds to transmit.

19 9 We use multi-pulse QPPM (M-QPPM) in our system, which means that in every symbol the lasers turn on for w time slots out of a possible Q time slots. Of course, w = 1 represents conventional QPPM described above. Instead of sending a single pulse as in traditional QPPM, w pulses are sent in certain ( symbol ) slots Q to transfer a digital message. Every symbol represents L = log 2 bits. So w ( ) Q T s = T b log 2 where T b is the bit duration. After establishing slot and symbol w synchronization, the receiver detects the un-coded M-QPPM symbols by determining which w out of the Q slots contains the laser pulses, and performs the inverse mapping operation to recover ( the ) bit stream. If E b is the energy per bit, then the symbol Q energy is E s = E b log 2. w In this thesis, we assume repetition coding across all lasers, that is, each of the M lasers transmits the same w-pulse symbol at the same time. While this constraint restricts the permissible bit rate, relative to an unconstrained set of patterns, the receiver processing is simple, performance analysis is more direct and, as we shall see, performance is remarkably good on the MIMO channel. 2.2 Free Space Channel Model Atmospheric turbulence can degrade the performance of free-space optical communication systems, particularly over ranges longer than 1 km. Inhomogeneities in the temperature and pressure of the atmosphere lead to variations in the refractive index along the transmission path. These index inhomogeneities cause fluctuations in both the intensity and the phase of the received signal. As a consequence, these fluctuations lead to an increase in the system error probability, limiting the performance of

20 10 the communication system. Atmospheric turbulence has been studied by many scientists, and various theoretical models have been proposed to describe the intensity fluctuations (i.e, the signal fading). None is universally accepted due to the difference of atmospheric conditions. Among these models, log-normal and Rayleigh models are widely used. Due to the turbulence of the atmosphere, the field strength received at the detector becomes a random field. We adopt both log-normal and Rayleigh models - which are the most accurate among them. In the log-normal model the path gain is A = e X where X is Gaussian distributed with mean µ X and variance σx 2. By definition, the logarithm of A follows a normal distribution. Strohbehn first showed that for optical atmospheric channels the gain can be assumed to be log-normal distributed and the phase has a uniform distribution [2]. The p.d.f. of A is f A (a) = 1 (2πσX 2 ) 1 2 a exp( (log e a µ X ) 2 ), a > 0 (2.2.1) We restrict the mean path intensity to unity, i.e. E[A 2 ] = 1. This requires µ X = σ 2 X. The scintillation index, used to measure the strength of fading, is defined as 2σ 2 X E[A S.I. = 4 ] E 2 [A 2 ] 1 (2.2.2) Typical values of S.I. are in the range of for log-normal distributions. The Rayleigh model is also widely used to describe the channel gain. It is used less often in the literature than log-normal fading to analyze free-space optical systems, but has some nice mathematical properties that make it an attractive model to use. First of all, the Rayleigh fading case exhibits deeper fading than log-normal fading

21 11 because of the higher concentration of low-amplitude path amplitudes and it can be considered the worst case. Furthermore, with Rayleigh fading, the diversity order - which means the number of independently fading propagation paths - of the MIMO system becomes apparent by analyzing the slopes of the symbol error probability curves. The p.d.f of A under the Rayleigh distribution is f A (a) = 2ae a2, a > 0 (2.2.3) 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. Figure 2.3 shows the probability density functions for Rayleigh and log-normal distributions. The density function of Rayleigh is more concentrated at low (deeply faded) values. We also assume that the spatial coherence distance of the field at the detector is large relative to the size of one photodetector. Spatial coherence distance in the turbulent atmosphere are reported to range between 10 centimeters and 1 meter [1] for a wavelength of 1.55µm as we use in our system. 2.3 Transmitter and Receiver Our study is based on a semi-classical treatment of photodetection, where the incident field is treated as a wave, and this wave produces a modulated Poisson point process of photoelectrons at the receiver end that contributes to the detector current at the output of the photodetector. An ideal photon-counting model with a typical quantum efficiency is assumed. Figure 2.3 illustrates the receiver model for our system. The aggregate optical field from all the lasers is detected by each photodetector

22 log nomral fading, S.I.=0.4 log nomral fading, S.I.=0.6 log nomral fading, S.I.=1.0 Rayleigh fading f A (a) a Figure 2.3: Probability density functions of channel gain A under Rayleigh and lognormal distribution Optical field Photodetector Integral Z1, Z2,...,Zq Receiver Aperture Figure 2.4: Optical detection model of free-space communication system

23 13 and we denote the total incident signal power at one photodetector for a non-fading channel from all the lasers as P r when a pulse is transmitted. Z nq is the number of photoelectrons at slot q collected by photodetector n. Z nq is a Poisson random variable with mean value depending on the receiving power, background power, receiver efficiency and receiver time slot duration. The average number of signal photoelectrons generated in a QPPM slot in which a pulse is transmitted is denoted as λ s = ηp rt Q hf (2.3.1) where η is the detector s quantum efficiency factor, defined as the ratio of generated photoelectrons to incident photons, assumed to be 0.5 here. T Q is the slot duration equal to T s /Q. h is Planck s constant, and f is the optical center frequency. In addition to the signal, background radiation is received. The average number of photoelectrons due to the background field is denoted as λ b = ηp bt Q hf (2.3.2) where P b is the incident background power on one photodetector. At the receiver end, in a slot a pulse is sent (what we call an on slot ), the photodetector receives both incident power and background noise. The probability mass function for the number of counts in an on slot is P r (Z nq = k) = (λ s + λ b ) k exp ( (λ s + λ b )), k = 0, 1, 2, (2.3.3) k! In the slot where no pulse is sent ( what we call an off slot ), the photodetector receives only background noise. The probability mass function for the number of counts in an off slot is

24 14 P r (Z nq = k) = (λ b) k exp ( (λ b )), k = 0, 1, 2, (2.3.4) k! In the fading case, we denote the path gain from the mth laser to the nth photodetector as a nm. The mean number of the photoelectrons at the nth detector in the signal on slot is derived from the sum of incident powers from all M lasers plus background noise; the mean number becomes λ on = 1 M M m=1 a2 nmλ s + λ b. To be fair in our comparisons, we keep the total laser power constrained by assuming the transmit power is equally shared among M lasers, so P r is the power received form all M lasers to one photodetector. This parallels the standard assumption for the microwave MIMO system. We designate the collection of slot-by-slot photoelectron counts for the nth photodetector at the qth slot in a symbol as [Z nq, n = 1,, N, q = 1,, Q], where n describes the photodetector number and q describes the slot number. Then Z = [Z nq ] is the received observation matrix. Suppose X is the whole set of possible symbols, i.e, every x X is a Q slot symbol with w slots on and (Q w) slots off. For every fading matrix A with entries a nm, the maximum likelihood detector is given by ˆx = arg max f (Z x, A) (2.3.5) x X Since the Z nq are all independent of each other, 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 Z nq.

25 15 ˆx = arg max x X = arg max x X N Q n=1 q=1 N n=1 q=1 ( exp λ s x M q ( Q exp λ s M x q ) ( M m=1 a2 λ nm + λ s b x ) Znq M M q m=1 a2 nm + λ b M m=1 Z nq! a 2 nm + λ b ) ( λ s M x q ) M a 2 nm + λ b Znq(2.3.6) We define the set of the on slots as Q on and the set of all the off slot as Q off. m=1 Their sizes are w and Q w respectively. The sets Q on and Q off depend on the symbol x. These elements are conditioned on whether they are in Q on, or in Q off. ˆx = arg max x X N exp n=1 q Q on ( q Q off exp ( λ b ) (λ b ) Znq λ s M where x X is possible sending symbol. M m=1 a 2 nm + λ b ) ( λ s M ) Znq M a 2 nm + λ b m=1 (2.3.7) We take the logarithm of the entire quantity to find the log-likelihood function. The ML detector becomes ˆx = arg max x X + N n=1 N n=1 q Q on which can be rewritten as ( λ s M ) M a 2 nm + λ b + Z nq log m=1 ( λ s M ) M a 2 nm + λ b m=1 q Q off ( λ b + Z nq log (λ b )) (2.3.8)

26 16 ˆx = arg max x X N n=1 N = arg max x X n=1 ( N + n=1 = arg max x X q Q q Q on Z nq log q Q on Z nq log Z nq log (λ b ) ( ) λ s M a 2 nm + λ b M m=1 ( ) λ s M a 2 nm + λ b M m=1 ) Z nq log (λ b ) q Q on ( N λs M ) M m=1 Z nq log a2 nm + λ b λ b q Q on n=1 + Z nq log (λ b ) q Q off (2.3.9) Therefore, the ML detector would make a decision based on a weighted sum over the on slots. If background noise can be ignored, the ML detector chooses the slots that photons are received and makes a free guess if there are some slots pulses were sent but nothing received. In case there is no channel fading, the ML detector does not need to weight Z nq. The ML detector just compares the sum of the photoelectrons counts of all N photodetector in every slots and chooses the largest w slots. When channel fading and background noise are both present, we propose to use an equal-gain combiner instead of the ML detector since monitoring the channel gains increases the complexity of the receiver and has only a slight benefit, as we show in Section 3.4. An equalgain combiner simply adds the output of every detector without weighting them. In summary, in all cases we form the sum over all detector counts slot by slot and choose the slots with the largest counts. ˆx = arg max x X q Q on n=1 N Z nq (2.3.10)

27 Link Budget Assume we want to establish a free space optical communication system between two buildings separated by 2 kilometers to transmit a data stream of 100 Mbps. To minimize the eye hazard, we use a wavelength of 1.55 µm. The transmitter laser power P t is 100 mw. The full laser transmit beam angle θ is 10 mrad (about 0.6 degrees). The receiver aperture diameter D R is 1 cm and we assume all photodetectors have the same size. We also assume that the whole path experiences no link fading and perfect alignment. At the receiver end, the power flux density is P (d) = P t 2πd 2 (1 cos( θ 2 )) 4P t πd 2 θ 2 mw/m 2 (2.4.1) where d is the distance between the laser and detector. The power intercepted by the receive aperture is P r = P (d)a rec = P td 2 R d 2 θ 2 = W (2.4.2) Assuming binary PPM without considering any background radiation, the slot time is 5 nanoseconds. In every signal on slot the average number of received signal photons is where the quantum efficiency factor η is 0.5. λ s = (ηp r /hf)t Q 244 photons (2.4.3) In addition to the desired source power, a receiver also collects undesirable strong background radiation falling within the spatial and frequency ranges of the detector. The background power levels can be calculated as P b = W A rec λω fv = watts (2.4.4)

28 18 where W is the spectral radiance function defined as the power radiated at the wavelength of interest per unit of bandwidth into a unit solid angle per unit of source area [1], which in our system is the background noise from the sky. λ is the received wavelength bandwidth assume to be 10 9 m. Ω fv is the receiver field of view. The field of view angle is 100 mrad so Ω fv = π sr here. For wavelength of 1.55 µm, W = W/(cm 2 µm sr) [1]. For a binary PPM system, in every signal off slot the average number of received photons is λ b = (ηp b /hf)t Q 36 photons (2.4.5)

29 Chapter 3 Performance Analysis We consider four cases: without or with background radiation, and non-fading or fading links. We discuss a general theory, and illustrate with specific results for the most interesting cases. The situation without background radiation and non-fading links is the easiest and is treated first. 3.1 Case I: No Channel Fading, No Background Radiation Error Probability Analysis With no loss of generality, we assume the symbol with the first w of total Q slots on is sent. At the receiver end, we receive a matrix Z with elements [Z nq, n = 1,, N, q = 1,, Q] where n indicates the receiver number and q indicates the slot number. Since there is no background radiation, then λ b = 0. If slot q Q off Z nq will be zero. The channel gain is the same for all paths with a nm = 1. The 19

30 20 maximum likelihood detector becomes ˆx = arg max x X = arg max x X = arg max q N n=1 N n=1 N n=1 q Q on Z nq log q Q on Z nq log ( λs M ) M m=1 a2 nm + λ b λ b ( ) λs + λ b λ b q Q on Z nq (3.1.1) In this case, an error will only occur when one or more of the w on slots register zero counts at all N detector outputs and likelihood ties represents the only mechanism for decision error. When Q = 2 and w = 1 this is equivalent to a binary erasure channel. Specifically, suppose i of the w on slots (i w) produce a column of zeros in the Z matrix where non-zero counts are expected. Then, a likelihood tie occurs among ) candidates and tie-breaking errors have probability ( Q w+i i P [making an error] = ( Q w+i i ( Q w+i i ) 1 ) = t(q, w, i) (3.1.2) By the Poisson property and independence we have that the probability of exactly i of w columns registering zero counts is P [i of w columns = 0] = ( ) w p i (1 p) w i (3.1.3) i where p = e λs from (2.3.3), and λ s = ηp r T Q /hf = ηp r T s /hfq from (2.3.1). Putting this altogether we can derive the symbol error probability in no background radiation, for a non-fading channel

31 21 P s = w i=1 ( ) w t(q, w, i)p i (1 p) w i (3.1.4) i By expanding the last term using a binomial expansion, i.e. w i ( ) w i (1 p) w i = ( 1) l p l, l l=0 we can combine terms to get a finite series expansion for symbol error probability: P s = w w i ( )( ) w w i ( 1) l t(q, w, i)p i (1 p) w i e λ sn(i+l) i l i=1 l=0 (3.1.5) This says that for a fixed total transmitter energy, the probability of symbol error is independent of M, i.e., there is no phased-array gain attached to the multiple sources, since these are non-coherent sources. The effective received power does increase linearly with N, the effect of increasing receiving aperture size Peak Power Constraint and Average Power Constraint To cast this symbol error probability in terms of peak power and a common information rate from the bit-symbol relation (T Q = T s /Q and T s = T b log Q ( 2 w) ) we get ( log Q ) 2 w P r T Q = P ave T b (3.1.6) w or ( log Q ) 2 w P r T Q = P peak T b (3.1.7) Q The symbol error probability P s is shown in Figure 3.1 for non-fading links without background radiation versus P ave T b and P peak T b. P represents either P ave for average power or P peak for peak power in this and all subsequent plots.

32 P s P peak, Q = 2 P peak, Q = 8, w = 4 P peak, Q = 8, w = 1 P ave, Q = 2 P ave, Q = 8, w = 4 P ave, Q = 8, w = PT b in dbj Figure 3.1: System error probability for non-fading, no background radiation case.

33 23 In this figure, we compare binary PPM and 8-ary PPM, showing both classic QPPM and M-QPPM with w = Q/2 = 4 for Q = 8. In the non-fading regime, the choice of M has no impact since we fix the total laser array power. We choose M = N = 1. Notice that the multipulse case with w = Q/2 exhibits a superior energy efficiency under a peak power constraint. On the other hand, if average energy per bit is the criterion, classic PPM with w = 1 is much superior. In optical system using semiconductor laser, peak power is the most logical comparison Modulation Efficiency Modulation efficiency of various system designs depends on whether one considers the peak or average power; efficiencies provide different result. By studying (3.1.5) for the non-fading case above, the dominant term in P s at large values of P T b (or small P s ) is found to be the term that one signal slot receives zero photoelectron count, i.e, i = 1 and l = 0 in (3.1.5). Thus, we may derive an approximation of efficiency as the multiplier of P T in (3.1.6) and (3.1.7). This approximation gives the asymptotic relative efficiency. The results are and γ ave = log ( Q 2 w w ) (3.1.8) γ peak = log ( Q ) 2 w (3.1.9) Q For purposes of comparison, we have γ ave = 1 in the binary PPM case, (Q, w) = (2, 1). For any fixed (Q, w) pair, the peak energy efficiency is always w/q times the average

34 24 energy efficiency. We may also measure the effect of w > 1 using these expressions. For example, comparing Q = 8 with w = 1, 4 we obtain that γ peak = γ ave /2 when w = 4 and γ peak = γ ave /8 when w = 1. We can draw the following conclusion: in case of average power constraint, large Q with w = 1 can get best performance and in case of peak power constraint, large Q with w = Q/2 is the best choice. In general, if we keep the energy per bit fixed, the average power efficiency improves with increasing Q, and decreasing w. On the other hand, the peak power efficiency is best when w = Q/2, and in this case, as Q grows, the efficiency approaches 1 from below. Since a large Q may increase the difficulty in QPPM decoding and receiver synchronization, from an engineering perspective, Q = 8, w = 4 is an attractive choice for peak-power-limited design, and we emphasize this choice in the rest of the thesis. Spectral efficiency is of lesser concern in free-space optical systems, but due to the required clock speed and the relative difficulty of receiver synchronization, spectral efficiency also affects the relative difficulty of implementation. If we measure bandwidth as proportional to bit rate 1/T b, then the spectral efficiency in bps/unit bandwidth is proportional to β = log ( Q ) 2 w (3.1.10) Q For a given Q, β is maximized for w = Q/2, and when we choose w = Q/2, as Q grows large, β monotonic approaches 1. In the binary PPM case, β = 0.5. For Q = 8 and w = 4, β =

35 Case II: Fading Channel, No Background Radiation First, we assume the channel gain of every laser-detector pair is fixed over a symbol duration. Letting a mn denote the amplitude fading on the path from laser m to photodetector n, we define the channel gain matrix as A with element [a nm, n = 1,, N, m = 1,, M]. The probability of symbol error conditioned on the fading variables is P s A = w w i ( )( w w i ( 1) l i l i=1 l=0 ) t(q, w, i)e λs M n m a2 mn (i+l), (3.2.1) where again λ s = ηp T Q /hf from (2.3.1). To extend the analysis of non-fading link and no background radiation case to the case of link fading, we can simply average the (conditional) symbol error probability of (3.2.1), with respect to the joint fading distribution of the A nm variables. We emphasize that this produces the symbol error probability averaged over fades. Formally, we find P s by evaluating P s = P s A f A (a)da (3.2.2) where the integral is interpreted as an MN-dimensional integral. Since the A nm variables are assumed independent, the above averaging leads to P s = w w i ( )( ) w w i ( 1) l t(q, w, i) i l i=1 l=0 ( 0 e λs M (i+l)a2 f A (a)da) MN (3.2.3)

36 26 which is a function of received energy per slot, number of lasers and photodetectors, and fading distribution. If the channel is under Rayleigh fading, the averaging in (3.2.3) may be done analytically, and produces a simple form P srayleigh = w w i ( )( [ ] MN w w i t(q, w, i) )( 1) l 1 (3.2.4) i l 1 + (λ s /M)(i + l) i=1 l=0 In case of log-normal fading, we can at least evaluate (3.2.3) numerically. Figure 3.2 presents results for Rayleigh fading and log-normal fading with varying values of scintillation index by using a single laser and photodetector. 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. A study of (3.2.4) as a function of P T b reveals that P s is an inverse-mn-power function of the signal energy in the large signal regime, which we take as the definition of the system achieving full diversity, MN. In contrast to a similar microwave system, notice that attainment of full transmit diversity, M, is obtained without resort to exotic space-time constructions here. The diversity gains are quite large in the Rayleigh case, though smaller for the log-normal model. However, there is a power penalty due to power sharing in the denominator of (3.2.4), but the diversity order is nonetheless MN. As in the non-fading case, the symbol error probability expressions can be plotted versus either P ave T b or P peak T b. Figure 3.3 illustrates the performance versus peak power for Q = 8, M {1, 2, 4}, N = 1, showing a diversity order M is attained in the Rayleigh case. Similar conclusions pertain to average power as were made for the non-fading case.

37 P s No fading S.I. = 0.4 S.I.= 0.6 S.I. = 1.0 Rayleighfading PT in dbj b Figure 3.2: System error probability for fading, no background radiation case with M = 1 and N = 1

38 Rayleigh fading P s no fading M = 1 M = 2 M = 4 log-normal fading PT in dbj b Figure 3.3: Symbol error probability for Rayleigh and log-normal fading, no background radiation, Q = 8, w = 4, M (1, 2, 4), N = 1, and peak power definition.

39 29 From the Rayleigh fading curves in this figure we can see 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. Just as in the Rayleigh fading case, a considerable performance gain is also achievable by increasing only the number of lasers in the log-normal cases. It may also be noted that the interchange of M and N is not symmetric, due to the power division by M at the transmitter (note the c/m factor in (3.2.4)) Thus, though they have the same diversity order, (M, N) = (2, 1) and (1, 2) cases are 3 db different in favor of the latter. Actually, if we fix the total receive aperture as in [18], then M and N are interchangeable. Figure 3.4 shows the advantage of using multiple photodetector. Notice that besides the diversity effect that we are seeking, adding detectors also improves efficiency due to large total aperture. 3.3 Case III: No Channel Fading, with Background Radiation For the case of background radiation, the evaluation of error probability is more complicated. At the receiver end, we receive a matrix Z with elements [Z nq, n = 1 N, q = 1 Q]. We assume λ b is the Poisson count random variable parameter due to the background radiation, and if slot q is an off slot, Z nq will be also a Poisson distributed random variable with parameter λ b. For signal on slot, the Poisson count random variable parameter is λ s +λ b. The channel gain is the same for all paths with a nm = 1.

40 no fading M = 1, N = 1 M = 2, N = 1 M = 4, N = 1 M = 1, N = 2 M = 2, N = 2 M = 4, N = P s PT b in dbj Figure 3.4: Symbol error probability for log-normal fading (S.I = 1.0), no background radiation, Q = 2, w = 1, M (1, 2, 4), N (1, 2), and peak power definition.

41 31 Again, we assume without lose of generality that the symbol with the first w slot on is send. The maximum likelihood detector becomes ˆx = arg max x X = arg max x X ( ) N λ s M Z nq log a 2 + λ b M q Q on m=1 N Z nq (3.3.1) n=1 q Q on n=1 where Q on = [1,, w] is the set of all on slot that pulses are sent and Q off = [w + 1,, Q] is the set of all off slot that only background noise is received. Detection is correct only if all of the noise slot counts Z q are less than all the signal slot counts. Thus, we can upper bound the symbol error probability. P s 1 P (all signal slot counts greater than noise slot counts) (3.3.2) Adding the tie-break part, we can get the exact error probability. Letting Z on be the set of the N n=1 Z nq for all slots q Q on and Z off be N n=1 Z nq in all slots q Q off, for any (Q, w) pair, the symbol error probability will be P s P [min(z on ) < max(z off )] = 1 (Poisson pmf (N (λ s + λ b ), i + 1)) i=1 (1 (Poisson cmf (N (λ s + λ b ), i))) w 1 (Poisson cmf (N (λ b ), i)) Q w 1 (3.3.3) where P oissonpmf(x, y) represent the Poisson probability density function at value y using the corresponding parameter x and P oissoncmf(x, y) represent the Poisson probability cumulative function at value y using the corresponding parameter x.

42 32 For the exact error probability we can get a precise form P s = P [min(z on ) < max(z off )] Q w w ( ) k + s + P [min Z on = max Z off ] (3.3.4) s k=1 s=1 where s minimal signal slots have the same counts as k maximum noise slots. Then the exact error probability can be written as P s = 1 (Poisson pmf (N (λ s + λ b ), i)) i=1 (Poisson cmf (N (λ s + λ b ), i)) w 1 (Poisson cmf (Nλ b, i ) Q w ) Q w w ( ) k + s (Poisson pmf (N (λ s + λ b ), i)) k s i=1 k=1 s=1 (Poisson cmf (N (λ s + λ b ), i)) w k (Poisson cmf (Nλ b, i 1) Q w s ) Q w s=1 (Poisson pmf (Nλ b, i) s ) (3.3.5) where k is the number of on slot which have count i and all the other on slots have counts greater that i while, at the same time, there are s off slots that have count i and all other off slot have smaller counts. In this case, the probability of error is also independent of the number of transmitters. An increase in the number of receivers provides gain due to the increase in receiving aperture size. Figure 3.5 shows the symbol error probability under different background radiation levels. Comparing Figure 3.5 to Figure 3.1, background radiation shifts the curves to the right by amounts ranging from 4 to 7 db without changing the shape of curve heavily.

43 Binary PPM, noise 170dBJ Binary PPM, noise 160dBJ 8 ary PPM, noise 170dBJ 8 ary PPM, noise 160dBJ 10 4 Binary PPM 8 ary PPM P s E b in dbj Figure 3.5: Symbol error probability in different background radiation levels without channel fading, for binary PPM and 8-ary PPM with w = 1, M = 1, N = 1

44 34 Figure 3.6 shows the symbol error probability under a background radiation with energy P b T b = joules - which is close to the background level we calculate in Chapter 2 - fixed for binary PPM and QPPM. We notice that the shape of the curve is slightly different from Case I, the new results being steeper essentially some minimal level of signal power is required to overcome the background noise, and once this level is exceeded, performance improves sharply. 3.4 Case IV: Fading Channel, with Background Radiation This case is the most general in practice, and there is no simple expression for the symbol error probability. Here some incorrect symbol can have higher likelihood (an incorrect set of w slots have larger weighted column sums) same as in Case III. One can formally sum conditional probabilities over the error region and correctly handle ties, but we have resorted to Monte Carlo simulation using importance sampling instead. Our simulation uses an equal-weight combiner instead of the ML detector, which is slightly suboptimal only in this case of fading and background radiation, and only when N > 1. In cases where it is an issue, Figure 3.7 shows that the loss of the equal-gain combiner compared to the ML detector is very small, prompting us to use it in simulations. The result shows that the curves of equal-weight combining has the same shape as the curves of ML detection with less than 1 db shift. By using equal-gain-combiner, the upper bound on P s conditioned on fading path gain matrix A is

45 P s P ave, Q = 2 P ave, Q = 8, w = 4 P ave, Q = 8, w = 1 P peak, Q = 2 P peak, Q = 8, w = 4 P peak, Q = 8, w = PT b in dbj Figure 3.6: Symbol error probability under background radiation without channel fading, for binary PPM and 8-ary PPM with w (1, 4), M = 1, N = 1, P b T b = 170 dbj

46 P s log normal, equal gain Rayleigh, equal gain log normal, ML detection Rayleigh, ML detection PT b in dbj, noise = 170 dbj Figure 3.7: Simulation result of symbol error probability for optimal combining and equal-gain combining, Rayleigh and log-normal fading, and background radiation, M = 1, N = 4, P b T b = 170 dbj

47 37 P s A 1 ( ( N (Poisson pmf i=1 1 (Poisson cmf M n=1 m=1 N M n=1 m=1 a 2 λ s nm M + Nλ b, i + 1 a 2 λ s nm M + Nλ b) ) w 1 (Poisson cmf (N (λ b ), i 1)) Q w 1 (3.4.1) ) ) We can get the overall symbol error probability by averaging the conditional symbol error probability. Numerical integration is a prohibitively slow process due to the large number of fading variables combined with the infinite summation. In this case, using numerical integration is not an efficient way to calculate the symbol error probability. We use simulation instead to analyze the performance of the MIMO system. Monte Carlo methods are a way of using random numbers to perform numerical integrations. By way of example consider the one variable integral P s = P s a f(a)da (3.4.2) For a given background and signal power, a normal Monte Carlo simulation does not select sampling points but instead it chooses points at random, then perform the detection and count errors. In our system, the symbol error probability could be evaluated as P s = 1 N I(a i ) (3.4.3) N where I(a i ) is the indicator function and in our system it is defined as an error event i=1

48 38 for fading sample a i. { 1 error occur I(a i ) = (3.4.4) 0 detection correct In simple, we count the number of errors and divide it by the total number of symbol we send. This procedure takes a very long simulation time if low error probabilities are sought. Importance sampling, also called biased sampling, is one of the variancereducing techniques in Monte Carlo methods. Monte Carlo calculations can be carried out using sets of random points picked from a different channel gain probability distribution in our system. The choice of distribution obviously makes a difference to the efficiency of the method. In most cases, Monte Carlo calculations carried out using uniform probability distributions give very poor estimates of high-dimensional integrals and are not a useful method of approximation. In 1953, however, Metropolis introduced a new algorithm for sampling points from a different probability function. This algorithm enables the incorporation of importance sampling into Monte Carlo integration. Instead of choosing random variables from a uniform distribution, they are now chosen from a distribution which concentrates the points where the function being integrated is large. The equation (3.4.2) can be written as P s = P s a f(a)da = f(a) P s a g(a)da (3.4.5) g(a) where the function g(a) is chosen to be a distribution different from f(a). The integral can be estimated numerically by choosing the random points from the probability distribution g(a) and evaluating f(a i )/g(a i ) at these points. The average of these evaluations gives an estimate of I. The Monte Carlo estimate by using importance sampling of the integral is then,

49 39 P s = 1 N N i=1 I(a i ) f(a i) g(a i ) (3.4.6) The method of importance sampling applied to our system produces a much higher frequency of errors than normal Monte Carlo simulation; weighting the error counts appropriately we obtain an unbiased estimate of P s. If the biased distributed random variable is correctly chosen, the variance of the estimate can be greatly reduced relative to that of the Monte Carlo procedure with the same number of trials. Since biasing the Poisson parameter give us non-monotonic curves due to the discontinuity of the Poisson distribution, in our procedure we bias the fading distribution. We used a one-sided exponential for the amplitude variable, which has the effect of decreasing the mean signal counts, i.e, exaggerating the fades. A sample result is shown in Figure 3.8, for both Rayleigh and log-normal fading. Background power remains fixed at P b T b = 170 dbj. In our simulation, normal Monte Carlo simulation was used for P s greater than 10 3, that means, a large number of Poisson and Rayleigh (or log-normal) fading variables are generated. For P s smaller than 10 3, we use an exponential distribution to generated random variables of path gain instead of the Rayleigh or log-normal distribution, to achieve more errors. The symbol error probability is the sum of weighted errors divided by number of trails. Again full diversity is observed. As seen from Figure 3.8 and Figure 3.2, the MIMO system clearly exhibits superior performance to the single-input single-output system over fading channels, for environment with or without background radiation.