Outage Probability of the Gaussian MIMO. Free-Space Optical Channel with PPM

Transcription

1 Outage Proaility of the Gaussian MIMO 1 Free-Space Optical Channel with PPM Nick Letzepis, Memer, IEEE and Alert Guillén i Fàregas, Memer, IEEE arxiv: v1 [cs.it] 1 Apr 2008 Astract The free-space optical channel has the potential to facilitate inexpensive, wireless communication with fier-like andwidth under short deployment timelines. However, atmospheric effects can significantly degrade the reliaility of a free-space optical link. In particular, atmospheric turulence causes random fluctuations in the irradiance of the received laser eam, commonly referred to as scintillation. The scintillation process is slow compared to the large data rates typical of optical transmission. As such, we adopt a quasi-static lock fading model and study the outage proaility of the channel under the assumption of orthogonal pulse-position modulation. We investigate the mitigation of scintillation through the use of multiple lasers and multiple apertures, therey creating a multiple-input multiple output (MIMO) channel. Non-ideal photodetection is also assumed such that the comined shot noise and thermal noise are considered as signal-independent additive Gaussian white noise. Assuming perfect receiver channel state information (CSI), we compute the signal-to-noise ratio exponents for the cases when the scintillation is lognormal, exponential and gamma-gamma distriuted, which cover a wide range of atmospheric turulence conditions. Furthermore, we illustrate very large gains, in some cases larger than 15 db, when transmitter CSI is also availale y adapting the transmitted electrical power. I. INTODUCTION Free-space optical (FSO) communication offers an attractive alternative to the radio frequency (F) channel for the purpose of transmitting data at very high rates. By utilising a high carrier frequency in the optical range, digital communication on the order of gigaits per second is N. Letzepis is with Institute for Telecommunications esearch, University of South Australia, SPI Building - Mawson Lakes Blvd., Mawson Lakes SA 5095, Australia, nick.letzepis@unisa.edu.au. A. Guillén i Fàregas is with the Department of Engineering, University of Camridge, Camridge CB2 1PZ, UK, guillen@ieee.org.

2 2 possile. In addition, FSO links are difficult to intercept, immune to interference or jamming from external sources, and are not suject to frequency spectrum regulations. FSO communications have received recent attention in applications such as satellite communications, fier-ackup, F-wireless ack-haul and last-mile connectivity [1]. The main drawack of the FSO channel is the detrimental effect the atmosphere has on a propagating laser eam. The atmosphere is composed of gas molecules, water vapor, pollutants, dust, and other chemical particulates that are trapped y Earth s gravitational field. Since the wavelength of a typical optical carrier is comparale to these molecule and particle sizes, the carrier wave is suject to various propagation effects that are uncommon to F systems. One such effect is scintillation, caused y atmospheric turulence, and refers to random fluctuations in the irradiance of the received optical laser eam (analogous to fading in F systems) [2 4]. ecent works on the mitigation of scintillation concentrate on the use of multiple-lasers and multiple-apertures to create a multiple-input-multiple-output (MIMO) channel [5 13]. Many of these works consider scintillation as an ergodic fading process, and analyse the channel in terms of its ergodic capacity. However, compared to typical data rates, scintillation is a slow timevarying process (with a coherence time on the order of milliseconds), and it is therefore more appropriate to analyse the outage proaility of the channel. To some extent, this has een done in the works of [6, 10, 12 14]. In [6, 13] the outage proaility of the MIMO FSO channel is analysed under the assumption of ideal photodetection (i.e. a Poisson counting process) with no andwidth constraints. Wilson et al. [10] also assume perfect photodetection, ut with the further constraint of pulse-position modulation (PPM). Lee and Chan [12], study the outage proaility under the assumption of on-off keying (OOK) transmission and non-ideal photodetection, i.e. the comined shot noise and thermal noise process is modeled as zero mean signal independent additive white Gaussian noise (AWGN). Farid and Hranilovic [14] extend this analysis to include the effects of pointing errors. In this paper we study the outage proaility of the MIMO FSO channel under the assumptions of PPM, non-ideal photodetection, and equal gain comining (EGC) at the receiver. In particular, we model the channel as a quasi-static lock fading channel wherey communication takes place over a finite numer of locks and each lock of transmitted symols experiences an independent identically distriuted (i.i.d.) fading realisation [15, 16]. We consider two types of CSI knowledge. First we assume perfect CSI is availale only at the receiver (CSI case), and the

3 3 transmitter knows only the channel statistics. Then we consider the case when perfect CSI is also known at the transmitter (CSIT case). 1 Under this framework we study a numer of scintillation distriutions: lognormal, modelling weak turulence; exponential, modelling strong turulence; and gamma-gamma [17], which models a wide range of turulence conditions. For the CSI case, we derive signal-to-noise ratio (SN) exponents and show that they are the product of: a channel related parameter, dependent on the scintillation distriution; the numer of lasers times the numer of apertures, reflecting the spatial diversity; and the Singleton ound [18 20], reflecting the lock diversity. For the CSIT case, the transmitter finds the optimal power allocation that minimises the outage proaility [21]. Using results from [22], we derive the optimal power allocation suject to short- and long-term power constraints. We show that very large power savings are possile compared to the CSI case. Interestingly, under a long-term power constraint, we show that delay-limited capacity [23] is zero for exponential and (in some cases) gamma-gamma scintillation, unless one codes over multiple locks, and/or uses multiple lasers and apertures. The paper is organised as follows. In Section II, we define the channel model and assumptions. In Section III we review the lognormal, exponential and gamma-gamma models. Section IV defines the outage proaility and presents results on the minimum-mean squared error (MMSE). Then in Sections V and VI we present the main results of our asymptotic outage proaility analysis for the CSI and CSIT cases, respectively. Concluding remarks are then given in Section VII. Proofs of the various results can e found in the Appendices. II. SYSTEM MODEL We consider an M N MIMO FSO system with M transmit lasers an N aperture receiver as shown in Fig. 1. Information data is first encoded y a inary code of rate c. The encoded stream is modulated according to a Q-ary PPM scheme, resulting in rate = c log 2 Q (its/channel use). epetition transmission is employed such that the same PPM signal is transmitted in perfect synchronism y each of the M lasers through an atmospheric turulent channel and collected y N receive apertures. We assume the distance etween the individual lasers and apertures is 1 Given the slow time-varying scintillation process, CSI can e estimated at the receiver and fed ack to the transmitter via a dedicated feedack link.

4 4 sufficient so that spatial correlation is negligile. At each aperture, the received optical signal is converted to an electrical signal via photodetection. Non-ideal photodetection is assumed such that the comined shot noise and thermal noise processes can e modeled as zero mean, signal independent AWGN (an assumption commonly used in the literature, see e.g. [3 5, 12, 14, 24 29]). In FSO communications, channel variations are typically much slower than the signaling period. As such, we model the channel as a non-ergodic lock-fading channel, for which a given codeword of length BL undergoes only a finite numer B of scintillation realisations [15, 16]. The received signal at aperture n, n = 1,..., N can e written as ( M ) y n [l] = p x [l] + z n [l], (1) m=1 h m,n for = 1,..., B, l = 1,..., L, where y n [l], zn [l] Q are the received and noise signals at lock, time instant l and aperture n, x [l], Q is the transmitted signal at lock and time instant l, and h m,n denotes the scintillation fading coefficient etween laser m and aperture n. Each transmitted symol is drawn from a PPM alphaet, x [l] X ppm = {e 1,..., e Q }, where e q is the canonical asis vector, i.e., it has all zeros except for a one in position q, the time slot where the pulse is transmitted. The noise samples of z n [l] are independent realisations of a random variale Z N (0, 1), and p denotes the received electrical power of lock at each aperture in the asence of scintillation. The fading coefficients h m,n of a random variale H with proaility density function (pdf) f H(h). are independent realisations At the receiver, we assume equal gain comining (EGC) is employed, such that the entire system is equivalent to a single-input single-output (SISO) channel, i.e. y [l] = 1 N N n=1 y n [l] = p h x [l] + z [l], (2) where z [l] = 1 N N n=1 zn [l] N (0, 1), and h, a realisation of the random variale H, is defined as the normalised comined fading coefficient, i.e. h = c MN M N m=1 n=1 h m,n, (3)

5 5 where c = 1/(E[ H] 1 + σ 2 I /(MN)) is a constant to ensure E[H2 ] = 1. 2 instantaneous received electrical power at lock is p received SN is snr E[h p ] = E[p ]. Thus, the total = M 2 N p /c, and the total average For the CSI case, we assume the electrical power is distriuted uniformly over the locks, i.e., p = p = snr for = 1,..., B. Otherwise, for the CSIT case, we will allocate electrical power in order to improve performance. In particular, we will consider the following two electrical power constraints Short-term: Long-term: 1 B p P (4) B =1 [ ] 1 B E p P. (5) B Throughout the paper, we will devote special attention to the case of B = 1, i.e., the channel does not vary within a codeword. This scenario is relevant for FSO, since, due to the large data-rates, one is ale to transmit millions of its over the same channel realisation. We will see that most results admit very simple forms, and some cases, even closed form. This analysis allows for a system characterisation where the expressions highlight the roles of the key design parameters. =1 III. SCINTILLATION DISTIBUTIONS The scintillation pdf, f H(h), is parameterised y the scintillation index (SI), σi 2 Var( H), (6) (E[ H]) 2 and can e considered as a measure of the strength of the optical turulence under weak turulence conditions [17, 30]. The distriution of the irradiance fluctuations is dependent on the strength of the optical turulence. For the weak turulence regime, the fluctuations are generally considered to e 2 For optical channels with ideal photodetection, the normalisation E[H] = 1 is commonly used to keep optical power constant. We assume non-ideal photodetection and work entirely in the electrical domain. Hence, we chose the normalisation E[H 2 ] = 1, used commonly in F fading channels. However, since we consider only the asymptotic ehaviour of the outage proaility, the specific normalisation is irrelevant and does not affect our results.

6 6 lognormal distriuted, and for very strong turulence, exponential distriuted [2, 31]. For moderate turulence, the distriution of the fluctuations is not well understood, and a numer of distriutions have een proposed, such as the lognormal-ice distriution [4, 17, 32 34] (also known as the Beckmann distriution [35]) and K-distriution [32]. In [17], Al-Haash et al. proposed a gamma-gamma distriution as a general model for all levels of atmospheric turulence. Moreover, recent work in [34] has shown that the gamma-gamma model is in close agreement with experimental measurements under moderate-to-strong turulence conditions. In this paper we focus on lognormal, exponential, and gamma-gamma distriuted scintillation, which are descried as follows. For lognormal distriuted scintillation, f lñ H (h) = 1 hσ 2π exp ( (log h µ) 2 /(2σ 2 ) ), (7) where µ and σ are related to the SI via µ = log(1 + σ 2 I ) and σ2 = log(1 + σ 2 I ). For exponential distriuted scintillation f exp (h) = λ exp( λh) (8) H which corresponds to the super-saturated turulence regime, where σ 2 I = 1. The gamma-gamma distriution arises from the product of two independent Gamma distriuted random variales and has the pdf [17], f gg H α+β 2(αβ) 2 (h) = Γ(α)Γ(β) h α+β 2 1 K α β (2 αβh), (9) where K ν (x) denotes the modified Bessel function of the second kind [36, Ch. 10]. The parameters α and β are related with the scintillation index via σ 2 I = α 1 + β 1 + (αβ) 1. IV. INFOMATION THEOETIC PELIMINAIES The channel descried y (2) under the quasi-static assumption is not information stale [37] and therefore, the channel capacity in the strict Shannon sense is zero. It can e shown that the codeword error proaility of any coding scheme is lower ounded y the information outage proaility [15, 16], P out (snr, ) = Pr(I(p, h) < ), (10)

7 7 where is the transmission rate and I(p, h) is the instantaneous input-output mutual information for a given power allocation p (p 1,..., p B ), and vector channel realisation h (h 1,..., h B ). The instantaneous mutual information can e expressed as [38] I(p, h) = 1 B B I awgn (p h 2 ), (11) =1 where I awgn (ρ) is the input-output mutual information of an AWGN channel with SN ρ. For PPM [24] I awgn (ρ) = log 2 Q E where Z q N (0, 1) for q = 1,..., Q. [log 2 ( 1 + Q q=2 e ρ+ ρ(z q Z 1 ) )], (12) For the CSIT case we will use the recently discovered relationship etween mutual information and the MMSE [39]. This relationship states that 3 d dρ Iawgn (ρ) = mmse(ρ) log(2) where mmse(ρ) is the MMSE in estimating the input from the output of a Gaussian channel as a function of the SN ρ. For PPM, we have the following result Theorem 4.1: The MMSE for PPM on the AWGN channel with SN ρ is ρ( mmse(ρ) = 1 E e2 ρ+z1 ) + (Q 1)e 2 ρz 2 ) 2, (14) (e ρ+ ρz 1 + Qk=2 e ρz k (13) where Z i N (0, 1) for i = 1,..., Q. Proof: See Appendix I. Note that oth (12) and (14) can e evaluated using standard Monte-Carlo methods. V. OUTAGE POBABILITY ANALYSIS WITH CSI For the CSI case, we employ uniform power allocation, i.e. p 1 =... = p B = snr. For codewords transmitted over B locks, otaining a closed form analytic expression for the outage proaility is intractale. Even for B = 1, in some cases, for example the lognormal and gammagamma distriutions, determining the exact distriution of H can e a difficult task. Instead, as 3 The log(2) term arises ecause we have defined I awgn (ρ) in its/channel usage.

8 8 we shall see, otaining the asymptotic ehaviour of the outage proaility is sustantially simpler. Towards this end, and following the footsteps of [20, 40], we derive the SN exponent. Theorem 5.1: The outage SN exponents for a MIMO FSO communications system modeled y (2) are given as follows: d ln (log snr) 2 = MN 8 log(1 + σ 2 I ) (1 + B (1 c) ) (15) d exp (log snr) = MN (1 + B (1 c ) ), 2 (16) d gg (log snr) = MN min(α, β) (1 + B (1 c ) ), 2 (17) for lognormal, exponential, and gamma-gamma cases respectively, where c = / log 2 (Q) is the rate of the inary code and d (log snr) k Proof: See Appendix II. log P out (snr, ) = lim k = 1, 2. (18) snr (log snr) k Proposition 5.1: The outage SN exponents given in Theorem 5.1, are achievale y random coding over PPM constellations whenever B (1 c ) is not an integer. Proof: The proof follows from the proof of Theorem 5.1 and the proof of [20, Th. 1]. The aove proposition implies that the outage exponents given in Theorem 5.1 are the optimal SN exponents over the channel, i.e. the outage proaility is a lower ound to the error proaility of any coding scheme, its corresponding exponents (given in Theorem 5.1) are an upper ound to the exponent of coding schemes. From Proposition 5.1, we can achieve the outage exponents with a particular coding scheme (random coding, in this case), and therefore, the exponents in Theorem 5.1 are optimal. From (15)-(17) we immediately see the enefits of spatial and lock diversity on the system. In particular, each exponent is proportional to: the numer of lasers times the numer of apertures, reflecting the spatial diversity; a channel related parameter that is dependent on the scintillation distriution; and the Singleton ound, which is the optimal rate-diversity tradeoff for ayleighfaded lock fading channels [18 20]. Comparing the channel related parameters in (15)-(17) the effects of the scintillation distriution on the outage proaility are directly visile. For the lognormal case, the channel related parameter is 8 log(1 + σ 2 I ) and hence is directly linked to the SI. Moreover, for small σ2 I < 1,

9 9 8 log(1+σ 2 I ) 8σ2 I and the SN exponent is inversely proportional to the SI. For the exponential case, the channel related parameter is a constant 1/2 as expected, since the SI is constant. For the gamma-gamma case the channel related parameter is min(α, β)/2, which highlights an interesting connection etween the outage proaility and recent results in the theory of optical scintillation. For gamma-gamma distriuted scintillation, the fading coefficient results from the product of two independent random variales, i.e. H = XY, where X and Y model fluctuations due to large scale and small scale cells. Large scale cells cause refractive effects that mainly distort the wave front of the propagating eam, and tend to steer the eam in a slightly different direction (i.e. eam wander). Small scale cells cause scattering y diffraction and therefore distort the amplitude of the wave through eam spreading and irradiance fluctuations [4, p. 160]. The parameters α, β are related to the large and small scale fluctuation variances via α = σ 2 X β = σ 2 Y. For a plane wave (neglecting inner/outer scale effects) σ2 Y > σ2 X, and as the strength of the optical turulence increases, the small scale fluctuations dominate and σ 2 Y and 1 [4, p. 336]. This implies that the SN exponent is exclusively dependent on the small scale fluctuations. Moreover, in the strong fluctuation regime, σ 2 Y 1, the gamma-gamma distriution reduces to a K-distriution [4, p. 368], and the system has the same SN exponent as the exponential case typically used to model very strong fluctuation regimes. In comparing the lognormal exponent with the other cases, we oserve a striking difference. For the lognormal case (15) implies the outage proaility is dominated y a (log(snr)) 2 term, whereas for exponential and gamma-gamma scintillation it is dominated y a log(snr) term. Thus the outage proaility decays much more rapidly with SN for the lognormal case than it does for the exponential or gamma-gamma cases. Furthermore, for the lognormal case, the slope of the outage proaility curve, when plotted on a log-log scale, will not converge to a constant value. In fact, a constant slope curve will only e oserved when plotting the outage proaility on a log-(log) 2 scale. In deriving (15) (see Appendix II-A) we do not rely on the lognormal approximation 4, which has een used on a numer occasions in the analysis of FSO MIMO channels, e.g. [5, 12, 29]. Under this approximation, H is lognormal distriuted (7) with parameters µ = log(1 + 4 This refers to approximating the distriution of the sum of lognormal distriuted random variales as lognormal [41 44].

10 10 σ 2 I /(MN)) and σ2 = µ, and we otain the approximated exponent 1 d (log snr) 2 8 log(1 + σ2 I ) (1 + B (1 c) ). (19) MN Comparing (15) and (19) we see that although the lognormal approximation also exhiits a (log(snr)) 2 term, it has a different slope than the true SN exponent. The difference is due to the approximated and true pdfs having different ehaviours in the limit as h 0. However, for very small σ 2 I approximately equal. < 1, using log(1 + x) x (for x < 1) in (15) and (19) we see that they are For the special case of single lock transmission, B = 1, it is straightforward to express the outage proaility in terms of the cumulative distriution function (cdf) of the scintillation random variale, i.e. where F H (h) denotes the cdf of H, and snr awgn ( ) snr awgn P out (snr, ) = F H snr (20) = I awgn, 1 () denotes the SN value at which the mutual information is equal to. Tale I reports these values for Q = 2, 4, 8, 16 and = c log 2 Q, with c = 1, 1, 3. Therefore, for B = 1, we can compute the outage proaility analytically when the distriution of H is availale, i.e., in the exponential case for M, N 1 or in the lognormal and gamma-gamma cases for M, N = 1. In the case of exponential scintillation we have that P out (snr, ) = Γ ( MN, (MN(1 + MN) snrawgn snr ) ) 1 2, (21) where Γ(a, x) 1 x Γ(a) 0 ta 1 exp( t) dt denotes the regularised (lower) incomplete gamma function [36, p.260]. For the lognormal and gamma-gamma scintillation with M N > 1, we must resort to numerical methods. This involved applying the fast Fourier transform (FFT) to f H to numerically compute its characteristic function, taking it to the MNth power, and then applying the inverse FFT to otain f H. This method yields very accurate numerical computations of the outage proaility in only a few seconds. Outage proaility curves for the B = 1 case are shown on the left in Fig. 2. For the lognormal case, we see that the curves do not have constant slope for large SN, while, for the exponential and gamma-gamma cases, a constant slope is clearly visile. We also see the enefits of MIMO, particularly in the exponential and gamma-gamma cases, where the SN exponent has increased from 1/2 and 1 to 2 and 4 respectively.

11 11 VI. OUTAGE POBABILITY ANALYSIS WITH CSIT In this section we consider the case where the transmitter and receiver oth have perfect CSI knowledge. In this case, the transmitter determines the optimal power allocation that minimises the outage proaility for a fixed rate, suject to a power constraint [21]. The results of this section are ased on the application of results from [22] to PPM and the scintillation distriutions of interest. Using these results we uncover new insight as to how key design parameters influence the performance of the system. Moreover, we show that large power savings are possile compared to the CSI case. For the short-term power constraint given y (4), the optimal power allocation is given y mercury-waterfilling at each channel realisation [22, 45], p = 1 { Q 1 mmse (min 1 h 2 Q, η h 2 }), (22) for = 1,..., B where mmse 1 (u) is the inverse-mmse function and η is chosen to satisfy the power constraint. 5 From [22, Prop. 1] it is apparent that the SN exponent for the CSIT case under short-term power constraints is the same as the CSI case. For the long-term power constraint given y (5) the optimal power allocation is [22], B =1 p = s 0, otherwise, where = 1 { }) Q 1 mmse (min 1 h 2 Q, 1, = 1,..., B (24) ηh 2 [ ] 1 and s is a threshold such that s = if lim s E B (s) B =1 P, and { } (s) h B + : 1 B s, (25) B =1 [ ] 1 otherwise, s is chosen such that P = E B (s) B =1. In (24), η is now chosen to satisfy the rate constraint 1 B B =1 (23) ( { })) Q 1 I awgn mmse (min 1 Q, 1 = (26) ηh 2 5 Note that in [22, 45], the minimum in (22) is etween 1 and η. For QPPM, mmse(0) = Q 1 (see (14)). Hence we must h 2 Q replace 1 with Q 1. Q

12 12 From [22], the long-term SN exponent is given y d st (log snr) d lt 1 d (log snr) = st (log snr) where d st (log snr) d st (log snr) < 1 d st (log snr) > 1, (27) is the short-term SN exponent, i.e., the SN exponents (15)-(17). Note that d lt (log snr) = implies the outage proaility curve is vertical, i.e. the power allocation scheme (23) is ale to maintain constant instantaneous mutual information (11). The maximum achievale rate at which this occurs is defined as the delay-limited capacity [23]. From (27) and (15)-(17), we therefore have the following corollary. Corollary 6.1: The delay-limited capacity of the channel descried y (2) with CSIT suject to long-term power constraint (5) is zero whenever 2 (1 + B (1 c ) ) 1 exponential MN. (28) 2 (1 + B (1 min(α,β) c) ) 1 gamma-gamma For lognormal scintillation, delay-limited capacity is always nonzero. Corollary 6.1 outlines fundamental design criteria for nonzero delay-limited capacity in FSO communications. Single lock transmission (B = 1) is of particular importance given the slow time-vary nature of scintillation. From (28), to otain nonzero delay-limited capacity with B = 1, one requires MN > 2 and MN > 2/ min(α, β) for exponential and gamma-gamma cases respectively. Note that typically, α, β 1. Thus a 3 1, 1 3 or 2 2 MIMO system is sufficient for most cases of interest. In addition, for the special case B = 1, the solution (24) can e determined explicitly since Therefore, η = ( h 2 mmse(i awgn, 1 ()) ) 1 = ( h 2 mmse(snr awgn ) ) 1. (29) opt = snrawgn. (30) h 2 Intuitively, (30) implies that for single lock transmission, whenever snr awgn /h 2 s, one simply transmits at the minimum power necessary so that the received instantaneous SN is equal to the SN threshold (snr awgn ) of the code. Otherwise, transmission is turned off. Thus an outage snr occurs whenever h < awgn s and hence ( ) snr awgn P out (snr, ) = F H (31) γ 1 (snr)

13 13 where γ 1 (snr) is the solution to the equation γ(s) = snr, i.e., where ν γ(s) snr awgn ν f H (h) h 2 dh, (32) snr awgn s. Moreover, the snr at which P out (, snr) 0 is precisely lim s γ(s). In other words, the minimum long-term average SN required to maintain a constant mutual information of its per channel use, denoted y snr, is Hence, recalling that snr awgn snr awgn = snr awgn of PPM) is 6 Cd(snr) = I awgn ( 0 f H (h) h 2 dh. (33) = I awgn, 1 (), the delay-limited capacity (under the constraint 0 snr f H (h) h 2 dh ). (34) In the cases where the distriution of H is known in closed form, (32) can e solved explicitly, hence yielding the exact expressions for outage proaility (31) and delay-limited capacity (34). For lognormal distriuted scintillation with B = M = N = 1, we have that ( γ ln (s) = 1 3 log(1 + σ 2 2 snrawgn (1 + σi) 2 4 erfc I ) + 1 log snrawgn 2 1 log s ) 2, (35) 2 log(1 + σ 2 I ) and C ln d (snr) = I awgn ( snr (1 + σ 2 I )4 where we have explicitly solved the integrals in (32) and (34) respectively. ), (36) For the exponential case with B = 1, we otain, ( ) γ exp (s) = snr awgn MN(1 + MN) (MN 1)(MN 2) Γ MN 2, MN(1 + MN) snrawgn, (37) s and C exp d (snr) = ( I awgn (MN 1)(MN 2) MN(1+MN) ) snr MN > 2 0 otherwise. (38) 6 Note that a similar expression was derived in [23].

14 14 For the gamma-gamma case with B = M = N = 1, γ gg (s) can e expressed in terms of hypergeometric functions, which are omitted for space reasons. The delay-limited capacity, however, reduces to a simpler expression 7 ( ) C gg d (snr) = I awgn (α 2)(α 1)(β 2)(β 1) snr α, β > 2 (αβ)(α+1)(β+1) 0 otherwise. Fig. 2 (right) compares the outage proaility for the B = 1 CSIT case (with long-term power constraints) for each of the scintillation distriutions. For MN = 1 we see that the outage curve is vertical only for the lognormal case, since C d = 0 for the exponential and gamma-gamma cases. In these cases one must code over multiple locks for C d > 0, i.e. from Corollary 6.1, B 6 and B 4 for the exponential and gamma-gamma cases respectively (with c = 1/2). Comparing the CSI and CSIT cases in Fig. 2 we can see that very large power savings are possile when CSI is known at the transmitter. These savings are further illustrated in Tale II, which compares the SN required to achieve P out < 10 5 (denoted y snr ) for the CSI case, and the long-term average SN required for P out 0 in the CSIT case (denoted y snr, which is given y (33)). Note that in the CSIT case, the values of snr given in the parentheses is the minimum SN required to achieve P out < 10 5, since C d = 0 for these cases (i.e. snr = ). From Tale II we see that the power saving is at least around 15 db, and in some cases as high as 50 db. We also see the comined enefits of MIMO and power control, e.g. at MN = 4, the system is only 3.7 db (lognormal) to 5.2 db (exponential) from the capacity of nonfading PPM channel (snr awgn 1/2 = 3.18 db). (39) VII. CONCLUSION In this paper we have analysed the outage proaility of the MIMO Gaussian FSO channel under the assumption of PPM and non-ideal photodetection, for lognormal, exponential and gamma-gamma distriuted scintillation. When CSI is known only at the receiver, we have shown that the SN exponent is proportional to the numer lasers and apertures, times a channel related parameter (dependent on the scintillation distriution), times the Singleton ound, even in the cases where a closed form expression of the equivalent SISO channel distriution is not availale 7 Note that since we assume the normalisation E[H 2 ] = 1, then 0 and f gg H (h) is defined as in (9) such that E[ H] = 1. f H (h) h 2 dh = 1 c 2 0 f gg H (u) du, where c = 1/ p 1 + σ 2 u 2 I

15 15 in closed-form. When the scintillation is lognormal distriuted, we have shown that the outage proaility is dominated y a (log(snr)) 2 term, whereas for the exponential and gamma-gamma cases it is dominated y a log(snr) term. When CSI is also known at the transmitter, we applied the power control techniques of [22] to PPM to show that very significant power savings are possile. APPENDIX I POOF OF THEOEM 4.1 Suppose PPM symols are transmitted over an AWGN channel, the non-fading equivalent of (2). The received noisy symols are given y y = ρx + z, where x X ppm (we have dropped the time index l for revity of notation). Using Bayes rule [46], the MMSE estimate is ˆx = E [x y] = Q q=1 e q exp( ρy q ) Q k=1 exp( ρy k ). (40) From (40) the ith element of ˆx is ˆx i = exp( ρy i ) Q k=1 exp( ρy k ). (41) Using the orthogonality principle [47] mmse(ρ) = E [ x ˆx 2 ] = E[ x 2 ] E[ ˆx 2 ]. Since e q 2 = 1 for all q = 1,..., Q, then E[ x 2 ] = 1. Due to the symmetry of QPPM we need only consider the case when x = e 1 was transmitted. Hence, mmse(ρ) = 1 ( E[ˆx 2 1] + (Q 1)E[ˆx 2 2] ). (42) Now y 1 = ρ + z 1 and y i = z i for i = 2,..., Q, where z q is a realisation of a random variale Z q N (0, 1) for q = 1,..., Q. Hence, sustituting these values in (41) and taking the expectation (42) yields the result given the theorem. APPENDIX II POOF OF THEOEM 5.1 We egin y defining a normalised (with respect to SN) fading coefficient, ζ m,n which has a pdf f ζ m,n(ζ) = = 2 log h m,n log snr, log snr ) e 1 2 ζ log snr f 2 H (e 1 2 ζ log snr. (43)

16 16 Since we are only concerned with the asymptotic outage ehaviour, the scaling of the coefficients is irrelevant, and to simplify our analysis we assume E[Ĥ2 ] = 1. Hence the instantaneous SN for lock is given y for = 1,..., B. Therefore, where ζ = (ζ 1,1 ( ρ = snrh 2 1 = MN lim snr Iawgn (ρ ) = M N m=1 n=1 0 if all ζ m,n > 1 log 2 Q snr 1 2(1 ζ m,n )) 2 (44) at least one ζ m,n < 1 = log 2 Q (1 11{ζ 1}),..., ζ M,N ), 11{ } denotes the indicator function, 1 = (1,..., 1) is a 1 MN vector of 1 s, and the notation a for vectors a, k means that a i > i for i = 1,..., k. From the definition of outage proaility (10), we have P out (snr, ) = Pr(I h (snr) < ) = A f(ζ)dζ (45) where ζ = (ζ 1,..., ζ B ) is a 1 BMN vector of normalised fading coefficients, f(ζ) denotes their joint pdf, and A = { ζ BMN : } B 11{ζ 1} > B (1 c ) =1 is the asymptotic outage set. We now compute the asymptotic ehaviour of the outage proaility, i.e. A. Lognormal case (46) lim log P out(snr, ) = lim log f(ζ)dζ. (47) snr snr A From (7) and (43) we otain the joint pdf, ( f(ζ) =. (log snr)2 exp 8σ 2 B M N =1 m=1 n=1 ) (ζ m,n ) 2, (48) where we have ignored terms of order less than (log snr) 2 in the exponent and constant terms independent of ζ in front of the exponential. Comining (47), (48), and using Varadhan s lemma [48], { B } lim log P (log snr)2 M N out(snr, ) = inf (ζ m,n snr 8σ 2 ) 2 A =1 m=1 n=1

17 17 The aove infimum occurs when any κ of the ζ vectors are such that ζ 1 and the other B κ vectors are zero, where κ is a unique integer satisfying Hence, it follows that κ = 1 + B (1 c ) and thus, lim snr log P out(snr, ) = κ < B (1 c ) κ + 1. (49) (log snr)2 8σ 2 MN (1 + B (1 c ) ). (50) Dividing oth sides of (50) y (log snr) 2 the SN exponent (15) is otained. B. Exponential case From (8) and (43) we otain the joint pdf, ( f(ζ) =. exp log snr MN 2 B M N =1 m=1 n=1 ζ m,n ), (51) where we have ignored exponential terms in the exponent and constant terms independent of ζ in front of the exponential. Following the same steps as the lognormal case i.e. the defining the same asymptotic outage set and application of Varadhan s lemma [48], the SN exponent (16) is otained. C. Gamma-gamma case Let us first assume α > β. From (9) and (43) we otain the joint pdf, f ζ m,n(ζ) =. exp ( β2 ) ζ log snr, ζ > 0 (52) for large snr, where we have used the approximation K ν (x) 1 2 Γ(ν)( 1 2 x) ν for small x and ν > 0 [36, p. 375]. The extra condition, ζ > 0, is required to ensure the argument of the Bessel function approaches zero as snr to satisfy the requirements of the aforementioned approximation. For the case β > α we need only swap α and β in (52). Hence we have the joint pdf f(ζ). = exp ( min(α, β) log snr 2 B M N =1 m=1 n=1 ζ m,n ), ζ 0. (53) Now, following the same steps as in the lognormal and exponential cases, with the additional constraint ζ 0, the SN exponent (17) is otained

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21 FIGUES 21 Transmitter eceiver PPM Laser 1 1 Aperture PD Encoder. Atmospheric Turulence. Decoder PPM Laser M N Aperture PD Fig. 1. Block diagram of an M N MIMO FSO system.

22 FIGUES M N = M N = Pout(snr, ) M N = 4 Pout(snr, ) M N = LN Exp GG snr (db) LN Exp GG snr (db) Fig. 2. Outage proaility curves for the CSI (left) and CSIT (right) cases with σi 2 = 1, B = 1, Q = 2, c = 1/2, snr awgn 1/2 = 3.18 db: lognormal (solid); exponential (dashed); and, gamma-gamma distriuted scintillation (dot-dashed), α = 2, β = 3.

23 TABLES 23 MINIMUM SIGNAL-TO-NOISE ATIO snr awgn TABLE I (IN DECIBELS) FO ELIABLE COMMUNICATION FO TAGET ATE = c log 2 Q. Q c = 1 4 c = 1 2 c =

24 TABLES 24 TABLE II COMPAISON OF CSI AND CSIT CASES WITH B = 1, = 1/2, Q = 2 σ 2 I = 1, α = 2, β = 3. BOTH snr AND snr AE MEASUED IN DECIBELS. lognormal exponential gamma-gamma MN snr snr snr snr snr snr (56.2) 65.6 (24.5) (17.8)