UBC Faculty Research and Publications

Free-Space Optical Communications Over Lognormal Fading Channels Using OOK With Finite Extinction Ratios Yang, Luanxia; Song, Xuegui; Cheng, Julian; Holzman, Jonathan F. 2016

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1Free-Space Optical Communications OverLognormal Fading Channels Using OOK WithFinite Extinction RatiosLuanxia Yang, Xuegui Song, Julian Cheng, Senior Member, IEEE, and Jonathan F. Holzman, Member, IEEEAbstract—Free-space optical communication links operatingover lognormal turbulence channels using on-off keying (OOK)are studied in this work. Such systems can suffer from irreducibleerror floors that result from the use of demodulation with fixedand unoptimized detection thresholds. The resulting error floorsare analyzed for the general case of low and high state offsets(i.e., finite extinction ratios). An electrical signal-to-noise ratio(SNR) optimized detection system is applied. The system usesthe electrical SNRs to implement adaptive detection thresholdsand eliminate the error floors. The system can accommodateoperation with finite extinction ratios, as it uses the methodof moments and maximum likelihhod estimation techniques toestimate the low and high state offsets and electrical SNR.Numerical results show that the SNR gap between the electrical-SNR-optimized detection system and the adaptive detectionsystem is 2:3 dB at a bit-error rate of 10−5 without a stateoffset. The SNR gap increases to 4:5 dB with a state offset of = 0:2.Index Terms—Atmospheric turbulence, optical wireless com-munications, on-off keying.I. INTRODUCTIONFree-space optical (FSO) communication links have impor-tant advantages over radio frequency links. Such FSO systemsoffer broadband operation, high link security, and freedomfrom spectral license regulations. But optical signals that aretransmitted over free-space are subject to amplitude and phasedistortion due to transient inhomogeneities of atmospherictemperature and pressure [1], [2]. The resulting scintillationor fading is a major cause of performance degradation forFSO systems. The performance degradation is especially pro-nounced for FSO systems using irradiance modulation anddirect detection (IM/DD) with on-off keying (OOK) and fixeddetection thresholds that are non-adaptive and inherently unop-timized [3], [4]. This can produce irreducible error floors whenoperation is extended to high signal-to-noise ratios (SNRs) [5],[6].Attempts to overcome the irreducible error floors of OOKIM/DD systems have focused on the application of adaptivedetection thresholds. Adaptive detection of OOK signal canbe classified into three categories. The first category, idealizedadaptive detection, applies bit-by-bit adaptations to the detec-tion threshold on the (typically) nanosecond timescale of thebit interval [2], [7]. The second category, quasi-static adaptiveL. Yang, X. Song, J. Cheng and J. F. Holzman are with the School ofEngineering, The University of British Columbia, Kelowna, BC, Canada. (e-mail: fluanxia.yang, xuegui.song, julian.cheng, jonathan.holzmang@ubc.ca)This work was supported by an NSERC Discovery Grant.detection, applies adaptations to the detection threshold onthe (typically) millisecond timescale of the turbulence coher-ence time [8], [9], [10]. The third category, electrical-SNR-optimized detection [11], [12], [13], unlike the prior two cate-gories, does not require knowledge of the instantaneous chan-nel state information (CSI), and it applies adaptations to thedetection threshold on the second- or minute-long timescaleover which the turbulence exhibits stationary statistics [14].Operation with electrical-SNR-optimized detection offers thepractical advantages of operation with fixed detection thresh-olds, as only slow adaptations are needed to define the de-tection thresholds1. Operation with electrical-SNR-optimizeddetection also offers the performance advantages of operationwith idealized adaptive detection thresholds, as it avoidsirreducible error floors. Unfortunately, existing electrical-SNR-optimized systems must assume perfect knowledge of theelectrical SNR and turbulence probability distribution function(pdf).While perfect knowledge of the turbulence distribution canbe difficult to realize, it is possible to determine and makeuse of the statistical moments of the turbulence. The authorsin [15]- [18] have, for example, made use of moments withPearson curves, John curves and saddlepoints to approximatepdfs, albeit with somewhat restrictive conditions [19]. Sim-ilarly, the authors in [20] have recently derived unified pdfformulas based on the generalized Laguerre polynomial seriesexpansion, although its parameters depend on the type offading distribution.In light of above systems and limitations, the contributionsof this paper are as follows:1) Novel analytical error floor expressions are derived for ageneral representation of practical FSO links with finitelow and high state offsets (i.e., finite extinction ratios)operating over various turbulence channels.2) The electrical-SNR-optimized system is implementedwithout requiring perfect knowledge of the instantaneousCSI and turbulence pdf. The turbulence pdf is approxi-mated by a sum of Laguerre polynomials. With perfectknowledge of the turbulence pdf and no state offsets, theoptimum detection rule reduces to that shown in [11],[12].3) Method of moments estimation (MoME) and maximumlikelihhood estimation (MLE) are used to estimate the1On the timescale of stationary statistics, the electrical SNR is constant,and the detection threshold is a fixed detection threshold.2state offsets and electrical SNR, and the electrical-SNR-optimized system is then employed to operate withoutirreducible error floors.The reminder paper is organized as follows. Section IIdescribes the system and channel models (for a finite extinc-tion ratio). Section III derives the irreducible error floors ofOOK with fixed detection thresholds. Section IV introducesour electrical-SNR-optimized detection system for operationwith unknown turbulence model. Section V addresses theestimation of unknown state offsets for operation with finiteextinction ratios and electrical SNRs. Section VI presentsnumerical results and discussions. Section VII makes someconcluding remarks.II. SYSTEM AND CHANNEL MODELSIn an OOK IM/DD system, the transmitted intensity is apositive quantity that can be expressed assˆ(t) = 1 +  +∑ivig(t− iip) (1)where vi ∈ {−1P 1} is the data, and ip is the symbol duration.In (1), pulse shaping is defined as g(t) = 1 for 0 Q t Q ip,and g(t) = 0 otherwise, and the positive parameter  is thelow and high state offset that results from operation with afinite extinction ratio2 [21], i.e., extinction ratio = (2 + )R.Finite extinction ratios are due to practical considerations forsemiconductor laser transmitters, which often operate withfinite power levels for the low and high states. Typical valuesof  are between 0:1053 and 0:2857 [22]. When  ̸= 0, the lowand high states of the received electrical signal are affectedby turbulence. When  = 0, the received electrical signalspecializes to the classical model discussed in [5].The signal sˆ(t) is transmitted through an atmospheric tur-bulence channel and is distorted by a multiplicative intensityprocess I(uP t). The received electrical signal after photode-tection can be written asr(t) = g[(1 + )I(uP t) +∑iI(uP t)vig(t− iip)] + n(uP t):(2)The photodetector responsivity, without loss of generality, isg = 1, and I(uP t) is assumed to be a normalized stationaryrandom process for signal scintillation caused by atmosphericturbulence and is modeled as lognormal distribution in thiswork, where u is an event in the sample space. The termn(uP t) is additive white Gaussian noise process due to thermalnoise and/or ambient shot noise. Using a p-i-n photodiode andfollowing [11], the shot noise is assumed to be dominatedby ambient shot noise. (Both ambient shot noise and thermalnoise are statistically independent of the desired signal.) Thetotal noise power is 2g = 2s + 2i , where 2s and 2i denotethe respective ambient shot noise power and the thermal noisepower.The received signal is sampled at time ip. The sampleI(uP t = ip) is a random variable (RV) I , and the sample2Extinction ratio, when used to describe the performance of an opticaltransmitter used in digital communications, is simply the ratio of the powerused to transmit a logic state \1", to the power used to transmit a logic state\0".n(uP t = ip) is a RV c having zero mean and variance2g = c0R2, where c0 is the noise power spectral density.If “0” is transmitted, s0 is true and the laser is in the lowstate, so the sample for demodulation is r|s( = I + c: If“1” is transmitted, s1 is true and the laser is in the high state,so the sample for demodulation is r|s) = (2 + )I + c . Itis important to note that the nonzero state offset  leads toturbulence dependence for the received signal when s0 or s1is true.The common statistical models that are used to character-ize atmospheric turbulence channels are the lognormal, K,negative exponential, and Gamma-Gamma models [23]. Thelognormal distribution characterizes weak turbulence and issuitable for characterizing FSO communications in clear skylinks over several hundred meters [24]. The K-distribution issuitable for describing strong turbulence over links that areseveral kilometers in length [25]. The negative exponentialdistribution describes the limiting case of saturated scintillation[26]. The Gamma-Gamma distribution is a generalized modelthat can be applied to a wide range of turbulence conditions[27]. In this work, a lognormal turbulence channel is em-phasized, but the developed detection algorithm is sufficientlygeneral and can be applied to any turbulence models.For the lognormal channel model, the optical irradiance Iis given byI = exp(m) (3)where m is a Gaussian RV with mean  and variance 2.Consequently, I follows a lognormal distribution with a pdfgiven by [24]fI(I) =1√2.Iexp(− (ln I − )222): (4)Normalizing the mean, i.e., E[I] = 1, where E[·] is theexpectation operation, the pdf of I can be written asfI(I) =1√2.Iexp(− (ln I + 2R2)222): (5)The parameter  is the scintillation level [28]. Turbulenceeffects on the performance are minimal when scintillationlevels are below 0:1, so the electrical-SNR-optimized detectionsystem is characterized for scintillation levels ranging from 0:1to 0:5. This is the typical range for scintillation levels [11],[29]. In this paper, the case for which the statistical details of(4) or (5) are completely unknown at the receiver is considered.III. OOK WITH FIXED AND UNOPTIMIZED DETECTIONTHRESHOLDSIn the low state, the received signal (r = I +c ) is a sumof two RVs, c and Is, where Is = I . Since c and Is areassumed to be independent, the pdf of the received low statesignal is the convolution of the marginal pdfs of Is and c3according tof(r|s0) =1fI(r)∗ fc (r)=∫ ∞01√2.xexp−(ln x +22)222× 1√2.gexp(− (r − x)222g)yx(6)where ∗ denotes the convolution operation, and fc (r) =1√2gexp(− r222g)denotes the noise pdf.In the high state, the pdf of the received signal (r = (2 +)I +c ) can be defined in a similar manner according tof(r|s1) = 12 + fI(r2 + )∗ fc (r)=∫ ∞01√2.xexp−(ln x2+ +22)222× 1√2.gexp(− (r − x)222g)yx:(7)For a given fixed detection threshold ith, the probability offalse alarm eF and probability of miss eM can be respectivelywritten aseF =∫ ∞ithf(r|s0) yr =∫ ∞ith1fI(r)∗ fc (r) yr (8)andeM =∫ ith−∞f(r|s1) yr =∫ ith012 + fI(r2 + )∗fc (r) yr:(9)Assuming that p1 represents the a priori probability that “1”is sent, i.e., p1 = 12 means “0”s and “1”s are equally likely tobe sent, one can write the bit-error rate (BER) for OOK usinga fixed detection threshold ith asee = (1− p1)eF + p1eM=(1− p1) exp(−28)√2.∫ ∞0√x3=2exp(−ln2 x22)×f(ith − xg)yx+p1 exp(−28)√2.∫ ∞0√2 + x3=2× exp(−ln2 x2+22)f(x− ithg)yx=(1− p1) exp(−28)√2.∫ ∞0√x3=2exp(−ln2 x22)×f (√(ith − x)) yx+p1 exp(−28)√2.∫ ∞0√2 + x3=2× exp(−ln2 x2+22)f (√(x− ith)) yx(10)TABLE IERROR FLOOR EXPRESSIONS FOR FSO SYSTEMS EMPLOYING FIXEDDETECTION THRESHOLDS OF Tth = (1 ) )E[I[ OVER A LOGNORMALFADING CHANNEL WITH . = 0M03 0M13 0M16 0M0 0M03Theoretical error floor 0M0022 0M0027 0M0032 0M0043Simulated error floor 0M0023 0M0027 0M0032 0M0044where f(x) = 1√2∫∞xz−t22 yt is the Gaussian f-function,and we have denoted the electrical SNR by  = (E[I])2Rc0[11], or simply  = 1Rc0 under a normalized mean assump-tion.In large SNR regimes, when  approaches infinity orequivalently when 2g = c0R2 approaches zero, the Gaussiandistribution approaches a Dirac delta function (·). Hence, onecan havelim→∞ fc (r) = (r) (11)andlim→∞1vfI( rv)∗ fc (r) = 1vfI( rv)(12)where v is a constant taking either  or 2 + . When theelectrical SNR is asymptotically large (i.e.,  → ∞), using(8), (9), and (12), one obtainslim→∞eF =∫ ∞ith1fI(r)yr = 1− FI(ith)(13)andlim→∞eM =∫ ith012 + fI(r2 + )yr = FI(ith2 + )(14)where FI(·) represents the cumulative distribution function(CDF) of the irradiance I . Therefore, the false alarm prob-ability and miss probability in large SNR regimes are deter-mined by the CDF of the irradiance evaluated at ithR andithR(2 + ), respectively. Substituting (13) and (14) into (10)giveslim→∞ee = lim→∞(1− p1)eF + p1eM=(1− p1)f(lnith − ln  + 2R2)+ p1f(ln(2 + )− lnith − 2R2) (15)which is the error floor for an OOK IM/DD system with a fixeddetection threshold through lognormal turbulence channels. Asshown from (15), the error floor depends on both ith and, and typically one chooses the fixed detection threshold asith = (E[r|s1] + E[r|s0])R2. When  = 0, ith = E[I],and the analytical error floor expression in (15) is equivalentto [5, eq. (20)], which was derived under an assumption ofnormalized second moment, i.e., E[I2] = 1. When  ̸= 0, itis simple to show that ith = (1 + )E[I].It is important to note that the error floor varies with theoffset . For a lognormal turbulence channel with  = 0:25and equal a priori data symbol probability, the predicted errorfloors are shown in Table I for different values of . It is4seen that an increase of  results in a higher error floor. Thetheoretical error floors are verified with simulated BER limitsin Table I.Following the same approach, one can predict the errorfloors for different turbulence channel models based on thecorresponding CDFs. The resulting error floors are summa-rized in Table II, where K−(·) denotes the modified Besselfunction of the second kind with order −, and the functionh(xP yP zP w) is defined ash(xP yP zP w)=(xyzw)x Γ(y − x)Γ(x+ 1)Γ(y)1F2(x;x+ 1P x− y + 1; xyzw)(16)where Γ(·) is the Gamma function and 1F2(·; ·P ·; ·) is thegeneralized hypergeometric function [30].IV. OOK WITH ELECTRICAL-SNR-OPTIMIZEDDETECTION THRESHOLDSA performance trade-off can be established between opera-tion with fixed detection thresholds (which can suffer fromirreducible error floors) and adaptive detection thresholds(which require knowledge of the instantaneous SNR for eachdata symbol). This is done with an electrical-SNR-optimizeddetection system [11], [12]. The approach considers the opti-mization problemargminithee = argminith[(1− p1)eF + p1eM ]: (17)From (10) and (17), it is clear that electrical-SNR-optimizeddetection requires knowledge of ith, , and the underlying tur-bulence model. To find the detection threshold that minimizesthe BER at a given electrical SNR, one can take the derivativeof (10) with respect to ith and set it to zero, i.e., @ee@ith = 0.This gives−(1− p1)f(ith|s0) + p1f(ith|s1) = 0 (18)where f(ith|s0) and f(ith|s1) are the likelihood functionsevaluated at ith. Assuming perfect knowledge of the pdf forthe turbulence model and  = 0, the optimum detection rulereduces to the rule discussed in [11], [12]. The location ofthe electrical-SNR-optimized detection threshold lies at theintersection of two scaled likelihood functions: (1−p1)f(r|s0)and p1f(r|s1). As shown in Fig. 1, when the electrical SNRapproaches infinity, the total area underneath the intersectedpdfs, i.e., (1 − p1)eF + p1eM , will become infinitely small.The electrical-SNR-optimized detection can therefore be usedto eliminate the error floors caused by a receiver using fixeddetection thresholds [13], [31].To accommodate the fact that the FSO receiver may notalways know the underlying turbulence model, the turbulencedistribution can be approximated by sample moments. Theapproximated turbulence distribution can then be used toderive the electrical-SNR-optimized detection threshold.The density functions of numerous statistical models onthe positive half-line can be approximated by a sum of−3 −2 −1 0 1 2 3 4 500.10.20.30.40.50.60.70.80.91Variable rLikelihood Functions  SNR = 2 dBSNR = 8 dBFig. 1. The likelihood functions f(rjs0) and f(rjs1) with . = 0M03 and = 0M0 when  = 0 dB and  = 6 dB. The likelihood functions are a resultof the convolution of the lognormal pdf and Gaussian pdf.Laguerre polynomials [19], [32]. Using this approach, one canapproximate the pdf of I as [33]fI(I) ≈ Iv exp(−IRx)xv+1Γ(v + 1)∞∑j=0jaj(vP IRx) (19)where aj (vP IRx) is a Laguerre polynomial of order j in IRxand is written asaj(vPIx)=j∑k=0(−1)kΓ(v + j + 1)k!(j − k)!Γ(v + j − k + 1)(Ix)j−k(20)andj =j∑k=0(−1)k j!Γ(v + 1)k!(j − k)!Γ(v + j − k + 1) Ic [j − k] (21)where the jth moment of I is denoted by I [j]. In (19), theparameters x = I [2]−2I [1]I [1]and v = I [1]x − 1 are chosen tohave the mean and variance of the Gamma RV I0, whose pdfis fI((I) =Iv exp(−I=x)xv+)Γ(v+1) , match those of RV I . From (19),the corresponding characteristic function (CF) and momentgenerating function (MGF) for RV I can also be obtained.The detailed derivations are given in the Appendix. Theseanalytical expressions can be used to estimate the performanceof an FSO system over the lognormal fading. Substituting (19)into (6) and (7) yields the likelihood functionsf(r|s0) = 1√2.gxv+1∞∑j=0j∫ ∞0(x)v−1exp(− xx)× exp (−(r − x)2)aj (vP x)yx(22)5TABLE IIPDF AND ERROR FLOOR EXPRESSIONS FOR VARIOUS TURBULENCE CHANNEL MODELSTurbulence Models PDF Error FloorsLognormal 1√2I(1− p1)f(lnith−ln +2=2)× exp{− (ln I+2)222 } +p1f(ln(2+)−lnith−2=2)K-distribution 2+)2Γ() I−)2 (1− p1){1− 12 [h(1P P ithP ) + h(P 1P ithP )]}×K−1(2√I) +p1{12 [h(1P P ithP 2 + ) + h(P 1P ithP 2 + )]}Gamma-Gamma 2()+2Γ()Γ() I+2 −1 (1− p1){1− 12 [h(P P ithP ) + h(P P ithP )]}×K−(2√I) +p1{12 [h(P P ithP 2 + ) + h(P P ithP 2 + )]}Negative Exponential 1 exp{− I} (1− p1) exp(ith)+ p1{1− exp[ith(2+)]}andf(r|s1) = 1(2 + )√2.gxv+1∞∑j=0j∫ ∞0(x2 + )v−1× exp(− x(2 + )x)exp(−(r − x)2)aj (vP x2 + )yx:(23)Substituting (22) and (23) into (18) yields− 1− p1∞∑j=0j∫ ∞0(x)v−1exp(−(x2 − 2xiop))× exp(− xx)aj(vPx)yx+p12 + ∞∑j=0j×∫ ∞0(x2 + )v−1exp(−(x2 − 2xiop))× exp(− x(2 + )x)aj(vPx2 + )yx = 0:(24)The detection threshold can be obtained numerically withrespect to a given offset  and electrical SNR from (24).On the timescale of stationary statistics, the electrical SNRis constant, and the detection threshold is a fixed detectionthreshold. A comparison of the electrical-SNR-optimized de-tection thresholds, acquired by the approximated and exactlognormal pdfs, are presented in Table III. The thresholdsare obtained by averaging 10 calculated detection thresholds.As shown from Table III, the approximated pdf can be usedto calculate the detection threshold with high accuracy whenthe electrical SNR is less than 16 dB. For higher values ofSNR, the calculated detection thresholds lose accuracy, andthe corresponding BER curve deviates from the BER curveobtained with perfect knowledge of the lognormal pdf. Thisdiscrepancy occurs because the Laguerre-polynomial-basedpdf approximation can not accurately describe the behaviors ofthe lognormal pdf near the origin. Fortunately, this inaccuracydoes not concern most practical FSO systems, as they typicallyoperate at relatively low SNR values [34].V. PARAMETER ESTIMATIONAs the electrical-SNR-optimized detection threshold intro-duced in Section IV requires knowledge of the state offset and electrical SNR , it is necessary to estimate  and .MoME and MLE are used for this estimation in this section.TABLE IIICOMPARISON OF DETECTION THRESHOLDS BY USING AN EXACT ANDAPPROXIMATED LOGNORMAL PDF WITH . = 0M03SNR (dB) Thresholds with Thresholds with Sampleexact PDF approximated PDF variance0 0M7275 0M7303 1M11 10−84 0M6411 0M6415 0M05 10−88 0M5306 0M5274 1M67 10−812 0M4100 0M4012 0M34 10−816 0M3065 0M2762 1M06 10−720 0M1761 0M2475 6M40 10−624 0M1014 0M3017 1M01 10−5With bit-by-bit interleaved fading channels3 [2], [11], it isassumed that there are 2a sampled signals during the observa-tion interval. The vectors R = [r[0] ::: r[2a− 1]]i , If =[I[0] ::: I[2a− 1]]i , and N = [n[0] ::: n[2a− 1]]irepresent the received signal vector, fading coefficient vec-tor, and noise vector, respectively. Assuming that a trainingsequence of length 2a is transmitted with a consecutive 1’sfollowed by a consecutive 0’s, one can write the receivedsignal at the lth bit interval when bit 1 is transmitted asr[l]|s1 = (2 + )I[l] + n[l]P l = 0P 1P :::P a− 1 (25)where I[l] and n[l] represent the fading coefficient and noiseduring the lth bit interval, respectively. Similarly, if a 0’s aretransmitted, the received signal at the kth bit interval can bewritten asr[k]|s0 = I[k] + n[k]P k = aPa+ 1P :::P 2a− 1: (26)A. Method of Moments EstimationUsing (25) and (26), one can obtain the estimation of  asˆ =1L∑2L−1k=L r[k]|s012L∑L−1l=0 r[l]|s1 − 12L∑2L−1k=L r[k]|s0: (27)3A typical Gbps FSO system operates in a quasi-static atmospheric tur-bulence channel with a coherence time on the order of milliseconds. Thiscoherence time is much longer than the nanosecond bit interval. Thus, thesame fading coefficient affects a block of information bits, and the systemperformance suffers from correlation. However, we can transform a quasi-static channel into a block fading channel by way of block interleaving [35].In such a system, we place each information bit in different blocks, such thateach block (i.e., information bit) experiences independent fading from that ofneighbouring blocks.6To assess the performance of the moment estimator ˆ,approximate expressions can be derived for the mean and vari-ance of ˆ when the sample size is asymptotically large. Assum-ing the statistics T = [i1 i2]i , where i1 = 1L∑L−1l=0 r[l]|s1and i2 = 1L∑2L−1k=L r[k]|s0, one can obtain the covariancematrix asCi =(Var[i1] Cov[i1P i2]Cov[i2P i1] Var[i2])=(1L [2g + (2 + )2Var[I]] 00 1L (2g + 2Var[I])):(28)Here, Var[·] denotes the variance, and Cov[·P ·] denotes covari-ance of two RVs. The estimator ˆ can be rewritten asˆ∆= <(T) =2i2i1 − i2 : (29)The estimator in (27) is consistent, i.e., ˆ er−→  as a −→∞,and is asymptotically Gaussian distributed, i.e.,√a(ˆ−) L−→N (0P 2ˆ). Performing a first-order Taylor expansion of <(·)about the point T = E[T] gives [36]ˆ ≈ <(T)∣∣∣∣∣T=E[T] +2∑i=1U<Uii∣∣∣∣∣T=E[T](ii − E[ii]) (30)where E[T] = [(2 + )E[I] E[I]]i . Taking the expectationof (30), one hasE[ˆ] ≈ <(T)∣∣∣∣T=E[T]=  (31)and the asymptotic variance of ˆ can be expressed as [37]2ˆ=Var[ˆ] =U<Uii∣∣∣∣iT=E[T]CTU<Uii∣∣∣∣T=E[T]=2g [2 + (2 + )2] + 22(2 + )2Var[I]4a(E[I])2:(32)Using (25) and (26), one can obtain the estimation of theturbulence mean m = exp(+2R2) and c0, respectively, asmˆ =12aL−1∑l=0r[l]|s1 − 12a2L−1∑k=Lr[k]|s0 (33)andcˆ0 = 2(mˆ|s1)2 ˆr[2]|s0 − (mˆ|s0)2 ˆr[2]|s1(mˆ|s1)2 − (mˆ|s0)2(34)where mˆ|s1 = 1L∑L−1l=0 r[l]|s1, ˆr[2]|s0 = 1L∑2L−1k=L r2[k]|s0,mˆ|s0 = 1L∑2L−1k=L r[k]|s0, and ˆr[2]|s1 = 1L∑L−1l=0 r2[l]|s1.Using (33) and (34), one can obtain the estimation of  asˆ =mˆ2cˆ0=(mˆ|s1 − mˆ|s0)2[(mˆ|s1)2 − (mˆ|s0)2]8[(mˆ|s1)2 ˆr[2]|s0 − (mˆ|s0)2 ˆr[2]|s1] : (35)B. Maximum Likelihood EstimationThe estimator based on the maximum-likelihood principle isthe most popular approach to obtaining practical estimators.Additionally, for most cases of practical interest, its perfor-mance is optimal for large data records and is approximatelythe minimum variance unbiased estimator due to its approxi-mate efficiency.For the MLE, with bit-by-bit interleaved fading channels[2], [11], we transmit a training sequence consisting of 2aconsecutive 1’s. Assuming the received signal model is thesame as (25), one can write the pdf of the received signal as[13]f(r[k]|s1;) =fI (r[k]) ∗ fc (r[k])=∫ ∞0exp(− (lnx− ln (2 + )− )222)× 1√2c0.xexp(− (r[k]− x)2c0)yx(36)where  = [ 2 c0 ]i denotes the unknown vector,and fc (r[k]) = 1√c( exp(− r2[k]c()is the noise pdf. Assum-ing that the components of the received signal vector R areindependent, we can write the pdf of the received signal whens1 is true asf(R;) =2L−1∏k=0f(r[k]|s1;)=2L−1∏k=0∫ ∞0exp(− (lnx− ln (2 + )− )222)× 1√2c0.xexp(− (r[k]− x)2c0)yx:(37)The MLE of the unknown vector  is obtained by maxi-mizing the log-likelihood functiona(R;) = ln f(R;)= ln2L−1∏k=0fI (r[k]) ∗ fc (r[k])=2L−1∑k=0ln∫ ∞0exp(− (lnx− ln (2 + )− )222)× 1√2c0.xexp(− (r[k]− x)2c0)yx:(38)Taking the derivative of (38) with respect to the unknownparameter and setting it equal to zero, we can obtain theMLE of the unknown vector . As it is difficult to obtaina closed-form expression for each unknown parameter, theexpectation-maximization (EM) algorithm can be implementednumerically to determine the MLE. This method, althoughiterative in nature, is guaranteed under mild conditions toconverge and produce a local maximum [37].In order to simplify the problem, we decompose the originaldata sets into the independent data sets y1[k] = I[k] and7y2[k] = n[k], where y1[k] and y2[k] are the complete data, andthey are related to the original data as r[k] = y1[k]+y2[k]: In-stead of maximizing ln f(R;), we can maximize ln f(Y;),where Y = [y1 y2]i , y1 = [y1[0] ::: y1[2a− 1]]i andy2 = [y2[0] ::: y2[2a− 1]]i . Since y1[k] = I[k], we haveln f(y1[k];)= ln(1√2.2y1[k]exp(− (ln y1[k]− ln (2 + )− )222))= − ln√2.2 − ln y1[k]− (ln y1[k]− ln (2 + )− )222:(39)Similarly, we haveln f(y2[k];)= ln(1√.c0exp(−y22 [k]c0))= − ln√.c0 − y22 [k]c0:(40)Assuming ˆ(j)= [ˆ(j) (ˆ2)(j) (cˆ0)(j) ()(j)]i is anestimate of  in the jth iteration, each iteration of the EMalgorithm can be written as follows.E-step: This step determines the conditional expectation of thecomplete dataj(P ˆ(j)) = EY|R;ˆ(j) [ln f(Y;)]= Ey)|R;ˆ(j) [ln f(y1;)] + Ey2|R;ˆ(j) [ln f(y2;)]=∫ln f(y1;)f(y1|R; ˆ(j))yy1+∫ln f(y2;)f(y2|R; ˆ(j))yy2:(41)where we havef(y1|R; ˆ(j)) =f(R|y1; ˆ(j))f(y1; ˆ(j))f(R; ˆ(j))(42)andf(y2|R; ˆ(j)) =f(R|y2; ˆ(j))f(y2; ˆ(j))f(R; ˆ(j))(43)and wheref(R|y1; ˆ(j)) =2L−1∏k=01√2.(ˆ2)(j)(r[k]− y1[k])× exp(− (ln(r[k]− y1[k])− ln (2 + ()(j))− ˆ(j))22(ˆ2)(j))(44)andf(R|y2; ˆ(j)) =2L−1∏k=01√.(c0)(j)exp(− (r[k]− y2[k])2(c0)(j)):(45)0 5 10 15 20 25 30 3510−410−310−210−1100Electrical SNR, γ (dB)Bit Error Rate  Tth=1Tth=1.20Electrical−SNR−optimized detection Adaptive detection Fig. 2. BERs of OOK modulated systems using fixed detection thresholdsTth, electrical-SNR-optimized detection thresholds and adaptive detectionthresholds over a lognormal turbulence channel with . = 0M03 and  = 0.M-step: This step maximizes (41) with respect to (j+1) = argmaxj(P ˆ(j)) (46)where ˆ(j+1)is the new estimate of . For the EM algorithm,the conditional expectation of the complete data is nondecreas-ing until it reaches a fixed point. This fixed point is the MLEof , i.e., ˆML = [ˆML ˆ2ML cˆ0PML ˆML]i . Based onthe invariance property of the MLE, we obtain the MLE ofI as ˆIPML = exp(ˆML +ˆ2ML2). The MLE of  can beobtained asˆML =(ˆIPML)2cˆ0PML=(exp(ˆML +ˆ2ML2))2cˆ0PML: (47)The Crame´r-Rao lower bound (CRLB) of ˆ can be calcu-lated using [37]Var[ˆ]≥[@@@@2@@c(@@]I−1()[@@@@2@@c(@@]i(48)where I() is the Fisher information matrix.VI. NUMERICAL RESULTSFigures 2 and 3 show the BERs versus electrical SNR whenthe OOK modulated system uses fixed detection thresholds ofith = 1 and ith = 1:2 with  = 0:25 for the lognormalchannel. For expository purposes, the parameter  is set to be = 0 for Fig. 2 and  = 0:2 for Fig. 3. It is observed that theBER curves obtained by using Monte Carlo simulation showexcellent agreement with the derived error floors in large SNRregimes and the error floors decrease for lower fixed detectionthresholds.To eliminate the error floors and improve the performance,the system with electrical-SNR-optimized detection thresholds80 5 10 15 20 25 30 35 4010−410−310−210−1Electrical SNR, γ (dB)Bit Error Rate  Tth = 1Tth = 1.20Electrical−SNR−optimized detectionAdaptive detectionFig. 3. BERs of OOK modulated systems using fixed detection thresholdsTth, electrical-SNR-optimized detection thresholds and adaptive detectionthresholds over a lognormal turbulence channel with . = 0M03 and  = 0M0.is used. The BERs for the system with the electrical-SNR-optimized detection thresholds are shown in Figs. 2 and3, along with the BERs for the system with the adaptivedetection thresholds. Both electrical-SNR-optimized detectionthresholds are obtained by using the approximated lognormalpdf with J = 3 sample moments. Both the electrical-SNR-optimized and adaptive detection threshold results exhibit noerror floors for increasing electrical SNR values. As expected,the system with electrical-SNR-optimized detection thresholdsdoes not perform as well as the optimum OOK system usingadaptive detection thresholds. For example, in Fig. 2, the OOKmodulated system using adaptive detection thresholds requiresan SNR of 13 dB to attain a BER of 10−5, while the systemusing electrical-SNR-optimized detection thresholds requiresan SNR of 15:3 dB to achieve the same BER performance.The corresponding SNR penalty factor in Fig. 2 for the systemusing an electrical-SNR-optimized detection threshold, is 2:3dB at BER of 10−5. The corresponding SNR penalty factorin Fig. 3 for the system using an electrical-SNR-optimizeddetection threshold, increases to 4:5 dB when  = 0:2.This performance difference can be factored into the ultimateFSO system design to offset the complexity of implementingsystems with adaptive detection thresholds (and their need forknowledge of the instantaneous SNR). It is also important topoint out that the BER performance achieved by the electrical-SNR-optimized system does not require rapid adjustment ofthe detection threshold. Since practical FSO systems typicallyoperate at constant transmit power, the detection threshold onlyneeds to be calculated once over durations of seconds or evenminutes. The electrical-SNR-optimized system can thereforereduce the implementation complexity, compared to that ofthe idealized system using adaptive threshold detection.In Fig. 4, the approximated lognormal pdf using J = 3sample moments is compared with the exact lognormal pdf, for = 0:25. The absolute errors between these pdfs are shownexplicitly in Fig. 5. The approximated lognormal pdf showsgood agreement with the exact lognormal pdf when  = 0:25:0 0.5 1 1.5 2 2.5 3 3.5 400.511.522.5Variable IProbability Density Function  Exact pdfApproximated pdfFig. 4. Comparison of an approximated pdf using J = 1 sample momentsand an exact pdf for a lognormal fading channel with . = 0M03.However, for higher  values ( > 0:75), the approximationof the lognormal pdf becomes inaccurate as integer momentscan not uniquely determine the lognormal pdf. Fortunately,such scintillation levels are not encounted in practice [38]. Acomparison of absolute errors from the pdf approximationsusing different numbers of sample moments is also givenin Fig. 5. Clearly, larger numbers of sample moments canreduce the absolute error, but this comes at the cost of highercomputational complexity. In general, a higher scintillationlevel  will require higher order sample moments and theresulting approximation can become increasingly inaccurate.The Laguerre-polynomial-based approximation is accurate forthe 0:1 to 0:5 range of  values that is of interest to FSOapplications [11], [29].In Fig. 6, we compare the BER performance between theapproximated lognormal pdf, for different values of  withJ = 3 sample moments, and the exact lognormal pdf. Thetwo simulated error rate curves show good agreement over awide range of SNR values. For the large SNR regime, theBER results from the approximate pdf have reduced accuracy,because the approximated pdf based on Laguerre polynomialsis unable to characterize the behaviors of the lognormal pdfnear the origin.In order to evaluate the estimator performance, the samplevariance of the electrical SNR estimator is compared with theCRLB. The variance of the electrical SNR estimator is givenbyˆ2ˆ =1M − 1M−1∑i=0(ˆi − ¯ˆ)2 (49)where ˆi is the estimation by using MoME or MSE at theith trail, M represents the total number of trails, and ¯ˆ isthe sample mean of the electrical SNR estimator. In orderto assess the estimator, Monte Carlo simulations are usedto obtain ˆ2ˆ . In the simulation, different training sequencelengths are used to estimate the mean and noise variance,M = 1 × 104 trials are used to calculate the variance of theelectrical SNR estimator, and  is set as 0:2. Figure 7 plots the90 1 2 3 4 5 600.0020.0040.0060.0080.010.0120.0140.0160.018Variable xAbsolute Error  10 moments6 moments3 momentsFig. 5. The absolute error between the approximated pdf and the exactlognormal pdf with . = 0M03 and J = 1; 4; 10 sample moments.0 5 10 15 2010−810−610−410−2100Electrical SNR, γ (dB)Bit Error Rate  Exact pdf, σ=0.25Approximated pdf, σ=0.25Exact pdf, σ=0.35Approximated pdf, σ=0.35Fig. 6. Comparison of BERs obtained by the approximated pdf and the exactlognormal pdf with  = 0 and J = 1 sample moments.0 5 10 15 20 25 3010−410−310−210−1100101102Electrical SNR, γ (dB)Normalized Variance of the Estimators  MoMEMLENormalized CRLBL=1,000L=10,000Fig. 7. Comparison of MoME and MLE normalized sample variance fordifferent training sequence lengths over a lognormal turbulence channel with. = 0M03. The normalized MSE is computed over M = 1 104 trials.normalized sample variance of the electrical SNR estimator,which is defined as the sample variance scaled by , versusthe average electrical SNR. It is shown that the normalizedsample variance for MLE attains the normalized CRLB, whichis obtained by scaling the CRLB by . However, there isa discrepency between the normalized sample variance forMoME and the normalized CRLB for SNR values greater than12 dB due to the inaccurate estimation of the noise variance. Itis shown that the discrepency between the normalized samplevariance for MoME and the normalized CRLB will disapearwhen  = 0. In this case, the received signal is the noise when0 is transmitted. Thus, the noise variance can be accuratelyestimated by transmitting a training sequence with consecutive0’s. When  ̸= 0 and 0 is transmitted, the received signal isthe noise as well as the fading coefficient term. This leadsto inaccurate estimation of the noise variance if a trainingsequence transmitted with consecutive 0’s.VII. CONCLUSIONIt is known that FSO systems operating with OOK andfixed detection thresholds can suffer from irreducible errorfloors and power inefficiency. With this in mind, the resultingerror floors are analyzed here (and validated with simulations)for lognormal turbulence channels and quantified for thegeneral case having low and high state offsets, i.e., withfinite extinction ratios. It is shown that the error floors canbe eliminated by using electrical-SNR-optimized detectionthresholds that minimize the average BER. The electrical-SNR-optimized system with the Laguerre-polynomials-basedapproximate pdf for the turbulence is found to be effectivefor typical FSO systems, which operate at relatively lowSNR values, as it yields near-optimal BER performance. Itis concluded that MLE is the preferred estimation techniquefor electrical-SNR-optimized detection with a finite extinctionratio, although such estimation comes at the cost of highercomputational complexity.APPENDIXThe CF of a RV I is the Fourier transform of its pdf, fI(I),and it is defined byΦI(!) =∫ ∞−∞fI(I) exp(j!I) yI (50)orΦI(!) = Re[ΦI(!)] + jIm[ΦI(!)] (51)where j2 = −1. In (51), Re[·] and Im[·] denote the real andimaginary parts, respectively. Both can be written, respectively,asRe[ΦI(!)] =∫ ∞0fI(I) cos(!I) yI (52)andIm[ΦI(!)] =∫ ∞0fI(I) sin(!I) yI: (53)10Using (19), one can approximate (52) asRe[ΦI(!)]≈∫ ∞0Iv exp(−IRx)xv+1∞∑n=0nan(vPIx)cos(!I) yI=1xv+1∞∑n=0n∫ ∞0Iv exp(−IRx)an(vPIx)cos(!I) yI:(54)Substituting (20) into (54), one hasRe[ΦI(!)] ≈ 1xv+1∞∑n=0nn∑k=0(−1)kΓ()k!(n− k)!Γ(− k)×∫ ∞0Iv exp(−IRx)(Ix)n−kcos(!I) yI=∞∑n=0nn∑k=0(−1)kΓ()k!(n− k)!Γ(− k)(1 + x2!2)−k2× cos((− k) arctan(x!))(55)where  = v+ n+1. In deriving the last equality of (55), anintegral identity [30, eq. 3.944(6)] has been used.Similarly, substituting (19) and (20) into (53) and using anintegral identity [30, eq. 3.944(5)], one obtainsIm[ΦI(!)] ≈∞∑n=0nn∑k=0(−1)kΓ()k!(n− k)!Γ(− k)(1 + x2!2)−k2× sin((− k) arctan(x!)):(56)The approximate CF is then found to beΦI(!) ≈∞∑n=0nn∑k=0(−1)kΓ()k!(n− k)!Γ(− k)(1 + x2!2)−k2× [cos((− k) arctan(x!)) + j sin((− k) arctan(x!))]:(57)Using an integral identity [30, eq. 3.326(2)], one can obtainthe MGF asMI(s) ≈∞∑n=0nn∑k=0(−1)kΓ(v + n+ 1)k!(n− k)!(1− sx)v+n−k+1 : (58)REFERENCES[1] V. W. S. Chan, “Free-space optical communications,” IEEE/OSA J.Lightwave Technol., vol. 24, pp. 4750-4762, Dec. 2006.[2] J. H. Shapiro and R. C. Harney, “Burst-mode atmospheric optical commu-nication,” in Proc.1980 Nat. Telecommun. Conf., 1980, pp. 27.5.1-27.5.7.[3] N. Chand, J. J. Loriz, A. J. Hunton, and B. M. 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