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Maximum Likelihood Estimation of the Lognormal-Rician FSO Channel Model Yang, Luanxia; Cheng, Julian; Holzman, Jonathan F. 2015

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. , NO. 1Maximum Likelihood Estimation of theLognormal-Rician FSO Channel ModelLuanxia Yang, Julian Cheng, Senior Member, IEEE, and Jonathan F. Holzman, Member, IEEEAbstract—In this work, the on-going challenges are addressedfor the application of the lognormal-Rician turbulence modelto free-space optical communication systems. Maximum likeli-hood estimation is applied to characterize the lognormal-Ricianturbulence model parameters, and the expectation-maximizationalgorithm is used to compute maximum likelihood estimates ofthe unknown parameters. The performance is investigated, byway of the mean square error, and it is found that the proposedtechnique can accurately characterize free-space optical commu-nication channels over a wide range of turbulence conditions,with reduced demand on the quantity of data samples.Index Terms—Free-space optical communications, lognormal-Rician turbulence model, maximum likelihood estimation.I. INTRODUCTIONFree-space optical (FSO) communication, also known asoutdoor optical wireless communication, is a broadband wire-less technology that provides high-speed data connectivitybetween nodes of optical line-of-sight links. High-speed datatransmission is facilitated by modulated optical beams throughthe atmosphere. Such FSO systems offer several benefits,including cost-effectiveness, rapid deployment, enhanced se-curity, protocol transparency, and freedom from spectral li-censing regulations [1]. However, optical signals transmittedover free-space are subject to distortion due to transientinhomogeneities of atmospheric temperature and pressure [2].The resulting scintillation or fading is a major cause ofperformance degradation for FSO systems.Current FSO systems often use intensity modulation anddirect detection (IM/DD) with on-off keying (OOK) and fixeddetection thresholds for reasons of practicality [3]. However,such fixed and unoptimized detection thresholds can result inirreducible error floors in large signal-to-noise ratio (SNR)regimes [4]. Attempts to overcome the irreducible error floorsof OOK IM/DD systems have focused on the application ofadaptive detection thresholds.Adaptive detection of OOK signal can be classified intothree categories. The first category, idealized adaptive detec-tion, applies bit-by-bit adaptations to the detection thresholdon the (typically) nanosecond timescale of the bit interval[2], [5]. The second category, quasi-static adaptive detection,applies adaptations to the detection threshold on the (typically)millisecond timescale of the turbulence coherence time [6].These adaptations can be difficult to implement—particularlyfor FSO receivers using numerous photodetectors. The thirdcategory, electrical-SNR-optimized detection [7], [8], unlikeThe authors are with the School of Engineering, The University ofBritish Columbia, Kelowna, BC, Canada (e-mail: {luanxia.yang, julian.cheng,jonathan.holzman}@ubc.ca).the prior two categories, does not require knowledge of theinstantaneous channel state information. Once the detectionthreshold is optimized with respect to the electrical SNR,the receiver operates with a fixed detection threshold, andthis continues while the turbulence channel exhibits station-ary statistics, i.e., constant channel model parameters, whichis typically on the timescale of several minutes [9]. Thus,electrical-SNR-optimized detection offers the practical advan-tages of operation with fixed detection thresholds (by avoidingthe need for rapid adaptations) and the performance advantagesof operation with idealized adaptive detection thresholds (byeliminating irreducible error floors). It must be noted, however,that the success of such systems relies heavily upon accuratestatistical models of the FSO fading channels. The workthat follows addresses this challenge of channel parameterestimation for OOK IM/DD systems.Many statistical models have been proposed to describe FSOfading channels. There exist models for use with a specificrange of turbulence conditions: the lognormal distributioncharacterizes FSO fading channels under weak turbulence con-ditions over meters (or longer, depending on the temperature,wind strength, altitude, humidity, and atmospheric pressure)[9], [10]; the K-distribution characterizes FSO fading channelsunder strong turbulence conditions over several kilometers[11]; the negative exponential distribution characterizes FSOfading channels in the limiting case of saturated scintillation[12]. There also exist generalized models for use over a broadrange of weak-to-strong turbulence conditions: the Gamma-Gamma distribution, which can underestimate effects of small-and large-scale scintillations and suffer from decreased accu-racy [13], and the lognormal-Rician distribution, which hasbeen found to offer two advantages. First, heuristic analy-ses of wave propagation through turbulence show that thelognormal-Rician fading distribution accurately characterizesexperimental data [14]. Second, the lognormal-Rician fadingdistribution is highly adaptable over a wide range of weak-to-strong turbulence conditions through its parameters. However,the application of the lognormal-Rician fading distributionhas been limited, as it does not have a tractable closed-formprobability density function (pdf).Given the potential of the lognormal-Rician distribution,for accurately characterizing FSO fading channels, there havebeen efforts to implement this distribution with estimatedshaping parameters. In [15], the authors applied a physicalmodel of turbulence-induced scattering, to estimate the shap-ing parameters of the lognormal-Rician fading distribution.It should be noted, however, that this approach dependsheavily upon estimated parameters in a physical model of theIEEE PHOTONICS TECHNOLOGY LETTERS, VOL. , NO. 2turbulence-induced scattering and such parameters are ofteneither unavailable or lacking in accuracy. For computationalsimplicity, the authors applied the Tatarskii model to charac-terize refractive index fluctuations and geometrical optics tocharacterize turbulent eddies, but the underlying assumptionsof this approach can lead to notable inaccuracies, as discussedin [14]. In [16], the authors introduced the generalized methodof moments approach to estimate the shaping parameters ofthe lognormal-Rician distribution. It should be noted, however,that this approach demands a large number of data samples,on the order of 106 data samples, and this impedes itsimplementation in FSO communications. For a standard FSOlink, experiencing quasi-static turbulence fading on a typicalmillisecond timescale, the system would exhibit latency on theorder of 1× 106 milliseconds = 1000 seconds. This durationis unacceptably long for FSO communications, as typical FSOchannels exhibit stationary statistics, i.e., constant channelmodel parameters, on the timescale of several minutes. Clearly,FSO systems applying channel estimation with a lognormal-Rician fading distribution need a more rapid estimation of theshaping parameters, to witness its benefits.In this paper, we propose the use of maximum likelihoodestimation (MLE), with the expectation-maximization (EM)algorithm, to estimate the parameters of the lognormal-Riciandistribution. The estimation approach is found to be highlyaccurate, and it operates with relatively low quantities of datasamples (on the order of one thousand data samples).II. LOGNORMAL-RICIAN PROBABILITY DENSITYFUNCTIONFor the FSO system of interest, there is an assumption ofperfect background noise rejection, from narrowband optical,electronic, and/or spatial filtering [17]- [19]. For the resultinglognormal-Rician channel model, the optical irradiance I canbe obtained by I = |UC + UG|2 exp(2χ), where UC is areal deterministic quantity, UG is a circular complex Gaussianrandom variable (RV) with zero mean, χ is a real GaussianRV, |UC+UG| is a Rician RV, |UC+UG|2 is a noncentral chi-square RV with a degree of freedom of two, and exp(2χ) isa lognormal RV. Consequently, I follows a lognormal-Riciandistribution with a pdf given by [15]fI(I) =(1 + r)e−r√2piσz∫ ∞0dzz2I0(2[(1 + r)rzI]1/2)× exp(−1 + rzI − 12σ2z(ln z +12σ2z)2) (1)where z represents exp(2χ), r = |UC |2/E[|UG|2] is thecoherence parameter, E[·] denotes the expectation operation,σ2z is the variance of the logarithm of the irradiance modulationfactor z, and I0(·) is the zero-order modified Bessel function ofthe first kind. As noted on [13], it is not generally known howto directly relate these two empirical parameters to the physicalcharacteristics of atmospheric conditions, but it is possible tocharacterize trends in the two parameters, with respect to theRytov variance, σ2R = 0.5k76L116 C2n. Here, k = 2pi/λ is thewavenumber for the wavelength of λ, L is the link length, andC2n is the altitude-dependent index of the refractive structureparameter, varying from 10−17 m−23 for weak turbulence to10−13 m−23 for strong turbulence. The characteristic trends inthe two parameters are seen in [15], for variations in the Rytovvariance, σ2R, i.e., variations in L and/or C2n. In the limit ofzero inner scale, it is shown that the parameter r decreases asσ2R increases, while the parameter σ2z is approximately equalto the Rytov variance for small σ2R, reaches a peak valueof approximately 0.58 for σ2R ≈ 8, and decreases slowly toapproximately 0.4 for large σ2R.III. MLE FOR THE LOGNORMAL-RICIAN SHAPINGPARAMETER ESTIMATIONThe maximum likelihood principle is the most popularapproach to obtaining practical estimators. Its performanceis optimal for large quantities of data, and it yields anapproximation of the minimum-variance unbiased estimator.The EM algorithm is applied to compute the MLE of thelognormal-Rician parameters.Assuming we have K independent and identically dis-tributed observations of the lognormal-Rician distribution,I = [I[0] ... I[K − 1]]T , with the unknown vectors θ =[σ2z r]T , the pdf of I can be written asfI(I;θ) =K−1∏l=0(1 + r)e−r√2piσz∫ ∞0I0(2[(1 + r)rzI[l]]1/2)× exp(−1 + rzI[l]− 12σ2z(ln z +12σ2z)2)dzz2.(2)The MLE of the unknown vector θ is then obtained bymaximizing the log-likelihood functionL(I;θ) = ln f(I;θ)=K−1∑l=0ln((1 + r)e−r√2piσz∫ ∞0I0(2[(1 + r)rzI[l]]1/2)× exp(−1 + rzI[l]− 12σ2z(ln z +12σ2z)2))dzz2.(3)It is difficult to obtain closed-form estimates of thelognormal-Rician parameters, due to the integral form of (12),so the following analysis uses the EM algorithm to findthe MLE of θ [21]. Given z = [z[0] ... z[L− 1]]T asthe unobserved data, we can write the complete-data log-likelihood function asL(I, z;θ) =K−1∑l=0ln f(z[l];σz) +K−1∑l=0ln f(I[l]|z[l]; r)= K{− ln√(2piσ2z) + ln(1 + r)−52KK−1∑l=0ln z[l]−σ2z8− 12σ2z1KK−1∑l=0(ln z[l])2 − 1KK−1∑l=01 + rz[l]I[l]−r + 1KK−1∑l=0ln I0(2[(1 + r)rz[l]I[l]]1/2)}.(4)IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. , NO. 3where f(z[l];σz) = 1√2piσzz[l] exp(− 12σ2z(ln z[l] + 12σ2z)2)and f(I[l]|z[l]; r) = 1+rz[l] I0(2[(1+r)rz[l] I[l]]1/2)exp (−r− 1+rz[l] I[l]). The resulting complete-data sufficientstatistics are T1(z) = 1K∑K−1l=0 ln z[l], T2(z) =1K∑K−1l=0 (ln z[l])2, and T3(z) = − 1K∑K−1l=01+rz[l] I[l] +1K∑K−1l=0 ln I0(2[(1+r)rz[l] I[l]]1/2).The initial value of σ2z is obtained by fitting the simple log-normal turbulence model, where σˆ2z(0)= − 2K∑K−1l=0 ln I[l].The initial value of the coherence parameter r can be obtainedby solving the polynomial equation [16, eq. (6)][(rˆ(0))2 + 4rˆ(0) + 2]3[1 + (rˆ(0))3][(rˆ(0))3 + 9(rˆ(0))2 + 18rˆ(0) + 6]=(1K∑K−1l=0 I2[l])31K∑K−1l=0 I3[l].(5)Iterations are then carried out with the expectation-step (E-step) and maximization-step (M-step).E-step: The E-step is carried out by computingT1(z; θˆ(j))=1KK−1∑l=0Ez|I[ln z[l]∣∣∣I; θˆ(j) ] (6)T2(z; θˆ(j))=1KK−1∑l=0Ez|I[(ln z[l])2∣∣∣I; θˆ(j) ] (7)T3(z; θˆ(j))= − 1KK−1∑l=0(1 + r)Ez|I[z−1[l]∣∣∣I; θˆ(j) ] I[l](8)+1KK−1∑l=0Ez|I[ln I0(2[(1 + r)rz[l]I[l]]1/2)∣∣∣∣∣ I; θˆ(j)]where θˆ(j)= [σˆ2z(j)rˆ(j)]T is an estimate of θ in the jthiteration. For the computations carried out as part of the E-step, it is noted that the expectation expressions in (8)-(10) areall functions of z[l], so the conditional expectations in (8)-(10)can be expressed asEz|I[g(z[l])∣∣∣I; θˆ(j) ] = ∫ g(z[l])f(z[l] ∣∣∣I; θˆ(j) )dz[l] (9)where f(z[l]|I[l]; θˆ(j)) = f(I[l]|z[l];θˆ(j))f(z[l];θˆ(j))f(I[l];θˆ(j))and wheref(I[l]|z[l]; θˆ(j)) = exp(−rˆ(j) − 1 + rˆ(j)z[l]I[l])× 1 + rˆ(j)z[l]I0(2[(1 + rˆ(j))rˆ(j)z[l]I[l]]1/2) (10)f(z[l]; θˆ(j)) =1√2piσˆ(j)z z[l]× exp− 12(σˆ(j)z)2 [ln z[l] + 12 (σˆ(j)z )2]2 (11)andf(I[l]; θˆ(j)) =(1 + rˆ(j))e−rˆ(j)√2piσˆ(j)z∫ ∞0I0(2[(1 + rˆ(j))rˆ(j)z[l]I[l]]1/2)× exp−1 + rz[l]I[l]− 12(σˆ(j)z)2 [ln z[l] + 12 (σˆ(j)z )2]2 dzz[l]2.(12)M-step: The M-step is carried out by computingσˆ2z(j+1)= T2(z; θˆ(j))−(T1(z; θˆ(j)))2(13)and finding rˆ(j+1) such that it maximizesrˆ(j+1) = arg maxr{T3(z; θˆ(j))}, where θˆ(j+1)=[σˆ2z(j+1)rˆ(j+1)]T is the new estimate of θ. For the EMalgorithm, the conditional expectation of the complete data isnondecreasing until it reaches a fixed point. This fixed pointis the MLE of θ, i.e., θˆML = [σˆ2z,ML rˆML]T .IV. NUMERICAL RESULTSTo evaluate the estimator performance, the mean squarederror (MSE) of the estimator θˆ is studied, the MSE isMSE[θˆ] = Var[θˆ] + (E[θˆ]− θ)2, where Var[θˆ] is the varianceof the estimator, i.e., Var[θˆ] = 1M−1∑M−1i=0 (θˆi− ¯ˆθ)2, ¯ˆθ is thesample mean of the estimator, and E[θˆ] = 1M∑M−1i=0 θˆi is themean of the estimator [22]. The simulation uses K = 1, 000data samples to estimate the lognormal-Rician parameters andM = 100 trials to calculate the MSE of the estimator.In Fig. 1, we present the simulated MSE and normalizedmean squared error (NMSE) performance of rˆ and σˆ2z whenσ2z = 0.25 and r ranges from 1 to 9. The NMSE is definedas the MSE scaled by the true value of the estimator. Theperformance trends at or above r = 2 are especially note-worthy. Increasing the value of r decreases the NMSE of rˆbut it does not change the NMSE of σˆ2z to any great extent.Thus, changes to the value of r have minimal effects on theestimation performance of σ2z , and the MLE is insensitive tothe value of r. The same conclusion can be seen for the MSEof rˆ and σˆ2z , which remain relatively flat as r increases.In Fig. 2, we present the simulated MSE and NMSEperformance of rˆ and σˆ2z when r = 4 and σ2z ranges from0.1 to 0.8. From the figure, we note that the MSE of σˆ2zincreases with the value of σ2z while the MSE performancecurve of rˆ stays flat with changing values of σ2z . It can be seenthat the MLE performance of the lognormal-Rician parameterσ2z is insensitive to the value of r but sensitive to the valueof σ2z , while the MLE performance of the lognormal-Ricianparameter r is insensitive to both the values of r and σ2z .Overall, the results for the MSE and NMSE in Figs. 1 and2 are indicative of accurate estimation. The MSE and NMSEperformance is comparable with that of the prior study [16],albeit with three orders of magnitude fewer data samples.However, for the implementation of the proposed estimationtechnique, it is worth noting the underlying assumption abovefor perfect background noise rejection. In general, the MSEperformance worsens with increasing background noise power,IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. , NO. 41 2 3 4 5 6 7 8 910−310−2rMSE or NMSE  MSE of rMSE of σz2NMSE of rNMSE of σz2Fig. 1. MSE and NMSE performance of the maximum likelihood estimatorsfor the lognormal-Rician parameters r and σ2z with σ2z = 0.25.so it is necessary to implement the technique in a system witha sufficient level of background noise rejection (to establishthe desired level of MSE performance). For example, usingthe Monte-Carlo simulation, it can be shown that the MSEfor the estimated parameters can be maintained below 0.1 orbelow 0.01, by rejecting the background noise power to levelsbelow 4% or below 0.5% of the signal power, respectively.Such background noise rejection can be achieved through es-tablished techniques of optical, electronic, and spatial filtering[17]- [19].V. CONCLUSIONIn this work, the challenges were addressed for the ap-plication of the lognormal-Rician turbulence model to FSOcommunications. The proposed technique used MLE, to es-timate the parameters of the lognormal-Rician fading chan-nels and the EM algorithm, to compute the MLE of theunknown parameters. The performance was simulated in termsof MSE. Numerical results showed that MLE with the EMalgorithm can effectively characterize FSO communicationsover lognormal-Rician fading channels, given a wide rangeof turbulence conditions. 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