RR EDUCTION OF THE B EAM P OINTING E RROR FOR I MPROVED F REE -S PACE O PTICAL C OMMUNICATION L INK P ERFORMANCE
I. N’Doye, W. Cai, A. Alalwan, X. Sun, W. G. Headary, M.-S. Alouini, B. S. Ooi, T.-M. Laleg-Kirati ∗ Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE)King Abdullah University of Science and Technology (KAUST)Thuwal 23955-6900, Saudi Arabia [email protected]; [email protected]; [email protected]
February 10, 2021 A BSTRACT
Free-space optical communication is emerging as a low-power, low-cost, and high data rate alternativeto radio-frequency communication in short- to medium-range applications. However, it requires aclose-to-line-of-sight link between the transmitter and the receiver. This paper proposes a robust H ∞ control law for free-space optical (FSO) beam pointing error systems under controlled weakturbulence conditions. The objective is to maintain the transmitter-receiver line, which means thecenter of the optical beam as close as possible to the center of the receiving aperture within aprescribed disturbance attenuation level. First, we derive an augmented nonlinear discrete-timemodel for pointing error loss due to misalignment caused by weak atmospheric turbulence. We theninvestigate the H ∞ -norm optimization problem that guarantees the closed-loop pointing error is stableand ensures the prescribed weak disturbance attenuation. Furthermore, we evaluate the closed-loopoutage probability error and bit error rate (BER) that quantify the free-space optical communicationperformance in fading channels. Finally, the paper concludes with a numerical simulation of theproposed approach to the FSO link’s error performance. Keywords
Free-space optical (FSO) communications · H ∞ pointing error control · Weak turbulence · Lognormaldistribution · Linear Matrix Inequality (LMI).
Free-space optical (FSO) communication systems have emerged as a viable technology that offers a large capacityusage (data, voice, and video) in short to medium-range applications. The range of applications include ﬁxed-locationterrestrial communication (Willebrand and Ghuman, 2001), communication between mobile robots (Kerr et al., 1996),underwater wireless optical communication (UWOC) (Oubei et al., 2015), airborne communication (Maynard, 1987),and inter-satellite communication (Chan, 2003). FSO is a line of sight communication network with a free-space oratmosphere as a channel. This channel may be turbulent, causing absorption and scattering of the optical signal due tothe presence of many factors, including fog, rain, snow, and temperature variations, resulting in its deterioration (Singhand Sappal, 2019). Due to the temperature variations in the atmosphere, the refractive index changes creating Fresnelzones of different densities that scatter the laser beam from its projected path to travel diverse directions.Atmospheric turbulence is a random phenomenon caused by the variation of temperature or humidity and the atmo-sphere’s pressure along the propagation path. Speciﬁcally, atmospheric turbulence is highly variable and unpredictabledue to weather effects (Henniger and Wilfert, 2010). It would make the optical beam ﬂuctuated when propagating ∗ This work has been supported by the King Abdullah University of Science and Technology (KAUST), Base Research Fund(BAS/1/1627-01-01) to Taous Meriem Laleg. a r X i v : . [ ee ss . S Y ] F e b rXiv Template
REPRINT through the channel and ﬁnally results in misalignment due to diffraction from particulates present in the channel,resulting in enlarging the beam’s size to become more signiﬁcant than the receiver aperture size. Misalignment can leadto intolerable signal fades and can signiﬁcantly degrade system performance. In other words, atmospheric turbulencemay lead to a signiﬁcant degradation in the performance of the FSO communication systems (Alheadary et al., 2015).In addition to this, the signal propagating through the FSO channel is also perturbed by building vibrations, sways, andthermal expansions result in degradation of link performance (Shin and Chan, 2002), (Farid and Hranilovic, 2007). Thismisalignment can lead to pointing errors, causing the optical beam’s displacement along with horizontal and verticaldirections. Hence, FSO links require accurate pointing (Borah and Voelz, 2009), which means the pointing error needsto be very small to reduce the loss due to misalignment between the transmitter-receiver line.Since FSO systems require precise pointing as the light signals are highly directional, the effect of pointing errorson link performance is a great interest for many potential applications. Several approaches have been proposed toaddress the LOS (Line-of-Sight) requirement in optical communication systems. In (Pontbriand et al., 2008), large-areaphotomultiplier tubes are used to increase the receiver’s ﬁeld of view. Multiple LEDs and multiple photodiodes havealso been used to avoid the need for active pointing during optical-communication (Rust and Asada, 2012), (Simpsonet al., 2012). However, these systems achieved the LOS through redundancy in transmitters and receivers, whichresulted in a larger footprint, higher cost, and higher complexity.Furthermore, different pointing strategies for FSO links have been proposed in (Arnon et al., 2002; Komaee et al., 2007;Yuksel et al., 2005; Liu, 2009; Cai et al., 2019). However, these methods focus on combining existing components ofthe pointing assembly and atmospheric turbulence effects using manual and special detection techniques. Additionally,these methods use statistical performance analysis tools to mitigate pointing error effects but did not consider thecontroller design aspect. Many of them did not include the inﬂuence of vibration levels and atmospheric turbulence. Toalleviate these shortcomings and arrive with an accurate pointing error solution in FSO links, we propose a robust controlstrategy for maintaining the optical link between free-space communication stations engaged in a laser communicationchannel.To the best of the authors’ knowledge, designing a beam pointing error control for improved FSO link performance hasnot been thoroughly investigated. One of this paper’s motivations is to study and characterize the lognormal turbulencefading theoretically and experimentally to construct fully auxiliary control subsystems for robust FSO links.The contributions of this paper to the existing body in the literature in pointing error control for FSO systems are asfollows.• We propose an experimental setup and analyze the lognormal fading of the weak turbulence FSO channel.• We derive a discrete-time nonlinear model based on a predeﬁned autocorrelation function using the implicitMilstein scheme to simulate the lognormal optical channel state.• Using the discrete-time model, we propose the H ∞ -norm optimization problem that guarantees the closed-looppointing error is stable and ensures a prescribed disturbance attenuation level.• We evaluate the quality of the closed-loop pointing error control, which shows that the proposed control lawcan maintain the optical beam’s center at the center of the receiving aperture.• Finally, we perform numerical simulation tests of the open-loop and closed-loop outage probability error andbit error rate (BER) that quantify the free-space optical communication’s performance in fading channels.This paper is an extension of (Cai et al., 2018), with the following signiﬁcant new contributions.• A discrete-time nonlinear model based on a predeﬁned autocorrelation function is adapted to capture thelognormal fading process and provide a comprehensive treatment of the optical beam model. Indeed, thelognormal random process is represented as a solution of a stochastic differential equation (SDE), which isapproximated by a general and effective discrete-time model.• A robust H ∞ control law is proposed to reduce the pointing error and maintain the line-of-sight link of theoptical beam under controlled lognormal weak-turbulence conditions.• The communication performance metrics using the outage probability error and the bit error rate (BER) areevaluated and analyzed.The outline of the paper is organized as follows: In Section 2, we describe the experimental setup of the FSO channel inwhich we characterize the lognormal fading and derive the discrete-time lognormal optical channel state. In Section 3,we formulate the pointing error problem to maintain the centroid of the optical beam as close as possible to the centerof the photodetector. In Section 4, the main results of the pointing error problem based on the H ∞ -norm optimization2 rXiv Template
Figure 1: Block diagram of the experiment FSO setup.method under controlled weak-turbulence conditions are derived. In Section 5, we evaluate the quality of the pointingerror control and the communication performance metrics through numerical simulation tests. Finally, concludingremarks of the proposed robust pointing error control are presented in Section 6.
Notations. M T is the transpose of M . In symmetric block matrices, the symbol ( (cid:63) ) in any matrix represents for anyelement that is induced by transposition. (cid:107) . (cid:107) is the induced -norm. and I stand for the null matrix and the identitymatrix of appropriate dimensions, respectively. The FSO link consists of a transmitter and receiver separated by the atmospheric channel. Here we set up anexperimental system, the schematic diagram of the experimental line-of-sight FSO link is shown in Fig. 1. The opticalsignal amplitude through the FSO channel is ﬂuctuated due to the atmospheric turbulence. Many statistical modelsof the intensity ﬂuctuation through FSO channels have been proposed in the literature for distinct turbulence regimes.For weak turbulence conditions, the most widely used model is the log-normal distribution, which has been validatedthrough studies (Farid and Hranilovic, 2007), (Majumdar and Ricklin, 2008), (Zhu and Kahn, 2002). It is a well-knownmodeling approach and has been adopted in many calculations for the turbulence channel. This paper will focus on theweak turbulence; therefore, the lognormal model will be used throughout.
The laboratory atmospheric channel is a closed glass chamber with a dimension of × × cm as depicted in Fig.2 with the aim of observing the effect of atmospheric turbulence on the laser beam propagating through the channel.The main parameters of the FSO link are given in Table 1. The probability density function (PDF) of the receivedirradiance I due to the turbulence is derived by (Osche, 2002), (Ghassemlooy et al., 2012a) p ( I ) = 1 √ πσ I exp (cid:26) − (ln( I/I ) + σ / σ (cid:27) (1)where I is the irradiance when there is no turbulence and σ is the log-amplitude variance or scintillation index in thechannel.The measured eye-diagrams for the received signal are depicted in Fig. 3 without turbulence and in Figs. 4 and 5with weak turbulence. We observe that the eye-opening is smaller in the presence of turbulence, which results in aconsiderable level of signal intensity ﬂuctuation and also reduces the FSO performance link.Fig. 7 shows the histogram and the curve ﬁtting plots of the received intensity signal without turbulence. As we can see,the PDF distribution is nearly Gaussian for lower values of σ . The experimental snapshot intensity sensed by the PDcan be ﬁtted with a nearly Gaussian distribution, as illustrated in Fig. 6.Figs. 8 and 9 illustrate the histograms and the curve ﬁtting plots of the received intensity signal with turbulence. It isclear that the PDF has a good ﬁtting with the lognormal distribution, and the estimated scintillation index σ falls within3 rXiv Template
Figure 2: Experimental laboratory turbulence chamber.Table 1: Parameters of the FSO link.
Description Parameter Value
Data Format OOK NRZPRBS length − Signal intensity V peak-to-peak . VData rate . Mb/sLaser diode Type LP642-SF20Peak wavelength nmOptical Output Power mWOperating current/voltage . A/ . VPhotodetector Type APD210Spectral range − nmMaximum gain . × V/WDetector diameter . mmRise time . nsLens Type LA1417-ADiameter . mmFocal length mmTransmitter Type N4903B J-BERTReceiver Type 86100C-DCA-JSampling time . nsChamber Dimension × × cm the range of [0 , . is characterized by weak turbulence regime (Ghassemlooy et al., 2012a). The results also showthe theoretical red-dotted-lines ﬁt well with the simulation solid-blue-lines, which demonstrate the close resemblancebetween the PDFs fading statistics based on lognormal distributions of the theoretical predeﬁned autocorrelationfunction (see (Primak et al., 2005; Kontorovich and Lyandres, 1995; Bykhovsky et al., 2015; Neuenkirch and Szpruch,2014) and the simulation for short-range turbulent-channel communication experiment. As σ increases, the distributionis more skewed with a long tail toward the inﬁnity and reduced peak of probability density as a result of signal fading.4 rXiv Template
Figure 3: Measured screen shot eye-diagramof received intensity signal without turbulence Figure 4: Measured screen shot eye-diagramof received intensity signal under weak turbu-lence: σ = 0 . .Figure 5: Measured screen shot eye-diagram of received intensity signal under weak turbulence: σ = 0 . . I n t e n s it y [ m v ] x [mm] y [mm] Figure 6: Experimental snapshot intensity of the laser beam obeying to Gaussian distribution.
The most important property of the optical beam is the PDF of gain samples, so we use a modeling approach to get theoptical beam position. The modeling approach is based on 1D lognormal distributed samples with a correspondingcorrelation function, as illustrated in Figs. 7, 8 and 9. The ﬁrst step is to efﬁciently approximate the weak turbulencelevel of the FSO chamber, which means to emulate the variance of the atmospheric turbulence. We verify the theoreticalresults of the lognormal process with the predeﬁned autocorrelation function using the implicit Milstein scheme forthe channel states, which converges to a simple discrete-time differential equation (Neuenkirch and Szpruch, 2014),(Bykhovsky, 2015). The implicit discrete-time Milstein scheme for the lognormal distribution describing the simulatedlognormal optical beam channel state and its relative position is generated by the following nonlinear discrete-time5 rXiv
REPRINT P r ob a b ilit yd e n s it y Irradiance (mv)
Figure 7: Gaussian PDF received distributionwithout turbulence (the curve ﬁtting is shownby solid lines) P r ob a b ilit yd e n s it y Irradiance (mv)
Figure 8: Log-normal PDF received distributionunder weak turbulence: σ = 0 . (the curveﬁtting is shown by solid lines). P r ob a b ilit yd e n s it y Irradiance (mv)
Figure 9: Log-normal PDF received distribution under weak turbulence: σ = 0 . (the curve ﬁtting is shown bysolid lines).state-space equations (Neuenkirch and Szpruch, 2014), (Kontorovich and Lyandres, 1995), (Bykhovsky, 2015) (cid:40) x pk +1 = a p x pk + ϕ ( x pk ) + b p u pk + r p w pk ,θ k = c p x pk , (2)where ϕ ( x pk ) = − K σ x pk [ln( x pk /I )] , r p = √ K × ∆ t, (3) K is given by K = 2 I exp( σ )[exp( σ ) − τ c , (4)where k ∈ Z + is the set of all nonnegative integers, x pk ∈ IR is the simulated optical channel state which can beconsidered as a moving object, θ k ∈ IR is the position of the optical beam transmitter, w pk ∈ IR are samples of thewhite Gaussian noise and τ c is a predeﬁned correlation time, u pk is the bounded control input through which the opticalchannel state and transmission angle are changed. ∆ t is the sampling time, a p ∈ IR , b p ∈ IR , c p ∈ IR and r p ∈ IR areconstant values. The nonlinearity ϕ ( x pk ) ∈ P ⊆ IR + represents here the full signal strength model. It is differentiablewith ϕ (0) = 0 , locally Lipschitz and also monotonically increasing in P . The time-correlated position of the transmittedoptical beam that was generated by (2) with σ = 0 . , τ c = 0 . , a p = 1 , b p = 1 and c p = 1 is illustrated in Fig. 10.6 rXiv Template
REPRINT θ - d i s p l ace m e n t [ m m ] Time [ s ec] Figure 10: Position of the transmitted optical beam motion versus time. α - d i s p l ace m e n t [ m m ] Time [ s ec] Figure 11: Position of the 1D receiving aperture motion versus time.
Although the photodetector’s receiver aperture is ﬁxed, it still suffers some random physical vibrations due to thermalexpansion, voltage jitter, etc (see (Cai et al., 2019)). So, the receiving aperture motion is assumed similar to theBrownian motion of a particle subjected to excitation, as showed in Fig. 11. The Brownian motion is given by thegeneralized differential equation (Volpe and Volpe, 2013; Amari et al., 2013, 2014) m d x ( t )d t = − γ d x ( t )d t − k d x ( t )d t + (cid:112) k B T γ W ( t ) , (5)where x ( t ) is the trajectory of the particle with respect to the center, m is the particle mass, γ is the friction exertedby the surrounding medium on the particle, k is the optical trap stiffness, k B T is the thermal energy unit, k B is theBoltzmann constant, T is the absolute temperature and W ( t ) is white Gaussian noise. The discrete-time state space ofthe receiving aperture model is derived from the discretized particle Brownian motion (5). It is given as follows (Volpeand Volpe, 2013; Amari et al., 2013, 2014) (cid:40) x lk +1 = a l x lk + r l w lk ,α k = c l x lk , (6)where x lk is the source position, α k is the measured position of the receiving aperture motion, w lk is a standard whiteGaussian noise, ∆ T is the discretization time step, a l = (cid:16) − k ∆ Tγ (cid:17) , r l = √ k B T γ and c l = 1 . We consider a one-way optical link that consists of an optical transmitter and an optical receiver. Both are subjectto relative motions. The emitted optical beam has a non-uniform intensity proﬁle, which is assumed to be Gaussian7 rXiv
REPRINT (Gagliardi and Karp, 1995) and can be considered as a moving object. The goal is to control how the position of theobject change in time. The receiver’s aperture is assumed to be smaller than the received optical beam so that thereceiver can collect only a fraction of the optical beam (Komaee et al., 2007). This captured fraction can be enlargedby active pointing whose objective is to maintain the optical beam’s center at the center of the receiving aperture. Aphotodetector is used at the receiver to measure the optical beam’s intensity proﬁle that strikes its aperture (Komaeeet al., 2007). The output is then sent as feedback through an optical link or low-bandwidth RF channel and used to adjustthe transmitter’s position. Fig. 12 illustrates the block diagram of this active pointing scheme under weak controlledturbulence. Figure 12: Active pointing scheme for a short range free-space optical channel.The discrete-time model considered in this study has been derived from the model structure that was introduced forthe stochastic state-space model (Komaee et al., 2007), (Bykhovsky, 2015). Indeed, the lognormal random process isrepresented as a solution of a stochastic differential equation (SDE), which is approximated by a general and effectivediscrete-time model. The model mainly describes the relative position’s effect between the transmitter and the receiveron the signal strength. We denote the transmitted optical beam position to a ﬁxed coordinate system by the vector θ k and the position of the receiving aperture of the stations to the same coordinate system by α k . We assume that thereceiving aperture is held perpendicular to the line-of-sight optical beam. The relative displacement of the optical beamcenter to the receiving aperture center is given by y k = d ( θ k − α k ) where d is the distance between the transmitter andreceiver. Fig. 13 illustrates the optical beam in the plane of the receiving aperture and the displacement vector y k .The pointing error y k = d ( θ k − α k ) is a linear function of x pk and x lk . It can be written as the following augmentedsystem form (cid:40) x k +1 = A x k + ϕ ( x k ) + B u k + R w k ,(cid:15) k = C x k , (7)where A = (cid:34) a p a l (cid:35) , B = (cid:34) b p (cid:35) , R = (cid:34) r p r l (cid:35) , C = (cid:104) r p − r l (cid:105) , and x k = (cid:34) x pk x lk (cid:35) ∈ IR is the augmented state vector, w k = (cid:104) w pk w lk (cid:105) is the augmented disturbance vector, y k = d(cid:15) k isthe pointing error and (cid:15) k = θ k − α k .Since ϕ ( x pk ) is Lipschitz, then the augmented nonlinearity ϕ ( x k ) is assumed to satisfy the following bound ϕ ( x k ) T ϕ ( x k ) (cid:54) x Tk H T H x k , (8)where H is a constant matrix.The robust control problem studied in this paper consists in minimizing the closed-loop pointing error while ensuringthe disturbance attenuation level. The objective is to maintain the centroid of the optical beam as close as possible tothe center of the photodetector. This pointing problem can be interpreted as ﬁnding a set-point u k = − K x k dependingon x k such that the following H ∞ norm of the pointing error y k with respect to disturbance w k is satisﬁed i.e : (cid:107) y k (cid:107) (cid:54) ε (cid:107) w k (cid:107) , (9)with ε being the smallest positive real to be minimized. H ∞ Pointing Error Control for FSO
In this section, we consider the H ∞ -norm optimization problem that guarantees the closed-loop pointing error y k isstable and ensures the disturbance attenuation level (cid:107) y k (cid:107) (cid:54) ε (cid:107) w k (cid:107) for a prescribed attenuation level ε > .8 rXiv Template
REPRINT y k Receiving apertureOptical beam
Figure 13: Optical beam, receiving aperture and the displacement vector y k .The following theorem provides the stability and the absolute pointing error of the augmented system (7). Theorem 1
If there exist matrices Y = Y T > , S and scalar ε such that the following LMI condition min S , Y > ε subject to − Y YA T − S T B YC T YH T ( (cid:63) ) − ε I 0 R T ( (cid:63) ) ( (cid:63) ) − δ I Y ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − Y ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − I 0 ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − δ − I < . (10) has a feasible solution with K = SY − , then i) the augmented closed-loop error system (7) is stable for w k = 0 . ii) and the pointing error y k satisﬁes the disturbance attenuation condition for w k (cid:54) = 0 and a speciﬁc attenuationfactor ε > , (cid:107) y k (cid:107) (cid:54) ε (cid:107) w k (cid:107) . (11) Proof 1
Let us deﬁne the Lyapunov function V k = x Tk P x k with P = P T > and the H ∞ cost (Boyd et al., 1994) inequation (11) as follows J (cid:44) N − (cid:88) k =0 (cid:0) y Tk y k − ε w Tk w k (cid:1) , (12) over the time interval [0 , N − , N ∈ IN . If V = 0 , i.e, all the initial conditions are null then, inequality (11) holds ifthe following satisﬁes along the trajectories of system (7) J < N − (cid:88) k =0 (cid:0) x Tk C T C x k − ε w Tk w k + V k +1 − V k (cid:1) . (13) A sufﬁcient condition to fulﬁll the inequality (13) is to guarantee for all k ∈ Z (cid:62) x Tk C T C x k − ε w Tk w k + V k +1 − V k < . (14) Inequality (14) can be written as follows x Tk (cid:2) ( A − BK ) T P ( A − BK ) − P (cid:3) x k + x Tk ( A − BK ) T PR w k + x Tk ( A − BK ) T P ϕ ( x k )+ w Tk R T P ( A − BK ) x k + w Tk R T PR w k + ϕ T ( x k ) P ( A − BK ) x k + ϕ T ( x k ) P ϕ ( x k )+ ϕ T ( x k ) PR w k + w Tk R T P ϕ ( x k ) + x Tk C T C x k − ε w Tk w k < . (15)9 rXiv Template
REPRINT − P + C T C + ν H T H ( (cid:63) ) − ε I 0 ( (cid:63) ) ( (cid:63) ) − ν I + ( A − BK ) T PR T PP P − (cid:104) P ( A − BK ) PR P (cid:105) < , (19) Inequality (15) can be expressed as x k w k ϕ ( x k ) T Ξ ( A − BK ) T PR ( A − BK ) T P ( (cid:63) ) − ε I + R T PR R T P ( (cid:63) ) ( (cid:63) ) P x k w k ϕ ( x k ) < , (16) where Ξ = − P + ( A − BK ) T P ( A − BK ) + C T C .Using the fact that constraint (8) is equivalent to the following quadratic inequality for any δ > δ x k w k ϕ ( x k ) T H T H ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − I x k w k ϕ ( x k ) (cid:62) , (17) Combining together (16) and (17) gives x k w k ϕ ( x k ) T Ξ ( A − BK ) T PR ( A − BK ) T P ( (cid:63) ) − ε I + R T PR R T P ( (cid:63) ) ( (cid:63) ) P − δ I x k w k ϕ ( x k ) < , (18) where Ξ = − P + ( A − BK ) T P ( A − BK ) + C T C + δ H T H .Using the well-known Schur complement, we obtain inequality (19) .Applying again the Schur complement to (19) , then we obtain the following sufﬁcient condition − P ( A − BK ) T P C T H T ( (cid:63) ) − ε I 0 R T P ( (cid:63) ) ( (cid:63) ) − δ I P ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − P ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − I 0 ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) ( (cid:63) ) − δ − I < . (20) Pre- and post-multiplying inequality (20) by diag[ Y , I , I , Y , I , I ] and using Y = P − and S = KY which yields to the LMI (10) .The fulﬁllment of inequality (10) implies the fulﬁllment of the optimality condition: (cid:107) y k (cid:107) (cid:54) ε (cid:107) w k (cid:107) , u k (cid:54) = 0 , w k (cid:54) = 0 . (21) This completes the proof.
This section evaluates both the quality of the pointing error performance and the communication performance metricsto ensure accurate pointing and improve the optical communication link performance.
To evaluate the quality of the pointing error obtained by the LMI condition given in (10). The YALMIP interface(Lofberg, 2016) to MATLAB 8.5 with SDPT3 optimization toolbox (Toh et al., 1999) was used to provide solutions.10 rXiv
REPRINT po i n ti ng e rr o r y k [ m m ] Time [ s ec] Figure 14: Closed-loop pointing error versus Time. P r ob a b ilit yd e n s it y Distance [mm]
Figure 15: Open-loop output-error displacement. -3 -2 -1 0 1 2 300.10.20.30.188.8.131.52 Closed-loop displacement in X-axis P r ob a b ilit yd e n s it y Distance [mm]
Figure 16: Closed-loop output-error displacement.The performance of the pointing error control strategy is visualized through Fig. 14 where ε = 0 . and K = (cid:104) . − . (cid:105) . We observe clearly that the closed-loop pointing maintains a small alignment error between theoptical beam transmitter and the receiving aperture system with a relatively error amplitude less than ± . m fromthe detector diameter, which is ± . m. Therefore the maximum relative error is ± .Statistical simulated values provide more analytical insight. Figs. 15 and 16 depict histogram data of the open-loop andclosed-loop output-error displacements, respectively. The open-loop error displacement varies from +2 m m to +7 m m.The performance of open-loop with an output-error variance σ = 0 . as expected due to bias and drift terms isinadequate and do not satisfactorily stabilize the beam at the center. For the closed-loop output-error feedback, thebeam is stabilized at the center of the sensing device, the closed-loop error displacement varies from − m to +3 m mand the output-error variance is σ = 0 . which is ± reduced from the open-loop output error.11 rXiv Template
30 31 32 33 34 35 3610 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 σ = 0.0231 σ = 0.03801.5 (dBm)Closed-loop Open-loop O u t a g e P r ob a b ilit y Power Margin ( d Bm)
Figure 17: Open-loop and closed-loop outage probability errors.
The outage probability error and bit error rate (BER) are metrics for quantifying communication systems’ performancein fading channels. FSO system with a good average BER can temporarily suffer from increases in pointing error ratedue to fading effects (Farid and Hranilovic, 2007; Alouini and Simon, 2002; Ghassemlooy et al., 2012b; Yang et al.,2014). The outage probability is given as follows (Ghassemlooy et al., 2012b) P o ( I ) = (cid:90) I /m √ πσ I exp (cid:26) − (ln( I/I ) + σ / σ (cid:27) d I, (22)where m is the power margin which is introduced to account for the extra power needed to cater for turbulence-inducedsignal fading. Using Chernoff upper bound on (22), an approximate power margin, m , needed to obtain P o can beobtained as follows (Ghassemlooy et al., 2012b) m ≈ exp (cid:16)(cid:112) − P o σ + σ / (cid:17) . (23)The evaluation of this outage probability error for the FSO link in open-loop and closed-loop conditions is depicted inFig. 17. The closed-loop outage probability error which output-error variance σ = 0 . is reduced of . dBm fromthe open-loop outage probability error which output-error variance σ = 0 . . For example, to achieve an outageprobability of − about . Bm of extra power is needed in open-loop condition at σ = 0 . . This is reduced to . Bm in closed-loop condition as the scintillation strength decreases to σ = 0 . .The average BER of OOK-based FSO in atmospheric turbulence is deﬁned as (Yang et al., 2014)BER = (cid:90) ∞ p ( I ) Q (cid:18) ηI √ N (cid:19) d I, (24)where Q ( x )= (cid:90) + ∞ x exp( − t /
2) d t , η is the optical-to-electrical conversion coefﬁcient, I represents the received opticalintensity signal, N is the additive white Gaussian noise power spectral density. The integration in (24) can be efﬁcientlycomputed by Gauss-Hermite quadrature formula (Navidpour et al., 2007; Ghassemlooy et al., 2012b; Alouini andGoldsmith, 1999; Osche, 2002; Yang et al., 2014). Fig. 18 shows the BER plots of OOK-based FSO in atmosphericturbulence corresponding to the open-loop and closed-loop data rate values at various levels of output-error variance.As we can see, the effect of turbulence strength on the amount of signal-to-noise ratio (SNR) is required to maintain agiven error performance level. From Fig. 18, it can be inferred that atmospheric turbulence can causes SNR penalty,which might affect the pointing error, for example, around
B of SNR is needed in open-loop condition to achieve aBER of − due to the very weak scintillation of strength σ = 0 . . It decreases by over dB with the closed-loopcontrol as the scintillation strength decreases to σ = 0 . , which implies that pointing error control strategy mightbe required to avoid a BER ﬂoor in the system performance.12 rXiv Template
REPRINT -3 -2 -1 σ = 0.0231: Closed-loop σ = 0.0380: Open-loop 2 dB B E R SNR ( d B) Figure 18: Open-loop and closed-loop BER performances of OOK-based FSO in atmospheric turbulence.
In this paper, the link performance of the presented FSO link under the inﬂuence of the weak atmospheric turbulencehas been investigated. The atmospheric turbulence chamber has been characterized theoretically and experimentally fora valid comparison. We found that the fading statistics follow the well-known lognormal distribution that is used forweak turbulence characterization. Based on that, a deterministic nonlinear discrete-time model for pointing error lossdue to misalignment has been derived. We then investigate the H ∞ norm optimization problem that guarantees theclosed-loop pointing error is stable and ensures the prescribed disturbance attenuation level. The closed-loop pointingerror from the numerical simulation shows the center of the optical beam close enough to the receiving aperture center,verifying the efﬁciency of our proposed robust pointing error control for FSO communication systems. References
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