Statistical Model And Performance Analysis Of A Novel Multilevel .
hvphotonicsArticleStatistical Model and Performance Analysis ofa Novel Multilevel Polarization Modulation inLocal “Twisted” FibersPierluigi Perrone *, Silvello Betti and Giuseppe Giulio RutiglianoDepartment of Electronics Engineering, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome,Italy; betti@ing.uniroma2.it (S.B.); rutigliano@ing.uniroma2.it (G.G.R.)* Correspondence: pierluigi.perrone@uniroma2.it; Tel.: 39-06-7259-7446Received: 6 November 2016; Accepted: 20 January 2017; Published: 26 January 2017Abstract: Transmission demand continues to grow and higher capacity optical communicationsystems are required to economically meet this ever-increasing need for communication services.This article expands and deepens the study of a novel optical communication system for high-capacityLocal Area Networks (LANs), based on twisted optical fibers. The complete statistical behaviorof this system is shown, designed for more efficient use of the fiber single-channel capacity byadopting an unconventional multilevel polarization modulation (called “bands of polarization”).Starting from simulative results, a possible reference mathematical model is proposed. Finally,the system performance is analyzed in the presence of shot-noise (coherent detection) or thermalnoise (direct detection).Keywords: Brownian motion; optical fiber circular birefringence; optical transmission; polarizationmodulation; twisted optical fiber1. IntroductionThe optical fiber constitutes the optimal communication channel for its efficiency, capacity, andtransmission rate. It forms the high-capacity transport infrastructure that enables global broadbanddata services and advanced Internet applications. For this reason, countless studies have focused theirattention on how it could be possible to exploit all the characteristics of this medium in terms of thetransmission capacity. As it often happens, certain features, initially considered as drawbacks, can berevealed to have advantages.One of these natural characteristics of the optical fiber is the birefringence. It has beenwidely studied, especially for the negative effects that it causes, such as the PMD (PolarizationMode Dispersion). It is important to distinguish between linear and circular birefringence. Linearbirefringence has several origins: bending, geometrical imperfections, and stress-induced anisotropies.All of them act randomly along optical fibers. Circular birefringence may be generated, instead, byan external magnetic field aligned with the axis of propagation, or by twisting the fiber itself [1,2].Basic models of the optical fiber birefringence have been described in [3,4]. These works only take intoaccount the linear birefringence, neglecting the circular one. This approach is justified by the fact that inmost of the fibers used in optical communications, circular birefringence can be considered negligible.A successive work [5] focused on the development of a complete model of the birefringence thatalso included the circular component. In this general model, during the production phase, a twistingprocess of the fiber generates an induced circular birefringence. This production process has theadvantage of a PMD decrease [6]. Another advantage of twisted fiber is the opportunity to betterexploit the multilevel polarization modulations.Photonics 2017, 4, 5; hotonics
Photonics 2017, 4, 52 of 13The multilevel polarization modulations (M-PolSK—Multilevel-Polarization Shift Keying) exploitthe additional degrees of freedom provided by the use of the State of Polarization (SOP) of a fullypolarized light wave as a “modulation parameter” in a three-dimensional [7] and a four-dimensionalEuclidean space [8]. Birefringence causes SOP changes during the propagation of the signal alongthe optical fiber. Therefore, M-PolSK modulations, despite a better exploitation of the single channelbandwidth [9], require a complex receiver, able to track the birefringence for the correct estimationof the transmitted symbols. In [10] it is shown that by applying a novel M-PolSK modulation in atwisted fiber, it is possible to confine the SOP evolution of the transmitted symbols within specificphysical “polarization bands” (on the surface of the Poincaré sphere). In this way, at the receiverend, there is no need to implement a complex mechanism for tracking the birefringence because it issufficient to identify the band of polarization of the received SOP to estimate the transmitted symbol.This modulation also offers the advantage of a “fluid” constellation of symbols that no longer need tobelong to a rigid geometric structure. The advantage of a “fluid” constellation lies in the fact that thedecision regions may be associated directly with physical regions. The structure of the constellationis such that the modulator must only change the value of the S3 component. The drawback is thepresence of a limited number of polarization bands (physical tracks), conditioned by the twistingprocess that is possible to be introduced in the optical fiber. Moreover, the performance of the proposedsystem is compatible with those systems with a similar number of symbols, but with the advantage ofa simpler receiver structure.This paper, starting from the model proposed in [10], analyzes the statistical properties of thisnovel multilevel polarization modulation for twisted fibers that can be used in a Local Area Network(LAN) environment. In fact, the proposed system would fit the LAN environment very well, suchas the systems mentioned in [11,12]. In this case, the advantage is not related to the growth of thetotal throughput, but rather to the complexity reduction of the transmitter and receiver. Moreover,a mathematical model [13] for the evolution of the SOP along the twisted fiber is proposed andcompared with the simulative results. Finally, the performance of this system is compared with that ofthe standard M-PolSK modulations both in the case of coherent detection (shot-noise limited) [7] anddirect detection (thermal noise limited) [14].2. Theoretical Background2.1. Birefringence ModelsIn the past, many studies focused on a possible mathematical model that could be adopted todescribe the phenomenon of birefringence in optical fibers. This physical feature of the fiber can becharacterized by means of β (β1 , β2 , β3 )T , which represents the local birefringence vector in anypoint of the optical fiber propagation axis (z-axis).The components β1 and β2 characterize the linear birefringence, while β3 takes into account thecircular birefringence. The above-mentioned works [3,4] described models for optical fibers with linearbirefringence (considering a negligible β3 component). This assumption is valid for most of the fibersused in telecommunications. Linear birefringence is a stationary stochastic process [15], and accordingto the Wai-Menyuk Model (WMM) [3], the components β1 and β2 are independent Langevin processes(i 1, 2), β i / z ρβi (z) ση i (z)(1)with η 1 (z) and η 2 (z) as independent white noise processes with the following statistical properties(i 1, 2)E [ηi (z)] 0, E[ηi (z)ηi (z υ)] δ(υ)(2)
Photonics 2017, 4, 53 of 13with δ(ν) being the Dirac distribution. The terms ρ and σ represent the statistical properties of thebirefringence [3,5]s1π4ρ , σ (3)hfiberL B h f iberwhere hfiber and LB are, respectively, the fiber autocorrelation length and the mean fiber beat length.The fiber autocorrelation length is the length over which an ensemble of fibers, all of which initiallyhave the same orientation of the axes of birefringence, lose memory of this initial orientation (hfiber isthe distance over which the autocorrelation of the birefringence vector decays to 1/e). The fiber beatlength is the length required for a complete SOP rotation. Another important parameter is hE,local [3],equal to hfiber , which represents the length scale over which the field, measured with respect to thelocal axes of birefringence, loses memory of its own orientation with respect to those axes [16].In [10], in order to study the spatial evolution of the birefringence (and, consequently, of the SOP),a model is presented in which the component β3 (induced circular birefringence) of the localbirefringence vector β is no longer considered negligible. The simulations have been developedaccording to the assumption of using an optical fiber subjected to a twisting process, for distances thatcan be compared to those of a LAN. With this model, the local birefringence vector can be expressed inthe following way [5] β 1 (z) β ( z ) T( z ) β 2 ( z ) (4)0gτ (z)where T(z) is the rotation matrix of the cross-sectional plane z originated by the twisting process cos 2τ (z) T(z) sin 2τ (z)0 sin 2τ (z) 0 cos 2τ (z) 0 01(5)while τ(z) is the twist measured in radians and τ 0 (z) is the twist-rate expressed in rad/m. The parameterg refers to the optical fiber coupling parameters and represents the proportionality coefficient betweenthe twist-rate and the induced circular birefringence, that is β 3 (z) gτ 0 (z). Typically, experimentalresults lead to the value g 0.14 [5].2.2. M-PolSK ModulationsIt is known that the M-PolSK modulations use the SOP of a fully polarized light wave as a“modulation parameter”. Considering a reference plane (x, y), normal to the z propagation axis,the complex components of an electromagnetic field that travels along the z-axis are given byEx u( x, y) a x (t)e j(ωt φx (t)) x̂ Ex x̂Ey u( x, y) ay (t)ej(ωt φy (t))ŷ Ey ŷ(6)where ω is the angular frequency, ax , ay , φx , and φy are respectively the amplitude and phase of the x, yfield components, and u(x, y) is the transversal mode profile. It is possible to measure and to identifyunivocally the SOP of the signal through the Stokes parametersS0 a2x a2y S1 a2x a2yS2 2a x ay cos δ S3 2a x ay sin δ(7)
Photonics 2017, 4, 54 of 13with S02 S21 S22 S32 and δ φx φy . S0 is proportional to the power of the optical field travellingalong the fiber. A powerful way to visualize the SOP is represented by the Poincaré sphere of radiusS0 , in which any point represents a SOP with a different longitude (ψ) and latitude (χ)tan 2ψ S2 /S1 , sin 2χ S3 /S0(8)In the Stokes space, the differential equation of motion, which describes the spatial response ofthe SOP to the local birefringence, is S/ z βxS(9)where S (S1 , S2 , S3 ) is the three-component Stokes vector. Equation (9) describes the spatialevolution of the Stokes vector at a fixed angular frequency. Therefore, for Equation (9), the birefringencecauses random SOP fluctuations depending on the β behavior along the propagation z-axis. M-PolSKmodulation formats are possible because the rate of polarization changes along the propagation in theoptical fiber is very low and no significant variation can take place within a time interval comparablewith the symbol time.M-PolSK modulation schemes can be mapped onto the Stokes space. For higher order modulationschemes, M-PolSK shows an increased efficiency with respect to the more conventional coherentmodulation schemes, based on amplitude and/or phase modulation [9]. These systems must provideat the receiver end a tracking mechanism of the polarization changes due to the birefringence. M-PolSKmodulations are usually related to fixed and rigid constellations of symbols represented in the Stokesspace. Symbol constellations can be regular and symmetric polyhedra [7] inscribed into the Poincarésphere, or asymmetric polyhedra [9].The M-PolSK modulator has to change the input SOP in such a way that the corresponding SOPpoint in the Stokes space matches one of the symbols belonging to the signal constellation.The Stokes receiver, in addition to the Stokes parameters extraction, must be able to compensatethe SOP fluctuations. Specific receiver schemes described in [7,9] are able to track time changesof the SOP due to fiber birefringence. In the presence of slow fluctuations, the reference SOPs,which are associated with the transmitted symbols, have to be updated every TUP seconds, with1/W TUP TSOP , where W is the signal bandwidth and TSOP is the characteristic time of theSOP fluctuations.3. Statistics of the Proposed ModelThe work in [10] proposed a new type of multilevel-polarization modulation (“bands ofpolarization” modulation) that goes beyond the classical concept of symbols belonging to a rigidconstellation in the Euclidean space. This modulation model takes advantage of an intrinsiccharacteristic of optical fibers such as the birefringence. As a matter of fact, with a suitable twistingprocess, the induced circular birefringence β3 becomes predominant with respect to the linearbirefringence components β1 and β2 . In this case, the evolution of the SOP during its spatialpropagation along the fiber is confined latitudinally within specific physical tracks (called “bands ofpolarization”). Figure 1 shows the spatial evolution in the Poincaré sphere of five different SOPs intheir own “bands of polarization”; the simulated twisted fiber has a twist rate of 6 rad/m ( 1 turn/m).A fundamental benefit of this system consists of the reduced complexity of the receiver.Its simplicity derives from the simple need to detect only the S3 component of the received SOP.Therefore, there is no need to implement a specific circuit to track the birefringence’s variations.
constellation in the Euclidean space. This modulation model takes advantage of an intrinsiccharacteristic of optical fibers such as the birefringence. As a matter of fact, with a suitable twistingprocess, the induced circular birefringence β3 becomes predominant with respect to the linearbirefringence components β1 and β2. In this case, the evolution of the SOP during its spatialpropagation along the fiber is confined latitudinally within specific physical tracks (called “bands ofPhotonics 2017,4, 5polarization”).Figure 1 shows the spatial evolution in the Poincaré sphere of five different SOPs intheir own “bands of polarization”; the simulated twisted fiber has a twist rate of 6 rad/m ( 1 turn/m).5 of 131. Spatial evolution of the States of Polarization (SOPs) in a twisted fiber.FigureFigure1. Spatialevolution of the States of Polarization (SOPs) in a twisted fiber.A fundamental benefit of this system consists of the reduced complexity of the receiver. Its3.1. SimulationModelsimplicityderives from the simple need to detect only the S3 component of the received SOP.Therefore, there is no need to implement a specific circuit to track the birefringence’s variations.Starting from the simulative results described in [10], we analyzed the statistical behavior ofPhotonics 2017, 4, 55 of 13the different transmitted SOPs during their propagation along a twisted fiber. All of the statistical3.1. Simulationresults shownin thisModelpaper have been obtained using the model and the physical parameters reportedin [5]. In ordertoachievesignificantresults,have beenrepeatedStarting from thestatisticallysimulative resultsdescribedin [10],thewe simulationsanalyzed the statisticalbehaviorof the500 times(cycles),differentfor eachtransmittedvalue of twistand foreachdistancealongof propagation.SOPs duringtheirpropagationa twisted fiber. All of the statistical resultsin this paperhave beenobtainedusing the modeland thephysicalparameters reportedin [5].Theshownsimulationsoftwareusedwas MATLABR2016awithan AcademicLicense.In order toIn order to achieve statistically significant results, the simulations have been repeated 500 timesperform, in a reasonable time, the onerous statistical calculations required by the adopted mathematical(cycles), for each value of twist and for each distance of propagation.model, we implementeda software code that could exploit all the available hardware of the workstationThe simulation software used was MATLAB R2016a with an Academic License. In order ndependencethe parametersutilized in theperform, in a reasonable Giventime, thestatisticalcalculationsofrequiredby the adoptedwetheimplementeda software codethat couldall nmodel,cycles,serial calculationmethodwasexploitreplacedwitha etersmethod, thanks to the simultaneous use of eight independent logical processes that could run theutilized in the different simulation cycles, the serial calculation method was replaced with a parallel500 cyclesin parallel. For this choice, the software code implemented in [10] was adapted andcomputing method, thanks to the simultaneous use of eight independent logical processes that couldconfiguredfor an efficient use of parallel computing.run the 500 cycles in parallel. For this choice, the software code implemented in [10] was adapted andconfigured for an efficient use of parallel computing.3.2. Statistical Analysis3.2. Statistical AnalysisThe five transmitted symbols were chosen in such a way that the relative “bands of polarization”The five aroundtransmittedwerechosensucha way thatTotheachieverelative “bandsof polarization”were this objective,we analyzedwere symmetrical around the starting value of the latitude. To achieve this objective, we analyzedthe behavior of different SOPs that belonged to the same band of polarization; the chosen test bandthe behavior of different SOPs that belonged to the same band of polarization; the chosen test bandwas the ons.Figure2 showsthe evolutionspatial evolutionthe equatorialincludedthelinearlinear polarizations.Figure2 showsthe spatialof differentlinear linearpolarizations.It canthatthethecycloidalcycloidalspatialtrajectoryof theSOP is notof differentpolarizations.It canbebeseenseen thatspatialtrajectoryof the SOPis notsymmetricalwith respectto equatorialthe equatorialplaneififthethe startingof Sfacingfacingdownwardssymmetricalwith respectto theplanestartingvaluevalueof2 isS2null,is null,downwards ifif S1 is positive2a) upwardsand upwardstheoppositeopposite case(Figure2b); 2b);conversely,the trajectoryisS1 is positive(Figure(Figure2a) andininthecase(Figureconversely,the trajectoryissymmetrical with respect to the equatorial plane if the starting value of S2 is equal to one (Figure 2c).symmetrical with respect to the equatorial plane if the starting value of S2 is equal to one (Figure 2c).Moreover, the trajectories in Figure 2a,b are prolate cycloids while that in Figure 2c is a curtateMoreover,the trajectories in Figure 2a,b are prolate cycloids while that in Figure 2c is a curtate cycloid.cycloid.Figure 2. Spatial evolution of the SOPs in a twisted fiber.Figure 2. Spatial evolution of the SOPs in a twisted fiber.To enhance the visualization of the cycloidal patterns, we chose a weak twist rate of 1.5 rad/m.In fact, with a low twist rate, the spatial trajectory is a prolate cycloid, while when increasing the twistrate, it becomes first an ordinary cycloid and then a curtate cycloid. The same behavior holds true forthe elliptical polarizations. On the contrary, circular polarization is flattened towards the pole becauseof the presence of strong spatial constraints (Figure 3).The first objective is to study the dependency of the transmitted SOP spatial evolution from the
Photonics 2017, 4, 56 of 13To enhance the visualization of the cycloidal patterns, we chose a weak twist rate of 1.5 rad/m.In fact, with a low twist rate, the spatial trajectory is a prolate cycloid, while when increasing the twistrate, it becomes first an ordinary cycloid and then a curtate cycloid. The same behavior holds true forthe elliptical polarizations. On the contrary, circular polarization is flattened towards the pole becauseof the presence of strong spatial constraints (Figure 3).The first objective is to study the dependency of the transmitted SOP spatial evolution from thepropagation distance of the optical field along the fiber for different values of the twisting process.For each simulation cycle, we calculated the probability density function (hereinafter referred toas the pdf) of the third Stokes vector component S3 relative to the transmitted symbols. In fact,as demonstrated by Equation (8), S3 depends directly on the latitude angle and its variance is closelyrelated to the width of its associated “band” (Figure 1).Photonics 2017, 4, 5Photonics 2017, 4, 56 of 136 of 13FigureFigure 3.3. TopTop viewview ofof thethe spatialspatial evolutionevolution ofof aa circularcircular SOP.SOP.Figure 3. Top view of the spatial evolution of a circular SOP.Afterwards, in order to consider all the cycles’ contributions, we derived the average pdf as theAfterwards, inin orderorder toto considerconsider allall thethe cycles’cycles’ contributions,contributions, wewe derivedderived thethe averageaverage pdfpdf asas thetheAfterwards,mean curve of all the executed simulations. In Figure 4, the behavior of the above-described ,thebehavioroftheabove-describedmeanpdfmean curve of all the executed simulations. In Figure 4, the behavior of the above-described meanpdf for different propagation distances is shown.for differentpropagationdistancesis shown.pdffor differentpropagationdistancesis shown.Figure 4. Mean S3 pdf for different distances.FigureFigure 4.4. MeanMean SS33 pdfpdf forfor differentdifferent distances.distances.These plotted functions were obtained with different fiber distances but with the same value ofThese plotted functions were obtained with different fiber distances but with the same value ofthe twist(6 rad/m).Theweretransmittedhad a 45 btainedsymbolwith differentfiberdistancesbut withSOPthe withsame avalueofthe twist rate (6 rad/m). The transmitted symbol had a 45 linear polarization SOP with a Stokesvectorequalto rad/m).[0,1,0]. TheS3 variance,symboland consequentlythe polarizationwidth of theSOPbandspolarization,the twistrate (6The transmittedhad a 45 linearwithofa Stokesvectorvector equal to [0,1,0]. The S3 variance, and consequently the width of the bands of polarization,widensincreasingthe distancepropagation.theThisvariancegrowtha linear dependenceequal towith[0,1,0].The S3 variance,andofconsequentlywidthof thebandshasof polarization,widenswidens with increasing the distance of propagation. This variance growth has a linear ,whichshowsacomparisonbetweenthesimulatedwith increasing the distance of propagation. This variance growth has a linear dependence on theon the propagation distance as shown in Figure 5, which shows a comparison between the simulatedvalues and a distancelinearly fittedcurve.propagationas shownin Figure 5, which shows a comparison between the simulated valuesvalues and a linearly fitted curve.and a linearly fitted curve.
These plotted functions were obtained with different fiber distances but with the same value ofthe twist rate (6 rad/m). The transmitted symbol had a 45 linear polarization SOP with a Stokesvector equal to [0,1,0]. The S3 variance, and consequently the width of the bands of polarization,widens with increasing the distance of propagation. This variance growth has a linear dependenceon2017,the propagationdistance as shown in Figure 5, which shows a comparison between the simulated7 of 13Photonics4, 5values and a linearly fitted curve.Figure 5. S3 variance versus the propagation distance.Figure 5. S3 variance versus the propagation distance.Figure 6, instead, shows the dependency of the S3 pdf on different values of the twisting process.Figure6, instead,showstheobtaineddependencyof the S3apdfonvaluedifferentof thedistancetwistingprocess.These plottedfunctionswereby consideringfixedof the valuespropagationequalThese plotted functions were obtained by considering a fixed value of the propagation distance equalPhotonics 2017, 4, 57 of 13to 500 m. In this case, the transmitted symbol also had a 45 linear polarization SOP with a StokesPhotonics 2017, 4, 57 of 13to500 m.this case,also had a 45 linearpolarizationwithofa Stokesvector equalto In[0,1,0].ThetheS3transmittedvariance,symboland consequentlythe widthof theSOPbandspolarization,equalto [0,1,0].TheS3 variance,andconsequentlythe widthofofthethetwistingbands etwistingvalue.Therefore,increasegenerates atowith500 m.In this case,transmittedsymbolalso hadana 45 linear polarizationSOP with a Stokesnarrows with increasing the twisting value. Therefore, an increase of the twisting process generatespotentialvectorthroughputwithsame availablebandwidth,thisof multilevelpolarizationequal to rise,[0,1,0].ThetheS3 variance,and consequentlythe forwidthof typethe bandsof polarization,a potential throughput rise, with the same available bandwidth, for this type of multilevelnarrowswith increasingthe fortwistingvalue. Therefore,an increaseof the ,polarization modulation, because it allows for the presence of a greater number of bands (anda onpotentialthroughputrise, with the same available bandwidth, for this type of multilevelsymbols)the Poincarésphere.consequently,symbols)on the Poincaré sphere.polarization modulation, because it allows for the presence of a greater number of bands (andconsequently, symbols) on the Poincaré sphere.Figure 6. Mean S3 pdf for different twisting values.Figure 6. Mean S3 pdf for different twisting values.Mean S3 pdf for different twisting values.The variance decrease Figurehas an6.exponentialdependence on the twist rate, as shown in Figure 7,Thevariancehas anexponentialdependencerate, as shownin Figure 7,whichshowsdecreasea comparisonbetweenthe simulatedvalues onandtheantwistexponentiallyfitted curve.The variance decrease has an exponential dependence on the twist rate, as shown in Figure thepolarizationbandshasadependenceonwhich showscomparisonbetweenthe simulatedvaluesvaluesand anexponentiallyfittedfittedcurve.Therefore,which ashowsa comparisonbetweenthe simulatedandan (linear).the bandshasadependenceonthetwistingTherefore, the statistical results show how the width of the polarization bands has a dependence onprocess thethattwistingis muchstrongerthan that ofthanthe thatpropagationdistance(linear).processthat is(exponential)much stronger (exponential)of the propagationdistance(linear).Figure 7. S3 variance versus the twist rate.Figure 7. S3 variance versus the twist rate.The data originating fromtheseshow that the twisting process gives rise to aFigure7. Ssimulations3 variance versus the twist rate.physical track even tighter for circular polarization than for those equatorial and ellipticalThe data originating from these simulations show that the twisting process gives rise to apolarizations. Considering the same values of distance (500 m) and twist rate (6 rad/m), Figure 8aphysical track even tighter for circular polarization than for those equatorial and ellipticalshows the comparison between a linear and an elliptical SOP, while in Figure 8b a circular SOP ispolarizations. Considering the same values of distance (500 m) and twist rate (6 rad/m), Figure 8aadded. It is clear from Figure 8b, how large the difference is between the pdf curves of circularshows the comparison between a linear and an elliptical SOP, while in Figure 8b a circular SOP ispolarization on one side and those of the equatorial and elliptical polarizations on the other side.added. It is clear from Figure 8b, how large the difference is between the pdf curves of circularAnother important result that can be deduced by Figure 8a,b is that the width of the “bands”polarization on one side and those of the equatorial and elliptical polarizations on the other side.
Photonics 2017, 4, 58 of 13The data originating from these simulations show that the twisting process gives rise to a physicaltrack even tighter for circular polarization than for those equatorial and elliptical polarizations.Considering the same values of distance (500 m) and twist rate (6 rad/m), Figure 8a shows thecomparison between a linear and an elliptical SOP, while in Figure 8b a circular SOP is added. It isclear from Figure 8b, how large the difference is between the pdf curves of circular polarization on oneside and those of the equatorial and elliptical polarizations on the other side.Another important result that can be deduced by Figure 8a,b is that the width of the “bands”decreases, starting from the equator to the pole. This behavior proves how a transmitted circular SOPis physically advantaged with respect to the other SOPs, in terms of a less probable deviation from itsinitial position, during the spatial propagation in a twisted optical fiber. It is reasonable to assumethat this behavior is determined by the greater strength of the circular polarization with respect to thesymmetry,Photonics2017,also4, 5 circular, of the fiber core.8 of 13Figure 8. Mean S3 pdf: (a) comparison between equatorial and elliptical SOPs, (b) comparison betweenFigure 8. Mean S3 pdf: (a) comparison between equatorial and elliptical SOPs, (b) comparison betweenequatorial, elliptical and circular SOPs.equatorial, elliptical and circular SOPs.4. Mathematical Model4. Mathematical ModelThe statistical analysis performed by simulation can be matched with the mathematical modelThe statistical analysis performed by simulation can be matched with the mathematical modelproposed by Perrin [13], which characterizes the Brownian motion of a particle on the surface of aproposed by Perrin [13], which characterizes the Brownia
Photonics 2017, 4, 5 3 of 13 with d(n) being the Dirac distribution.The terms r and represent the statistical properties of the birefringence [3,5] r 1 h fiber, s 4 LB s p h fiber (3) where h fiber and LB are, respectively, the fiber autocorrelation length and the mean fiber beat length. The fiber autocorrelation length is the length over which an ensemble of fibers, all of which .
Module 5: Statistical Analysis. Statistical Analysis To answer more complex questions using your data, or in statistical terms, to test your hypothesis, you need to use more advanced statistical tests. This module revi
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Lesson 1: Posing Statistical Questions Student Outcomes Students distinguish between statistical questions and those that are not statistical. Students formulate a statistical question and explain what data could be collected to answer the question. Students distingui
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In addition to the many applications of statistical graphics, there is also a large and rapidly growing research literature on statistical methods that use graphics. Recent years have seen statistical graphics discussed in complete books (for example, Chambers et al. 1983; Cleveland 1985,1991) and in collections of papers (Tukey 1988; Cleveland
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