performance monitoring and measurement techniques for coherent optical systems

648 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 4, FEBRUARY 15, 2013

Performance Monitoring and MeasurementTechniques for Coherent Optical Systems

Bogdan Szafraniec, Senior Member, IEEE, Senior Member, OSA, Todd S. Marshall, Member, IEEE, andBernd Nebendahl

(Invited Tutorial)

Abstract—Modern spectrally efficient optical communicationsystems utilize polarization-multiplexed coherent transmissionin complex modulation format. Coherent receivers used in thesesystems measure the amplitude and phase of the optical signals forboth orthogonally polarized components carrying information.Knowledge of the amplitude and phase of the optical field, incombination with digital signal processing, gives the receiver aninherent metrology and performance monitoring capability. Asthe optical signal propagates from the transmitter over opticalfiber to the receiver, a signal transformation and degradation isexpected. The receiver observes the properties of the transmittedoptical signal as degraded by the impairments of the transmissionmedium. The details of monitoring optical signal parameters andlink impairments are the focus of this paper. The optical signalparameters include polarization state and residual carrier phase;optical link impairments include chromatic dispersion and polar-ization mode dispersion. Two distinct techniques are presented:one based on Stokes space analysis, and the other on Kalmanfiltering. The Stokes space techniques are modulation-formatindependent and do not require demodulation. The Kalman fil-tering provides optimal estimation of the physical quantities thatdescribe the optical signal and the optical medium.

Index Terms—Chromatic dispersion (CD), optical fiber disper-sion, optical fiber testing, optical modulation, optical polarization,Stokes parameters.

I. INTRODUCTION

O PTICAL performance monitoring estimates parametersof the optical signal or of the optical channel in order to

maintain the operation and to predict the performance of the op-tical transmission system [1]. Among multiple parameters thatare often monitored, the signal parameter that is of great interestis the optical signal-to-noise ratio (OSNR). The OSNR changesas the signal is attenuated and noise is accumulated. The OSNRcan be used to predict the bit error rate (BER) under the assump-tion of additive white Gaussian noise dominated channels. The

Manuscript received June 06, 2012; revised August 01, 2012; accepted Au-gust 02, 2012. Date of publication August 08, 2012; date of current versionJanuary 16, 2013.B. Szafraniec and T. S. Marshall are with the Measurement Research

Laboratory, Agilent Technologies, Santa Clara, CA 95051 USA (e-mail:[email protected]; [email protected]).B. Nebendahl is with the Photonic Test and Measurement Division, Agilent

Technologies, 71034 Böblingen, Germany (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JLT.2012.2212234

BER serves as a direct measure of the system performance. Analternative measurement to the OSNR is the Q-factor. However,the latter is becoming more challenging to estimate in the sys-tems that utilize complex modulation formats with complicatedeye diagrams. Recently, in coherent systems, efforts were madeto use error vector magnitude (EVM) as an alternative way ofpredicting the BER [2], [3]. The EVM is estimated from thedemodulated signals often illustrated in constellation plots. Im-pairments of the optical link that lead to degradation of perfor-mance are equally important. Among them are chromatic dis-persion (CD), polarization mode dispersion (PMD), and polar-ization-dependent loss. The classical approaches for measuringoptical signal and optical link (channel) include spectrum anal-ysis, polarimetry, AM and PM pilot tones injection, and mea-surement [1]. Often the optical performance monitor is associ-ated with an external device that is not a part of the transmis-sion system. The external monitor is capable of estimating oneor more parameters of interest.The concept of performance monitoring has dramatically

changed since the adoption of coherent optical transmission.In coherent systems, many impairments such as CD and PMDcan be estimated and compensated within the receiver itself[1], [4], [5]. Thus, the optical receiver can be considered toserve as an optical performance monitor in addition to itsprimary function of receiving data. A conceptual diagram ofthe coherent receiver is shown in Fig. 1. This diagram includesmany functions that are commonly performed in the processorof the receiver, including transformation of the received signal(e.g., polarization alignment) and compensation of impairments(e.g., CD and PMD compensation). The coherent receiver hasalso become part of some test instruments that are designedto analyze the quality of optical signals and to estimate theparameters of an optical link. These test instruments, unlike thereceivers in communication systems, are designed to performprimarily metrological functions. Consequently, they employtools that are optimized from the metrology point of view. Inmetrology, the estimation of the optical signal parameter or ofthe impairment is often as important as its compensation. Thisdiffers from the classical receiver where accomplishment ofequalization is often sufficient. Therefore, the techniques thatare described in this paper are selected to match the needs ofmetrology of coherent optical signals and optical links ratherthan of data reception alone. This is the key reason justifyingthe use of Stokes space and the Kalman filter, the two subjectsof this study.

0733-8724/$31.00 © 2012 IEEE

SZAFRANIEC et al.: PERFORMANCE MONITORING AND MEASUREMENT TECHNIQUES 649

Fig. 1. Conceptual diagram of the coherent receiver. The polarization multi-plexed optical signal incident on the receiver’s polarizing beam splitter is sepa-rated into orthogonal polarization components and combined with the commonoptical local oscillator (LO) in both 90 optical hybrids. The combined signalsare converted to electrical signals and digitized for further processing to mon-itor and compensate impairments in addition to demodulation.

Stokes space and Stokes vectors, unlike Jones vectors, do notcontain information about phase of the received optical signals.Thus, at the first glance, they may not seem very attractive forprocessing optical signals that are modulated in amplitude andphase. However, Stokes space contains information about po-larization state and it is frequently used for visualization of po-larization phenomena, often on the Poincare sphere. It is shownin this paper that Stokes space is especially useful for observingpolarization-multiplexed signals. It can serve to visualize andsolve the polarization alignment problem even in the presenceof CD. Furthermore, Stokes space can be used in estimationof clock frequency and phase, CD and PMD. Section II of thispaper is dedicated to Stokes space.Since its introduction in 1960, the Kalman filter has been used

as an optimal estimator in many different disciplines from nav-igation to the analysis of the stock market. The Kalman filteris a general purpose estimation technique. Thus, the examplesgiven in this paper do not address all possible applications ofthe Kalman filter but merely illustrate the process that allowscasting some of the common tasks that are performed in the re-ceiver into a form that is suitable for applying the Kalman filter.The three Kalman filtering examples of this paper include car-rier phase tracking, polarization tracking, and estimation of thefirst-order PMD. It is shown that, when using the Kalman filter,it is often possible to formulate models of physical phenomenathat lead to the estimation of some specific physical parameters,in this case optical parameters of the signal or link. Section IIIis dedicated to Kalman filtering.

II. STOKES SPACE ANALYSIS

A. Polarization and Birefringence

The polarization-multiplexed optical signal is convenientlydescribed by two complex numbers represented by the well-known Jones vector [6]. The first component of the Jones vectorrepresents a horizontally polarized component of the op-

tical wave, while the second component represents the verticallypolarized component, :

(1)

The vector components and are proportional to the trans-verse electric fields in the optical fiber. In the coherent receiver,the optical signals are mixed down in frequency and measuredas proportional electrical signals. The Jones vector representa-tion (1) corresponds to a transmitter that takes two polarizedmodulated optical waves, and , and combines them inorthogonal polarization states. The same representation corre-sponds to the receiver that separates the arriving optical wavewithin the polarizing beam splitter into two linearly polarizedoptical waves and . In each case, the system of coordinatesis different and defined by either the transmitter’s polarizingbeam combiner or the receiver’s polarizing beam splitter. Fur-thermore, the signal is affected by the properties of the opticalmedium. Thus, as the optical signal propagates from the trans-mitter to the receiver, a signal undergoes some kind of transfor-mation. The transmitted optical wave is typically different thanthe received optical wave; the nature of transformation changeswith time. The details of monitoring the optical signal at the re-ceiver are explored in this paper.Since Jones vectors are commonly used to describe the po-

larization state of monochromatic light, it is natural to enquireabout the polarization state of the combined (polarization-mul-tiplexed) optical waves. A convenient means of visualizing po-larization state is to observe it in Stokes space, often on thePoincare sphere [7]. The conversion from the Jones vector tothe Stokes vector is obtained from the following equation:

(2)

Unlike the Jones vector, the Stokes vector is real. The firstcomponent of the Stokes vector represents power while theremaining components , and represent content of 0linear, 45 linear, and circularly polarized light, respectively.These three components arranged in the vectordefine a 3-D space, here referred to as Stokes space. Forfully polarized light, the components of the Stokes vector aredependent and related by the equation .To relate modulated optical signals to the Poincare sphere

, it is convenient to select the sphere radiusequal to the average power. Thus, in this paper, the Poincare

sphere size represents the average power of the signal unless thesignals are purposely normalized to fit into a sphere of radius

. This scaling with power differs from a conventionof reserving the inside of the normalized sphere for partiallypolarized light. In this paper, light is always assumed to bepolarized. The bandwidth of the system is considered to bebroad enough to support this assumption.To give a quick overview of some conventions that are used

in connection with the Poincare sphere, we consider here se-lected polarization states and polarization state transformations


Fig. 2. Stokes space showing horizontal and vertical polarization states (and ) associated with Stokes parameter . The axis – defines 0 linearbirefringence. Rotations by angles and demonstrate transformation of hor-izontal polarization state into polarization state .

resulting from birefringence. Fig. 2 shows selected polarizationstates in 3-D Stokes space as represented on the Poincare sphere.Linear polarization states are all located on the equator of thesphere; circular polarization states are located at the poles. Ingeneral, orthogonal polarization states are always located onthe opposite sides of the sphere. In the case of optical medium,orthogonal polarization states define a birefringence axis andrepresent birefringence. A birefringent element represented bythis axis supports propagation of two eigenstates of polarizationwithout changing them; these eigenstates define the axis. Allother polarization states that enter such a medium evolve aroundthe birefringence axis, tracing a circular trajectory around it [8].This circular trajectory defines a typical polarization state trans-formation that takes place within the birefringent medium, oris observed at the output of that medium when the optical fre-quency (wavelength) is swept.In Fig. 2, the linear polarization states include the linear hor-

izontal polarization state and the linear vertical polarizationstate that are located on the opposite sides of the Poincaresphere and define the axis – (the axis associated with Stokesparameter ). The axis – represents the 0 linear birefrin-gence. Similarly, circularly polarized polarization states at thepoles of the sphere define an axis of the circular birefringence(the axis associated with Stokes parameter ) and the corre-sponding polarization evolution. It is evident from the Fig. 2that it is possible to transform the linear horizontal polarizationstate into any arbitrary state by combining rotation aroundthe circular birefringence axis (rotation along equator) and rota-tion around the 0 linear birefringence axis – . The rotationangles and are associated with the normalized form of theJones vector:

(3)

The rotations are defined by the Jones matrices. The circularbirefringence takes the form of a simple rotation matrix:

(4)

The linear birefringence is

(5)

The rotation angles and are expressed in terms of the com-ponents of the Stokes vector:

(6)

The transformations described here are used in the next sectionto align an arbitrary polarization state with linear horizontal po-larization state of the receiver.

B. Complex Modulation in Stokes Space

After the brief review of polarization and birefringence in theprior section, this section focuses on interpretation of complexpolarization-multiplexed signals in Stokes space. In today’soptical communication systems, the orthogonally polarizedoptical waves that are combined into a polarization-multiplexedsignal carry information by being independently modulatedin phase, or in both amplitude and phase. For instance, thequadrature phase-shift keying (QPSK) format that is populartoday, applies only phase modulation. However, the higherorder formats like quadrature amplitude modulation (QAM),for example, QAM16 or QAM64, are expected to becomewidespread in the future and involve both amplitude and phasemodulation. There are a number of other formats that are beingconsidered to optimize immunity to the phase noise. Thissection aims to develop a general description of complex polar-ization-multiplexed signals in Stokes space that is independentof modulation format. The result is a powerful method ofaligning the polarization state at the receiver independently ofthe format. The additional benefit of using Stokes space is thatdemodulation is not required to achieve polarization alignment.It is initially assumed that the properties of the optical

medium remain nearly the same over the spectral width of themodulated signal. Thus, the same polarization transformationapplies to all frequency components of the signal. It is alsoassumed that the modulated components and of thepolarization-multiplexed signal (1) remain confined to a unitcircle in the complex plane after appropriate normalization.This assumption is not in any way restrictive as optical signalpowers are always limited and any constellation may in fact benormalized to fit into a bounded area in the complex plane. Thecircular regions of modulated signals and are shown inFig. 3; each signal represented in a complex plane. The shapeof the circle is selected for ease of computation. It is initiallyassumed that the optical signal is in a linear horizontalpolarization state while the optical signal is in a linearvertical polarization state . The amplitude of signals in thecomplex plane is measured by the distance from the origin.Thus, the larger the modulus is, the larger is the power of thesignal. The similar principle is applicable to Stokes space whenconversion (2) is applied. The larger the amplitudes of theconstituent components and are, the larger is the distancefrom the origin in the 3-D space, and the larger is the powerof the resulting combined signal. Thus, it is expected that inorder to find the boundaries of the region that is occupied bymodulated polarization-multiplexed signals in Stokes space, it


Fig. 3. Two complex variables representing components of the Jones vector,each in a complex plane. An arbitrary complexmodulation format is bounded bya circle in the complex plane. Hence, a general polarization-multiplexed signalis represented by two circles showing independently modulated signals in hor-izontal and vertical polarization states ( and ).

is necessary to consider the points of highest amplitudes in thecomplex planes.A simple illustrative example of the polarization-multiplexed

optical signal is that of QPSK modulation. For QPSK modula-tion, all symbols have equal amplitude but differ in phase only.In Jones vector notation, this can be captured by the followingexpression:

(7)

where is used for normalization purposes and andrepresent discrete phases taken by QPSK symbols. In a normal-ized form (3), the Jones vector becomes

(8)

where and , or 3. Using theinterpretation of Jones vectors given in the previous section and(3), it can be readily determined that . Thus, as shownin Fig. 4, polarization multiplexing of two QPSK modulatedsignals results in four points in Stokes space. All four points arelocated on the meridian defined by the linear polarizationstates and the circular polarization states. Here, it is assumed forsimplicity that the phasing of the modulation (clock) is the samefor both components. The normalized power and the radius ofthe sphere are equal to 1. It is instructive to consider the casewhen one of the multiplexed signals, at some point in time, takesa value of zero. For the case of taking zero value, the Jonesvector is

(9)

Since the vertical polarization component is not present, thesignal is horizontally polarized. Also, the power drops to 1/2 ofthe power of (8) and, consequently, as shown in Fig. 4, the pointrepresenting pure horizontal polarization state is no longerlocated on the surface of the sphere but halfway between thesurface and the origin. This illustration of scaling with powerdiffers from the commonly used convention of using the insideof the sphere for partially polarized light. In this paper, all lightis assumed to be fully polarized. To summarize the preceding il-lustrative example, the distance from the origin in the complexplane is a measure of amplitude, the distance from the origin inStokes space is a measure of power.The general case of an arbitrary modulation format that is

normalized to fit into a unit circle is now considered. To stay

Fig. 4. Characteristic lens-like object in Stokes space is defined by mappingthe circle bounded constellations in Fig. 3 to 3-D Stokes space. The lens axisuniquely locates polarization states of transmission, in this case, and . Thisprovides a means for polarization state alignment.

at the boundary of the region occupied by a complex modu-lated polarization-multiplexed signal in Stokes space, a pointof largest amplitude is selected from the polarization state.Since the phase of that point does not change the transformationto Stokes space, it is convenient to choose 1. For the polar-ization state, all the points within a unit circle of the complexplane are considered. Mathematically, this is represented by thefollowing Jones vector:

(10)

where represents the amplitude and represents the phaseangle. Using (2), the expression for the Stokes vector is found

(11)

The last three components of this Stokes vector parametricallydescribe one of the paraboloidal surfaces shown in Fig. 4. Theother paraboliodal surface results from selecting 1 for the po-larization state and considering the full circle in the complexplane for the polarization state. Together the two paraboloidsdefine a lens-like object or a lens. The lens shown in Fig. 4 isinscribed into the Poincare sphere. It contains within its bound-aries all the points that comprise an arbitrary polarization-mul-tiplexed modulated signal. Furthermore, the axis of the lensuniquely identifies the polarization states and that wereused to examine formation of the lens-like object. In practicaltransmission systems, polarization states never remain the same;they evolve over distance and over time. Thus, this lens-like ob-ject never remains stationary; it evolves over the length of thefiber and over time. However, the orientation of the lens andof its axis can be always identified, for example, by finding thebest fit plane and its normal. Based on that normal vector’s ori-entation, it is always possible to align the lens and its axis withthe coordinate system of the receiver [9], [10]. The alignmentinvolves an estimation of the polarization states that define thelens axis and rotations that align these states with local hori-zontal and vertical polarization states. Two exemplary rotationsthat accomplish the alignment are provided in (4) and (5).


Fig. 5. Measured polarization-multiplexed QPSK modulated data at 40 Gb/s isplotted in Stokes space. The – axis locates the lens axis which is then ro-tated to align data with the receiver and thereby achieve polarization alignment.The inset shows data after polarization alignment.

Fig. 5 shows 40 Gb/s experimental data viewed in Stokesspace. Data represent polarization-multiplexed signals modu-lated in QPSK format and propagated over several meters ofSM fiber. Upon arrival at the receiver, the data are polarizationmisaligned with the lens axis pointing to polarization statesand that are rotated away from the horizontal and verticalpolarization states of the receiver. The plane of the lens and– axis can be located using the best fit plane. The inset of

the figure shows polarization aligned data that resemble someof the features considered in the preceding discussion. In partic-ular, the data contain four clusters that correspond to the QPSKmodulation format that are located near the surface of the lens.The remaining points that define all the transitions fill the re-maining volume of the lens. The lens-like object is visibly flatand defines a plane in 3-D space; the plane defines the normal tothat plane. It is important to mention that determining the ori-entation of the lens serves the function of a polarization statemonitor. As long as the sampling frequency is higher than theevolution rate of the lens, the polarization state can be estimated,tracked, and aligned. Thus, this method is highly insensitive tothe sampling frequency; indeed, asynchronous sampling maybe used. Furthermore, the polarization tracking is modulationformat insensitive and does not require demodulation of the op-tical signal. For completeness, the corresponding demodulateddata are shown in Fig. 6. It is important to mention that after po-larization alignment each polarization channel can be demodu-lated independently using techniques well known for the non-multiplexed signals [9], [11], [12].

C. Clock Recovery and CD Estimation

The squaring synchronizer is a method of clock recoverythat is frequently described in the literature [11], [12]. Themethod utilizes a nonlinear element that leads to squaring

Fig. 6. Demodulated polarization-multiplexed 40 Gb/s data from Fig. 5. AfterStokes space polarization alignment two polarization components ( and )are demodulated independently. The correctly demodulated symbols are high-lighted in the four corners of each constellation.

of the complex signal and to the recovery of the clock tone.Interestingly, transformation of Jones vectors into Stokesspace also involves the operation of squaring. The first Stokesparameter of (2) contains the powers of two polarizationcomponents: and . Both expressions contain clockinformation. Thus, after polarization alignment as describedin the prior section, the clock can be separately estimatedfor each polarization component. If the clock is the same forsignals in both polarization states, then spectral analysis ofis sufficient. Fig. 7(a) shows the spectrum of a complex signalprior to the operation of squaring. The signal represents 40Gb/s polarization-multiplexed QPSK transmission with a clockfrequency equal to 10.153 GHz. A notch filter was used tosuppress the clock tone in the complex signal before squaring.The emergence of the clock tone after the operation of squaringis illustrated in Fig. 7(b). The clock tone can be used to estimateboth the frequency and phase of the clock. This method ofclock recovery is used in the Kalman filter section of this paperin the examples that require knowledge of the clock.The clock tone can be further used to estimate CD [13]. In

the presence of CD, the strength of the clock is reduced. Thus,by subjecting the detected signal to CD compensation and ob-serving the strength of the clock tone, the CD can be estimated.The CD compensation can be accomplished in the time domainor in the frequency domain. The frequency-domain compen-sation relies on the fundamental CD property of group delaychanging linearly with optical frequency. This in turn yields aphase that changes quadratically with optical frequency. Thus,the compensating function implements an all-pass filter withquadratic phase. To recapture the described properties mathe-matically, the all-pass filter transfer function is defined as anexponential function

whose phase is quadratic as defined by the following expression:

(12)

The first derivative of phase determines the group delay

(13)

This contains a term that is linearly dependent on the opticalfrequency , and so the second derivative determines CD:

(14)


Fig. 7. Clock recovery by a squaring synchronizer. (a) Spectrum of the nonsquared complex signal after a notch filter shows that the clock tone is suppressed.(b) Spectrum of the squared complex signal with the desired clock peak clearly distinguished.

Here, it is important to mention that CD is commonly definedin the literature with respect to wavelength as opposed to an-gular optical frequency. Dispersion that is defined with respectto wavelength may be denoted by . Based on the relation-ships and , where denotesthe speed of light in vacuum, the dispersions and are re-lated by . The definition of is chosenfor the sake of mathematical simplicity in defining . Inthe coherent receiver, optical frequency is mixed down bythe optical local oscillator to an electrical frequency; however,(12) and (13) are still applicable. Thus, frequency can equiva-lently be interpreted as electrical frequency. For computationalsimplicity, and in (12) and (13) are typically set to zero.Since CD acts equally on both polarization componentsof the signal, the correction can be applied to the Jones vectorafter transformation to the frequency domain. Consequently, thecompensated signal is transformed back to the time domain andexamined for the strength of the clock. The range of the CD thatis examined is initially predefined based on the knowledge ofthe optical link.The example of a CD compensation search is shown in Fig. 8

for a compensating fiber known to have dispersion of about1700 ps/nm. At 1550 nm, this corresponds approximately to

ps/GHz. The examined dispersion values are steppedbetween and 60 ps/GHz in 100 steps. The maximum clockstrength is found for a compensating dispersion of approxi-mately 13.36 ps/GHz, thus, in agreement with the known targetvalue. The polarization-multiplexed QPSK modulated 40 Gb/sdata are shown in Fig. 9(a) and (b) before and after CD com-pensation, respectively. The insets show data in the complexplane viewed in a traditional manner. The complex plane imagein the inset of Fig. 9(b) shows that the data were properlycompensated. However, it is also useful to examine the datain Stokes space. When formulating the polarization alignmentprocedure in the prior section the only assumption was thatthe normalized data fit into a unit circle in the complex plane.Data subjected to CD experience optical frequency-dependentdelays that translate into optical frequency-dependent phaseshifts. This smears the data in the complex plane as shown in

Fig. 8. Clock strength is plotted as a function of compensating dispersionfor 1700 ps/nm dispersion compensating fiber. At the peak location of13.36 ps/GHz, CD is compensated.

the inset of Fig. 9(a). However, since CD operates in the samemanner on both polarization components there are no effects ofpolarization cross-coupling. Consequently, the original outlineshape of the lens is not altered. Therefore, the polarizationalignment procedure in Stokes space is fully applicable to thechromatically dispersed data. In Fig. 9(a), the data in the Stokesspace are scattered; however, they still fit into a lens shape andare polarization aligned without demodulation. Furthermore,by examining the clock strength, the CD can also be estimatedwithout demodulation. Not only are the compensated dataimproved in the complex plane, but also, the data are betterconfined in Stokes space. Fig. 9(b) shows a well-confined lenswith four distinct data clusters that are characteristic of QPSKmodulation. The correlation between the confinement of thedata in the complex plane and in Stokes space suggests that it ispossible to use Stokes space for compensation of impairments.This behavior will be further explored in sections dedicated toPMD.


Fig. 9. QPSK modulated polarization-multiplexed 40 Gb/s data in Stokes space and in the complex plane. (a) Before CD compensation. (b) After CD compensa-tion. Optimal CD compensation is determined from the peak location in Fig. 8. The compensation is verified by the distinct symbols of the constellation plot andthe lens-like shape in Stokes space with four characteristic data clusters.

D. Effects of PMD

In the initial analysis of Section II, it was assumed that all ob-jects in Stokes space retain their shape; although the orientationof the objects may evolve, the shape of a lens always remain alens. It was also shown that CD distorts the constellation in thecomplex plane; however, CD does not alter the boundaries of thelens. In this section, we consider polarization effects that maychange the shape of objects in Stokes space and consequentlylimit applicability of the Stokes space polarization alignmentmethod. The shape change of objects implies that the relativeposition of points in Stokes space is not preserved. This be-comes possible when rotations in Stokes space are dependent onoptical frequency. To understand this process, an illustrative ex-ample of 45 linear birefringence is first considered. The Jonesmatrix that represents 45 linear birefringence (PMD) can bederived by rotating a 0 linear birefringence matrix [14] or byusing matrices [15], [16]. The Jones matrix is describedby the following equation:

(15)

where denotes a 45 component of the PMD vectorwith and equal to zero, and denotes a

deviation from the optical carrier frequency. Thus, the carrierfrequency serves as a reference with the matrix being theidentity matrix at that carrier frequency. As the frequency devi-ates from the carrier, the off-diagonal terms of the matrix are no

longer zero. Then, the polarization state begins to evolve alonga circular trajectory around the 45 linear birefringence axis ina manner similar to rotations shown in Fig. 2. The evolution is afunction of the optical frequency and PMD. For example, whenthe frequency deviation and are large enough mayreach the value of or depending on the sign of .Then, the matrix takes a simplified form

(16)

The above matrix transforms a linear horizontal polariza-tion state (1, 0) into either a right circular polarization state

or into a left circular polarization state .Thus, the same polarization state may be transformed into twodifferent polarization states depending on the optical frequency.Furthermore, it is also possible that the linear horizontal po-larization state (1, 0) and the linear vertical polarization state(0, 1) are both transformed into, for example, the same rightcircular polarization state . Thus, the relationshipbetween different polarization states in Stokes space is nolonger maintained and the distortion of objects in that spaceis possible. Generally speaking, once the clock period thatdetermines the spectral width of the modulated optical signalsbecomes comparable to the differential group delay (DGD)the objects in Stokes space may change shape. This criterionalso applies to the lens-like shape that serves for polarizationalignment purposes. Based on our experimental work, the


Fig. 10. QPSK modulated polarization-multiplexed 40 Gb/s data in Stokes space. Before PMD compensation, as the misalignment between the polarization-mul-tiplexed signal and the birefringent element increases, the characteristic lens-like shape becomes increasingly distorted: (a) alignment, (b)–(c) small misalignment,(d) large misalignment. After PMD compensation, the lens-like shape is restored independently of the misalignment; (e)–(f) small misalignment; (g) large mis-alignment.

technique of polarization alignment in Stokes space is reliablewhen the DGD is smaller than half of the clock cycle.Fig. 10 illustrates distortions that are observed in Stokes

space in the presence of a large PMD. This experiment is per-formed for 200 ps DGD and a 40 Gb/s polarization-multiplexedQPSK modulated signal with a clock period of approximately100 ps. Thus, the DGD used in the experiment exceeded themaximum allowed value by a factor of 4. In Fig. 10(a), thepolarization multiplexed signal is aligned with the birefrin-gent element. The undesired scattering is minimal as eachpolarization component of the polarization-multiplexed signalpropagates within an eigenmode of linear birefringence. Asmisalignment increases [see Fig. 10(b)–(d)], so does the scat-tering of the points in Stokes space. For large misalignment[see Fig. 10(d)], the familiar outline of the lens vanishes andpolarization alignment in Stokes space is no longer possible.Since the power of the optical signal is preserved (PMD isrepresented by a unitary transformation), the outline of thePoincare sphere remains the same in size. However, the dis-tribution of points in Stokes space changes. For the alignedbirefringence [see Fig. 10(a)], the points are well confined intoa lens and contained within the sphere. For the misalignedbirefringence [see Fig. 10(d)], many points are outside of thesphere. The average distance from the origin remains the samefor all cases.It is not unexpected that the compensated data shown in

Fig. 10(e) and (f) is again well confined into a lens with fourdata clusters characteristic for the QPSK signals. The illustratedcompensation was accomplished by a search for a PMD vectorthat resulted in the best Stokes space data confinement. In gen-eral, data that have an exemplary constellation of symbols andtransitions in the complex plane for two orthogonal polarizationcomponents also have the distinct lens-like shape in Stokesspace and vice versa. This is a consequence of the mappingthat exists between Jones and Stokes spaces. This behavior was

observed for CD compensated data in the previous section andis reconfirmed in this section for PMD. As a matter of fact,it is possible to construct a Kalman filter that estimates PMDin Stokes space by reducing the scattering of points. This isillustrated in the next section when PMD is revisited.

III. KALMAN FILTERING

The problem of signal estimation in the presence of noise wasfirst solved by Wiener in the frequency domain in the 1940sand subsequently in the time domain by Kalman in 1960 [17],[18]. Thus, the two approaches are intimately related. The re-cursive nature of the Kalman filter and its direct applicability tosampled data made it more popular than the Wiener filter [19].Nevertheless, it is important to note that the very popular leastmean squares (LMS) algorithm converges to the Wiener solu-tion [20]. The popularity of the LMS algorithm results from itssimplicity. The Kalman filter tends to be more complex; how-ever, it typically converges about an order of magnitude fasterthat the LMS algorithm [20]. The Kalman filter is especially at-tractive in multi-input and multi-output applications [19]. Wewere attracted to the Kalman filter in the metrology of signalsbecause of its ability to model physical phenomena and to es-timate physical parameters. The Kalman filter can be used tomodel both linear and nonlinear problems. The application ofthe Kalman filter to nonlinear problems is known as the ex-tended Kalman filter. At the top level the LMS algorithm andthe Kalman filter are similar as discussed in more detail below.Conceptually, the operation of the LMS algorithm may be

described by the following equation:


The tap vector that is estimated may, for example, be an FIRfilter that optimizes the quality of the received data. The inputvector contains complex input data, the error signal depends onthe nature of the algorithm, and the learning rate is a constantthat controls the rate of convergence. For a decision directedalgorithm, the error signal may be simply the distance of the fil-tered data point from the nearest symbol. For the constant mod-ulus algorithm (CMA), the error is proportional to the distanceof the data point from the desired constant modulus circle [21],[22]. The recursive operation of the algorithm starts with a guessfor the tap vector that is subsequently updated by observing theincoming data while calculating the error signal. It is desirableto find the solution quickly; however, selecting a learning rate(a step size) that is too large may cause undesirable noise in theestimated filter or algorithm instabilities. Thus, the step size hasto be suitably small.Conceptually, the operation of the Kalman filter is similar:

The sate vector contains state variables that are adaptively esti-mated. In optical communications, the state variables may havea well-defined physical meaning related to the properties of theoptical signal or of the optical transmission medium (opticalfiber). For example, state variables may represent carrier phase,polarization transformation, CD, or a PMD vector [23], [24].The Kalman gain plays the role of a learning rate. Unlike theLMS algorithm’s learning rate, the Kalman gain is not equal toa preset numerical value, but instead, it is recursively estimatedby the Kalman filter algorithm. Finally, the innovation vectorplays the role of the error signal; in that it compares the pre-diction made by the Kalman filter with the measurement. Theprocess of estimation is iterative and adapts to the new data.The full description of the Kalman filter including the ex-

tended Kalman filter is provided in the Appendix with (A6)defining the state vector update step described above. The othertwo iterated quantities are the Kalman gain and the error co-variance matrix . Some aspects of the Kalman filter equationsare discussed in more detail in the examples that are given in thesections to follow. In these examples, the process (A1) takes aparticularly simple form because the matrix is an identity ma-trix and the matrix is a zero matrix. The matrix being equalto zero implies that there is an absence of a deterministic signalthat acts on the state vector. The matrix being an identitymatrix implies that the estimated quantities are nominally con-stant and that there is no interdependence between them. Thefollowing examples chosen to demonstrate the Kalman filter in-clude carrier phase tracking, polarization tracking, and PMDestimation.

A. Carrier Phase Tracking

The design of a Kalman filter starts with the question of whatare the quantities of interest that need to be estimated. Thesedetermine the state variables of the state vector. Although thestate variables often cannot be observed directly, they can berelated to some other quantities that can be measured. In addi-tion, both the state variables and the measurements are affectedby noise. The Kalman filter provides an optimal estimate of the

state variables based on measurements of the related quantitiesin the presence of noise [25]. Noise is assumed to be white withnormal probability distribution. If the state variables are relatedto the measurements in a nonlinear way the solution is approxi-mated by the extended Kalman filter. Since the Kalman filter isiterative in nature, all Kalman filter equations, as described inAppendix A, contain an iteration subscript . The subscriptwill be shown in equations of this section but omitted later forthe sake of simplicity of the notation. All the variables, whethershown with the iteration subscript or without, have exactly thesame meaning.Phase tracking is the simplest example of the Kalman filter

described in this paper. This implementation of the Kalman filteris complex and all the variables, including noise, are also com-plex. The carrier phase is assumed to be contained within a com-plex variable that changes with time slowly in comparison tothe time scale of the clock that defines the arrival times of thenew symbols. In this example, the state vector contains only asingle state variable, i.e., . The state variable is affectedby noise. This leads to the following process equation:

(17)

where represents complex process noise. The preceding (17) isgeneral and may contain amplitude noise and phase noise, but,in the scenario considered in this example, the phase noise isdominant. By inspecting (17) and comparing it with (A1) fromthe Appendix, the matrix is simply equal to 1. The processnoise is described by the process covariance matrix . Sincethere is only one state variable, the matrix , just like matrix ,is reduced to a scalar. The selection of the value for dependson the expected variations of the estimated quantity; in the sce-nario considered, that is predominantly on the phase noise of thetransmitter. Here, three different values are considered:

, and .Important relevant information is that the data are modulated

in a known format, in this case, QPSK. Thus, the data that arriveto the receiver after correction for the carrier phase noise mustbelong to the QPSK set of symbols. This results in the followingequation:

(18)

In Kalman filter terminology, this is the measurement equation.It is assumed here that the data are properly sampled at the clockfrequency at the locations corresponding to symbols. The esti-mation of the clock frequency could for example be based onthe concept of the squaring synchronizer implemented in Stokesspace as described in Section II-C. The left side of (18) repre-sents symbols, with being the nearest symbol to the currentdata point. The alignment between the measured data pointand the symbol is enforced by the complex multiplier, andthe state variable, that is being estimated by the Kalmanfilter. All measurements are affected by complex noise thatis described by the measurement noise covariance matrix .The measurement noise and the process noise in (17) areindependent. In this case of a single measurement equation, thecovariance matrix measurement is a scalar. The selection ofthe value of must reflect the noise of the measurement. Since


the data are normalized and the distance of the QPSK symbolsfrom the origin is set to 1, it is expected that, after the removal ofthe phase noise, the demodulated symbols fall within a fractionof 1 from the desired symbol locations. This is not a rigorouslydefined value but an initial guess that can be later refined basedon the observations of the Kalman filter. Thus, it is initially as-sumed that is equal to .To run the Kalman filter algorithm as described in (A3)–(A7),

all the matrices need to be defined. By comparing (A2) and(18), it is apparent that the matrix is simply equal to thecurrent measured data point . In the general case of the ex-tended Kalman filter, the matrix is the Jacobian matrix ofpartial derivatives of the right side of the measurement equa-tion with respect to the state variables comprising . The lastmatrix that needs to be defined is the error covariance matrix .The Kalman filter algorithm is typically very forgiving of thefirst guess as the covariance matrix is reestimated in everyiteration of the filter together with the Kalman gain and thestate vector . Here, the initial guess for the error covariancematrix is an identity matrix. At this point, all matrices aredefined and the Kalman algorithm can be exercised. It is worthmentioning that the form of the Kalman filter presented here issomewhat unconventional. Typically, the left side of the mea-surement equation contains the measured data. In the model pre-sented here, it was more convenient to select symbols on the leftside of the equation as it is expected that only the symbols arereceived. The measured complex signal is on the right side ofthe equation; however, it is modified by an unknown complexmultiplier to enforce reception of the symbols. Since it is knownthat the data is in QPSK format, the knowledge of the format isused to distinguish between data symbols and noise. This leadsto the estimation of the phase noise while the signal is being de-modulated.The results of Kalman filtering are shown in Fig. 11 for three

different values of the process covariance matrix . The datacome from a transmitter using a DFB laser; the clock frequencyis 11.454 GHz. Due to the phase noise of the laser, the trans-mitted symbols are smeared into arcs. Symbol smearing is es-pecially apparent for equal to when the slow filter dy-namics do not allow tracking of the phase noise. As the value ofthe noise variance is increased to , the phase noise of thelaser is removed and a constellation with round symbols is dis-played. The phase noise that is estimated by the Kalman filteris also shown in the figure. As the noise variance is further in-creased to , the symbols become narrow showing overcor-rection of noise. This filtering example demonstrates how theKalman filter dynamics are controlled by the value of the co-variancematrix . The example also shows how a simplemodelcontained in (18) can be used in solving the relatively compli-cated problem of carrier phase tracking.An interesting question to pose is whether the presented

model can be easily applied to other modulation formats, inparticular to QAM16. QAM16 deserves attention as it may bea suitable format for 400 Gb/s data rate transmission [26]–[28].The simplest experiment is to take the Kalman filter as de-scribed in (18), including the QPSK symbol set, and to applyit to QAM16 data. The results are shown in Fig. 12 for a 112Gb/s polarization-multiplexed signal having 14 GHz clock

Fig. 11. Kalman filter residual carrier phase tracking for different values of theprocess covariance . (a) Constellation plots. (b) Unwrapped estimated phase.The covariance matrix controls the rate at which the Kalman filter can trackcarrier phase. The preferred value of results in round symbol regions in theconstellation plot .

frequency. Despite relatively noisy data, the Kalman filter iswell locked to the constellation, as shown in Fig. 12(a). Note,however, that it is locked, not to the individual symbols, butrather to the four quadrants of the square occupied by QAM16format. The carrier phase, relatively well behaved in thisexperiment (low phase noise), is shown in Fig. 12(b).A simple generalization from QPSK to QAM16 might be to

include all 16 symbols on the left side of (18). However, it turnsout that this simple implementation may occasionally result inundesirable solutions. For instance, the QAM16 constellationmay be reduced in size to fit just the inner four symbols as il-lustrated conceptually in Fig. 12(a). Intuitively, a simple solu-tion to this problem is an appropriate normalization. One elegantmethod of normalization is the use of two measurement equa-tions: one locking to QPSK symbols and preventing undesiredreduction of the complex signal, and another locking to all 16symbols of QAM16 and leading to the demodulation. The twomeasurement equations that describe this procedure are

(19)

The symbol set contains QPSK symbols. In every iter-ation of the Kalman filter, the nearest symbol is selected fromthis set to prevent the undesired reduction of the complex signal. Thus, the presence of the QPSK symbols provides a coarse

alignment with the square-shaped QAM signal. The symbol setcontains QAM16 symbols. The nearest symbol that is

selected from this set provides the demodulated symbol. Useof multiple measurement equations is natural for any Kalmanfilter that simply tries to satisfy multiple inputs [19]. Havingtwo measurement equations changes the dimensionality of thecovariance matrix . The two diagonal terms of the matrix maybe set to different values to reflect the fact that the first equa-tion contains on average larger errors. The data demodulated bythe filter (19) are shown in Fig. 13. The constellation plot inFig. 13(a) shows the two symbol sets that were used by the twomeasurement equations in addition to the demodulated data. Theestimated phase noise is shown in Fig. 13(b). Interestingly, theestimates of the phase noise based on (18) and (19) are nearly


Fig. 12. QAM16 112 Gb/s polarization-multiplexed data. (a) Constellation tracked by the QPSK Kalman filter. The four red points represent the symbols usedin the Kalman filter definition. The black square identifies the region of possible misconvergence when QAM16 symbols are used. (b) Measured residual carrierphase.

Fig. 13. QAM16 112 Gb/s polarization-multiplexed data. (a) Constellation demodulated and tracked by the Kalman filter with two measurement equations ac-counting for QPSK and QAM16 symbol sets. (b) Measured residual carrier phase.

the same [see Figs. 12(b) and 13(b)]. This suggests that, fromthe point of view of carrier phase estimation, in some cases, itmay be sufficient to track QAM16 signals with a simple QPSKsymbol set.

B. Polarization State Tracking

A natural extension of the ideas from the last section is po-larization state tracking. Intuitively, the simplest approach is toseek a complex matrix that converts two received signals inorthogonal polarization states into two decoupled polarizationchannels each containing symbols from the QPSK modulationformat. The QPSK format is chosen for this example becauseof the ease of its demodulation. However, this approach justlike the CMA [29], may experience singularity. To counteractthis, an explicit notation is used that excludes the singularityby using real state variables to enforce the format of the polar-ization transformation matrix [23]. The two measurement equa-tions, expressed in a convenient matrix notation, are defined asfollows:

(20)

where represents symbols that belong to QPSK modula-tion format, is the residual carrier phase that is assumed to becommon to the two polarization channels, de-notes the Jones vector, and are the two complex signalsreceived in the horizontal and vertical linear polarization states,and the matrix is the nonsingular polarization transformationmatrix:

(21)

The matrix is defined in terms of real parameters ,and . The use of real parameters not only provides a simpleway of ensuring that the polarization transformation is not sin-gular, but also allows explicit tracking of the residual carrierphase . The state vector contains five real variables, i.e.;

. It is worth noting that the phase trackingin this model is nonlinear; therefore, the formulation is consis-tent with the extended Kalman filter. The Jacobian matrix iscalculated by differentiating the right side of (20) with respectto the individual state variables. The matrix is

(22)


Fig. 14. Measured 40 Gb/s dual polarization-multiplexed QPSK signal; (a) before (b) after the extended Kalman filter is applied. The Kalman filter simultaneouslyaligns the polarization state and tracks the residual carrier phase. The plots show about 4000 symbols.

Fig. 15. Kalman filter convergence is illustrated by plotting each of the estimated state variables as a function of iteration number: (a) polarization state transfor-mation parameters and (b) residual carrier phase. The Kalman filter converges well within 200 iterations.

where the last column denoted by contains the deriva-tives with respect to . The Kalman filter iteration subscript isomitted for simplicity of notation. The clock is assumed to bethe same for both of the polarization components.This Kalman filter is applied to 40 Gb/s polarization-mul-

tiplexed QPSK modulated data. Fig. 14(a) shows the data inone of the received polarization channels before Kalman filterpolarization alignment. The data are sampled at the clock,which is recovered by a squaring synchronizer, as described inSection II. The figure indicates some drift of the carrier phase(manifested by radial arcs in the complex plane) and some po-larization cross-coupling (manifested by two distinct moduli ofthe complex data). It is expected that data with proper polariza-tion alignment would be located on a single circle of constantmodulus. Note that the expected single circle is the basis forthe CMA polarization alignment procedure [22]. Fig. 14(b)shows the data in one of the polarization channels after theKalman filter is applied. The vast majority of the demodulatedsymbols belong to the QPSK constellation. The constellationis slightly deformed, indicating that the modulator bias driftedoff the optimum point. The symbols that are scattered in thecomplex plane correspond to the initial convergence of theKalman filter.This filter convergence is illustrated in Fig. 15. Fig. 15(a)

shows the four real state variables that define the polarizationtransformation. The variables reach their approximately con-stant values after about 200 symbols. Fig. 15(b) shows the car-rier phase. After the initial instability, the Kalman filter locks to

the data and the carrier phase is tracked. It is important to notethat the Kalman filter allows the dynamics of polarization stateand phase tracking to be defined independently. In this calcula-tion, it is assumed that the process noise covariance matrix isdiagonal: the first four diagonal elements represent the variancesof the real parameters that define the polarization transformationmatrix and the last element represents the variance of the phasenoise. If the polarization state is known to evolve slowly, smallvalues of the corresponding diagonal elements can be selected.Similarly, depending on the phase noise of the transmitter, thelast diagonal element can be appropriately adjusted. In this ex-ample, the values of and were used for polarizationevolution and phase noise, respectively.The Kalman filter approach is known to converge faster than

the LMS approach at the expense of complexity [20]. Fast con-vergence allows the Kalman filter to track rapid changes of thepolarization state. It has been shown that the implementationdescribed here can track polarization evolution at an angularvelocity, as defined on the Poincare sphere, of 6 to 8 Mrad/s[23]. Last but not least, one additional comment needs to bemade about the complex measurement equations and the realstate variables. The complex measurement equations lead to acomplex formulation of the Kalman filter, but this would resultin complex estimates of state variables. In order to avoid com-plex solutions, the real and imaginary parts of the measurementequations are considered separately. This allows concise formu-lation of equations, but yields real solutions for state variablesas desired.


Fig. 16. Polarization prealigned PMD scattered data in Stokes space. The ar-rows show the desired direction of data compression along the – axis. Thecompression of data is used by the Kalman filter to estimate PMD.

C. First-Order PMD Compensation

The previous two examples of the Kalman filter illustratedits operation on properly sampled complex variables corre-sponding to symbols. The prior knowledge of the modulationformat was essential as it allowed the measurement equationsto be defined solely in terms of the discrete symbols of the con-stellation. In this way, demodulation was effectively an integralpart of filter operation. This, in turn, led to the estimation ofstate variables, like carrier phase and polarization state, thatwere changing in a continuous manner.The Kalman filter defined in this section operates in a very

different manner. It does not require knowledge of the mod-ulation format nor sampling at the clock, thus allowing morefreedom. The proposed filter examines the scatter of points inStokes space due to PMD and compresses the scattered pointsby applying a simple PMD compensation to all the data in-cluding symbols and transitions between symbols. This task isillustrated in Fig. 16. As discussed in Section II, compressionof the sampled points and PMD compensation are equivalent.The compression of points is preceded by polarization align-ment in Stokes space. The PMD is assumed to be small enoughso that such polarization alignment can be performed, i.e., DGDmust be smaller than half of the clock period. Before the Kalmanfilter is formulated in detail, a simple PMD model will now bediscussed.The first-order PMD described by a vector

can be conveniently expressed by using -matrices, originallyproposed by Jones [15], [16]. The -matrices that correspondto the 0 linear, 45 linear, and circular component of PMD are

(23)

The -matrices are added together to describe an arbitraryPMD. Jones denoted the sum of the -matrices by an -ma-trix. In his work, Jones considered eight different -matrices;however, only three of them are required to describe PMD

(24)

The related Jones matrix is found from the -matrix byusing the matrix exponential, i.e., . This yieldsthe following Jones PMD matrix:

(25)

where is an identity matrix and can be expressedin terms of the components of the PMD vector:

(26)

The simple form of (25) is a consequence of theCayley–Hamilton theorem that makes any function of a 2 2matrix expressible by the first-order matrix polynomial [30].This formula becomes especially simple for small PMD whenthe trigonometric functions can be approximated for smallas and . Then, the Jones matrixbecomes

(27)

Since the Jones vector , that describes an optical polariza-tion-multiplexed signal in the frequency domain, is transformedby this simple PMDmatrix, it is possible to find the Jones vectorin time in terms of the -matrix from the inverse Fourier trans-form of :

(28)

The preceding time-domain-based equation of the first-orderPMD provides the needed theoretical model for the formula-tion of the Kalman filter which operates in the time domain.The steps that lead to this formulation are now described. Thereceived Jones vector , after Stokes space polarization align-ment, still contains PMD distortions. However, the distortionscan be compensated by subjecting the received vector to theunknown PMD according to (28). Since the received signal issampled at some sampling time , it is convenient to normalizethe PMD vector to the sampling time. The normalized PMDvector contains three normalized components, i.e.,

. The same three normalized PMD componentsare used to express the normalized -matrix, .The sampled Jones vector is denoted by and its derivativeis expressed by the ratio , where is the differencebetween adjacent samples. Using these newly defined variables,(28) is rewritten in the following form:

(29)

where denotes a compensated Jones vector. In the full form,with the sought variables , and explicitly shown andthe iteration subscript implied but omitted for notational sim-plicity, (29) is rewritten as

(30)

where the subscripts and denote the two polarization com-ponents. By using (2), the Jones vector can be visualized in theStokes space as illustrated in Fig. 16. It is expected that the


Fig. 17. Measured 40 Gb/s polarization-multiplexed QPSK data scattered inStokes space by 24 ps DGD. The inset shows the constellation plot for one of thepolarization components. The transitions between symbols are not well defineddue to distortions resulting from PMD.

data to some degree resembles the shape of the lens with addi-tional scatter of points caused by the PMD. For the polarizationaligned data, it is further expected that the thickness of the lensobject is the smallest along the – axis. The Stokes vectorcomponent in this direction, , is expressed by the second equa-tion of (2):

(31)

The Kalman filter can enforce minimization of scatter in thisdirection by constraining to be zero. The resulting measure-ment equation is

(32)

The state vector of the filter under consideration con-tains the normalized components of the PMD vector, i.e.,

. The Jacobian matrix is defined bythe partial derivatives . Thus, the formulation of theKalman filter is complete.The operation of the filter is illustrated in Figs. 17 and 18.

Fig. 17 shows 40 Gb/s polarization-multiplexed data modulatedin QPSK format before compensation. The side view of thePoincare sphere indicates that the data are polarization alignedas the flattened lens-like object axis is aligned with the –axis. However, the outline of the lens is somewhat distorted dueto the scatter of the points by the birefringent element (PM fiber)used in the experiment. The DGD of the PM fiber is known tobe about 24 ps. The distortions are also visible in the constella-tion plot, shown in the figure inset, especially in the region oftransitions between symbols.Fig. 18 shows the compensated data. The PMD vector is es-

timated by the Kalman filter with the diagonal elements of thestate covariance matrix set to and the measurement co-variance matrix equal to . The object in Stokes space isnow closer to the ideal shape of the lens. Compression of the

Fig. 18. Compensated 40 Gb/s polarization-multiplexed QPSK data. Theinset shows the constellation plot for one of the polarization components. AfterPMD compensation by the Kalman filter, the lens-like shape in Stokes space isrestored. The constellation plot shows four QPSK symbols and well-definedtransitions.

data points in Stokes space results in a much cleaner constel-lation, shown in the figure inset, with very well-defined transi-tions that are confined to the outline and the diagonals of theQPSK constellation. The DGD estimated by the Kalman filteris 20.5 ps; the estimate also includes the direction of the PMDvector. It is important to mention that it is not always possibleto accurately estimate the PMD. If the polarization modulatedsignal is aligned with the axes of PMD, the individual polar-ization states of the signal travel in two eigenmodes of the fiber.Thus, the PMD leads to the shift in clock phase between the twopolarization components. If it is known that the clock phase wasaligned to start with, the clock shift can be interpreted as PMD.However, the shift of the clock is not clearly visible as an in-creased scatter of points in the Stokes space. Thus, the presentedmethod, always leads to compensation of the received signalbut the PMD estimation may not be accurate. Furthermore, itis important to emphasize that the Kalman filter described hereis modulation format independent, it does not require knowl-edge of carrier or clock, and that the sampling rate is not critical.Thus, the filter has all the benefits of working in Stokes spaceas described in Section II.

IV. CONCLUSION

This paper presented details of performance monitoring andmeasurement techniques for coherent systems based on Stokesspace analysis and Kalman filtering. These techniques are appli-cable to coherent systems with polarization-diverse coherent re-ceivers capable of reconstructing the optical field of the receivedpolarized optical wave. The measured optical wave, representedby the Jones vector, contains information about the received op-tical signal and about the optical transmission medium (opticalfiber). This paper dealt with the estimation of the optical signalparameters and impairments of the optical link. The techniques


of Stokes space analysis and Kalman filtering were selected tomatch the needs of metrology rather than simply those of dataequalization and demodulation.The Stokes space technique is based on the mathematical

mapping of the Jones vector into Stokes space. In the generalcase, with an arbitrary modulation format, it was shown that alens-like object is produced for dual-polarization complex-mod-ulated signals. Further, the axis of the lens-like object identifiesthe polarization states of transmission thereby serving the func-tion of a polarization state monitor. Knowing the polarizationstates of transmission allows one to construct an inverse trans-formationmatrix and to align the multiplexed polarization statesof themeasured signal with the receiver. This polarization align-ment was verified with a measured 40 Gb/s QPSK dual-polar-ization transmitter. Under the condition of Stokes space polar-ization alignment, clock recovery was demonstrated to be sep-arately estimated for each polarization state using the squaringsynchronizer technique. It was further shown that the recoveredclock tone could be used to estimate and compensate CD. Inparticular, an optical link with a dispersion of 1700 ps/nm wassuccessfully measured and compensated. Of important note, theStokes space method was proven to be independent of the mod-ulation format and to not require demodulation.The Kalman filter is an optimal estimator. The Kalman filter

dynamically estimates physical quantities contained in the statevector and described by the Kalman filter’s state equation andmeasurement equation. In this paper, the physical quantities suc-cessfully estimated included residual carrier phase, polarizationstate, and the first-order PMD. For carrier phase and polariza-tion state tracking, knowledge of the modulation format and theclock were required. Carrier phase tracking was successfullydemonstrated for a QPSK 11.454 GBaud (45.8 Gb/s) dual-po-larization transmitter with a DFB laser and for a QAM16 14GBaud (112 Gb/s) dual-polarization transmitter with an externalcavity laser. In addition, the extended Kalman filter was used totrack polarization state for a 10-GBaud dual-polarization QPSKtransmitter. PMD was estimated in Stokes space and thereforeprior knowledge of the modulation format, clock, or carrier wasnot required. The Kalman filter was shown to estimate 20.5 psof DGD in an optical link used for transmission of a 10 GBaudQPSK dual-polarization optical signal.

APPENDIX

The Kalman filter is designed to estimate the state vectordefined by the following process equation:

(A1)

where the current state is related to the prior state bya square matrix represents a deterministic signal thatalters the state vector through matrix , and representsprocess noise. The state estimation is performed based on themeasurement defined by the following measurement equa-tion:

(A2)

where matrix relates the state vector and the measurementvector and represents the measurement noise. The randomvariables and are assumed to be independent, white, anddescribed by process and measurement covariance matricesand , respectively.The Kalman filter algorithm estimates the state vector

through an iterative procedure typically grouped into the timeupdate equations:

(A3)

(A4)

and the measurement update equations:

(A5)

(A6)

(A7)

where the superscript minus denotes the a priori estimate, isthe a posteriori estimate of the state vector is the error co-variance matrix, and is the Kalman gain. If the state vector isreal, the conjugate transpose operator is reduced to a transpose.The algorithm can be extended to nonlinear forms of process

andmeasurement equations. It is then referred to as the extendedKalman filter. In this paper, we are using a nonlinear form of themeasurement equation:

(A8)

where denotes some function of the state vector. Matrixis then expressed as the Jacobian matrix of partial derivativesof function with respect to state variables comprising thestate vector. This is illustrated in the examples contained in thispaper. In the case of the extended Kalman filter, measurementupdate (A6) changes form to

(A9)

For more detailed reviews of the Kalman filter, the reader isreferred to numerous works on the Kalman filter [18]–[20], [25].

ACKNOWLEDGMENT

The authors would like to sincerely thank R. Van Tuyl,Dr. G. Owen of Agilent Technologies, and Prof. J. M. Kahn ofStanford University for their comments and suggestions.

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Bogdan Szafraniec (SM’01) was born in Sokolow Podlaski, Poland, in 1956.He received the M.S. and Ph.D. degrees in electrical engineering from the Illi-nois Institute of Technology, Chicago, in 1982 and 1988, respectively.He has worked as an Electronics Designer of crystal and LC filters, matrix

video switches and analog fiber-optic links. In 1989, he joined the HoneywellTechnology Center, Phoenix, AZ, where he worked on fiber-optic gyroscopes.He played a major role in the development of depolarized gyros at Honeywell.He focused on investigation of various polarization effects including polariza-tion cross-coupling, polarization evolution, and magnetic sensitivity, all leadingto nonreciprocal errors. In 1999, he joined the Measurement Research Lab-oratory, Agilent Technologies, Santa Clara, CA. He investigates technologiesof new optical test instruments. His work is focused on coherent metrologyand covers high-resolution optical spectrum, network, and signal analysis. Hemade significant contributions to productization of first coherent instruments atAgilent.Dr. Szafraniec is a senior member of the Optical Society of America.

Todd S. Marshall (S’97–M’00) received the B.S. degree in engineeringphysics, the M.S. degree in electrical engineering, and the Ph.D. degree inelectrical engineering from the University of Colorado, Boulder, in 1992, 1996,and 2000, respectively.From 1992 to 1995, he was contracted by the National Renewable Energy

Laboratory, Golden, CO, to develop and implement computational models oftextured solar cell light-trapping effects with the goal of efficiency optimization.In 2000, he joined the Central Research Laboratory, Agilent Technologies, PaloAlto, CA, where he worked on microwave design for 40 Gb/s test equipmentincluding bit error rate measurement, InP clock recovery, and phase detectorcircuits used in 50 GHz sampling oscilloscopes, optical electroabsorption mod-ulator-based high-impedance 26 GHz electrical probes, and optical spectrumanalysis. He is currently a Research Engineer at the Agilent Measurement Re-search Laboratory, Santa Clara, CA, where he conducts research in the field ofhigh-speed test and measurement for lightwave instrumentation. His current re-search interests include both microwave design and the Kalman filter focusedon coherent optical communications.

Bernd Nebendahl was born in Stuttgart, Germany, in 1966. He received theDiploma and Ph.D. degrees in physics from theUniversity of Stuttgart, Stuttgart,in 1992 and 2000, respectively.He then joined Agilent Technologies’ Optical Communication and Measure-

ment Division in Böblingen, Germany, as an R&D Engineer where he has beenworking on various R&D projects including tunable external cavity lasers, op-tical attenuators, coherent optical all parameter testers, distributed temperaturesensing, and the optical modulation analyzer in positions from optics designerto project lead. He currently focuses on all topics around coherent transmission.

performance monitoring and measurement techniques for coherent optical systems

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