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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 7, APRIL 1, 2009 901

Phase Estimation Methods for Optical CoherentDetection Using Digital Signal Processing

Michael G. Taylor, Member, IEEE

Abstract—The advent of digital signal processing (DSP) tooptical coherent detection means that more phase estimationoptions are available, compared to the earlier generation wherephase-locked loops (PLLs) were invariably deployed in syn-chronous coherent receivers. Several phase estimation methodsare numerically modeled: the maximum a posteriori (MAP) phaseestimate, decision directed estimate, power law-Wiener filterestimate and power law-PLL estimate. An asynchronous coherentdetection case is also modeled. The phase estimates are evaluatedwith respect to their tolerance of finite laser linewidth and theirsuitability for implementation in a parallel digital processor.Laser phase noise causes transmission system performance to bedegraded by excess bit errors and cycle slips. The optimal phaseestimate is the MAP estimate, and it is also included as a baseline.The power law-Wiener filter phase estimate is found to performonly marginally worse than the MAP estimate. It must be recastusing a look-ahead computation to be implemented in a paralleldigital processor, and the impact is investigated of the increase inthe number of computations required. Differential logical detec-tion is often used to reduce the impact of cycle slip events, and theimplications of this operation on the bit error rate are studied.It is found that by choosing the correct FEC scheme differentiallogical detection does not increase the Q-factor penalty.

Index Terms—Coherent detection, optical communications,phase estimation, synchronous detection.

I. INTRODUCTION

C OHERENT detection of optical signals is once again ofinterest for applications in fiber optic communications.

The latest coherent receiver realization employs real-time dig-ital signal processing (DSP) technology. The mixing product ofsignal and local oscillator is sampled by analog-to-digital (A/D)converters at typically twice the symbol rate. A digital processormakes computations on the sample values in real time to calcu-late the electric field envelope of the signal and hence obtain theinformation encoded on it. The earlier incarnations of coherentreceivers were complicated, and typically included high speedanalog electronics for demodulation from the carrier or for op-tical phase locking, and also active optical components for po-larization control [1]. In comparison, in the new realization thefunctions of these components are performed in the digital do-main within the DSP. Provided that the DSP hardware and also

Manuscript received February 6, 2008; revised June 2, 2008. Current versionpublished April 17, 2009.

The author is with Atlantic Sciences, Laurel, MD 20725 USA. Part of thispaper was prepared while the author was a visitor with the Optical NetworksGroup, Department Electronic & Electrical Engineering, University CollegeLondon, U.K (e-mail: [email protected]).

Digital Object Identifier 10.1109/JLT.2008.927778

the necessary optical passive components can be produced involume at low cost, the new version of the coherent receiverwill be cost-effective compared to direct detection, and is likelyto be widely deployed.

In the past, the primary motivation for studying coherentdetection was because it offered optical gain. Today there areseveral optical amplifier technologies that provide optical gain,and we are more interested in the features of coherent detectionthat were previously of secondary importance. A coherentreceiver responds only to light in the spectral neighborhoodof the local oscillator (LO), and so it is equivalent to havingan ultranarrow wavelength division multiplex (WDM) opticalfilter in front of the receiver. Furthermore, if the LO laseris tunable in wavelength, then it is equivalent to a tunableultranarrow WDM filter. Recently a total information spectraldensity of 2.5 b/s/Hz was demonstrated using coherent detec-tion [2], and an information spectral density in one quadratureof 1 b/s/Hz [3]. Chromatic dispersion can be compensated foreffectively by electrical signal processing of the intermediatefrequency signal. This was done originally using a length ofmicrostrip line chosen to have dispersion opposite to that of theoptical fiber path [4]. In fact, it is more effective to implementpropagation impairment compensation in the digital domain,since the required signal processing function can be preciselygenerated. Exact compensation of 1500 ps/nm of chromaticdispersion at 10.7 Gb/s has been demonstrated [5], as hascompensation for other propagation impairments such as selfphase modulation [6] and polarization mode dispersion [7].Coherent detection is sensitive to the phase of an optical signal,and so it can detect phase encoded modulation formats such asphase shift keying (PSK) and quadrature amplitude modulation(QAM) formats. These formats have better sensitivity thanthe amplitude modulated formats having the same numberof levels. Binary phase shift keying (BPSK) and quadraturephase shift keying (QPSK) formats both offer the best possiblesensitivity in terms of optical SNR when detected by opticalcoherent detection, provided that synchronous demodulation(also referred to as carrier recovery or carrier synchronization)is used. A receiver sensitivity just 2.5 dB from the theoreticallimit was demonstrated using a 10.7-Gb/s BPSK signal [5]with coherent detection followed by digital domain processing.There have been several impressive fiber optic transmissionsystem demonstrations using coherent detection of a measure-ment burst with offline digital signal processing, includingtransmission over transoceanic distances and compensation ofmultiple fiber propagation impairments at 40 Gb/s [8]–[10] and111 Gb/s [11]. Real-time digital signal processors employingfield programmable gate arrays [12], [13] and application

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902 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 7, APRIL 1, 2009

Fig. 1. Diagram of coherent detection apparatus using DSP.

specific integrated circuits [14] have been built at data ratesup to 10 Gbaud. All these experiments used synchronous de-modulation. However the phase estimation process associatedwith synchronous demodulation can be hard to implement in adigital processor, and that is the subject of this paper.

There have been other studies recently of how to estimate thephase in a digital processor. Noé’s feedforward estimate [15]will be discussed later in the paper. The Wiener filter-basedphase estimate of this paper was tested in a burst-mode exper-iment [16]. Ip and Kahn also identified the Wiener filter as theoptimal phase estimate, and evaluated it in a different structurefor QPSK and QAM formats [17]. Kazovsky et al. studied aphase estimate using a first-order phase-locked loop (PLL) [18].

Fig. 1 shows the configuration of a coherent receiver usingdigital domain processing. The optical signal is mixed with localoscillator light in a 90 hybrid. The upper and lower output armsof the hybrid differ in that the local oscillator path is longer by aquarter cycle in the lower arm than the upper arm, which meansthat the balanced photodetector in the upper arm effectively seesan inphase beat product while the one in the lower arm sees aquadrature beat product. The 90 hybrid can be implementedin a waveguide platform following the topology shown in Fig. 1[19]–[21]; or as a polarization beamsplitter where the signal andLO polarizations are set appropriately [2], [19]; or as a passivecoupler with correctly dimensioned coupling region [22], [23].The photodetectors at the outputs of the 90 hybrid each see thesignal and LO light adding as electric field. The powers at thetwo balanced outputs of the upper arm are

and the balanced photodetector output is, therefore, proportionalto

where

complex envelope of electric field of opticalsignal;angular frequency of optical signal;

phase of optical signal;

complex envelope of LO electric field;

angular frequency of local oscillator;

phase of local oscillator;

time.

is a constant given that the local oscillator is continuouswave. The angular frequencies and are written as con-stants because they drift slowly, due to laser aging, for example.The optical phases and are functions of time containingthe phase noise of the lasers. The quarter wave shift in LO pathof the lower arm of the 90 hybrid means that the balanced pho-todetector in the lower arm responds to

Hence, the electric field envelope of the signal is obtained byapplying the mathematical relationship

(1)

The frequency difference and phase difference must be esti-mated from and to deduce , which is the subjectof this paper. A pair of 90 hybrids connected in a polarizationdiverse configuration can be used to obtain the Jones vector ofthe optical signal, a two-element complex vector, using a math-ematical relationship similar to (1). The electric field envelopeof the signal can then be obtained from the Jones vector even asthe signal’s state of polarization (SOP) changes over time, whilethe local oscillator SOP is fixed.

The A/D converters in Fig. 1 may sample at times corre-sponding to the centers of the digital symbols encoded on the op-tical signal. Alternatively they may sample at a rate sufficientlyhigh to fully characterize the optical signal in accordance withthe Nyquist criterion and the lowpass response of the photode-tector, and then the values at the center of the symbols calculatedby sample rate conversion. Either way it is assumed that the dig-ital processor has available symbol center samplesand , where is the symbol period and

(For compactness, all functions of in this document will bewritten as functions of . Later, in the paper the -transforms ofsome discrete-time functions will be discussed, and these will bewritten as functions of . The dependent variable or willindicate the nature of a parameter, continuous-time, discrete-time, or -transform.)

The task of processing the samples to derive the informationon the optical signal is the same mathematically as if it werea radio signal, and we can use the techniques developed forradio. There is a considerable body of work on phase estima-tion [24]–[26]. However, the high levels of phase noise that aretypical of laser light sources are unique to optical coherent de-tection, and there is no solution in radio to draw upon for thisaspect of the problem. The lasers used for coherent detectionmay have linewidths of , much higher than that of a re-alistic radio carrier. The solution is to use a Wiener filter that isoptimal for the phase noise characteristics of the lasers.

TAYLOR: OPTICAL COHERENT DETECTION USING DSP 903

The digital processor in the coherent receiver is very likelyto have a parallel architecture. The signaling rate of the op-tical signal, perhaps 10 Gbaud, is larger than the clock speedof any suitable integrated circuit, probably CMOS, so that theincoming samples from the A/D converters must be demulti-plexed and processed in parallel streams at the lower rate. Thisleads to an additional constraint on the signal processing algo-rithms which can be used. No algorithm is allowed which uti-lizes feedback of an immediately preceding result. If the numberof parallel paths in the processor is , then the earliest result thatis available at the start of the th calculation is the result of oper-ation (assuming the calculation is completed in one clockcycle). Most phase estimation algorithms are of the type thatuse feedback of the immediately preceding result. The resolu-tion to this issue is to either avoid using such an algorithm, or toadapt it so that it no longer uses feedback to operation butstill yields the same behavior. It is easy to adapt the algorithmby substituting result in place of result , but thisapproach means that a lower laser linewidth can be tolerated.There are methods of adapting algorithms for VLSI processorsthat leave the signal processing behavior unchanged [27], andone of these methods will be used.

Many of the analog hardware-based experiments on opticalcoherent detection used synchronous demodulation. They wereeither experiments where the local oscillator laser was phaselocked to the received signal [28]–[30], or heterodyne detec-tion experiments with the phase estimation process performedin radio frequency electronics [31]. Usually it was found neces-sary to employ external cavity lasers having low linewidth. Theimpact of the delay in the feedback path of the PLL on the allow-able laser linewidth has been studied in detail [32]–[36]. Nori-matsu evaluated the upper limit of to be forBPSK and for QPSK [36], where is the com-bined laser linewidth and the number of symbols of delay. Hiscriterion was dB penalty at a bit error rate (BER) of .In the hardware-based PLLs, the delay was due to the finite dis-tance between components. The parallel architecture DSP con-straint is of course equivalent to the existence of a delay in thefeedback path, and we can expect the same restrictions on laserlinewidth to apply if a digital PLL is used as part of the phaseestimation process.

The approach taken in this paper is to try to find an estimate ofthe phase which is an optimal estimate. Although we can studythe true optimal estimate, the maximum a posteriori (MAP) es-timate, it is not feasible to calculate it in a real-time DSP, andit is shown that the power law-average estimate using a Wienerfilter is a practical alternative which is a near-optimal estimate.Then the issue is addressed of adapting the algorithm to be ex-ecutable on a parallel digital processor. The performance of thedifferent phase estimates are compared to one another by MonteCarlo simulations. The comparison includes the optimal (MAP)estimate, so it is clear how far each estimate deviates from theoptimum. The core contribution of this paper is to present in de-tail the Wiener filter-based phase estimate, already introducedby the author in [16], and its implementation in a parallel pro-cessor. In addition, the new phase estimate is put in context bycomparing it to existing phase estimation methods, the decisiondirected phase estimate and the feedforward phase estimate. As

well as increasing the number of bit errors, laser phase noisecan cause cycle slips. While earlier papers have not studied theoccurrence of cycle slips quantitatively, in this paper the cycleslip probability is determined from numerical modeling for thevarious phase estimates. The usual way to deal with cycle slipsis by the use of differential logical detection, and some of theimplications of this feature are explored.

The paper is organized as follows. Section II describes howthe MAP phase estimate is calculated. Section III discusses thedecision directed phase estimate, which is an alternative to thepower law-average phase estimate. The conditions are identifiedwhere the decision directed estimate outperforms the power-lawaverage phase estimate, but it is shown that for the receiverscontemplated in this paper the power-law average is the betterchoice. Section IV describes the structure of the power law-av-erage phase estimate, and its Wiener filter transfer function isderived. The reorganization of the Wiener filter using the look-ahead computation is described in Section V, so that it may beimplemented in a parallel digital processor. In Section VI thesimulation results of -factor penalty versus linewidth are pre-sented for the different phase estimation methods. Section VIIexplains how cycle slip events affect the recovered data signal,and the use of differential logical detection to reduce the nega-tive impact. Section VIII presents numerical simulation resultsof the cycle slip probability for different phase estimates. Fi-nally, Section IX reviews the main conclusions of the paper.

II. MAP PHASE ESTIMATE

The optical signal electric field envelope arriving at the re-ceiver is the sum of the modulated transmitter laser and additivenoise. Assuming the transmit pulse shape and receiver impulseresponse are chosen so there is no intersymbol interference

(2)

comprises digital values that represent the information en-coded on the signal. Two -ary PSK modulation formats willbe considered in this paper: binary phase shift keyingand quadrature phase shift keying . For a BPSK-mod-ulated signal is drawn from the set , and for QPSKfrom the set . is a constant. Thesecond term in (2) is a Gaussian noise sequence representingthe additive noise. Amplified spontaneous emission is the dom-inant additive noise source for a transmission system using op-tical amplifiers, and receiver thermal noise is dominant for apoint-to-point link. They both have Gaussian distributions, andso the noise term in (2) can represent either noise source or acombination of the two. is a complex Gaussian noise se-quence where each complex part has variance . In the sim-ulations described here for a baseline -factor of9 dB, corresponding to .

The phase diverse coherent receiver observes the quantity

(3)


where

When the signal and local oscillator lasers have Lorentzian line-shape, the phase noise is a Wiener process, that is a Gaussianrandom walk function

(4)

where is a (real) Gaussian noise sequence of variance ,and

, the combined linewidth, is the sum of the full width halfmaximum linewidths of the signal and local oscillator lasers.

The angular frequency difference can be estimated readilyby existing methods [26], [37]. is treated as a constant herebecause it varies slowly in a real transmission system, and arelatively small amount of digital signal processing resourcescan estimate it with accuracy. The frequency difference can thenbe removed by multiplying by a rotating phase factor withinthe DSP, or by using the estimate to control the LO opticalfrequency so that frequency difference is set to zero. (In thisdocument, denotes the estimate of a quantity whose actualvalue is .) For this reason it will be assumed that the frequencydifference is zero in the sections that follow.

The best possible estimate of the phase that can be made giventhe observed values is the MAP estimate. Given the knownstatistical behavior of the additive noise and phase noise, theMAP phase estimate is the sequence of values that maxi-mizes the log-probability function

(5)

and is a joint estimate with the data . It is not possible tosolve for by any method that can be contemplated for areal-time digital processor, but the maximization of canbe performed numerically to a high degree of accuracy.

A series of numerical simulations were performed for dif-ferent (different laser linewidths) to evaluate the MAP phaseestimate and the other phase estimates that will be describedlater. A sequence of symbols of was generated in accor-dance with (2). The data values were taken from apseudorandom sequence. At least symbols were simulated,making the statistical uncertainty in the -factor dB. TheMAP data and phase estimates were made using a per survivormethod based on groups of five symbols. The process for cal-culating the th symbol was as follows. was deter-mined at the previous iteration. The different possible casesof were each evaluated. For each case, thevalues of that maximize according to(5) were calculated using successive Newton’s approximation.

Fig. 2. Decision directed phase estimator.

To conclude the estimate of the th symbol, the values ofand were chosen by comparing for the cases.A value of penalty was determined for the simulation based onthe -factor of the BER associated with the sequence com-pared to the 9-dB baseline -factor.

The results of the penalty for the MAP phase estimate aregiven in Figs. 6–9, alongside the results for the other phase es-timation methods. These results will be discussed together inSections VI and VIII.

III. DECISION DIRECTED PHASE ESTIMATE

It is better for a real-time digital processor to estimate thephase separately from the data in order to minimize computa-tion resources. Most phase estimates comprise two steps. Firstthe effect of data modulation is removed, and then the phase isaveraged to reduce the impact of additive noise while trackingthe varying phase. The PLL is one way to perform the aver-aging operation. Some PLL implementations, such as the Costasloop, effectively merge the data modulation removal and the av-eraging steps together. There are two ways to do the first step ofremoving the effect of data modulation: by making a decisiondirected (decision feedback) phase estimate, or by applying apower law nonlinearity.

The decision directed approach is illustrated in Fig. 2. Thedecided symbol values are assumed to be correct and arefed back to remove the effect of modulated data on the incomingobserved electric field values . The presence of a feedbackpath means that we must expect performance to be compro-mised when implemented on a parallel digital processor. How-ever, the decision directed phase estimate typically uses fewerlogic gates than the phase estimates using the power law non-linearity, as it does not have the th root stage described inSection IV. Also, the estimate is less affected by additive noise(there is no squaring noise term). Because of these advantagesit is worth evaluating the decision feedback phase estimationmethod.

The decision directed phase estimate may be expressed math-ematically as follows. In one branch of the processor, isbuffered for a duration of symbols until the estimated datavalue becomes available, and then is multiplied by theconjugate of


Given that the bit error rate is low, for most symbols

and so

is a different complex Gaussian noise sequence fromhaving the same variance . The values of

are smoothed over time, such as by a PLL, to give a phaseestimate . The smoothing reduces the impact of additivenoise and of the infrequent symbol error events where

. In a second branch the converse ofthe phase estimate , which is an old phase estimate,is applied to the newest observed electric field value , andthen a decision is made

is the decision (slicing) function, that decides which of theset of allowed values most resembles. For BPSK modulationformat

and for QPSK

Substituting in (3)

(6)

is a different complex Gaussian noise sequence fromhaving the same variance . The data estimate will beunaffected by laser phase noise provided is a goodphase estimate and the phase has not drifted significantly during

symbols.It is a substantial problem to model the decision feedback

loop accurately because of the occurrence of error extensionevents. If a short burst of data errors occurs naturally due tothe additive noise, then that causes the phase estimate to bein error, which in turn causes more data errors. The short errorburst can become significantly longer so that it leads to bit errorseven after forward error correction (FEC) decoding. To success-fully model the decision directed phase estimate, a large numberof symbols must be simulated in order to capture error extensionevents, and assumptions have to be made about the FEC code.These issues are avoided by use of a simplification which ig-nores the impact of a poor . The phase estimate is assumed tobe perfectly accurate.

This is clearly overly optimistic and so the result of the simu-lation can be considered to be a lower bound on the -factorpenalty. The impact of the symbol delay in the feedback loopis included in the model. Referring to (6),

Fig. 3. �-factor penalty for decision feedback phase estimate for BPSK. Thecurves are for different values of feedback delay, � symbols.

Fig. 4. �-factor penalty for decision feedback phase estimate for QPSK. Thecurves are for different values of feedback delay, � symbols.

The results of the lower bound of the -factor penalty asfunction of symbol time-linewidth product for different valuesof are shown in Figs. 3 and 4. In fact the penalty is a func-tion only of . The penalty reaches 1 dB at

for BPSK, and for QPSK.Comparing the decision directed phase estimate lower boundwith the penalty for the power law-average phase estimates ofFigs. 8 and 9, they are about the same only for a short feedbackdelay of about . A realistic real-time digital processor isexpected to have a degree of parallelism of 10 to 100, and so itmust be concluded that the decision directed phase estimate willperform considerably worse than the power law-average phaseestimate.

IV. POWER LAW-AVERAGE PHASE ESTIMATE

The alternative way to remove the effect of data modulationis by raising the -ary PSK signal to the th power [38].

(7)


where

(8)

The high-order terms mean that is not exactlya Gaussian noise process. It will be treated as Gaussian, as anapproximation. The variance of the high order terms is includedin the variance of each complex part of . For BPSKmodulation format

and for QPSK

It is clear from (7) that then can be obtained from the phaseangle of . Defining

(9)

It is necessary define as the unwrapped phase angle becauseis an unwrapped angle, extending from to , according

to (4). The argument function normally provides the wrappedphase angle, in the interval to , and to unwrap the phaseangle of must be incremented or decremented by everytime crosses the negative real axis. Assuming that the additivenoise is small, can be expanded using a small angleapproximation

(10)

is a real Gaussian noise sequence whose variance is .The task of making a phase estimate that this paper addresses isto estimate from the observable quantity .

The best linear estimate of that can be made is by applyinga Wiener filter to [39]. There are two kinds of Wiener filter thatmay be considered: a zero-lag filter which makes an estimate of

based on all up to and including the th symbol ,and a finite-lag (smoothing-type) filter which looks up to symbol

, where is a positive integer. Both are lowpass filters,which smooth out the additive noise while tracking the varyingphase. The finite-lag filter is expected to give the best result be-cause it effectively looks forward in time by symbols, as wellas looking at the infinite past, to perform its smoothing function.It is trivial to cope with the noncausal nature of the finite-lagfilter by buffering the observed electric field values for anextra symbols. The -transfer function of the zero-lag filteris calculated by standard procedures [38] based on the Gaussiannoise nature of and the fact that is a Wiener process,as detailed in the Appendix.

(11)

where

(12)

Fig. 5. Impulse response of (a) zero-lag Wiener filter, and (b) finite-lag Wienerfilter having � � ��.

The impulse response of the zero-lag filter is a negative expo-nential response, as shown in Fig. 5(a). The finite-lag filter has

-transfer function

(13)

Fig. 5(b) shows an example of the impulse response forand . The best result is obtained for high , and theimpulse response tends to a Laplacian, a symmetrical two sidedexponential, as . However, the number of feedforwardtaps is , so it is pragmatic to choose an intermediate valuefor . The noncausal term is included in (13) because of thegroup delay of symbols associated with the remaining partsof the Wiener filter.

Having found the optimal transfer function in phase angle,within the two approximations that have been made so far, thereare many ways to apply this transfer function. The direct methodis to calculate in the digital processor, via (9). The phaseangle may be calculated by a look-up table, and the unwrappingprocess described below can be followed. Next the linear filterof (11) or (13) is applied. Finally the exponential of is calcu-lated, again by a look-up table, to be applied in the discrete-timeversion of (1). Aside from its complexity, this method has thedisadvantage that it suffers from cycle slips even when the phasenoise is low. The additive noise causes errors in the cycle countoperations, that is cycle slips, which become bit errors after dif-ferential logical detection. The number of cycle slips is not re-duced by the smoothing operation because the smoothing oper-ation happens after the cycle count operation.

A second method of applying the Wiener filter transfer func-tions is mathematically less direct than the first, but outperformsit in all respects. By applying a small angle approximation to (9)once more, it is clear that it is approximately the same to applythe transfer functions of (11) and (13) to complex variableas to its phase angle . Thus, a planar filtered intermediatevariable is formed by applying a digital filter to .

(14)

(15)


for the zero-lag and finite-lag Wiener filter cases respectively.The divide-by- operation on the phase angle is then executedas taking an th root, to give

(16)

In fact the absolute value in the denominator may be omitted andan approximate th root may calculated whose amplitude is notcorrect, because the decision function is not affected bychanges to . There are possible th roots of . The throot in (16) refers to the principal root. The function is thecycle count function, described below, so thatresolves the -way ambiguity correctly. Finally, the data esti-mate is made

(17)

The cycle count function keeps count of every time thephase of crosses the boundary (the negative realaxis).

(18)

where

is the wrapped phase angle lying between and . Anequivalent expression for the function which can be im-plemented more readily in a digital processor is

V. LOOK AHEAD COMPUTATION

While the power law-Wiener filter phase estimation methodlaid out in Section IV provides a good phase estimate, it is notyet suitable for implementation in a parallel architecture digitalprocessor. There are two places where feedback of the immedi-ately preceding result is used. The term in the denominatorof (14) and (15) indicates feedback of to calculate

. Also the cycle count function of (18) depends on.

A resolution to this issue is found by recasting the algorithmsusing a look-ahead computation [27]. The look-ahead computa-tion can be described as follows. A function may depend onresult and older, as well as on inputs .

The problematic term is eliminated by substituting inan expression using .

We now have a more complicated function containing nestedcalls of function , but the newest result it depends on is now

. Repeating this operation yields an expression forthat depends on results and older.

There is a condition that function must satisfy in order forthe look-ahead computation to be applied. Suppose that hasbeen recast to depend on result , and the time taken to com-pute one iteration of the algorithm is , which is longerthan . This means that the recast algorithm cannot be exe-cuted in a parallel digital processor. The next attempt might beto further recast using the look-ahead computation so thatdepends , and comprises nested calls function .If possesses no special property, for example if it is executedas a look-up table, then it will take to compute oneiteration, which will inevitably be longer than . No matterhow large is chosen it is not possible to execute the recastalgorithm on a parallel digital processor. To be implementablein a parallel processor, the function must satisfy the condi-tion that grows more slowly than linearly with . Anylinear digital filter function satisfies this requirement. The recastdigital filter contains the sum of terms, which is executed assequential pairwise additions. The computation time for one it-eration grows as , which is indeed slower than linear. Onthe other hand, the PLL and the decision feedback operation areexamples of functions which cannot be made suitable for im-plementation in a parallel digital processor via the look-aheadcomputation.

The simplest way to present the look-ahead computation fora digital filter is by multiplying both numerator and denomi-nator of the -transfer function by the same polynomial. For the

-transfer functions of (14) and (15) the appropriate polynomialis

and the zero-lag and finite-lag digital filter functions become(19)-(20), as shown at the bottom of the next page. Althoughwritten as a product of two sums, the numerator of the finite-lagtransfer function can be expanded to give feedforwardtaps.

The cycle count function of (18) is recast using a look-aheadcomputation by repeated substitution of the most recent result

(21)

Like a digital filter, the recast algorithm comprises the sum ofterms, which is executed as pairwise additions, and so


a value for can always be found which permits the expressionof (21) to be implemented in a parallel digital processor.

After recasting using the look-ahead computation, the Wienerfilters require substantially more digital processor resources toimplement, having taps instead of taps. Theamount of resources can be reduced by using incremental blockprocessing [27]. When the phase estimation is executed inparallel paths, only one path employs feedback and implements(19) or (20). The remaining -1 paths calculate their resultsfrom the first path or from one another. For example for the

case, the path employing feedback is for ,where is an integer, then

in accordance with (19), and

...

The average number of taps is 3–1/ . Recasting the algorithmand using incremental block processing for the zero-lag casewhen , for example, increases the number of taps by amodest 47%, from 2 to 2.94. This compares to a 750% increaseif incremental block processing were not used.

For the case where , further computation savingsare available by writing the sum in the numerator of (15) as apower-of-2 decomposition [27]. The sum of terms becomes aproduct of terms. [See (22) at the bottom of the page.]The product of terms is realized as a sequence of digital filteroperations. It is most hardware-efficient to use the algorithmof (20) in the one path that employs feedback, and the algo-rithm of (22) in the other -1 paths within the digital processor.The average number of multiplication operations per symbol isthen . For example when

and , the average number of multiplications isincreased marginally from 7 to 8.63 by recasting using the look-ahead computation and using incremental block processing and

Fig. 6. �-factor penalty for different phase estimation methods on a BPSKsignal.

power-of-2 decomposition. The application of look-ahead com-putation, incremental block processing and power-of-2 decom-position to the Wiener filter algorithms makes them suitable fora parallel digital processor but does not change the behavior ofthe algorithms.

Since the 1-dB penalty point associated with the Wiener fil-ters occurs at the high values of seen in Figs. 6 and 7,it may be desirable to further save on the computation require-ment by approximating the Wiener filter function and sacrificingsome performance. A simple approximation is to replace theideal Laplacian impulse response of Fig. 5(b) with a rectangularimpulse response. This is implemented by summing a suitablychosen number of neighboring values of in place of(14) or (15)

No multiplications are required for the filtering stage, which re-duces the computational resources needed, and it does not needto be recast using the look-ahead computation because there isno feedback. The filter has a group delay of . Thissolution was proposed by Noé [15], and has been implementedin burst-mode and real-time experiments [2], [6]–[8], [10]–[12].The method is known as the feedforward phase estimate because

(19)

(20)

(22)


Fig. 7. �-factor penalty for different phase estimation methods on a QPSKsignal.

it does not use feedback. Note that the approach is different fromrecasting the Wiener filter using the look-ahead computation,which produces an algorithm that uses feedback of a distant pastresult.

VI. COMPARISON OF PHASE ESTIMATION METHODS

The -factor penalty as a function of symbol time-linewidthproduct was obtained by Monte Carlo simulations of differentphase estimation methods. The results are plotted in Fig. 6 forBPSK modulation format and Fig. 7 for QPSK. For the syn-chronous detection methods differential logical detection is ap-plied. The penalty is obtained from the bit error rate associatedwith compared to the true data .The power law-Wiener filter phase estimation method describedin Section IV was modeled for the zero-lag case andthe finite-lag case where . The MAP phase estimateis the best possible phase estimate as described in Section II.Also included is the penalty for a phase estimate comprising an

th power stage followed by a PLL. The PLL cannot be im-plemented in a parallel digital processor, but of course it canbe modeled, and is included for comparison. The small angletransfer function of the PLL is set to be the same as the zero-lagWiener filter. Finally, there is a curve for differential field vectordetection, which is an asynchronous detection method. The dataestimate is , which is different from the othercases, the synchronous detection cases, where differential log-ical detection is used. The differential field vector case is equiv-alent to differential detection performed in the optical domainby a passive optical delay network [40], [41], although here

is calculated within the digital processor.The curve for the finite-lag Wiener filter phase estimate is

close to the MAP estimate for both BPSK and QPSK. Recallthat three approximations were made: 1) neglecting high ordernon-Gaussian contributions to the noise in (8), and the vari-ance of includes some squaring noise; 2) small angle ap-proximation given that additive noise is small in (10); and 3) afurther small angle approximation so that the Wiener filter canbe applied as a planar filter in (14) and (15). The small devia-tion from the penalty of the MAP estimate confirms that these

approximations are minor approximations. The QPSK case isfarther from the MAP curve than the BPSK case, which is con-sistent with approximation 1) being less valid for QPSK becauseof more high order terms. The point of 1 dB excess penalty forthe finite-lag Wiener filter is at for BPSKand for QPSK. This means that for a 10 Gbaud QPSKtransmission link, for example, a linewidth of 8 MHz for each ofthe signal and local oscillator lasers can be tolerated. This is suf-ficiently high to allow distributed feedback (DFB) lasers to beused. The application of the algorithms in an experiment usingDFB lasers for the transmitter and local oscillator was demon-strated in [16].

The curve for the zero-lag Wiener filter is significantly worsethan for the finite-lag Wiener filter and the MAP curve, whichis because the latter two benefit from seeing forwards as well asbackwards in time. The PLL is worse than the zero-lag Wienerfilter overall, though it is close for small values of , whichis consistent with the two having the same behavior for smallphase excursions.

In the absence of phase noise , the differentialfield vector detection case shows a substantial penalty comparedto any of the synchronous detection cases, as is expected. Theasynchronous detection method can be said to be less sensitiveto laser linewidth in that the slope of the penalty curve is lowerthan any of the synchronous cases, but the value of the penaltyis always higher for asynchronous detection.

VII. DIFFERENTIAL DETECTION TO AVOID IMPACT OF

CYCLE SLIPS

Any phase estimate may make a cycle slip error due to phasenoise. If a cycle slip occurs at andis in error by 1, then following through (18), (16) and (17)

(23)

All symbols following the cycle slip are in error, and so thereare continuous bit errors after FEC decoding. Typically the highBER triggers a reset of the receiver so that it functions correctlyonce again, but the short term loss-of-signal event is not consid-ered acceptable.

The standard method to avoid the persistent nature of cycleslips is to use differential logical detection, known in the fieldof radio as coherently differential detection. The data sequenceis precoded at the transmitter. The actual symbol values trans-mitted are calculated from the desired symbol values

by

(24)

includes FEC and any other overhead, and also usesGray coding in the case of QPSK. Equation (24) employs feed-back of the immediately preceding result, and there are methodsincluding recasting using the look-ahead computation that allowthe calculation to be implemented in a parallel digital processor.


At the receiver differential logical detection is executed on thedecided symbol value as follows:

(25)

Differential logical detection transforms a cycle slip from a per-sistent impairment on the received signal into a single symbolerror. Combining (23) and (25) reveals that after the differentiallogical operation there is a single isolated symbol error

The disadvantage of differential logical detection is thatsymbol errors become multiplied. Suppose a single symbolerror occurs at because of additive noise, and

After differential logical detection

The single symbol error has become a pair of consecutivesymbol errors. For both BPSK and QPSK, assuming Grayencoding is used, a single bit error is transformed into a pair ofbit errors. For QPSK the bit errors may be either consecutiveor separated by two nonerrored bits. In all cases, it is accurateto say that the differential logical decoding operation hasconverted a single bit error into a short burst of bit errors. Thesynchronous detection cases of Figs. 6 and 7 show a finitepenalty of 0.8 dB for because of the doubling of biterror rate to .

It may be considered pessimistic to treat the excess bit er-rors as if they were random bit errors, since many forward errorcorrection codes effectively have higher gain when presentedwith short bursts of bit errors than with random bit errors. Forexample, an cyclic block code can correct thesame number of error bursts in a block as isolated errors, pro-vided the bursts are not longer than bits [42].Most bursts resulting from QPSK symbol errors are thereforecorrected by a cyclic code having . SomeFEC schemes employ column-to-row interleaving before FECdecoding, in order to spread out a long burst of errors amongseveral FEC blocks so that it does not result in residual bit er-rors. The G.975 FEC standard is an example [43]. Such an in-terleaving stage should not be applied first as it would also dis-tribute short bursts of errors, and so would negate the abilityof the FEC code to inherently correct short bursts. A solutionis to employ a short burst-correcting code first such as a cycliccode, followed by an interleaving stage, and then followed by ahigh gain code. The true penalty associated with the conversionof single symbol errors into pairs of symbol errors, then, is the

Fig. 8. �-factor penalty for different phase estimation methods on a BPSKsignal, where a pair of bit errors is counted as a single error.

Fig. 9. �-factor penalty for different phase estimation methods on a QPSKsignal, where a burst of up to four bit errors is counted as a single error.

difference between the gain of an FEC code that corrects shorterror bursts and the best possible code that could be deployedin the same computational hardware. This penalty is probablymuch less than 0.8 dB in practice.

The simulation results of the different phase estimationmethods were reevaluated by counting the number of shortbursts of bit errors, and treating a burst as equivalent to oneerror to calculate the penalty. A short burst is defined as up tofour bits in length for QPSK and two bits for BPSK. The newcurves are shown in Fig. 8 for BPSK and Fig. 9 for QPSK.These penalties can be considered to be more realistic thanthose of Figs. 6 and 7 since they take into account the extra gainexperienced by the FEC decoder which follows the coherentreceiver. The penalty is now zero for (error burstprobability of ) for all of the synchronous phaseestimates. The differential field vector case also benefits fromthe use of short burst error correcting code, since many of thebit errors occur in bursts for that detection mode. However it isstill significantly worse than the synchronous phase estimates.For QPSK the asynchronous case is 2.2 dB worse at(error burst probability ).


Some simplifications are possible to the power law-averagephase estimate when differential logical detection is used.The slicing function has the property that a factor of theoperand which is an integer power of can be takenoutside the function without changing the result. Ignoring thedenominator of the th root also, this means that (17) can berewritten

and so

Substituting in (18)

With differential logical detection it is not necessary to evaluateby (18), and the sum of terms in the recast version of (21)

may also be avoided. This simplification has been applied inphase estimates that work in conjunction with differential log-ical detection [2], [6]–[8], [10]–[12], [15].

VIII. CYCLE SLIP PROBABILITY

The choice of whether to use differential logical detection ornot is driven by the occurrence of cycle slip events. Without dif-ferential logical detection, when a cycle slip occurs it causes allbits to be in error until the receiver is reset. A customer will re-quire a much lower probability for this undesirable kind of eventthan the specification on the bit error rate after FEC decoding.An acceptable cycle slip probability might be . If the laserlinewidths are so large that the cycle slip probability is too highthen differential logical detection is mandated, together with thedeployment of data precoding and perhaps a nonstandard FECcode.

The number of cycle slips was counted for the Monte Carlosimulations of the synchronous phase estimation methods. Asimple rule was used where 11 or more consecutive bits in errorwere treated as being due to a cycle slip, while 10 consecutivebits in error were determined to be simply bit errors. In factthe threshold level of number of consecutive bit errors does notaffect the overall cycle slip count very much. The cycle slipprobability is plotted in Fig. 10 for BPSK and Fig. 11 for QPSK,for three phase estimation methods. The PLL result is betterthan that for the phase estimate using a zero-lag Wiener filter,a change of places compared to the -factor penalty. The bestperformer is still the phase estimate with the finite-lag Wiener

Fig. 10. Cycle slip probability for BPSK signal. The dashed vertical line is the1 dB �-factor penalty point.

Fig. 11. Cycle slip probability for QPSK signal. The dashed vertical line is the1 dB �-factor penalty point.

filter. All three sets of points were a good fit to curves of theform

cycle slip probability

where and are positive constants. Heuristic argumentscan be made as to why this relationship should apply. Thefitted curves are also included in Figs. 10 and 11. For the fi-nite-lag Wiener filter, the cycle slip probability occurs at

for BPSK and for QPSK. Thedashed vertical lines indicate the 1–dB penalty point inferredfrom Figs. 6 and 7. For both modulation formats the conditionof low cycle slip probability is more stringent than the conditionof low -factor penalty, by about two orders of magnitude inlaser linewidth.

IX. CONCLUSION

While the new generation of optical coherent detection usesdigital signal processing to derive the data content of the signal,it is still true that the effects of finite laser linewidth and delays in


feedback paths on the phase estimate need to be managed care-fully. Laser phase noise combined with additive noise causesbit errors and cycle slips. In the older generation of hardwarethe PLL was the only phase estimate that was considered, buttoday a wide range of signal processing operations may be ap-plied and must therefore be evaluated.

The optimal phase estimate is the MAP phase estimate. The-factor penalty associated with the MAP phase estimate has

been evaluated as a baseline to compare with other phase esti-mates. For BPSK and QPSK modulation formats a phase esti-mate comprising a power law nonlinearity followed by a Wienerfilter gives a result that is almost as good as the MAP estimate.The laser linewidth for dB -factor penalty is high enoughto accommodate DFB lasers, assuming 10 Gbaud signaling rate.The Wiener filter is designed to be either zero-lag or finite-lag,and the finite-lag version gives the better result because it effec-tively makes an estimate looking both forwards and backwardsin time. The decision feedback operation is an alternative to thepower law nonlinearity, but it gives a higher -factor penaltywhen there is a significant delay in the feedback path. How-ever, although not studied in this paper, the decision feedbackestimate may be the best choice for modulation formats wherethe power law nonlinearity cannot be applied, such as QAM,in which case the feedback path delay and consequent low laserlinewidth must be tolerated. Also, the decision directed PLL is agood choice for all modulation formats when narrow linewidthlasers are readily available, because it requires a small numberof computations to implement.

When the digital processor has a parallel architecture it isequivalent to imposing a delay on any feedback paths. TheWiener filters use feedback of the immediately preceding result,and so cannot tolerate a delay in the feedback path. However,the issue is resolved by recasting the Wiener filter algorithmsusing a look-ahead computation, so that they employ feedbackfrom a distant past result. The recast algorithm has identicalbehavior to the original, but requires more computations toimplement. Further techniques may be applied such as iterativeblock processing and power-of-2 decomposition to bring thenumber of computations down to a level close to the original.Thus, while the issue of delay in feedback paths was a key con-straint on the laser linewidth for the old generation of coherentreceivers, it is not important for today’s DSP-based receiversdetecting PSK modulation formats.

The probability of a cycle slip occurring was evaluated asa function of combined laser linewidth for several phase esti-mates. The constraint on laser linewidth to achieve a tolerablylower cycle slip probability is about two orders of magnitudebelow the value to obtain a low -factor penalty. The persistentnature of a cycle slip event is avoided by applying differentiallogical detection following the data estimate, but this also dou-bles the bit error rate. If an appropriate FEC scheme is chosenthen the excess bit errors are corrected, and the net penalty fromdifferential logical detection is very small.

APPENDIX

A. Derivation of Wiener Filter Transfer Functions

Equation (10) established that there is a linear relationshipbetween the desired phase and the observable phase . The

field of estimation theory tells us that the best linear estimateof is made by applying a Wiener filter to . The derivation ofthat Wiener filter will follow the procedure laid out by Proakisand Manolakis [38]. Variables and and the nota-tion of that reference will be used here, and will have differentmeanings from earlier in this paper.

A variable is defined as

(26)

so that (10) takes on the classical form of a signal embedded innoise

, the -transform of the autocorrelation of , is related to the-transforms of the autocorrelation of and by

is a real Gaussian noise sequence having variance , and so

is a random walk function driven by complex Gaussian se-quence . -transforming (26) substituted into (4) gives

and

Since

It is desirable to express in the form

Matching coefficients in the numerator yields the quadraticequation

Only one root meets the expected condition that for, and that is

(12)

Also

(27)

is related to the inverse transfer function of thewhitening filter by


and so

First consider the zero-lag Wiener filter. The desired outputis not delayed compared to .

The -transform of the cross-correlation of and is

The Wiener filter is given by

where indicates taking the causal part, that is rejecting termsof positive powers of .

Substituting in (27)

Expressing as partial fractions

So

which is the transfer function of (11) given that .Next, consider the finite-lag Wiener filter, where

To find the causal part, we need to express as a sum of par-tial fractions, each of which contains either positive or negativepowers of , but not a mixture of both. Try the following form:

(28)

Equating the numerators

There is a consistent solution

The final partial fraction is the only noncausal term in (28), so

and

which leads to the transfer function of (13).

ACKNOWLEDGMENT

The author thanks Prof. P. Bayvel for support of this workand for providing facilities to experimentally evaluate the phaseestimation algorithms.

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Michael G. Taylor (M’94) received the B.A. de-gree in physics from Oxford University, Oxford,U.K., in 1987 and the Ph.D. degree in electricaland electronic engineering from University CollegeLondon, U.K., in 1990. His doctoral thesis concernedmultiquantum-well modulator devices for opticalcomputing applications.

He joined STC Technology Ltd. (subsequentlyNortel), Harlow, U.K., in 1990 to work on the de-velopment of optical fiber communications systemsusing EDFAs. He contributed to the design of the

first transoceanic transmission systems using optical amplifiers, and worked onearly research projects in 1550 band WDM and the NTON all-optical networkdemonstration. In 1996, he joined Ciena Corporation, MD. He worked onthe design of several of that company’s dense WDM products, including the10-Gb/s product, and in management of fiber propagation effects in general.Since 2002, he has been an independent consultant and has conducted researchon coherent optical communications in collaboration with groups at UniversityCollege London and University of Central Florida, Orlando. He has publishedmore than 30 research papers and has been granted 17 U.S. patents.

Dr. Taylor is a regular reviewer for the IEEE PHOTONICS TECHNOLOGY

LETTERS and the JOURNAL OF LIGHTWAVE TECHNOLOGY.

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