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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 8, AUGUST 2007 2033

Digital Equalization of Chromatic Dispersion andPolarization Mode Dispersion

Ezra Ip and Joseph M. Kahn, Fellow, IEEE

Abstract In this paper, we consider a fractionally spacedequalizer (FSE) for electronic compensation of chromatic disper-sion (CD) and polarization-mode dispersion (PMD) in a duallypolarized (polarization-multiplexed) coherent optical communica-tions system. Our results show that the FSE can compensate anyarbitrary amount of CD and rst-order PMD distortion, providedthat the oversampling rate is at least 3/2 and that a sufcientnumber of equalizer taps are used. In contrast, the amount of CD and PMD that can be corrected by a symbol-rate equalizeronly approaches an asymptotic limit, and increasing the numberof taps has no effect on performance due to aliasing that causessignal cancellation and noise enhancement.

Index Terms Chromatic dispersion (CD), coherent detection,digital signal processing, equalizers, optical ber communication,polarization mode dispersion (PMD).

I. INTRODUCTION

C HROMATIC dispersion (CD) and polarization-mode dis-persion (PMD) are two important linear distortions thataffect the performance of optical systems. In traditional re-ceivers that employ direct detection, the phase of the opticalE -eld is lost, making exact equalization of the channel by alinear lter impossible [1], [2]. Coherent detection circumventsthis problem by combining the received signal with a local

oscillator (LO) laser and by using balanced detection to down-convert it into a baseband electrical output that is proportionalto the optical E -eld. The resulting signal can then be sampledand processed by digital signal processing (DSP) algorithms,providing a exible platform based on a software that is anattractive alternative to optical CD and PMD compensation[3][5].

In principle, any linear channel distortion can be compen-sated in DSP by a digital receiver operating at one sample persymbol [6]. This requires that, prior to sampling, the receiveremploys an analog matched lter (matched to the convolutionbetween the transmitted pulse shape and the channel). Also,the performance of a symbol-rate equalizer is sensitive tosampling time error. A fractionally spaced equalizer (FSE)can implement the matched lter and equalizer as a singleunit and can compensate for sampling time errors, providedthat the baseband signal is sampled above the Nyquist rate[6][9]. Thus, the FSE is widely employed in other areas of

Manuscript received January 3, 2007; revised April 27, 2007. This work was supported by the Defense Advanced Research Projects Agency under theTACOTA Program Prime Contract W911-QX-O6-C-0101 via a subcontractfrom CeLight, Inc.

The authors are with the Department of Electrical Engineering, StanfordUniversity, Stanford, CA 94305-9515 USA (e-mail: [email protected]; [email protected]).

Digital Object Identier 10.1109/JLT.2007.900889

Fig. 1. Canonical model of a baseband communications channel with a digitalreceiver.

digital communications such as mobile radio [10] and digital

subscriber lines [11]. Recent experiments in coherent opticaltransmission invariably use an oversampling rate of two for CDcompensation [12][14].

In single-mode ber (SMF) communication, PMD causes astatistical time-varying differential group delay (DGD) betweenthe two principal states of polarization (PSP). The impact of PMD on the performance of a symbol-rate equalizer is similarto the sampling time error. Thus, it should be possible to usea polarization diversity coherent receiver in conjunction withan FSE in suppressing any arbitrary amount of PMD-inducedgroup delay since PMD is a linear effect.

This paper is organized as follows. In Section II-A, we willreview the canonical digital transmission model for a singlechannel and show why symbol-rate sampling is susceptibleto the sampling time error and how this can be overcome byan FSE. In Section II-B, we will extend our analysis to apolarization-multiplexed optical channel. We will derive theoptimal settings for a nite-length FSE and compute its mean-square error (MSE) performance. We will use the example of anSMF with CD and rst-order PMD in demonstrating the abilityof the FSE to compensate these distortions. In Section III, wewill present numerical results and compare the performanceof the symbol-rate equalizer versus the FSE. We will alsocompare the performance of different optical pulse shapes inthe presence of CD and PMD. Further discussion is provided in

Section IV.

II. THEORY

A. Canonical Digital Receiver

The canonical model of a baseband communications channelemploying a digital receiver is shown in Fig. 1. The input isa set of complex-valued symbols xn modulating a transmittedpulse shape b(t). The transmission medium is assumed to belinear, having an impulse response h(t), the noise added by thechannel, and the receiver is modeled as n(t). We denote thereceived pulse shape as c(t) = h(t) b(t), where denoteslinear convolution.

0733-8724/$25.00 2007 IEEE

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2034 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 8, AUGUST 2007

The front end of the canonical digital receiver consists of aband-limiting lter whose impulse response is p(t). The outputy(t) is then sampled at rate 1/T = M/KT s , where T s is thesymbol interval, and M and K are integers. For symbol-ratesampling, M = K = 1 . We can write the sampled signal as

yk

= y(kT ) =n

xn q (kT nT s ) + n (kT ) (1)where q (t) = p(t) h(t) b(t) is the received pulse shapeafter ltering, and n = p(t) n(t). If the channels impulseresponse h(t) is known, the optimum receiver consists of a matched lter p(t) = c(t),1 followed by a symbol-ratesampler [6]. However, symbol-rate sampling has signicantdrawbacks. First, the exact channel impulse response may notbe known. Second, it may not be possible to synthesize theanalog function c(t) exactly. Third, a symbol-rate equalizeris sensitive to sampling time error. Consider the discrete-timeFourier transform (DTFT) of (1), where we ignore the noiseterm. We have

Y sig (ejT ) = X ejT s 1T m

Q 2m

T (2)

where Q() is the Fourier transform of q (t), and X (ejT s )and Y (ejT ) are the DTFTs of the sequences xk and yk ,respectively. The zero-forcing (ZF) linear lter that can undothe channels distortion has a DTFT given by the following:

W ZF (ejT ) = 1/T m

Q ( 2m/T )1

. (3)

However, in the presence of a sampling delay s , the sampledsequence becomes yk = y(kT s ). We can characterize theimpact on the system by dening the postltered pulse shape asq (t) = q (t s ). The DTFT of the signal term becomes

Y sig (ejT )= X (ejT s )

1T m

Q 2m

T ej (

2 mT ) s .

(4)

In comparison with (2), each term in the summation overm is multiplied by a phase shift. If sampling is performedbelow the Nyquist rate, spectral overlap between the Q(

2m/T ) exp( j ( 2m/T ) s ) terms will occur. It is pos-sible that, for some values of s , the phase shift will cause theoverlapping spectra to interfere destructively, leading to signalcancellation. Even if the equalizer is designed according tothe minimum MSE (MMSE) criteria, noise enhancement willstill happen. A digital receiver that employs sub-Nyquist-ratesampling is therefore susceptible to sampling time errors. Thisproblem is particularly acute when symbol-rate sampling isused because spectral overlap always occurs, unless q (t) is asinc function, which is not realizable in practice.

1

Throughout this paper, we shall use superscripts

, T

, and H

in denotingthe complex conjugate, the transpose, and the Hermitian transpose of matrixvariables, respectively. A scalar is simply a special case of a 1 1 matrix.

Fig. 2. (a) Dual-polarization coherent optical system employing homodynedetection. (b) Equivalent baseband model.

B. Polarization-Diversity Receiver

The canonical model of a dual-polarization (polarization-multiplexed) coherent optical system employing a homodynereceiver is shown in Fig. 2(a). The transmitter consists of two MachZehnder (MZ) modulators, where each modulatoris capable of generating symbols from any arbitrary signalconstellation [15]. The two optical data streams, which arepolarized in orthogonal directions, are then combined using apolarization beam splitter. After passing through an SMF, thereceived signal is split into two paths that are mixed with theLO through a network consisting of two 90 hybrids and, then,is coherently detected using four balanced photodetectors. Theelectrical outputs are baseband signals corresponding to the I and Q associated with the polarizations parallel and orthogonalto the LO. These signals are then passed through low-passantialiasing lters having an impulse response p(t) and aresampled at a rate of 1/T using analog-to-digital converters(ADCs).

The dual-polarization coherent receiver has a basebandequivalent model that is shown in Fig. 2(b). The transmittedsignals in the two input polarizations are of the form

x j (t) =n

x j,n b(t nT s ) (5)

where xj,n is the nth symbol transmitted in the j th polarizationof the ber, and b(t) is the transmitted pulse shape, which isassumed to be the same for both channels. We can also write thetransmitted signal in vector notation as x (t) = [x1 (t), x2 (t)]T .The received signal is then expressed as

r (t) = h (t) x (t) + n (t) (6)where

h (t) = h 11 (t) h 12 (t)h 21 (t) h 22 (t). (7)

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IP AND KAHN: DIGITAL EQUALIZATION OF CHROMATIC DISPERSION AND POLARIZATION MODE DISPERSION 2035

h (t) is the matrix representation of the ber impulse response,where hij (t) denotes the response of the ith output polar-ization to an impulse in the j th input polarization. n (t) =[n 1 (t), n 2 (t)]T and r (t) = [ r 1 (t), r 2 (t)]T are the noise and thereceived signal vectors, respectively.

We note that (7) is a complete description of any dually

polarized linear channel and can characterize both CD and anyorder of PMD. For instance, the frequency response H () =F {h (t)} of an SMF with CD and rst-order PMD has the form

H () = cos sin sin cosej/ 2 0

0 ej/ 2

cos sin

sin cos ej

12 2 L fiber

2

(8)

where 2 is the bers GVD parameter [16], Lber is the berlength, is the angle between the reference polarizations andthe PSP of the ber, and is the DGD between the PSPs.

In this paper, we shall assume that all ber nonlinear ef-fects, as well as the phase noises of the transmitter and LOlasers, are either negligible or compensated so that the bercan be modeled as a linear channel. The baseband electricalsignals recovered by the polarization-diversity receiver areexpressed as

r i (t) =n

2

j =1x j,n cij (t nT s ) + n i (t) (9)

where cij (t) = h ij (t) b(t), and ni (t) is an additive whiteGaussian noise (AWGN) 2 process with two-sided power-spectral density (psd) of N 0 .

Upon passing r i (t) through p(t), the sampled outputs are

yi (kT ) = yi,k =

n

2

j =1x j,n q ij (kT nT s ) + n i (kT ) (10)

where q ij (t) = p(t) h ij (t) b(t), and n i = p(t) n i (t).We now derive the optimal coefcients for a linear transver-sal lter W whose output is an MMSE estimates of the trans-mitted symbols. The equalizer computes

x k = x1 ,kx2 ,k = W

T11 W T21W T12 W

T22

y 1 ,ky 2 ,k

= W T y k (11)

where y i,k = yi, kM/K + L , yi, kM/K + L 1 , . . . ,yi, kM/K L T , and W ij = [W ij, L , W ij, L +1 , . . . , W ij,L ]Tare column vectors of length N = 2L + 1 , and x denotes thelargest integer less than or equal to x. It can be shown that theWiener lter coefcients are given by the following:

W opt = A 1 (12)

2Possible noise sources include LO-spontaneous beat noise for an ampliedsystem, shot noise, and thermal noise.

Fig. 3. For noninteger oversampling, the sample times relative to the symbolpeaks differ between neighboring symbols. In the 3/2-oversampling exampleshown, odd symbols have a sample near the optimum sampling instant, whereaseven symbols have two samples straddling the symbol peak.

where A = E yk yTk , and = E yk x Tk . We can partitionthese matrices as

A =A 11 A 12A 21 A 22

(13)

and

= 11 12

21 22. (14)

The submatrices A ij = E yi,k yTj,k are N N , whereas thecolumn vectors ij = E yi,k x j,k are of length N . Let A ij,lm

be the element at the intersection of the lth row and mthcolumn of submatrix A ij . Similarly, let ij,l be the elementat the lth row of ij . By assuming that the transmitted sym-bols are independent identically distributed (i.i.d.), satisfyingE x i,l xj,m = P x ijlm , we have

A ij,lm = P xn

2

s =1q is

kM K

T lT nT s q js

kM K

T mT nT s

+ N 0

+

p(t lT ) p(t mT )dt (15)

and

ij,l = P x2

s =1q is

kM K

T lT kT s . (16)

We observed that (15) and (16) can take on K differentvalues, depending on mod(kM,K ). This effect is shown inFig. 3 for the case of 3/2 sampling. It is observed that thesampling time relative to the symbol peak differs for odd andeven symbols. Hence, the Wiener lter that reconstructs the bestestimate of the kth transmitted symbol needs to have differentcoefcients for k even and for k odd. In general, a rate M/K system needs to have K different sets of coefcients forW opt . These are obtained by evaluating (15) and (16) for allpossible values of mod(kM,K ) and by substituting into (12).

A block diagram implementation of the FSE is shown inFig. 4(a). The inputs y1 ,n and y2 ,n are the outputs of Fig. 2(b),which were sampled at intervals of T = KT s /M . The W ijfunction blocks takes N = 2L + 1 consecutive samples of y1 ,k

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Fig. 4. (a) Block diagram of the FSE. The four signal processing blocks eachimplement the dot product between a compartment W ij of the FSE matrixwith a block of N = 2 L + 1 samples of y1 ,k or y2 ,k . The implementations of these are shown for FSEs having an oversampling rate of (b) M/K = N and(c) M/K = 3 / 2 .

or y2 ,k and evaluates the matrix multiplication in (11). We canimplement the W ij function blocks using Fig. 4(b) and (c) forthe cases where the FSE has an integer oversampling rate of M and for M/K = 3 / 2. Since the FSE only needs to producean output at the symbol rate 1/T s , the concatenation of the

nite impulse response (FIR) lter, followed by decimation(the dashed boxes [W ij ]M ), can, in fact, be implemented bya polyphase lter. Let k = x k x k be the error between thetransmitted symbols and the output of the FSE in (11). It isshown that using the Wiener lter coefcients for the FSEresults in the MSE matrix

E k Hk = P x I H A 1 . (17)We then have

MSE MMSE =2

s =1E |s,k |

2 = tr E k Hk . (18)

For an oversampling rate of M/K , where K = 1 , (17) needsto be averaged over all K phases of (15) and (16).1) Limiting Performance of the FSE: In the limit where the

rate of the FSE 1/T = M/KT s satises the Nyquists criterion,the frequency range || < /T bears all the information inthe analog received signal. We can write (10) in terms of itsDTFT as

Y = QX + N . (19)

For convenience, we have dropped the ejT in our DTFTs. Theerror between the transmitted symbols and the FSE output is

= X W T Y = ( I W T Q )X W T N . (20)

We can write Q in terms of its singular value decom-position Q = USV H . If the symbols are i.i.d., satisfyingE x i,l xj,m = P x ijlm , the MSE matrix is given by thefollowing:

E [H ] = W T U P x VS P x S 2 + S N N I 1

P x S 2 + S N N I

W T U P x VS P x S 2 + S N N I 1 H

+ P x P 2x VS P x S 2 + S N N I 1

SV H

(21)

where the psd of the ltered noise is

S N N (ejT ) = N 01T m

P 2m

T

2

. (22)

The goal of W is to minimize tr E [ H ], which is equal tothe sum of the eigenvalues of (21). Since (P x S 2 + S N N I ) ispositive denite, (21) is minimized by setting its rst term tozero. Thus, we have

W Topt = P x VS P x S2 + S N N I 1 U H . (23)

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We note that the Wiener solution given in (23) is a function of frequency. The FSE coefcients may be found by taking theinverse DTFT

w opt ,n = T

2

/T

/T W opt ejT ejnT d. (24)

The psd of the resulting MSE is then equal to the sum of the eigenvalues of the second term of (21). By using Parsevalstheorem, we have

MSE MMSE =2

i=1E |i,k |

2

= T 2

/T

/T

2

i=1

P x S N N (ejT )P x s2i (ejT ) + S N N (ejT )

d

(25)

where si (ejT ) is the frequency-dependent singular value of Q (ejT ).

2) CD/PMD Example: To see that the Wiener solution in(23) is indeed optimal, consider the ber channel given in (8).Since b(t) and p(t) are identical for the two polarizations, wehave Q (ejT ) = P (ejT )B (ejT )H (ejT ). As H has unitsingular values, we have s2i (ejT ) = |P (ejT )B (ejT )|2 =|Q(ejT )|2 , for i = 1 and 2. Therefore

W T opt (ejT ) = P x Q(ejT )

P x |Q(ejT )|2 + S N N (ejT )

H(ejT )

(26)

where we note that

H(ejT ) = cos sin sin cosej/ 2 0

0 ej/ 2

cos sin

sin cos ej

12 2 L fiber

2

. (27)

It is the lter that exactly cancels the CD and rst-order PMDof the channel over the frequency range || < /T . The scalingfactor in (25) is an artifact of minimizing MSE [17]. If thesystem is symmetric, we can dene the unbiased SNR for theFSE output as

SNRMMSE U = 2P x

MSE MMSE 1. (28)

Finally, we dene the input SNR per symbol to be

SNR in = P x T s

N 0=

E sN 0

. (29)

The ratio between (29) and (28) is the system power penalty.

Fig. 5. Intensity waveforms of isolated NRZ, 33% RZ, 50% RZ, and 67% RZpulses. Normalized pulses are shown with unit energy.

III. NUMERICAL RESULTS

We simulated the performances of the symbol-rate equalizerand the FSE for four commonly used optical pulse shapes.These are the nonreturn to zero (NRZ), 33% RZ, 50% RZ,and 67% RZ pulses. The analytical formulas for the RZ pulsesand their respective spectra were studied in [18]. For NRZ[19], we assumed that the electrical drive signals for theMZ modulator were produced by passing an ideal rectangularpulse train through a fth-order low-pass Bessel lter with a3-dB bandwidth of 0.8/T s . The normalized waveforms of thesepulses are shown in Fig. 5. For the receiver, we assumed thatthe antialiasing lter p(t) is a fth order of any of the following:1) Bessel lter or 2) Butterworth lter with a 3-dB bandwidthof 0.4/T , where 1/T = M/KT s is the sampling rate. We shall

compare the relative performances of the choices of p(t).In Figs. 6 and 7, we show the amount of CD that a symbol-rate equalizer and an FSE with N taps (corresponding to Wbeing of size 2N 2) can tolerate with less than 2-dB powerpenalty at an input SNR of 20 dB per symbol, as dened by (28)and (29). Figs. 6 and 7 consider Bessel and Butterworth lters,respectively. We observed that, while the correctable amountof CD grows almost linearly with N for the FSE, the resultfor the symbol-rate equalizer only approaches an asymptote.This is due to the effect of aliasing that was described inSection II-A, where signal cancellation by the overlappingspectra leads to noise enhancement. The effect of aliasing isalso apparent for an FSE with 3/2 times oversampling when33% and 50% RZs are used in conjunction with a Bessel lter.This is because 33% and 50% RZs have a higher proportionof its energy outside || < /T . The Bessel lter, with itsgradual roll-off, is less able to prevent aliasing. In contrast,the Butterworth lter can better suppress the received signalssidelobes. Even though the Butterworth lter does not havea linear phase, its phase distortion is corrected by the FSE.Hence, the Butterworth lter achieves superior performance atlow oversampling. Since the sampling rate of ADCs is often alimiting factor in current coherent optical systems, 3 the reducedoversampling requirement can be of signicance. For FSEsoperating at M/K

2, the CD performance of all pulse shapes

3The fastest ADC commercially available in April 2007 is around 20 GSa/s.

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Fig. 6. Maximum allowable CD versus the number of equalizer taps for a

2-dB power penalty at an input SNR of 20 dB per symbol, using a Besselantialiasing lter. o denotes transmission using NRZ pulses, x denotes33% RZ, denotes 50% RZ, and + denotes 67% RZ.

Fig. 7. Maximum allowable CD versus the number of equalizer taps for a

2-dB power penalty at an input SNR of 20 dB per symbol, using a Butterworthantialiasing lter. o denotes transmission using NRZ pulses, x denotes33% RZ, denotes 50% RZ, and + denotes 67% RZ.

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is similar, regardless of whether a Bessel or Butterworth lteris used.

The linearity between CD and the FSE length is intuitivelysatisfying since the delay difference between two packets of energy that are separated by in frequency is given by DD =

| 2 |Lber . As the bandwidth of the transmitted signal isproportional to Rs , we expect that the pulse spreading willassume the form DD 2| 2 |Lber R s , which corresponds toN DD 2| 2 |Lber R2s (M/K ) sampling intervals. In order tocancel the intersymbol interference (ISI) of a q (t) that has asignicant energy over N DD samples, we expect N N DDtaps are needed. Fig. 6(c) and (d) and Fig. 7(b)(d) conrmthe linearity between | 2 |Lber R2s and N , where the empiricalrelationship is

| 2 |Lber R2s (M/K ) 0.15N. (30)This formula is consistent with the results of the study

in [20], which considered rounded pulses with 35% excessbandwidth and an FSE with M/K = 2 .

In Figs. 8 and 9, the performance of the symbol-rate equal-izer and the FSE in the presence of a rst-order PMD isshown. Figs. 8 and 9 consider Bessel and Butterworth lters,respectively. We assume that the ber has no CD, and theangle between the input signal polarizations and the PSP is = 45 (worst case). Again, the correctable amount of PMDapproaches asymptotic limits for the symbol-rate equalizer,whereas for the FSE, the tolerable grows linearly with N .This is, again, intuitively satisfying since a PMD channel witha DGD of will have an impulse response consisting of twopeaks separated by N = M/KT s samples. This matches the

slopes observed in Figs. 8(b)(d) and 9(b)(d).The required lter length will typically be dictated by CD,

not by PMD. Pulse spreading that is caused by PMD is typicallyless than one symbol interval, while that caused by CD typicallyspans several symbol intervals. For example, consider a symbolrate Rs = 10 GHz and a ber of length Lber = 2000 kmhaving a PMD of 0.1 ps/ km and a CD of 17 ps/ (nm km).By considering PMD, the mean DGD is = 4 .5 ps, and a DGDof ve times the mean corresponds to /T s = 0 .22 symbolintervals. Suppose the CD is compensated to within 5% error,corresponding to an uncompensated length Lber = 100 km.We have 2| 2 |Lber R2s = 8 .5 symbol intervals. Thus, settingN to the value given by (30) is typically more than enoughto compensate PMD. The combined power penalty from botheffects should be close to 2 dB, which is caused almostentirely by CD.

To illustrate the tolerance of an FSE to sampling time error,we plot the MSE per channel versus s /T s in Figs. 10 and 11,assuming no CD or PMD. Figs. 10 and 11 consider Bessel andButterworth lters, respectively. The input SNR per symbol is20 dB. As expected, we observed performance degradation forthe symbol-rate sampler. Although some noise enhancementis present for the FSE with 3/2 times oversampling, the MSEbecomes insensitive to sampling time error for M/K 2.Finally, we recall that our theoretical analysis and simulationresults used MSE as the criterion for evaluating system perfor-mance. In practice, MSE incorporates the effects of AWGN,

Fig. 8. Maximum allowable PMD DGD versus the number of equalizer taps

for a 2-dB power penalty at an input SNR of 20 dB per symbol using a Besselantialiasing lter. o denotes transmission using NRZ pulses, x denotes33% RZ, denotes 50% RZ, and + denotes 67% RZ.

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Fig. 9. Maximum allowable PMD DGD versus the number of equalizer taps

for a 2-dB power penalty at an input SNR of 20 dB per symbol using aButterworth antialiasing lter. o denotes transmission using NRZ pulses, xdenotes 33% RZ, denotes 50% RZ, and + denotes 67% RZ.

Fig. 10. MSE at the output of the digital equalizer versus the samplingtime delay for a channel with no CD or PMD at an input SNR of 20 dB

per symbol using a Bessel antialiasing lter. o denotes transmission usingNRZ pulses, x denotes 33% RZ, denotes 50% RZ, and + denotes67% RZ.

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Fig. 11. MSE at the output of the digital equalizer versus the sampling timedelay for a channel with no CD or PMD at an input SNR of 20 dB persymbol using a Butterworth antialiasing lter. o denotes transmission usingNRZ pulses, x denotes 33% RZ, denotes 50% RZ, and + denotes67% RZ.

ISI, and crosstalk between the two polarization-multiplexedchannels. The ultimate parameter of interest is the system bit-error ratio (BER). However, optimization of the FSE withrespect to BER leads to intractable equations; hence, MMSEis the most widely used criterion. A potential aw of the MSE

Fig. 12. Probability of symbol error versus input SNR per symbol for a16-QAM transmission over an optical channel with CD and rst-order PMD,with the receiver employing an FSE with 15 taps at an oversampling rate of M/K = 2 . o denotes Monte Carlo simulation results for NRZ transmission,x denotes 33% RZ, denotes 50% RZ, and + denotes 67% RZ. Thesolid lines aretheoretical SERcurves,assumingthat theerror signalafter digitalequalization is Gaussian-distributed.

approach is that ISI and crosstalk may not have Gaussianstatistics, as it is unlikely that the residual impulse durationafter equalization is long enough for these effects to achieve

Gaussianity. Thus, the actual BER of the system may not bethe same as the one expected for a system with the output SNRdened in (28).

To show that the MSE characterization is valid, we plot thesymbol error rate (SER) against the input SNR per symbol for achannel with a CD of | 2 |Lber R2s = 1 and a PMD group delayof = T s / 5, where the PSP angle is 45 . We assumed p(t) tobe a fth-order Bessel lter with a 3-dB bandwidth of 0.4/T .Our simulation used 16-QAM transmission with N = 15 taps.From Fig. 6, we expect the system to have power penaltiesaround 2 dB. Fig. 12 shows the SER obtained by Monte Carlosimulations as well as the theoretical SER (solid curves) forGaussian noise

SERQAM 16 = 32

erfc SNRMMSE U 10 . (31)The agreement between the Monte Carlo simulations and

the theoretical SER formula shows that the MMSE criterionis valid.

IV. D ISCUSSION

In this paper, we have only considered xed equalizer tapsfor the FSE. In practice, it is possibleand often desirabletoimplement W as an adaptive lter. Such an approach is usefulwhen the exact frequency response of the channel is not known.

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TABLE IOSNR R EQUIREMENT FOR VARIOUS MODULATION FORMATS

In particular, PMD is statistically time varying, so that anadaptive algorithm allows W to continually adjust for the bestMSE performance. Since the system is linear and we haveshown that the MSE performance in (21) surface is parabolic,we expect an adaptive solution to be well behaved.

In terms of implementing the FSE, there are two key con-siderations: the number of bits required for the ADCs and thesystem complexity. Table I shows the optical signal-to-noiseratio (OSNR) per symbol required for 4-, 8-, and 16-QAMtransmissions. Suppose we allow a 2-dB power penalty forCD/PMD compensation and a 1.5-dB margin for other im-pairments. A 16-QAM requires an OSNR of 20 dB. ADCquantization adds noisewith variance 2 / 12 per sample, where is the quantization step size. Suppose we require quantizationnoise variance to be less than 10% of the AWGN variance perdimension, which is N 0 / 2T = N 0 M/ 2KT s . For M/K = 3 / 2,this gives < 0.095 P x , where P x is the transmitted power.To determine the output range of the required ADC, we notethat the outermost symbols in 16-QAM have a magnitude of 3 P x / 10 = 0 .95 P x per dimension, and from Fig. 3, thepeak amplitude of a unit energy 33%-RZ pulse is less thanthree. Hence, even for the worst case of 33%-RZ 16-QAM

transmission, the ADC output only needs to swing between2.85 P x , corresponding to 60 . Even when allowing foramplitude distortion due to CD/PMD distortion, an 8-b resolu-tion should be more than sufcient.

To determine the system complexity, consider the examplein Section III, where we transmit at 10 GHz over 2000 kmof SMF whose PMD and CD parameters are 0.1 ps/ kmand 17 ps/ nm km, respectively. Suppose CD is compensatedusing inline dispersion compensation ber to within 5% er-ror. From Section III, we have 2| 2 |Lber R2s = 8 .5 symbolintervals. By using (30), the length of the FSE that is re-quired for 3/2-time oversampling is 15 taps, which is roundedto the nearest odd integer. Thus, the equalizer operation in(11) requires 60 complex multiplication per symbol at 8-bresolution. Achieving such performance would require a verylarge-scale integrated circuit with extensive pipelining andparallelization.

V. CONCLUSION

We have demonstrated the ability of an FSE to compensatefor any linear channel impairment in a dually polarized opticalsystem, including CD and any arbitrary amount of rst-orderPMD. We derived the optimal tap settings for an FSE based onthe MMSE criterion, and we also evaluated the theoretical MSEperformance. Our simulations showed that MMSE is a validperformance criterion despite the potential non-Gaussianity of

crosstalk and ISI. For NRZ, 33% RZ, 50% RZ, and 67% RZpulses, we showed that the FSE is insensitive to sampling timeerrors and can correct an arbitrary amount of CD and rst-orderPMD with less than 2-dB power penalty if an oversamplingrate of M/K 3/ 2 is used in conjunction with a fth-orderButterworth antialiasing lter. We showed that the amount of CD distortion that can be compensated by an FSE with N tapsis approximately | 2 |Lber R2s (M/K ) 0.15 N, whereas theamount of rst-order PMD DGD that can be compensated is PMD M/K = N T s , where M/K is the oversampling ratio.

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[21] S. Fan and J. M. Kahn, Principal modes in multimode waveguides, Opt. Lett. , vol. 30, no. 2, pp. 135137, Jan. 2005.

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Ezra Ip received the B.E. (Hons.) degree in electrical and electronics engineer-ing from the University of Canterbury, Christchurch, New Zealand, in 2002 andthe M.S. degree in electrical engineering from Stanford University, Stanford,CA,in 2004, where he is currentlyworkingtoward thePh.D.degree in electricalengineering.

In 2002, he was a Research Engineer at Industrial Research Ltd, NewZealand. He is currently with the Department of Electrical Engineering,Stanford University. His research interests include single-mode optical bercommunications, free-space optical communications, and nonlinear optics.

Joseph M. Kahn (M90SM98F00) received the A.B., M.A., and Ph.D.degrees in physics from the University of California (UC), Berkeley, in 1981,1983, and 1986, respectively.

From 1987 to 1990, he was at AT&T Bell Laboratories, Crawford HillLaboratory, Holmdel, NJ. He demonstrated multigigabit-per-second coherentoptical ber transmission systems, setting world records for receiver sensitivity.From 1990 to 2003, he was with the Faculty of the Department of ElectricalEngineering and Computer Sciences, UC, performing research on optical and

wireless communications. Since 2003, he has been a Professor of electricalengineering at Stanford University, Stanford, CA. His current research interestsinclude single- and multimode optical ber communications, free-space opticalcommunications, and MEMS for optical communications.

Prof. Kahn received the National Science Foundation Presidential YoungInvestigator Award in 1991. From 1993 to 2000, he served as a Technical Editorof the IEEE Personal Communications Magazine. In 2000, he helped foundStrataLight Communications, where he served as Chief Scientist from 2000to 2003.

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