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PHOTONICS TECHNOLOGY LETTERS - SUBMISSION 1

Impact of Loop Delay on the Performance ofGardner Timing Recovery

Luca Barletta, Member, IEEE, Maurizio Magarini, Member, IEEE, Federica Scardoni, Arnaldo Spalvieri.

Abstract— Feedback timing recovery schemes suffer from thedelay in the feedback loop. Since this delay can be large in opticalsystems, especially when digital signal processing is implementedin FPGA, the performance of the phase-locked loop can becompromised when it has large loop bandwidth. In the letter,the performance of the digital feedback timing recovery schemebased on Gardner’s detector is analyzed taking into accountthe delay in the loop and the phase noise that affects the localoscillators used for clocking data converters. Numerical resultsreported in the letter show that simulations closely fit the analysis.

Index Terms— Synchronization, Timing jitter, Optical re-ceivers, Phase noise.

I. INTRODUCTION

Timing recovery is a classical topic in communicationtheory that is receiving renewed interest in the frameworkof coherent optical communications, see e.g. [1]–[3]. Themost popular digital feedforward timing recovery scheme forQuadrature Amplitude Modulation (QAM) and Phase ShiftKeying (PSK) modulation formats has been proposed in [4].This scheme requires that the signal processed in the timingrecovery branch of the receiver is sampled at four timesthe symbol frequency. Feedforward timing recovery with twosamples per symbol has been introduced in [5], analyzedin [6], and recently proposed for optical coherent QPSK in[7]. Timing recovery with half-baud spaced sampling can beperformed also by the scheme due to Gardner [8], which isbased on the Phase-Locked Loop (PLL) principle. Comparedto the feedforward scheme, the feedback scheme suffers fromdelay in the feedback loop, which can be large in opticalsystems, especially when Digital Signal Processing (DSP)is implemented in FPGA, compromising the performance ofthe PLL when it has large loop bandwidth [2]. The need oflarge bandwidth arises when timing recovery has to track anincoming timing reference affected by large phase noise, as [3]has put in light in the context of feedforward timing recovery.As pointed out in [1] phase noise is always inherently presentin the oscillator used for clocking the data converters, andtherefore it is always present in the synchronism to be tracked.More generally, the issue of delay in the loop is common toall the synchronization schemes based on the PLL principle,

Manuscript received Month DD, 2013; revised Month DD, 2013.L. Barletta is with the Institute for Advanced Study, Technische Universitat

Munchen, Germany (E-mail: luca.barletta@tum.de).M. Magarini, F. Scardoni, and A. Spalvieri are with the Diparti-

mento di Elettronica, Informazione e Bioingegneria, Politecnico di Mi-lano, Italy (E-mail: {maurizio.magarini,arnaldo.spalvieri}@polimi.it, feder-ica.scardoni@mail.polimi.it).

the impact of delay on system’s performance being analyzedin [9] for phase noise spectrum of first and second order.

In this letter, the transfer function of the PLL in Gardner’sscheme is optimized taking into account the phase noise that ispresent in the incoming timing reference and taking the delayin the loop as a constraint. The optimal transfer function turnsout to be difficult to implement, hence a suboptimal transferfunction is proposed and its performance is analyzed.

II. GARDNER TIMING DETECTOR

Consider a continuous-time QAM or PSK baseband signalplus complex Additive White Gaussian Noise (AWGN):

r(t) =∑

k

aku(t− kT + τ) + w(t), (1)

where T is the symbol repetition interval which is assumedto be known, τ is a random and unknown parameter, {ak}is the sequence of independent and identically distributedcomplex random constellation points with zero mean and unitvariance, u(t) is the impulse response of the transmit filter, andw(t) is complex AWGN with power spectral density N0. Weassume that the frequency response of the transmit filter is thesquare root of a Nyquist filter with unit energy, therefore theimpulse response of the Nyquist filter is 1 at t = 0. With thisassumption the resulting signal-to-noise ratio is SNR = N−1

0 .To extract τ from the received signal, r(t) is filtered by a

pre-filter, getting

y(t) =∑

k

akv(t− kT + τ) + n(t), (2)

withV (f) = U(f)P (f), (3)

where V (f) (U(f), P (f)) is used to indicate the Fourier trans-form of v(t) (u(t), p(t)), P (f) is the frequency response of thepre-filter whose impulse response is assumed to be real, andn(t) is the complex additive Gaussian noise filtered throughthe pre-filter. In the timing recovery scheme by Gardner [8]the filtered signal y(t) in (2) is sampled at twice the symbolfrequency 2/T and the Timing Error Detector (TED) is givenby

zk = <{(y((k+1)T−τ)−y(kT−τ))y∗(kT +T/2−τ)}, (4)

where τ is the estimate of the parameter τ produced by a PLLscheme based on the phase detector (4), <{·} is the real partof the argument, and ∗ denotes complex conjugation.

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PHOTONICS TECHNOLOGY LETTERS - SUBMISSION 2

We assume in what follows that U(f) is bandlimited toT−1. The sequence zk can be written as

zk = Aρk · sin(

2π(τ − τ)T

+ ψk

), (5)

where ρk and ψk are undesired amplitude and phase modula-tion, and [11]

A =

∣∣∣∣∣4T

∫ 1/T

0

e−jπβT V (β)V ∗(T−1 − β)dβ

∣∣∣∣∣ . (6)

The undesired phase modulation ψk that affects the detectedtiming wave (4) has zero mean, and its power spectral densitycan be analyzed by assuming that the amplitude modulation ρk

is negligible. With this assumption, the power spectral densityof ψk can be computed as the sum of three terms: “Noise ×Noise,” “Noise × Signal,” “Signal × Signal.” Following [11],one finds that, for V (f) and P (f) bandlimited to T−1, thepower spectral densities of “Noise × Noise” and “Noise ×Signal” at frequency zero are

Sn×n =2

∫ 1/T

0|P (β)P (T−1 − β)|2dβ

T · SNR2 ·A2, (7)

Sn×s =4

∫ 1/T

0|V (β)P (T−1 − β)|2dβ

T 2 · SNR ·A2. (8)

It is well known that the term “Signal × Signal” can besuppressed by properly filtering the signal before the TED[12], [13]. In what follows the term “Signal × Signal” isassumed to be negligible, while the other two contributions areassumed to be uncorrelated, leading to the following powerspectral density of the undesired phase modulation ψk atfrequency zero

Sψ = Sn×n + Sn×s. (9)

Example

Let U(f) be the square root of a raised-cosine Nyquist filterwith roll-off α and assume that it has unit energy. When P (f)is the filter matched to U(f) or the pre-filter of [13] one has

A =4 sin(πα/2)π(4− α2)

, (10)

Sψ =απ2(4− α2)2(1 + SNR)

16 sin2(πα2 ) · SNR2 . (11)

III. OPTIMIZATION OF THE LOOP TRANSFER FUNCTION

In the practice, τ is not a parameter, because the local oscil-lators that are used to clock the data converters are affected byphase-noise. We assume that the sampled-time phase-noise atfrequency T−1 is a discrete-time Wiener process whose powerspectral density in the z-domain is

Sθ(z) =γ2

(1− z)(1− z−1), (12)

where θk = 2πτk/T and z−1 indicates a delay of T seconds.1

For instance, the passively mode locked laser oscillator at

1The reader is referred to [3] for formulae that can be used to pass fromthe Lorentzian power spectral density of the continuous time phase noise to(12).

frequency T−1 = 40 GHz of [14] is characterized by γ2 =4 · 10−6, while the active mode-locked laser oscillator of [15]at the same frequency is characterized by γ2 = 10−8. Thetransfer function of the PLL should be designed to minimizethe mean-square phase error

E{φ2k} = E{(θk − θk)2}. (13)

Assuming that the phase modulation ψk is stationary andthat its power spectral density is white and equal to Sψ (atleast in the bandwidth of the PLL, whatever it is), then theoptimal transfer function of the PLL can be worked out bythe standard linear prediction theory, which is based on thespectral factorization

(1 + Q(z))β2(1 + Q∗(z−∗)) = Sθ(z) + Sψ, (14)

where the result of the spectral factorization is the transferfunction 1 + Q(z) that is causal, monic, and minimum phase,and z−∗ indicates the complex conjugate of z−1. One easilyfinds the following results

Q(z) =(1− zp)z−1

1− z−1, (15)

zp = ζ −√

ζ2 − 1, ζ =γ2 + 2Sψ

2Sψ, (16)

and

β2 =Sψ

zp=

γ2

(1− zp)2. (17)

By the analysis of [9] one finds that the low-pass transferfunction of the PLL, that is the transfer function between Θ(z)and Θ(z), that minimizes the mean-square phase error is

H(z) =(1− zp)z−δ

1− zpz−1, (18)

where δ ≥ 1 is the delay in the loop, and (1−zp) is commonlycalled loop gain. The normalized loop noise bandwidth is

BnT =T

2

∫ 1/T

0

|H(ej2πfT )|2df =1− zp

2(1 + zp). (19)

The so-called open-loop transfer function results

G(z) =H(z)

1−H(z)=

(1− zp)z−δ

1− zpz−1 − (1− zp)z−δ, (20)

leading to the block diagram of Fig. 1a. The mean-squarephase error with the constraint of delay δ in the loop obtainedusing the optimal open-loop transfer function (20) is

E{φ2k} =

γ2

1− zp+ γ2(δ − 1). (21)

For zp → 1, that is for small normalized bandwidth of theloop, from the second equality of (17) one gets

1− zp ≈√

γ2/Sψ, (22)

that, substituted in (21), leads to

E{φ2k} ≈

√γ2Sψ + γ2(δ − 1). (23)

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PHOTONICS TECHNOLOGY LETTERS - SUBMISSION 3

Fig. 1. (a) Block diagram of the optimal loop filter transfer function G(z)given by (20). (b) Block diagram of the approximated loop filter transferfunction G(z) given by (24).

Also, for z → 1, that is at low frequency, one can approximatethe optimal G(z) given in (20) as

G(z) =(1− zp)z−δ

1− z−1, (24)

where

1− zp =1− zp

zp + δ(1− zp), (25)

which is easier to implement than (20). Note that, for δ > 1and 0 ≤ zp < 1 one has

1− zp ≥ 1− zp ≥ 0, (26)

hence the proposed approximation leads to a first-order loopwith a loop gain lower than the optimal loop gain with unitdelay. The block diagram of G(z) is shown in Fig. 1b. Thephase margin in degrees is

PMo = 180(

1− 2δf0T − arg(1− e−j2πf0T )π

), (27)

where f0 is such that |G(ej2πf0T )| = 1. Substituting (25) into(24) one realizes that

2δf0T ≤ 1π

, (28)

the upper bound being achieved as δ → ∞. Sincearg(1− e−j2πf0T ) ≤ π/2 one concludes that

PMo ≥ 180(

π − 22π

)> 32, (29)

hence the proposed approximation to the optimal transferfunction guarantees a good phase margin for all the valuesof zp and δ. Figure 2 shows the phase margin of G(z) as afunction of γ2/Sψ computed by (27) for different values ofthe delay.

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

30

40

50

60

70

80

90

γ2/Sψ

PM

°

n=0

n=1

n=2

n=3

n=5n=4

n=6

Fig. 2. Phase margin in degrees versus γ2/Sψ with δ as a parameter. Fromthe upper to the lower line δ = 10n, n=0,1,. . . ,6.

16 18 20 22 24 26 28 30 32 3410

−5

10−4

10−3

10−2

10−1

100

BE

R

SNR (dB)

AWGNG(z) comp w/o CD

G(z) comp w/o CDG(z) sim w/o CD

G(z) sim w/o CDG(z) sim w CD

G(z) sim w CD

Fig. 3. Simulated (sim) and computed (comp) BER vs. SNR for 256-QAM with γ2 = 4 · 10−6 and δ = 800 using the optimal G(z) andapproximated G(z) with (w) and without (w/o) residual CD. The performancefor transmission over the AWGN channel with known τ is also shown forreference.

IV. SIMULATION RESULTS

In all the simulations we adopt as a transmit filter a squareroot Nyquist filter with roll-off α = 0.3, the pre-filter proposedin [13], the optimal G(z) and the approximated G(z) for thetransfer function of the loop filter.

Figure 3 reports the BER for 256-QAM with γ2 = 4 ·10−6,that characterizes oscillators with large phase noise bandwidthand symbol frequency of 40 GHz [14], and δ of 800 symbols,that is indicated in [2] as a realistic value of the loop delay. Thecomputed BER is obtained by using the mean-square phaseerrors associated with the optimal G(z) and the approximatedG(z) as described in [3], where the sum of channel noise andInterSymbol-Interference (ISI) caused by non-ideal samplingis approximated with a Gaussian random variable in eq. (25)of [3]. The BER with AWGN and known τ is also reported asa reference. The results of Figure 3 show that the performancewith the two filters deviate from the performance predicted bythe analysis at BER lower than 10−3 and that, as expected,the optimal G(z) outperforms G(z). Figure 3 also shows theBER with residual Chromatic Dispersion (CD). These results

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PHOTONICS TECHNOLOGY LETTERS - SUBMISSION 4

100

101

102

103

104

105

106

107

0

0.2

0.4

0.6

0.8

1

δ

SN

R P

enal

ty (

dB)

G(z) comp w/o CD

G(z) comp w/o CDG(z) sim w/o CD

G(z) sim w/o CDG(z) sim w CD

G(z) sim w CD16−QAM

256−QAM

16−QAM

256−QAM

γ2=4⋅ 10−6

γ2=10−8

Fig. 4. Comparison between computed (comp) and simulated (sim) SNRpenalty versus delay δ at BER = 4 · 10−3 for 16-QAM and 256-QAM withγ2 = 4 · 10−6 and γ2 = 10−8 with (w) and without (w/o) residual CD.

have been obtained with λ = 1.55 µm, symbol frequencyof 40GHz, and residual CD of 50ps/nm which, from Fig.4 of [16], seems to be a typical value. It is worth notingthat the results reported in the Figure take into account onlythe ISI due to non-ideal timing recovery, the residual ISIdue to residual CD being assumed to be ideally removed byan equalizer that follows timing recovery. Therefore, in thecomputer simulation, the BER is obtained from a Nyquist-shaped signal sampled by a timing instant that is extractedfrom a signal affected by CD. It appears that the performanceof the two loop filters with CD are closer compared to the casewithout CD, thus indicating that, with CD, the performanceof timing recovery is more affected by CD and less sensitiveto the transfer function of the loop filter.

The performance of timing recovery can also be assessedby the SNR penalty with respect to transmission over the pureAWGN channel. The SNR penalty is analyzed as describedin [3], and the analysis is compared to the simulation results.Figure 4 shows the SNR penalty versus loop delay for 16-QAM and 256-QAM for γ2 = 4 · 10−6 and γ2 = 10−8

at BER = 4 · 10−3. The BER value of 4 · 10−3 is atypical input BER for hard decision decoding of forward errorcorrecting codes based on continuously-interleaved versionsof Bose-Chaudhuri-Hocquenghem codes, see Table 1 of [17].Simulation results fairly fit the analysis, except for 256-QAMwith CD for δ = 10 and δ = 100, where the performance oftiming recovery is dominated by the residual CD, not by theloop delay.

V. CONCLUSION

The performance of the Gardner timing recovery with delayin the loop and with phase noise affecting the local oscillatorsused for clocking data converters has been analyzed. Theoptimal loop filter and the analytical expression that givesthe corresponding mean-square phase error have been derived.Since the resulting optimal transfer function can be difficultto implement, a suboptimal transfer function that is easierto implement has been introduced. It has been shown in the

letter that simulation results closely fit the analysis when localoscillators with linewidths of practical interest are considered,and that the performance of the suboptimal transfer functionis close to that of the optimal one in many cases of practicalinterest.

REFERENCES

[1] H. Sun and K.-T. Wu, “Clock recovery and jitter sources in coherenttransmission systems,” in Proc. OFC/NFOEC, Los Angeles, CA, 2012,pp. 1-3, paper OTh4C.1.

[2] D. Zibar, A. Bianciotto, Z. Wang, A. Napoli, and B. Spinnler, “Analysisand dimensioning of fully digital clock recovery for 112 Gb/s coherentpolmux QPSK systems,” in Proc. Eur. Conf. Exhibit. Opt. Comm., 2009,pp. 1–2, paper 7.3.4.

[3] L. Barletta, R. Disaro, M. Magarini, and A. Spalvieri, “Post-filteroptimization in timing recovery based on square-law detection,” IEEEPhoton. Technol. Lett. vol. 25, no. 9, pp. 821–824, May 1, 2013.

[4] M. Oerder and H. Meyr, “Digital filter and square timing recovery,”IEEE Trans. Commun., vol. 36, no. 5, pp. 605–612, May 1988.

[5] S. J. Lee, “A new non-data-aided feedforward symbol timing estimatorusing two samples per symbol,” IEEE Commun. Lett., vol. 6, no. 5, pp.205-207, May 2002.

[6] Y. Wang, E. Serpedin, and P. Ciblat, “An alternative blind feedforwardsymbol timing estimator using two samples per symbol,” IEEE Trans.Commun., vol. 51, no. 9, pp. 1451–1455, Sept. 2003.

[7] D. Wang, A. P. T. Lau, C. Lu, and S. He, “Optimization of a feedforwardsymbol timing estimator using two samples per symbol for optical coher-ent QPSK systems,” in Proc. Asia Commun. Photon. Conf., Guangzhou,P. R. China, 2012, pp. 1-3, paper ATh1C.7.

[8] F. M. Gardner, “A BPSK/QPSK timing error detector for sampledreceivers,” IEEE Trans. Commun., vol. 34, pp. 423–429, May 1986.

[9] A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-timephase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp.Briefs, vol. 55, pp. 596–600, June 2008.

[10] A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presenceof phase noise,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1966–1974,July 2011.

[11] A. N. D’Andrea and U. Mengali, “Performance analysis of the delay-line clock regenerator,” IEEE Trans. Commun., vol. 34, pp. 321–328,April 1986.

[12] A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-freeclock recovery scheme for QAM systems,” IEEE Trans. Commun., vol.41, pp. 1296–1299, Sept. 1993.

[13] A. Spalvieri and M. Magarini, “Minimum timing jitter in timing recoverybased on pre-filtered square-law timing detection,” in Proc. 2nd CTRQ,Colmar, France, 2009, pp. 46–51.

[14] C.-Y. Lin, F. Grillot, Y. Li, R. Raghunathan, and L. F. Lester, “Charac-terization of timing jitter in a 5 GHz quantum dot passively mode-lockedlaser,” Opt. Express, vol. 18, no. 21, pp. 21932–21937, Sept. 2010.

[15] K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenløwe, J.Mørk, D. Birkedal, J. M. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Tech. Lett.,vol. 16, no. 4, pp. 975–977, Apr. 2004.

[16] F. N. Hauske, M. Kuschnerov, B. Spinnler, and B. Lankl, “Optical per-formance monitoring in digital coherent receivers,” J. Lightw. Technol.,vol. 27, no. 16, pp. 3623–3631, Aug. 2009.

[17] F. Chang, K. Onohara, and T. Mizuochi, “Forward error correction for100 G transport networks,” IEEE Commun. Mag., vol. 48, no. 3, pp.S48–S55, March 2010.