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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6749–6763
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Performance analysis of blind timing phase estimators for digital coherent receivers

Lingchen Huang, Dawei Wang, Alan Pak Tao Lau, Chao Lu, and Sailing He  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6749-6763 (2014)
http://dx.doi.org/10.1364/OE.22.006749


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Abstract

The performances of various blind timing phase estimators (TPE) for digital coherent receiver are analyzed. The equivalence among four TPE algorithms is analytically presented, showing that two TPE algorithms applying squaring pre-filters are in fact identical. Three TPE algorithms applicable to Nyquist signals are proposed based on the equivalence analysis. In addition, the impact of receiver bandwidths, spectrum weighting bandwidths and signal timing phases on TPE performance are investigated. The definition of sampling diversity and the analysis of sampling diversity gain for four pulse shapes are presented. The effect of sampling diversity is observed and verified via both simulations and experiments.

© 2014 Optical Society of America

1. Introduction

Digital coherent receiver is a crucial part of an optical coherent transmission system, enabling techniques such as high-order modulation format and dual polarization multiplexing [1

1. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef] [PubMed]

, 2

2. M. Kuschnerov, F. Hauske, K. Piyawanno, B. Spinnler, M. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27(16), 3614–3622 (2009). [CrossRef]

]. In such receivers, fast and accurate acquisition of the timing phase is a critical issue for both long-haul transmission and short range access networks [3

3. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express 19(6), 5611–5619 (2011). [CrossRef] [PubMed]

6

6. N. Stojanovic, F. N. Hauske, C. Xie, and M. Chen, “Clock recovery in coherent optical receivers,” in Proceeding Photonic Networks 12.ITG Symposium, Leipzig, Germany (2011), Paper 9.

]. Specifically, after chromatic dispersion (CD) compensation, the timing phase error of the sampled signal should be correctly estimated and compensated prior to the subsequent DSP algorithms, in order to produce samples with optimal signal-to-noise ratio (SNR) and phase representation. The most common method to achieve this involves using a timing phase estimator (TPE) in conjunction with a digital interpolator. Various algorithms of TPE have been proposed for classical wireless communications and subsequently adopted in optical coherent systems, such as the well-known “square-and-filter” proposed by Oerder & Meyr [7

7. M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988). [CrossRef]

] (also known as the square law of nonlinearity (SLN)), and those proposed by Gardner [8

8. F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986). [CrossRef]

], Godard [9

9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]

] and Lee [10

10. S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002). [CrossRef]

]. The “O&M” requires at least three samples per symbol (SPS) in principle (usually takes four (4SPS SLN) for practical reasons) while the others require two. Constrained by the sampling speed of analog-digital convertors (ADC), however, the signal is commonly digitalized with a two-fold oversampling rate. Consequently, the signal samples have to be converted up from 2SPS to 4SPS via digital interpolation before entering the SLN TPE. The SLN and Lee TPE can be applied with a feedforward structure as shown in Fig. 1(a)
Fig. 1 Block diagram of the (a) feedback and (b) feedforward timing phase recovery (TPR) for an asynchronous coherent optical receiver.
, whereas Gardner’s and Godard’s TPEs are commonly running within a phase lock loop (PLL) as shown in Fig. 1(b). Note that Godard’s TPE can also be realized as in Fig. 1(a) if the timing phase estimation is performed alternatively to its original mode [9

9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]

], as adopted in this paper. On the other hand, to implement a real-time processor, parallelization has to be introduced as the symbol’s rate of signal is normally much higher than the clocking speed of silicon devices. Therefore, the feedforward estimators are intuitively appealing as they process data block-by-block and incur no feedback delay. Note that, throughout the paper, the all-digital implementation of timing phase recovery is assumed as shown in Fig. 1, meaning that the estimated timing phases are not used to adjust the ADC but to drive a digital interpolator. Consequently, the ADC will run with a nominal sampling rate continuously without locking to the optimal sampling phase, and the signal is recovered by digital interpolation based on the original signal and estimated timing phases.

The aforementioned estimators are considered as the basic algorithms and are well studied in the literature. Additionally, a variety of modified algorithms and techniques have been reported to improve the TPE performance under certain conditions. For instance, pre-filtering is introduced to SLN and Lee TPE to improve their performance at medium and high SNR [11

11. K. Shi, Y. Wang, and E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004). [CrossRef]

]. An alternative algorithm is reported to remove the asymptotic bias of Lee TPE [12

12. Y. Wang, E. Serpedin, and P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003). [CrossRef]

]. In addition to SLN, absolute value nonlinearity (AVN) [13

13. E. Panayirci and E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996). [CrossRef]

], fourth law nonlinearity (FLN) [14

14. T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991). [CrossRef]

], and logarithm nonlinearity (LOGN) [15

15. M. Morelli, A. N. D’Andrea, and U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997). [CrossRef]

] are also utilized to estimate the timing phase. Moreover, Sun et al contributed to improving the TPE tolerance to the residual CD and PMD by compensating CD and polarization rotation prior to the Godard’s TPE [16

16. H. Sun and K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5. [CrossRef]

].

With so many estimators, it is imperative to clarify their pros and cons and their applicability to different scenarios. In this paper, we limit ourselves to the class of 2SPS TPEs including Godard’s, Gardner’s, and Lee’s TPEs as well as the interpolation-based SLN. We investigate and compare their performance via computer simulations and experiments. By inspecting the frequency domain (FD) expression, we analytically show that the seemingly different algorithms are all equivalent with an explicable difference, thus leaving us greater confidence to focus more on the implementation issues. By applying this result, we propose three additional TPE algorithms applicable to the Nyquist signals. In addition, the effect of “sampling diversity” is demonstrated and the underlying principle is presented, i.e., performance gain can be obtained by over-sampling for signals with “flat” pulse shapes.

The rest of the paper is organized as follows: in Section 2, the equivalence and difference of various TPE is analytically presented and their computational complexities are compared. In Section 3, computer simulations are carried out to compare the timing jitter performance of these algorithms. The results are discussed and the oversampling diversity is investigated. Section 4 presents experimental results and the paper is concluded in Section 5.

2. Principle of different TPE and their equivalence

The expressions of the TPE algorithms considered in this paper are listed in the left column of Table 1

Table 1. Algorithms of timing phase estimator

table-icon
View This Table
. The parameter definitions are as follows: the data length is N during the observation window, xn denotes the nth sample entering the estimator, Xk is the kth element of the discrete Fourier transform (DFT) of the data block, and ε^ = τ/T∈(−0.5,0.5] is the estimation of the normalized symbol timing offset, where τ is the time delay from the optimal symbol timing instant and T denotes the symbol period. For SLN, a two-fold oversampling ratio is assumed such that the signal samples xn should be interpolated from 2SPS to 4SPS x'n by padding zeros in the frequency domain.

In order to prove the equivalence among these TPE algorithms, the frequency domain expressions are derived as following according to three properties [16

16. H. Sun and K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5. [CrossRef]

, 17

17. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

]:
n=0N1xnyn=1Nk=0N1XkYk
(1)
DFT[x2n]=Xk+Xk+N/22,n,k=0,1,...,N/21
(2)
DFT[x2n+1]=ej2πk/NXkXk+N/22,n,k=0,1,...,N/21
(3)
whereXkandYkare the DFTs ofxnandyn. Equations (2) and (3) are for down-sampling a T/2-spaced signal to a T-spaced signal in the frequency domain.

(I) Derivation of the frequency domain expression of SLN

ε^SLN=12πarg{n=02N1|xn|2ejπn/2}=12πarg{n=02N1xn(xnej2πn(N/2)/(2N))*}substitutingEq.(1)=12πarg{k=02N1Xk(XkN/2)*},
(4)

wherexnis the 4SPS sequence andXkis the DFT of it. As 2SPS is assumed for the received sequence xn, the frequency domain interpolation is conducted to up-sample the signal. Thus, we have
Xk|k=0,1,...,N/21={Xk|k=0,1,...,N/21,k=0,1,...,N/210,k=N/2,N/2+1,...,3N/21Xk|k=N/2,N/2+1,...,N1,k=3N/2,3N/2+1,...2N1.
Substituting it to Eq. (1) yields

ε^SLN=12πarg{k=0N/21Xk(Xk+N/2)*}=ε^Godard.
(5)

(II) Derivation of the frequency domain expression of Gardner:

ε^Gardner=Re{n=0N/21(x2nx2n+2)x2n+1}=Re{n=0N/21(x2nx2n+1x2n+2x2n+1)}substitutingEq.(1),(2)and(3)=2Nk=0N/21{sin(2πNk)Im(XkXk+N/2*)}
(6)

(III) Derivation of the frequency domain expression of Lee

ε^Lee=12πarg{n=0N1|xn|2ejnπ+n=0N1Re{xnxn+1}ej(n0.5)π}=12πarg{n=0N1|xn|2(1)n+jn=0N1Re{xnxn+1}(1)n}=12πarg{n=0N/21[|x2n|2|x2n+1|2]+jn=0N/21[Re{x2nx2n+1}Re{x2n+1x2n+2}]}=12πarg{ε^D+jε^Gardner},
(7)

where
ε^D=n=0N/21[|x2n|2|x2n+1|2]=n=0N/21|x2n|2n=0N/21|x2n+1|2substitutingEq.(1),(2)and(3)=2NRe{k=0N/21XkXk+N/2}.
(8)
By substituting Eq. (6) and Eq. (8) into Eq. (7), we get
ε^Lee=12πarg{k=0N/21[Re(XkXk+N/2)+jsin(2πNk)Im(XkXk+N/2*)]}.
(9)
Equations (5), (6) and (9) are listed in the right column of Table 1.

It is clear that, unlike the 4SPS SLN [7

7. M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988). [CrossRef]

], 2SPS TPEs explore the auto-correlation (AC) between the positive and negative signal band centered at a frequency of ± fb/2, respectively. From Table 1, the following conclusions can be drawn: (i) the interpolation-based SLN is equivalent to that of Godard’s, provided that FD zero padding is adopted; (ii) Gardner’s TPE is approximately equivalent to the imaginary part of Godard’s or to Godard’s original TPE in [9

9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]

]; (iii) Lee’s TPE is approximately equivalent to Godard’s and consists of Gardner’s as the imaginary part; (iv) the difference between Gardner’s and Godard’s original TPEs [9

9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]

] lies in the weighting of spectral auto-correlation, i.e., whether it is a sinusoidal or rectangular multiplicative window, as shown in Fig. 2(a)
Fig. 2 (a) Illustration of the different weighting of spectra auto-correlations for Gardner’s and Godard’s original TPE. (b) S-curves of Godard’s, the SLN, Gardner’s and Lee’s TPE.
for a 28GBaud/s NRZ-QPSK signal.

The estimator characteristics (S-curves) of the various TPEs are shown in Fig. 2(b), using 256 symbols with 2SPS of an NRZ-QPSK signal for each point. The timing phase ranges within a unit interval (UI), i.e., the symbol period. It can be seen that the S-curves of the SLN and Godard’s completely overlap and are slightly biased off the true value, similarly to Lee’s, which was pointed out by Wang et al [12

12. Y. Wang, E. Serpedin, and P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003). [CrossRef]

]. The S-curve of Gardner’s TPE exhibits a sinusoidal shape which is consistent with the fact that it is approximately equal to the imaginary part of Godard’s TPE. Based on this, one can expect close estimation accuracy among these algorithms. In addition, any pre-filtering claimed to enhance the estimation performance of one TPE should be equally applicable to others. One example is the equivalence between two TPEs put forward in [18

18. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7. [CrossRef]

, 19

19. N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898. [CrossRef]

]. The proof is derived as following.

(IV) Derivation of equivalence of TPEs for Nyquist system

The TPE in [18

18. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7. [CrossRef]

] is to apply Gardner’s TPE on the power received sequence,
ε^P=n=0N/21p2n(p2n+1p2n1)
(10)
where pn = |xn|2. Noticing the similarity betweenε^Gardnerandε^P, Eq. (10) can also be expressed in the frequency domain as
ε^P=2Nk=0N/21{sin(2πNk)Im(PkPk+N/2*)},
(11)
where Pk is the DFT of pn.

The other TPE is a 2SPS-based fourth power phase detector (4PPD) put forward in [19

19. N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898. [CrossRef]

]. The TPE is expressed as
ε^4PPD=Im{k=0N/21PkPk+N/2*}
(12)
where P = fft(xx*), which is to apply Godard’s TPE on pn = |xn|2. Thus, the equivalence is explicit except for spectrum weighting, as pointed out in Fig. 2(a), as well as a constant scale factor. Furthermore, by utilizing the equivalence among the TPEs we prove previously, another three feedforward based TPEs can be deduced to work on the Nyquist signals by applying Godard’s, Lee’s and interpolation based SLN TPEs on the power of received sequence:
ε^Godard,P=12πarg{k=0N/21PkPk+N/2*}
(13)
ε^SLN,P=12πarg{n=02N1|xn|4ejnπ/2}
(14)
ε^Lee,P=12πarg{n=0N1|xn|4ejnπ+n=0N1|xn|2|xn+1|2ej(n0.5)π}
(15)
where P = fft(xx*) is the DFT of power sequence of the received signal samples.

In terms of computational complexity, exponential functions in TPE calculations are simply reduced to 1, −1, j, and -j, and the computational complexity of addition and subtraction is insignificant relative to that of multiplication. In this case, the computational complexity is measured with the number of real multiplications consumed per symbol. Note that, for Godard’s TPE, additional FFT is required if the sample sequences are in the time domain, and for the SLN, interpolation from 2SPS to 4SPS is needed. Nevertheless, assuming the corresponding sample sequences are available for each TPE algorithm, it requires 4 real multiplications in each symbol for Godard’s, and 16 for the SLN, 2 for Gardner’s and 12 for Lee’s.

3. Simulation and discussion

We conduct simulations to corroborate the theoretical analysis. The simulation system is set up with no frequency offset, laser phase noise or fiber impairments except for random polarization rotation. The receiver bandwidth is 21 GHz. 217 symbols were used to calculate the BER for 28Gbaud/s PDM-NRZ-QPSK, 16-QAM and 64-QAM systems. The results are averaged over 10 independent trials. The loop bandwidth of the TPR, normalized to baud rate, is set to be 2E-3, i.e., the block-size for feedforward algorithms are 512 samples and the step-size for feedback algorithms are 2.9e-3. The bandwidth of the loop determines the quality of timing estimate. A narrowband loop provides more averaging over the additive noise and improves the quality of the estimate, whereas an increased bandwidth of loop is needed to provide better tracking performance if the channel response is changing and/or the transmitter clock is drifting with time [20

20. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)

]. After TPR, CMA equalization is adopted for polarization de-multiplexing and for equalizing the filtering effect caused by the linear interpolation process of resampling. One additional stage of the cascaded multi-modulus algorithm (CMMA) is used in the cases of 16-QAM and 64-QAM to achieve better convergence compared to CMA.

First, we use timing jitter in decibel (dB) as the metric to quantify the TPE accuracy. The normalized jitter variance is defined as the variance of zero-crossing positions of the TPE S-curve, i.e. jitter = 20log10(δt), where δt is the standard deviation of the zero-crossing positions normalized to symbol period [18

18. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7. [CrossRef]

]. Figure 3
Fig. 3 S-curves and zero-crossing phase histograms of (a) Gardner’s, (b) Godard’s, (c) SLN and (d) Lee’s TPE.
shows various TPE S-curves produced by scanning the sampling phases on bursts of received NRZ-QPSK signal with a block size of 512 samples at a 10dB optical signal-to-noise ratio (OSNR). The histograms of zero-crossing phases of the corresponding algorithms are also plotted. Timing jitter variance can be calculated and plotted against OSNR in 0.1 nm in Fig. 4
Fig. 4 Normalized jitter variance vs OSNR (in 0.1nm) for NRZ-QPSK, 16-QAM and 64QAM signals.
, together with the Modified Cramer Rao Bound (MCRB) [21

21. N. A. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994). [CrossRef]

]. The OSNR ranges for QPSK, 16-QAM and 64-QAM are 9-15dB, 16-22dB and 16-29dB, respectively. Clearly, the curves of Godard’s and the SLN overlap for all three modulation formats considered, and so are Gardner’s and Lee’s, though they exhibit about 1 dB lower jitter than the other two algorithms. This should be attributed to their sinusoidal weighting of the spectrum auto-correlation as pointed out in Fig. 2(a). In addition, the jitter performance seems insensitive to modulation format.

The evolution curves of timing phase estimation are shown in Figs. 6(a)
Fig. 6 Evolution of timing phase estimation for (a) feedback timing recovery using Gardner’s TPE and (b) feedforward timing recovery using Godard’s TPE.
and 6(b) for the feedback and feedforward schemes, respectively. The step-size of feedback loop is 2.9e-3 and the block-size of feedforward loop is 512 samples, both at a 10dB OSNR. No sampling frequency drift is considered for this result. Compared to the feedforward scheme, there is an additional acquisition stage for the feedback scheme, consuming about 250 symbols in this case. Note that instead of locking the ADC to the optimal strobe points, which will lead to a near zero timing phase upon the loop convergence, the digital feedback loop essentially lock the timing phase estimation to its corresponding real value upon convergence, as shown for sampling phases of + 0.25, 0 and −0.25 in Fig. 6.

The OSNR penalty at BER = 3.8E-3 (2E-2) with respect to the sampling phase for the PDM-NRZ-QPSK (16-, 64-QAM) system is shown in Fig. 7
Fig. 7 OSNR penalty with respect to sampling phase for 28GBaud/s (a) QPSK, (b) 16QAM and (c) 64QAM system.
. The reference OSNR is calculated according to the theoretical equation [20

20. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)

]. The “1SPS” stands for the case of down-sampling the received signal to symbol rate directly without applying any TPE algorithm. Clearly, the impact of different TPE algorithms on system performance is hardly distinguishable, mostly because the subsequent adaptive equalizer could effectively remove the residual timing error. In fact, Kikuchi has demonstrated that the fractionally spaced adaptive equalizer is capable of compensating the timing phase error even without using the TPE [3

3. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express 19(6), 5611–5619 (2011). [CrossRef] [PubMed]

]. On the other hand, it is observed that all the scenarios for two-fold sampling exhibit OSNR penalty reduction compared to the optimal condition of “1 SPS” case. This is something we called the effect of “sampling diversity”. It can be defined as using two or more received samples per symbol to improve the quality and reliability of a communication link. And the “sampling diversity gain” is defined as the OSNR penalty reduction from the optimal condition of “1 SPS” case, as is indicated in Fig. 7.

The sampling diversity is the results of signal oversampling and can be comprehended by analogy with the classical receiver diversity in wireless communications [20

20. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)

], where two or more antennas are employed to receive multiple copies of signal with uncorrelated noise for each antenna, leading to a signal-to-noise ratio (SNR) gain for the combined signal. Assuming the digital coherent receiver applies two-fold oversampling to the signal and the TPE applies linear interpolation to combine the two samples, the SNR of the received signal can be expressed as:
SNR=E[(1μ)s1+μs2]2(1μ)2n12+μ2n22.
(16)
wheres1ands2are the amplitude of the first and second signal sample in one symbol, respectively.n1andn2are the amplitude of the first and second noise sample in one symbol, respectively.μ=2ε^2ε^is the interpolation interval andε^is the timing phase estimation. Note that the noise samples are uncorrelated so thatn12=n22=n2. The SNR is further expressed as:

SNR=E[(1μ)s1+μs2]2(12μ+2μ2)n2.
(17)

Conclusion can be drawn from (17) that (c1) for cases of μ = 0 (ε^=0) and μ = 1 (ε^=0.5), the SNR will be the same as that in “1SPS” case with corresponding sampling phase; (c2) The SNR is maximized when μ is 0.5 (ε^=±0.25), provided that s1 = s2 = s for all possible sampling phases, i.e., the signal is perfectly “flat”. The SNR in this case will beSNR2sps=E[2s2/n2]=2SNR1sps, i.e., the two-fold oversampling in conjunction with linear interpolation will gain 3dB on the SNR at sampling phase of±0.25for such signal. The in-phase/quadrature part of a NRZ-QPSK signal with infinite bandwidth suits this case and can be expressed as:
sNRZ(t)=PmCmrect(tmT)
(18)
where P is the average signal power, C belongs to {-1, 1}, and
rect(t)={1,0,0.5Tt0.5Totherwise
(19)
Substituting (18) and (19) to (17) yields

SNRNRZ=Pn21(12|ε|)2+(2|ε|)2
(20)

Similar to the analysis of NRZ signal, the SNR expressions of RZ 33%, RZ 50% and RZ 67% signal can be written as
SNRRZ33=Pn2[(12|ε|)cos(π2sin(π|ε|))+2|ε|cos(π2sin(π(0.5|ε|)))]2(12|ε|)2+(2|ε|)2
(21)
SNRRZ50=Pn2[(12|ε|)sin[π4cos(2π|ε|)+π4]+2|ε|sin[π4cos(2π(0.5|ε|))+π4]]2(12|ε|)2+(2|ε|)2
(22)
and

SNRRZ67==Pn2[(12|ε|)sin(π2cos(π|ε|))+2|ε|sin(π2cos(π(|ε|+0.5)))]2(12|ε|)2+(2|ε|)2
(23)

4. Experiment and discussion

The simulation results were substantiated by experiments. The experimental setup is depicted in Fig. 12
Fig. 12 Block diagram of the experiment setup.
. The 10 Gbaud/s QPSK signal was generated by an I/Q modulator, driven by electrical signals synthesized via a pulse pattern generator (PPG) with 215-1 PRBS. At the receiver, ASE noise was added to the signal to set the OSNR. Then the combined signal was sufficiently detected and converted to the baseband electrical signal. A digital storage oscilloscope (DSO) is used to capture the signal at 50 G samples/s followed by offline DSP. The digital signal was first up-sampled to 10SPS to offer 10 effective sampling phases. Then, the data sequence was down-sampled to 2SPS and processed successively by timing phase recovery (TPR), adaptive equalization (AEQ) with constant modulus algorithm (CMA), carrier phase recovery (CR) with Viterbi-Viterbi phase estimation algorithm, data recovery (DR) and bit error counting.

First, the 10SPS signal is utilized to produce S-curves and to calculate the jitter variance of the TPE. The S-curves at 8.3 dB OSNR are shown in Fig. 13(a)
Fig. 13 (a) S-curves of (i) Gardner’s, (ii) Godard’s, (iii) SLN and (iv) Lee’s TEDs; (b) normalized jitter variance with respect to OSNR at 0.1nm resolution
, associated with the measured jitter variance as shown in Fig. 13(b). Similar to the simulation results, Gardner’s and Lee’s exhibit about 1 dB lower jitter variance than Godard’s and the SLN at the same OSNR. The BER curves are plotted in Fig. 14(a)
Fig. 14 (a) BER versus OSNR and (b) Required OSNR with respect to sampling phase for the four TPE algorithms.
, showing that ~0.8 dB OSNR gain at BER = 1E-3 can be obtained by 2SPS oversampling diversity. The OSNR penalty at BER = 3.8E-3 with respect to sampling phases are shown in Fig. 14(b), which agrees well with simulation.

5. Conclusions

In this paper, the performances of four timing phase estimator algorithms are analytically investigated. By expressing them in the frequency domain, we proved that the four TPEs are all actually equivalent with a slight but yet explicable difference. Thus, the four algorithms are interchangeable, subject to their applicable structures and computational complexity. The impact of the bandwidth of the pre-filter to the timing phase estimation accuracy is presented. The performance dependence on the sampling phase is observed and can be explained with the effect of “sampling diversity”. The analysis has been verified by simulated and experimental results.

Acknowledgments

The work is partially supported by the Program of Zhejiang Leading Team (2010R50007) of Science and Technology Innovation (the Science and Technology Department of Zhejiang Province). The authors would also like to acknowledge the generous support of Hong Kong Polytechnic University under project G-YN28.

References and links

1.

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef] [PubMed]

2.

M. Kuschnerov, F. Hauske, K. Piyawanno, B. Spinnler, M. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27(16), 3614–3622 (2009). [CrossRef]

3.

K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express 19(6), 5611–5619 (2011). [CrossRef] [PubMed]

4.

H. Sun and K. Wu, “Clock recovery and jitter sources in coherent transmission systems,” in Proceedings OFC/NFOEC, Los Angeles, CA (2012), Paper OTh4C.1. [CrossRef]

5.

T. Tanimura, T. Hoshida, S. Oda, H. Nakashima, M. Yuki, Z. Tao, L. Liu, and J. C. Rasmussen, “Digital clock recovery algorithm for optical coherent receivers operating independent of laser frequency offset,” in Proceeding ECOC, Brussels, Belgium (2008), Paper Mo.3.D.2. [CrossRef]

6.

N. Stojanovic, F. N. Hauske, C. Xie, and M. Chen, “Clock recovery in coherent optical receivers,” in Proceeding Photonic Networks 12.ITG Symposium, Leipzig, Germany (2011), Paper 9.

7.

M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988). [CrossRef]

8.

F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986). [CrossRef]

9.

D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]

10.

S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002). [CrossRef]

11.

K. Shi, Y. Wang, and E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004). [CrossRef]

12.

Y. Wang, E. Serpedin, and P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003). [CrossRef]

13.

E. Panayirci and E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996). [CrossRef]

14.

T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991). [CrossRef]

15.

M. Morelli, A. N. D’Andrea, and U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997). [CrossRef]

16.

H. Sun and K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5. [CrossRef]

17.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

18.

M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7. [CrossRef]

19.

N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898. [CrossRef]

20.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)

21.

N. A. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994). [CrossRef]

22.

L. Franks and J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974). [CrossRef]

23.

N. A. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993). [CrossRef]

24.

R. Kudo, T. Kobayashi, K. Ishihara, Y. Takatori, A. Sano, and Y. Miyamoto, “Coherent optical single carrier transmission using overlap frequency domain equalization for long-haul optical systems,” J. Lightwave Technol. 27(16), 3721–3728 (2009). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Optical Communications

History
Original Manuscript: January 31, 2014
Revised Manuscript: March 5, 2014
Manuscript Accepted: March 6, 2014
Published: March 14, 2014

Citation
Lingchen Huang, Dawei Wang, Alan Pak Tao Lau, Chao Lu, and Sailing He, "Performance analysis of blind timing phase estimators for digital coherent receivers," Opt. Express 22, 6749-6763 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6749


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References

  1. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef] [PubMed]
  2. M. Kuschnerov, F. Hauske, K. Piyawanno, B. Spinnler, M. Alfiad, A. Napoli, B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27(16), 3614–3622 (2009). [CrossRef]
  3. K. Kikuchi, “Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers,” Opt. Express 19(6), 5611–5619 (2011). [CrossRef] [PubMed]
  4. H. Sun and K. Wu, “Clock recovery and jitter sources in coherent transmission systems,” in Proceedings OFC/NFOEC, Los Angeles, CA (2012), Paper OTh4C.1. [CrossRef]
  5. T. Tanimura, T. Hoshida, S. Oda, H. Nakashima, M. Yuki, Z. Tao, L. Liu, and J. C. Rasmussen, “Digital clock recovery algorithm for optical coherent receivers operating independent of laser frequency offset,” in Proceeding ECOC, Brussels, Belgium (2008), Paper Mo.3.D.2. [CrossRef]
  6. N. Stojanovic, F. N. Hauske, C. Xie, and M. Chen, “Clock recovery in coherent optical receivers,” in Proceeding Photonic Networks 12.ITG Symposium, Leipzig, Germany (2011), Paper 9.
  7. M. Oerder, H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun. 36(5), 605–612 (1988). [CrossRef]
  8. F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986). [CrossRef]
  9. D. Godard, “Passband timing recovery in an all-digital modem receiver,” IEEE Trans. Commun. 26(5), 517–523 (1978). [CrossRef]
  10. S. Lee, “A new non-data-aided feedforward symbol timing estimator using two samples per symbol,” IEEE Commun. Lett. 6(5), 205–207 (2002). [CrossRef]
  11. K. Shi, Y. Wang, E. Serpedin, “On the design of a digital blind feedforward, nearly jitter-free timing-recovery scheme for linear modulations,” IEEE Trans. Commun. 52(9), 1464–1469 (2004). [CrossRef]
  12. Y. Wang, E. Serpedin, P. Ciblat, “An alternative blind feedforward symbol timing estimator using two samples per symbol,” IEEE Trans. Commun. 51(9), 1451–1455 (2003). [CrossRef]
  13. E. Panayirci, E. K. Bar-Ness, “A new approach for evaluating the performance of a symbol timing recovery system employing a general type of nonlinearity,” IEEE Trans. Commun. 44(1), 29–33 (1996). [CrossRef]
  14. T. T. Fang, “Analysis of self-noise in a fourth-power clock regenerator,” IEEE Trans. Commun. 39(1), 133–140 (1991). [CrossRef]
  15. M. Morelli, A. N. D’Andrea, U. Mengali, “Feedforward ML-based timing estimation with PSK signals,” IEEE Commun. Lett. 1(3), 80–82 (1997). [CrossRef]
  16. H. Sun, K. Wu, “A novel dispersion and PMD tolerant clock phase detector for coherent transmission systems,” in Proceeding OFC/NFOEC, Los Angeles, CA (2011), SPWC5. [CrossRef]
  17. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. (Prentice Hall, 1999).
  18. M. Yan, Z. Tao, L. Dou, L. Li, Y. Zhao, T. Hoshida, and J. Rasmussen, “Digital Clock Recovery Algorithm for Nyquist Signal,” in Proceeding OFC/NFOEC, Anaheim, CA (2013), paper OTu2I.7. [CrossRef]
  19. N. Stojanovic, N. G. Gonzalez, C. Xie, Y. Zhao, B. Mao, J. Qi, and L. M. Binh, “Timing recovery in nyquist coherent optical systems,” in Proceeding 20th Telecommunications Forum (TELFOR), Serbia, Belgrade (2012), pp. 895–898. [CrossRef]
  20. J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw Hill, 2007)
  21. N. A. D’Andrea, U. Mengali, R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(2–4), 1391–1399 (1994). [CrossRef]
  22. L. Franks, J. Bubrouski, “Statistical properties of timing jitter in a PAM timing recovery scheme,” IEEE Trans. Commun. 22(7), 913–920 (1974). [CrossRef]
  23. N. A. D’Andrea, M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993). [CrossRef]
  24. R. Kudo, T. Kobayashi, K. Ishihara, Y. Takatori, A. Sano, Y. Miyamoto, “Coherent optical single carrier transmission using overlap frequency domain equalization for long-haul optical systems,” J. Lightwave Technol. 27(16), 3721–3728 (2009). [CrossRef]

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