Blind symbol rate estimation using nonlinearity on sample correlation for digital coherent systems

Optics Communications 451 (2019) 246–254

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Blind symbol rate estimation using nonlinearity on sample correlation for digital coherent systems Junhao Ba a ,∗, Zhiping Huang a , Zhen Zuo a , Junyu Wei a,b a b

College of Intelligent Science, National University of Defense Technology, Changsha, 410073, China College of Electric and Information Engineering, Hunan University of Technology, Zhuzhou, 412007, China

ARTICLE

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Keywords: Correlation Symbol rate estimation Coherent detection Digital signal processing

ABSTRACT We propose a symbol rate estimation technique by using absolute value of correlation of received signal for single-carrier polarization division multiplexed (PDM) systems in coherent detection system. We analytically show that spectrum of absolute value of correlation contains a peak indicating the symbol rate and has a robust performance which is insensitive to accumulated chromatic dispersion (CD) and first-order polarization-mode dispersion (PMD), amplified spontaneous emission (ASE) noise and laser frequency offset. Simulation result shows that this technique is applicable to long-haul application.

1. Introduction Symbol rate estimation, or detection, is an important task for elastic optical network (EON) and passive signal sensing application such as optical performance monitor (OPM) [1]. Receivers in coherent system using architecture similar to software defined radio have potential to receive signal in EON systems, which require different symbol rates and modulation to maximize spectral efficiency. In addition, for application where transmission parameters such as modulation format, symbol rate, rolling factor are unknown a priori, symbol rate estimation become more essential as the receivers often synchronize and operate at symbol intervals. In such application where many parameters are assumed unknown, ‘blind’ or ‘flexible’ algorithm is preferable. The receiver has to work without pilot or training sequence. And the algorithm needs to be tolerant to fiber link distortion and receiving imperfection such as oscillator frequency offset. It would be desirable if the algorithm supports multiple modulation formats and has a large margin of symbol rate. For flexible transceivers in EON, it is desirable if all of the process could be done in digital domain and no external devices are required. Several methods have been proposed to solve this problem. However, most of these works could not satisfy all the requirements for passive sensing application. Spectral width measurement [2] is the most direct way to estimate the symbol rate. The idea is simple but not applicable to various pulse shapes. Asynchronous delay-tap signaling [3–6] has been considered as a popular method for symbol rate estimation. For those direct detection methods, auxiliary converter is needed and designing of accurate delay tap can be challenging. These techniques need recalibration as the pulse shapes and relative sample rates change. ∗

Traditional cyclostationarity based technique [7] such as cyclic spectrum and cyclic high order statics is also applicable to this problem since digital linearly modulated signal is a cyclostationary process. In addition, chromatic dispersion which suppress cyclostationarity is the main obstacle for these methods. Many frequency domain timing error detectors (TED), for example Godard’s [8] and Oerder’s [9] TED can be used to detect symbol rate. The tolerance of fiber impairment is basically the main issue and extra chromatic dispersion estimation and compensation is required. In this paper, we proposed a non-data-aided symbol rate estimation scheme for single-carrier digital coherent receivers. As the received signal is assumed to be PDM-QPSK or PDM-16QAM, the spectrum of absolute value of correlation contains a peak indicating the symbol rate. We analytically show that with sufficient data length, the algorithm is insensitive to laser frequency offset and chromatic dispersion. And we invested the PMD’s effect on the algorithm and proposed a solution to avoid worst case of PMD. The rest of paper is organized as follows: In section, we discussed the model of the signal and fiber link. And we provide the operation principle of our proposed techniques. In Section 3, we show the tolerance against various impairments. Section 4 provide the simulation and comparison with several similar techniques. Finally, the conclusion is provided in Section 5. 2. Operation principle Let 𝑠n be the information bit of independent and identically distributed random variable of M-ary symbols with equal possibility and

Corresponding author. E-mail address: [email protected] (J. Ba).

https://doi.org/10.1016/j.optcom.2019.06.016 Received 28 February 2019; Received in revised form 6 June 2019; Accepted 7 June 2019 Available online 10 June 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

J. Ba, Z. Huang, Z. Zuo et al.

Optics Communications 451 (2019) 246–254

Fig. 1. The periodogram of |̂ Rxy (m) | (a) and with high pass filter(b), the 𝑦-axis is converted to natural frequency. (28 GBaud 16 QAM signal transmitted through 3600 km Ultra Large Effective Area Fiber, Sample Rate 50 GSps, DFT size 65536). The rectangle shown in figure is the passband of filter.

The key task of symbol rate estimation is to use the discrete time signal to estimate the symbol interval T. The sample interval of ADC is used as a known parameter in estimating. We assume oversample rate = T∕Ts > 1. This assumption is plausible since most coherent receiver work at oversample rate = 2. And the sample correlation of two N-dimension vector from two polarization is defined as

let 𝑠x,n and 𝑠y,n be the information bits for each polarization in PDM system. We assume the expectation of 𝑠n satisfy { [ ] [ ∗] 0, 𝑛 ≠ 𝑚, . (1) 𝐄 𝑠n = 0, 𝐄 𝑠n 𝑠𝑚 = 1, 𝑛 = 𝑚. The baseband signal of transmitted signal can be expressed as, [ ] ∑ 𝑠 𝑥,𝑛 𝒔 (𝑡) = ⋅ 𝑝 (𝑡 − 𝑛𝑇 ) , (2) 𝑠𝑦,𝑛 𝑛

⎧𝑁−𝑚−1 ∑ ⎪ 𝑟𝑥 (𝑛 + 𝑚) 𝑟𝑦 (𝑛)∗ , ̂ Rxy (𝑚) = ⎨ 𝑛=0 ⎪ ̂ Rxy (−𝑚) , ⎩

where 𝑝(𝑡) is the pulse shape of the transmitter and T is the symbol duration. The linear fiber transmission model is character with CD element 𝐻𝐶𝐷 (𝜔) and the matrix of PMD 𝐌 (𝜔). The loss of fiber is compensated with fiber amplifier and neglected. 𝐇 (𝜔) = 𝐻𝐶𝐷 (𝜔) 𝐌 (𝜔) . The CD element is given by, ( ) 𝑗𝛽2 𝑧𝜔2 𝐻𝐶𝐷 (𝜔) = exp , 2

(9)

− (𝑛 − 1) < 𝑚 < 0.

It should be noted that, the sample correlation is a random vector rather than an expectation (which is deterministic). The distribution of sample correlation for stationary process has been studied in [11] and similar techniques can be applied to cyclostationary process. The distribution of ̂ Rxy (𝑚) is asymptotically joint normal with means 0 and periodic variance. Detailed proof and closed form of the distribution is given in Appendix A. Let s(m) = |̂ Rxy (m) | and s (m) follows a half-normal distribution. √ 𝐄 (s (m)) = 2𝐃(̂ Rxy (m))∕𝜋 and 𝐃 (s (m)) = (1 − 2∕𝜋)𝐃(̂ Rxy (m)). And the mean of s(m) is periodic since the variance of ̂ Rxy (𝑚) is periodic. So, this

(3)

(4)

where 𝛽2 is the dispersion parameter, z is the transmission distance. To characterize PMD, we use the model of principal states of polarization (PSP) for long distance fiber, which states that there exist two special orthogonal polarization states keep the in and out pulse undistorted to the first order PMD. The delay between two PSPs is the differential group delay. The PMD matrix 𝐌 (𝜔) is given by, ( ) ⎡exp j 𝜔𝛥𝜏 ⎤ 0 2 ( )⎥ 𝐌 (𝜔) = 𝐑2 ⎢ (5) 𝜔𝛥𝜏 ⎥ 𝐑1, ⎢ 0 exp −j ⎣ ⎦ 2

problem is equivalent to detecting sinusoidal wave in a colored noise. One of the most straight-forward method is periodogram. As shown in Fig. 1(a), the spectral line can be used to detect the symbol rate. With some algebra manipulations, a closed form proof is given in Appendix B to show that a spectral line is generate at N𝑇𝑠 ∕𝑇 when ( ) 𝑇𝑠 ∕𝑇 < 1∕2. For 1> 𝑇𝑠 ∕𝑇 > 1∕2, the line is generated at 1 − 𝑇𝑠 ∕𝑇 N. Basically, it is the cyclostationary property of the sampled signal produce such sinusoidal wave. We assume that following assumption is satisfied. (AS1) Oversample rate = T∕Ts > 1. It is obvious that with oversample rate less than 1 the sampled signal is independent and the signal will be stationary rather than cyclostationary. (AS2) The electric bandwidth is larger than half of the symbol’s frequency to preserve the cyclostationarity of sampled signal. The line for baud rate can be detected by simply finding the maximum value of the periodogram. However, the colored noise tends to have a large low frequency component which could lead to erroneous result. A simple 2-order IIR high pass filter can improve the detector’s performance. If we limit the oversample ratio to 4∕3 > 𝑇𝑠 ∕𝑇 > 1∕4, and then we can use a Butterworth high-pass filter with 𝜔c = 1∕4 (normalize frequency) and detect max value in [1∕4, 1]. The result is shown in Fig. 1(b). The summary of the proposed technique is shown in Fig. 2. The frequency offset is neglected since the sample correlation with sample lag m have the same frequency offset term and is canceled with the absolute operator. The proposed technique is found robust against various fiber impairments and we further show the robustness in Section 3.

where 𝛥𝜏 is DGD, 𝐑1 and 𝐑2 is the unitary matrix of polarization rotation with respect to PSP, which have the form of ( ) ( ) ⎡ ( ) ( ) 𝑗𝜓𝑘 𝑗𝜓𝑘 ⎤ sin 𝜃𝑘 exp ⎢cos 𝜃𝑘 exp − ⎥ 2 2 ⎥. Rk = ⎢ (6) ( ) ( ) ⎢ ( ) ( ) 𝑗𝜓 𝑗𝜓𝑘 ⎥ ⎢ sin 𝜃𝑘 exp − 𝑘 ⎥ cos 𝜃𝑘 exp ⎣ 2 2 ⎦ The received baseband signal can be expressed as [10] [ ] 𝑟 (𝑡) 𝐫 (t) = 𝑥 = [𝐬 (t) ∗𝐡 (t)] exp (j𝛥𝜔t) + nASE 𝑟𝑦 (𝑡)

𝑛 − 1 > 𝑚 ≥ 0,

(7)

The 𝐡 (t) is the impulse response of the fiber and 𝛥𝜔 is the radius frequency offset between transmitter and receiver laser. ‘∗’ denotes temporal convolution. nASE is the noise of amplifier. We use analog to digital converter (ADC) with sample intervals of Ts to get discrete time received signal [ ( )] [ ] ( ) 𝑟 𝑛𝑇 𝑟 (𝑛) 𝐫 (n) = 𝐫 nTs = 𝑥 ( 𝑠 ) ≜ 𝑥 . (8) 𝑟𝑦 𝑛𝑇𝑠 𝑟𝑦 (𝑛) 247

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Optics Communications 451 (2019) 246–254

chromatic dispersion is equivalent to discrete convolution of ℎ𝑘 given in (11). If 𝑁 > 𝑁𝐶𝐷 , then [ ] [ ] 1 (13) DFT 𝑟𝑥∕𝑦 (𝑛) = DFT 𝑠𝑥∕𝑦 (𝑛) DFT [ℎ (𝑘)] 𝑁 By substituting (13) into (12) [ ]∗ ] ] [ [ 1 ̂ (14) Rxy (𝑚) = IDFT DFT 𝑠𝑥 (𝑛) DFT 𝑠𝑦 (𝑛) 𝑁 By comparing the (12) and (14), we found that if (13) satisfy and input length is greater than 𝑁𝐶𝐷 , the sample correlation of transmitted signal and received signal will be the same. Suppose the 1550 nm wavelength transmission, for a 32 GBaud system transmitting over 16 000 km of standard fiber with D = 17 ps/nm/ km, 𝑁𝐶𝐷 = 8913. And this requirement is relatively easy to satisfy.

Fig. 2. Summary of the proposed technique.

3. Robustness against various fiber impairments

3.2. Effect of first-order PMD

Fiber impairments are the limit for high speed optical communication. While the fiber loss can be compensated by the amplifier, the most significant fiber impairment for optical communication is dispersion including CD and PMD. The proposed technique is insensitive to fiber impairments including CD and PMD distortion and one can estimation symbol rate after analog to digital conversion. The analysis of fiber impairments is carried out in a separate manner, i.e., when we focus on one fiber impairment and the others are neglected for simplicity. While it is reasonable since CD and PMD are linear impairment. We further show the robustness against all the impairments in simulation part. Fiber nonlinearity is another impairment worth noting if the launch power is high enough. For fiber communication system, the impact of fiber nonlinearity is small compare to the signal itself. Fiber nonlinearity induces complicated distortion and no analytic solution is present. To analyze the impact of fiber nonlinearity is quite difficult since different scenario could have totally different expression of the fiber nonlinearity. For dispersion uncompensated fiber link, the Gaussian Noise (GN) model [12–14] is a promising candidate to illustrate the fiber nonlinearity. At the end of the section, the robustness against the fiber nonlinearity which can be expressed as Gaussian noise is included.

PMD in optical fiber is the random couple between two polarization modes in PDM system caused by optical birefringence [17]. Most method based on correlation failed since the signal completely shifted in stokes spaces when DGD is half the symbol period [18]. The periodic mean of our proposed method is also deteriorated in this situation. Fig. 3(a) shows the clock tone completely disappear in worst case. Although this situation may be relieved as polarization angle rotates and DGD varies. For robust symbol rate estimation, a work-around is needed. [ ]𝑇 Let 𝐬 (𝑡) = 𝑠𝑥 (𝑡) 𝑠𝑦 (𝑡) be the transmitted baseband signal for [ ]𝑇 a PDM system. And let 𝐫 (𝑡) = 𝑟𝑥 (𝑡) 𝑟𝑦 (𝑡) be base band signal of received signal. And let 𝐬 (𝜔) and 𝐫 (𝜔) be the Fourier transform of 𝐬 (𝑡) and 𝐫 (𝑡). The baseband signal of a PDM system at the receiver can be expressed as [ ]𝑇 (15) 𝐫 (𝜔) = 𝐌 (𝜔) 𝐻𝐶𝐷 (𝜔) 𝑠𝑥 (𝜔) 𝑠𝑦 (𝜔) PMD matrix 𝐌 (𝜔) is given by 𝐌 (𝜔) = 𝐑2 𝐁 (𝜔, 𝜏) 𝐑1 [ =

3.1. Robustness against chromatic dispersion

cos 𝜃2 𝑒−𝑗𝜓2 ∕2 − sin 𝜃2 𝑒−𝑗𝜓2 ∕2 [

×

The transfer function for a fiber link with length z and group velocity dispersion coefficient 𝛽2 , the effect of chromatic dispersion can be characterized by the transfer function 𝐻𝐶𝐷 (𝜔) = exp (𝑗𝛽2 𝑧𝜔2 ∕2),

h𝑘 =

( 2 2) −𝑗𝑇 2 𝑇 𝑘 exp 𝑗 2𝜋𝛽2 𝑧 2𝛽2 𝑧

(10)

−

⌊

𝑁 2

⌋

≤𝑘≤

⌊

⎤ ⎥ 𝜔𝛥𝜏 ⎥ exp (−j )⎦ 2 0

(16)

where 𝐑𝟐∕𝟏 is the unitary matrices of polarization rotation. 𝜃2∕1 and 𝜓2∕1 are the angle and phase between the reference polarizations of the input and output of the fiber and the principal states of polarization (PSP). 𝛥𝜏 is DGD. Here we use the result of [19] to simplify the PMD transfer function. Let 𝑠′𝑥 and 𝑠′𝑦 be the signal projected on two PSPs. It is found by that 𝑠′𝑥 and 𝑠′𝑦 are uncorrelated random variables [20]. The time domain received signal is given by [ ]𝑇 𝐫 (𝑡) = 𝐑2 𝑠′𝑥 (𝑡 − 𝛥𝜏∕2) 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) ) ( )( ′ 𝑠𝑥 (𝑡 − 𝛥𝜏∕2) cos 𝜃2 𝑒−𝑗𝜓2 ∕2 sin 𝜃2 𝑒𝑗𝜓2 ∕2 = − sin 𝜃2 𝑒−𝑗𝜓2 ∕2 cos 𝜃2 𝑒𝑗𝜓2 ∕2 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) ) ) ( ( 𝑗𝜓2 𝑗𝜓 𝛥𝜏 𝛥𝜏 ⎞ ⎛ − 22 ′ 𝑠𝑥 𝑡 − + sin 𝜃2 𝑒 2 𝑠′𝑦 𝑡 + ⎟ ⎜ cos 𝜃2 𝑒 2 2 (17) =⎜ ( ) ( )⎟ 𝑗𝜓2 𝑗𝜓2 ⎜− sin 𝜃2 𝑒− 2 𝑠′ 𝑡 − 𝛥𝜏 + cos 𝜃2 𝑒 2 𝑠′ 𝑡 + 𝛥𝜏 ⎟ 𝑥 𝑦 ⎝ 2 2 ⎠

where 𝜔 is the radius frequency. In DSP-based receivers, chromatic dispersion is compensated by a finite impulse response (FIR) filter. The effect of chromatic dispersion can be approximated by an inverse FIR filter. The tap weight is given by [15] √

cos 𝜃1 𝑒−𝑗𝜓1 ∕2 − sin 𝜃1 𝑒−𝑗𝜓1 ∕2

𝜔𝛥𝜏 ]⎡ sin 𝜃2 𝑒𝑗𝜓2 ∕2 ⎢exp (j 2 ) cos 𝜃2 𝑒𝑗𝜓2 ∕2 ⎢ 0 ⎣ ] sin 𝜃1 𝑒𝑗𝜓1 ∕2 cos 𝜃1 𝑒𝑗𝜓1 ∕2

⌊| | ⌋ ⌋ 𝛽2 𝑧𝜋 𝑁 𝑎𝑛𝑑𝑁 = 2 | | 2 𝑇2 (11)

According to correlation theorems [16] of discrete Fourier transform (DFT), ̂ Rxy (𝑚) given by (9) can be obtained by [ [ ] [ ]∗ ] 1 ̂ (12) Rxy (𝑚) = IDFT DFT 𝑟𝑥 (𝑛) DFT 𝑟𝑦 (𝑛) 𝑁 [ ] [ ] where DFT 𝑟𝑥 (𝑛) and DFT 𝑟𝑦 (𝑛) is the DFT of N-dimension vector 𝑟𝑥 (𝑛) padding zero to length of 2𝑁 − 1. The padding is caused by the relationship of sample correlation and DFT. Chromatic dispersion is continuous time convolution caused by fiber transmission while here we use the equivalent discrete FIR filter for analysis. By using the result of [15]. We assumed that the effect of

By scanning through all possible 𝜃2 and 𝜓2 , we applied out proposed technique and get the result in Fig. 3. As the result shows, the clock tone is suppressed PMD. Here we applied a differential approach to solve this problem. When 𝜃2 is 𝜋∕4, the received signal of the worst case can be expressed by [ ] −𝑗𝜓 ∕2 ′ 𝑗𝜓 ∕2 ′ 1 𝑒 2 𝑠𝑥 (𝑡 − 𝛥𝜏∕2) + 𝑒 2 𝑠𝑦 (𝑡 + 𝛥𝜏∕2) 𝐫 (𝑡) = (18) 2 −𝑒−𝑗𝜓2 ∕2 𝑠′𝑥 (𝑡 − 𝛥𝜏∕2) + 𝑒𝑗𝜓2 ∕2 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) 248

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Optics Communications 451 (2019) 246–254

Fig. 3. Clock magnitude obtained from |̂ Rxy (𝑚) |(a) and |̂ Rx+y,x−y (𝑚) |(b). There is a large region where clock tone is vanished for both (a) and (b)

and the main issue of PMD is caused by the couple of 𝑠′𝑥 (𝑡 − 𝛥𝜏∕2) and 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) which change the distribution of autocorrelation. And the sum and difference of two polarizations is given by 𝑟𝑥 (𝑡) + 𝑟𝑦 (𝑡) = 𝑒𝑗𝜓2 ∕2 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2)

(19)

𝑟𝑥 (𝑡) + 𝑟𝑦 (𝑡) = 𝑒−𝑗𝜓2 ∕2 𝑠′𝑦 (𝑡 − 𝛥𝜏∕2)

(20)

where 𝑛𝑁𝐿 is the zero mean Gaussian noise induced by fiber nonlinearity and 𝑛𝑁𝐿 is the Gaussian noise induced by amplifier. We adopt the assumption that 𝑛𝑁𝐿 and 𝑛𝐴𝑆𝐸 are independent noise variable [14]. As the nonlinear interference is the Gaussian noise, the limit of the sample correlation can be expressed as lim ̂ Rxy (𝑚) = lim

when DGD = T∕2, 𝑟𝑥 (𝑡) + 𝑟𝑦 (𝑡) and 𝑟𝑥 (𝑡) − 𝑟𝑦 (𝑡) eliminate the coupled elements between 𝑠′𝑥 (𝑡 − 𝛥𝜏∕2) and 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2). | |̂ |Rx+y,x−y (𝑚)| = | | |

𝑁−𝑚−1 ∑

(

𝑒𝑗𝜓2 ∕2 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) 𝑒𝑗𝜓2 ∕2 𝑠′𝑥 𝑡 −

𝑛=0 𝑁−𝑚−1 ( ) ∑ 𝛥𝜏 ∗ = |𝑒𝑗𝜓2 𝑠′𝑦 (𝑡 + 𝛥𝜏∕2) 𝑠′𝑥 𝑡 − | 2 𝑛=0 𝑁−𝑚−1 ( ) ( ) ∑ 𝛥𝜏 ′ 𝛥𝜏 ∗ 𝑠′𝑦 𝑡 + = ||𝑒𝑗𝜓2 || | 𝑠𝑥 𝑡 − 2 2 𝑛=0 𝑁−𝑚−1 ( ) ( ) ∑ 𝛥𝜏 ′ 𝛥𝜏 ∗ 𝑠′𝑦 𝑡 + =| 𝑠𝑥 𝑡 − | 2 2 𝑛=0

𝛥𝜏 2

)∗

𝑁→∞

𝑁−𝑚−1 ∑

(𝑟𝑥 (𝑛 + 𝑚) + 𝑛𝑁𝐿 (𝑛 + 𝑚) + 𝑛𝐴𝑆𝐸 (𝑛 + 𝑚))

𝑁→∞

𝑛=0

× (𝑟𝑦 (𝑛) + 𝑛𝑁𝐿 (𝑛) + 𝑛𝐴𝑆𝐸 (𝑛))∗

|

= lim

𝑁−𝑚−1 ∑

𝑁→∞

𝑟𝑥 (𝑛 + 𝑚) 𝑟𝑦 (𝑛)∗

𝑛=0

This relationship shows with the assumption that fiber nonlinearity can be expressed as Gaussian noise, the effect of fiber nonlinearity is reduced as the sample length N increases. This result is compliant with the result of [21]. We further show the tolerance in the simulation part. For every simulation, fiber nonlinearity is included. The fourth simulation shows that the proposed method has a large margin of tolerance against Gaussian noise.

| (21)

And Fig. 3(b) shows the clock tone magnitude of periodogram of |̂ Rx+y,x−y (𝑚) |. When DGD = T∕2 and 𝜃2 is 0 and 𝜋∕2, the clock tone is suppressed too. On the other hand, the clock tone in original algorithm reaches maximum complementarily. Robust symbol rate against PMD is realized by adding two periodogram together, i.e., ) ) ( ( | | | | | | | | H (𝜔) = ||𝐹 𝐹 𝑇 |̂ Rxy (𝑚)| || + 0.5 ||𝐹 𝐹 𝑇 |̂ Rx+y,x−y (𝑚)| || (22) | | | | | | | | by scanning through all possible PMD, we found that the clock tone is approximately not affected by polarization angle of 𝐑2 . The result is shown in Fig. 4.

4. Simulation 4.1. Simulation setup We performed numerical simulation with VPI transmission maker to investigate the feasibility of our proposed technique. Five simulations for PDM coherent systems was carried out to test our proposed scheme. The fiber link consists of loops of 100 km ultra large effective area fiber (ULAF) and compensation with an EDFA amplifier with noise figure of 4.5 dB and 18 dB gain is used to fully compensate power loss. DGD is set to 0.1 ps km−1∕2 and follows the Maxwellian’s distribution. Addition noise loading is used to set the OSNR at the receiver’s endpoint. In all simulations, the fiber nonlinearity is taken into consideration and nonlinear parameter is set to 0.8 W−1 km−1 . We generated 219 symbols from 263 − 1 PRBS sequences and the transmitter modulates the signal with 194.2 THz laser and 1 MHz linewidth. The local laser’s linewidth is set to 100 kHz with random frequency offset within ±3 GHz. We resample the detected signal with 50 GSps and limit the electronic bandwidth to 20 GHz with 9th order Butterworth filters. Table 2 summarizes the parameters for transmission. Fig. 5 shows the block diagram of the simulation setup. In all simulations, the hybrid method of (22) is used to avoid worst case PMD. The length of input is given in each simulation. In the simulation, we evaluated different similar techniques, which can be categorized into two classes

3.3. The impact of GN-model fiber nonlinearity Reliability is one of the main tasks for fiber communication. Fiber nonlinearity induced by Kerr’s effect can cause complicated distortion to signal of the fiber link. For the communication signal without nonlinear compensation, the fiber nonlinearity is neglectable since the transmission is optimized with launch power. In the future, as nonlinearity compensation techniques can extend the fiber capacity. There is need for checking the performance of our proposed techniques in nonlinear transmission regime. For new-installed fiber link, dispersion uncompensated link is preferable for larger tolerance against fiber nonlinearity and dispersion compensation is easily done is DSP in the receiver. For dispersion uncompensated links, GN model is a simple model for analyzing the fiber nonlinearity. We adopt the basic assumption for GN model, nonlinearity is small enough for perturbation analysis [14]. Under this assumption, the received signal can be expressed as.

• Techniques using timing error detector: Gardner’s and Oeder’s timing error detector (TED) with chromatic dispersion compensation.

𝑟 (𝑛) = 𝑠 (𝑛) + 𝑛𝑁𝐿 + 𝑛𝐴𝑆𝐸 249

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Optics Communications 451 (2019) 246–254

Fig. 4. The clock tone obtained from |H (𝜔) |(a) and side view of |H (𝜔) |(b). As shown in (a), the minimum value is enough to give an accurate symbol rate estimation. As shown in (b), the magnitude of clock tone is approximately not affected by angle of the polarization. Table 2 Transmission parameters.

Fig. 5. Detailed Simulation Setup. Additional parameters are list in Tables 1 and 2. PBS stands for polarization beam splitter. PBC stands for polarization beam combiner. CW stands for continuous wave.

Simulation 1 NRZ-PDMQPSK

Simulation 2 NRZ-PDM-QPSK NRZ-PDM16QAM

Simulation 3 RC-PDM-QPSK with 𝜌 = 0 𝑡𝑜 0.5

Simulation 4 NRZ-PDMQPSK

Channels of WDM Span of fiber (N) Launch power WDM channel Grid Symbol rate ADC sample rate ADC bandwidth OSNR

10

10

10

10

0–25

140/36

8

0

−2 dB

0 dB/1 dB

−2 dB

−2 dB

50 GHz

50 GHz

50 GHz

50 GHz

32 Gbaud 50 GSps

28 Gbaud 50 GSps

28 Gbaud 50 GSps

28 Gbaud 50 GSps

20 GHz

20 GHz

20 GHz

20 GHz

12 dB (fixed)

Not limit

Not limit

fixed to [1 dB–8 dB]

As we discussed in the last section, the performance is hampered by data length. Fig. 6 depicts the successful rate with 16 384, 32 768 and 65 536 input sample. The power of the clock tone increase with the sample length. Basically, 65 536 samples are enough to ensure a distinct clock tone for uncompensated long-haul transmission. For system with even larger accumulated chromatic dispersion, we can furtherly increase the input length or use high-pass filter to extend the CD tolerance. Similar methods utilizing Gardner’s [8] and Oeder’s [9] timing error detector are evaluated using the same setup. 65 536 points is used in estimating the symbol rate. We tested without extra CD compensation since we focus on the CD tolerance of the symbol rate estimator itself. Comparing to these two methods, our proposed technique has much larger CD tolerance against these methods. We also tested the CAF and SCF method in the same condition. As a reference, the input length for CAF method is limit to 65 536 points while input length for SCF is 218 . The scanning step for both CAF and SCF method is 100 kHz. The weight method is used for CAF method [22]. The SCF is realize with frequency smoothing method described in [7]. The frequency smooth bandwidth optimized. As a remark, the parameter for CAF and SCF method is hard to optimize. For both methods, scanning step is crucial, the number of the scanning make the complexity grows linearly. For SCF method, the tolerance of CD is largely depending on the frequency smooth bandwidth, which may need few trials to get optimal result. In the second simulation, we tested the technique with PDM-16 QAM and PDM-QPSK. The launch power is higher than the optimal launch power to include nonlinearity. Filtered version approach is used

Table 1 Link parameters. ULAF Attenuation coefficient Dispersion parameter Polarization mod dispersion Nonlinear coefficient EDFA noise figure EDFA gain Span length

Modulation

0.18 db km−1 20 ps nm−1 km−1 0.1 ps km−1∕2 0.8 W−1 km−1 4.5 dB 18 dB 100 km

• Techniques blindly estimate the cyclic spectrum: cyclic autocorrelation function (CAF) and spectral correlation function (SCF) methods Our proposed method and the techniques being compared are peak finding method, the accuracy of the method only depend on the resolution of the technique. And we use the term successful rate as a benchmark of these method. The successful rate is calculated by the occurrence of ±0.1% of the true value in 200 times simulation. The random seed is changed in each run. 4.2. Results In the first simulation, we test the technique with 32 Gbaud NRZPDM-QPSK in 10-channel DWDM transmission with 0–50000 ps/nm accumulated CD. OSNR is fixed at 12 dB. High-pass filter is not used in this simulation to achieve wider range of symbol rate estimation. 250

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Optics Communications 451 (2019) 246–254 Table 3 Summary of the Nyquist pulse generation. Pulse shaping filter type

Square-root-cosine

Roll-off factor DAC sample per bit Filter length (M)

0.001 to 0.5 2 16 and 128

spectral correlation function (SCF) method [7], which performance is better than CAF in Nyquist transmission. In this simulation, the transmission length is fixed to 800 km as the performance of SCF degrades as CD increases. We show the data length required for 100% successful estimation. As calculation resource is limited for transmitter DSP, a filter tap of 16 is preferred in real-time generation [23]. SCF method need 219 samples for roll-off at roll-off factor of 0.1. This is mainly due to the effect of CD. While our proposed technique is not affected by CD, so that it is more flexible in optical transmission. The high-pass filter approach is also applicable for root-raised cosine pulse shaped signal to reduce sample size. The result is shown in dashed line in Fig. 9. Considering jitter tolerance and complexity, rolloff greater than 0.1 seems to be a more appropriate candidate as it is difficult for low roll-off Nyquist signal to regenerate the timing information. In the fourth simulation, we test the algorithm with different OSNR. As ASE noise in WDM system is bandlimited Gaussian process, it will not interrupt the clock tone for symbol rate estimation. We compared the performance in low OSNR scenario. All the input length is fixed to 65 536 points. Although in these situations the transmission is impossible as OSNR requirement is relatively high. We set this simulation to test the limit of our proposed technique. As a reference, we assume B2B transmission in order to test all the methods. The result is shown in Fig. 10. As the result shows, the filtered method may have advantage than unfiltered method in high OSNR scenario. The timing error method is preferable in extremely low SNR. The CAF and SCF method show similar performance.

Fig. 6. The successful rate against accumulated CD without high-pass filter.

here to enhance the performance in ultra-long transmission scenario. For PDM-QPSK signal, we compare the performance of 14 000 km ULAF fiber transmission and back to back(B2B) scenario. And for PDM16 QAM signal, the transmission length is 3600 km. The result is shown in Figs. 7 and 8. The color noise induced by 16 QAM signal is greater than QPSK signal. For QAM system, the main limit is the noise induced by signal itself rather than CD. Therefore, the require length of sample is longer for high-order modulation. Only CAF method is compared since in this condition since other methods fail to work in this condition. As a remark, we found that under 14 000 km transmission, both of our method and the CAF method satisfy 100% estimation accuracy as the length of 214 . However, our proposed method only needs one calculation while the time need for CAF calculation depends on the scanning step resolution. In the third simulation, we test the technique with root-raised cosine pulse shaping. The parameter for root-raised cosine generation is summarized in Table 3. For practical reasons, finite impulse is used in generating Nyquist pulse. Generally, the impulse response of a rootraised pulse shape filter is longer than NRZ pulse and the successive bandwidth is typically narrower. The length of filter and roll-off factor is affecting the performance of the symbol rate estimator. For low roll-off scenario, the CAF we compared in the second simulation fail to work. And we compare the proposed technique with

4.3. Comparison with similar techniques Several similar techniques have been tested in the same condition. We found the complexity is similar to TED based method since only one correlation is needed. While generally the performance against fiber impairment is similar to CAF method and the tolerance against Nyquist

Fig. 7. The successful rate against Log2 Nsample for PDM-QPSK system, a high-pass filter with 𝜔c = 1∕4 is used to reduce the sample needed for accurate symbol rate estimation.

251

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Optics Communications 451 (2019) 246–254

Fig. 8. The successful rate against Log2 Nsample for PDM-16QAM system, a high-pass filter with 𝜔c = 1∕4 is used to reduce the sample needed for accurate symbol rate estimation.

Fig. 9. Minimum sample length for 100% symbol rate estimation with different roll-off factors.

̂ 𝐱𝐲 Appendix A. Asymptotically distribution of 𝐑

pulse signal is similar to SCF method. TED based method has largest tolerance against Gaussian noise. 5. Conclusion

Suppose n is length of acquired data, 𝑞∕𝑝 is the Diophantine approxA novel symbol rate estimator that can be used in monitoring WDM system is proposed. The high-pass filter approach is proposed to enhance spectral line extraction. The proof of spectral line generation based on central limit theorem is given. The robustness against CD and PMD is given. An alternative approach to avoid worst case PMD is given. We show that 100% reliable symbol rate estimator is realized for PDM-QPSK and PDM-16QAM signal and is applicable to Nyquist superchannel. As the technique does not require external CD compensate or scanning, the calculation complexity is dramatically reduced.

imation of 𝑇𝑠 ∕𝑇 , i.e.,

Acknowledgments

imated received signal and the original information vector is 𝒄 =

𝑝, 𝑞 ∈ Z+ , 𝑝 < 𝑛, 𝑞 < 𝑛, min ||𝑞∕𝑝 − 𝑇𝑠 ∕𝑇 || ,

(A.1)

and the discrete sample signal can be approximated by (A.2)

𝑠 (𝑛) = 𝑐⌊𝑛𝑞∕𝑝⌋ .

Let vector 𝒔 = [𝑠 (0) 𝑠 (1) … 𝑠 (𝑛 − 1)] denotes n-dimensional approx[𝑐0 𝑐1 … 𝑐𝑚−1 ] where 𝑚 = 𝑛𝑞∕𝑝. And 𝒔 = 𝑨𝒄 where

This work was supported by Natural Science Foundation of China under Grant No. 51575517 and Natural Science Foundation of Hunan province, China under Grant No. 2019JJ50121.

[ ] 𝑨 = 𝑎𝑖𝑗 = 252

{

1, 0,

⌊𝑖𝑞∕𝑝⌋ = 𝑗, 𝑜𝑡ℎ𝑒𝑟𝑠.

(A.3)

J. Ba, Z. Huang, Z. Zuo et al.

Optics Communications 451 (2019) 246–254

Fig. 10. The successful rate against various different OSNR. NRZ-PDM-QPSK is used in this simulation.

⌊𝑛𝑞∕𝑝⌋ is a Beatty sequence of rational number, which has a period of p. If n>q, and 𝐀 is block diagonal matrix: ⎛𝐐 ⎜ 0 𝐀=⎜ ⎜0 ⎜0 ⎝

0 𝐐 0 0

0 0 ⋱ 0

0⎞ { ⎟ [ ] 0⎟ 1, , where, 𝐐 = 𝑎𝑖𝑗 = 0⎟ 0, 𝐐⎟⎠

⌊𝑖𝑞∕𝑝⌋ = 𝑗, 𝑜𝑡ℎ𝑒𝑟𝑠.

=

33 ∑

∗ 𝐬3n 𝐐T Σ3 𝐐𝐭3n+3 ≜

33 ∑

𝐘n,3

𝑛=0

𝑛=0

It is obvious that for each 𝐘n,k is 1-dependent, i.e., 𝐘n,k 𝑎𝑛𝑑 𝐘n+2,k is independent, and we can apply the result of Billingsley [24] directly. That is, ) ( ̂ R𝑠𝑡 (𝑘) ‖2 ‖ T , 𝐐 Σ 𝐐 ⇒ N 0, ‖ ‖ √⌊ 𝑘 ⌋ ‖𝐹 ‖ (A.9) 𝑁−𝑘

(A.4)

As a special case, if 𝑇𝑠 ∕𝑇 = 3∕5, and the matrix Q can be expressed as,

𝑞

⎛1 ⎜1 ⎜ 𝐐 = ⎜0 ⎜0 ⎜ ⎝0

0⎞ 0⎟ [ ]𝑇 [ ]𝑇 ⎟ 0⎟ , 𝒄 = 𝑐0 𝑐1 𝑐2 , 𝒔 = 𝑐0 𝑐0 𝑐1 𝑐1 𝑐2 , 0⎟ ⎟ 1⎠

0 0 1 1 0

where ‖ ⋅‖𝐹 denotes Frobenius norm. Since Σ𝑘 is periodic with period q and the variance of ̂ R𝑠𝑡 (𝑚) is periodic.

(A.5)

̂ 𝒔𝒕 (𝒎) | Appendix B. The fourier transform of mean of |𝐑

The sample correlation of two n-dimension vectors 𝐬 and 𝐭 is defined as ̂ R𝑠𝑡 (𝑚) =

𝑛−𝑘−1 ∑

𝑠 (𝑛) 𝑡∗ (𝑛 + 𝑚) .

√ ‖𝐐T Σ𝑘 𝐐‖2 and we showed that The mean of 𝐄|̂ R𝑠𝑡 (𝑘) | = ‖ ‖𝐹 T ‖2 ̂ 𝐄|R𝑠𝑡 (𝑘) | is periodic with period q. Since max ‖ ‖𝐐 Σ𝑘 𝐐‖𝐹 < 2 min ‖𝐐T Σ𝑘 𝐐‖2 , we can approximate the square function with linear func‖𝐹 ‖ 2 tion. For simplicity, we focus on the Fourier transform 𝐄|̂ R𝑠𝑡 (𝑘) | . ‖ T ‖2 T ‖2 ‖ T ‖2 𝐐 Σ𝑞−1 𝐐‖ ] = diag We found that [‖ ‖𝐐 Σ0 𝐐‖𝐹 ‖𝐐 Σ1 𝐐‖𝐹 … ‖ ‖ ‖𝐹 (𝐐𝐐T ⊗ 𝐐𝐐𝑇 , 𝟏), where diag(⋅, k) denotes the 𝑘th main diagonal of the matrix, ⊗ denotes 2D circular convolution of matrix. As a special case, if 𝑇𝑠 ∕𝑇 = 3∕5.

(A.6)

𝑚=0

When 𝑛 ≫ 𝑞, if we neglect the last items of (A.6), the correlation can be approximated by ⌊(𝑁−𝑚)∕𝑞⌋

∑

̂ R𝑠𝑡 (𝑚) ≈

∗ 𝐬nq 𝐐T Σ𝑘 𝐐𝐭nq+k ,

(A.7)

𝑛=0

where 𝐬nq = [𝑠 (𝑛𝑞 + 1) 𝑠 (𝑛𝑞 + 2) … 𝑠 (𝑛𝑞 + 𝑝)], 𝐭nq = [𝑡 (𝑛𝑞 + 1) 𝑡 (𝑛𝑞 + 2) … 𝑡 (𝑛𝑞 + 𝑝)] and Σ𝑘 denotes operator that left shift matrix for k columns. ( ) 0 𝑰 𝑛−𝑘 Σ𝑘 = , and 𝑰 𝑘 is a k-dimension identical matrix 𝑰𝑘 0

⎛7 ⎜4 ⎜ 𝐐𝐐T ⊗ 𝐐𝐐𝑇 = ⎜1 ⎜0 ⎜ ⎝4

As a special case continues, let s be the sample vector taken from x polarization and t be the sample vector taken from y polarization, 𝐬 = ]𝑇 [ [ 𝑐0𝑥 𝑐0𝑥 𝑐1𝑥 𝑐1𝑥 𝑐2𝑥 … 𝑐99𝑥 𝑐99𝑥 𝑐100𝑥 𝑐100𝑥 𝑐101𝑥 and 𝐭 = 𝑐0𝑦 𝑐0𝑦 𝑐1𝑦 𝑐1𝑦 𝑐2𝑦 ]𝑇 … 𝑐99𝑦 𝑐99𝑦 𝑐100𝑦 𝑐100𝑦 𝑐101𝑦 , and we can get some example of sample correlation of s and t is

33 ∑

∗ 𝐬3n 𝐐T 𝐐𝐭3n ≜

𝑛=0

33 ∑

𝐘n,0

1 4 5 4 2

0 0 4 9 4

4⎞ 1⎟ ⎟ 2⎟ , 4⎟ ⎟ 5⎠

(B.1)

‖ T ‖2 T ‖2 ‖ T ‖2 and [‖ 𝐐 Σ𝑞−1 𝐐‖ ] = [7 7 5 9 5] ‖𝐐 Σ0 𝐐‖𝐹 ‖𝐐 Σ1 𝐐‖𝐹 … ‖ ‖ ‖𝐹 As 𝐐𝐐T is a sparse matrix, it is relatively easy to figure out the diagonal of 𝐐𝐐T ⊗ 𝐐𝐐𝑇 , i.e.,

∗ ∗ ∗ ∗ ∗ ∗ ̂ R𝑠𝑡 (0) = 2𝑐0𝑥 𝑐0𝑦 + 2𝑐1𝑥 𝑐1𝑦 + 𝑐2𝑥 𝑐2𝑦 + 2𝑐3𝑥 𝑐3𝑦 + 2𝑐4𝑥 𝑐4𝑦 + 𝑐5𝑥 𝑐5𝑦 +⋯

=

4 7 4 0 1

𝑛−1 ∑ ( ) ( ) ( ) diag 𝐐𝐐T ⊗ 𝐐𝐐𝑇 = 2 diag 𝐐𝐐T , 𝑘 ∗𝑝 diag 𝐐𝐐T , 𝑘

(A.8)

𝑘=1

( ) ( ) + diag 𝐐𝐐T , 1 ∗𝑝 diag 𝐐𝐐T , 1

𝑛=0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ̂ R𝑠𝑡 (3) = 𝑐0𝑥 𝑐1𝑦 + 𝑐0𝑥 𝑐2𝑦 + 2𝑐1𝑥 𝑐3𝑦 + 𝑐2𝑥 𝑐4𝑦 + 𝑐3𝑥 𝑐4𝑦 + 𝑐3𝑥 𝑐5𝑦 + 2𝑐4𝑥 𝑐6𝑦 ∗ + 𝑐5𝑥 𝑐7𝑦 +⋯

where ∗𝑝 denotes circular convolution with period p. 253

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Optics Communications 451 (2019) 246–254

( ) ( ) When 2>p/q>1, diag 𝐐𝐐T , 1 ∗𝑝 diag 𝐐𝐐T , 1 = [𝑝 𝑝 … 𝑝] and ( ) when k>2 diag 𝐐𝐐T , 𝑘 = 0. And it can be simplified to

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