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Abstract
A versatile digital coherent receiver capable of handling optical signals with different kinds of pulse shaping schemes (PSSs) is indispensable for future flexible and heterogeneous coherent optical communication networks. Therefore, a low-complexity timing phase error detector (TPED) versatile for all PSSs is of particular interest. In this paper, we propose a TPED suitable for both Nyquist signals with different roll-off factors and non-Nyquist signals. It requires two samples per symbol and involves no multiplications. As far as we know, it has the lowest computation complexity compared with the existing TPEDs used in coherent systems, while incurring no receiver sensitivity penalty. Numerical simulations and experiments are carried out to demonstrate the superior timing performance and PSS versatility of the proposed TPED in both open-loop and closed-loop working conditions. We also implement the novel TPED in a field programmable gate array (FPGA) and verify its real-time clock recovery performance using the 10 Gbaud very low roll-off Nyquist and non-Nyquist quadrature phase shift keying (QPSK) signals.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical modulation formats adopting non-return-to-zero (NRZ), return-to-zero (RZ) or Nyquist pulse shapes have different transmitter complexity, spectral efficiency and tolerance to noise and nonlinear effects [1–3]. Therefore, optical signals with various pulse shaping schemes (PSSs) may co-exist in the future flexible coherent transmission systems to adapt to different data traffic, transmitter configurations, frequency grids and optical links [1,4,5]. For this reason, an ideal digital coherent receiver (DCR) should be versatile for various PSSs. As the transmitters and receivers are driven by different clock sources, a timing phase error detector (TPED) is indispensable for DCRs because it can estimate the timing phase error (TPE) so that the time-variant TPE can be corrected by a clock recovery algorithm (CRA) to provide optimal sampling instances and hence an optimal sensitivity [4,6]. For this reason, a versatile DCR necessitates a TPED suitable for all PSSs.
By now, various TPEDs have been proposed. They can be classified into two main categories. The first kind of TPEDs processes sampling instances in the time domain, while the second kind of TPEDs performs a Fourier transform and processes the signal spectra in the frequency domain. The well-known time-domain TPEDs include the Gardner’s [7], Lee’s [8], Alexander’s (also known as the early-late gate) [9], Oerder & Meyr’s (also known as the square-and-filter) [10] and Mueller & Müller’s [11] TPEDs. The Gardner’s and Lee’s TPEDs require two samples per symbol, while the Alexander’s and Oerder & Meyr’s TPEDs require more than two samples per symbol. As the hardware resources and power consumption of the algorithms increase proportionally with the sampling rate, the Gardner’s and Lee’s TPEDs are more preferable in practice. The Mueller & Müller’s TPED requires only one sample per symbol, but it is sensitive to carrier offset and works in a decision-directed manner, incurring significant latency [11,12]. The well-known frequency-domain TPEDs include the Godard’s [13], Barton and Al-Jalili’s (BAJ’s) [14,15] and the modified Godard’s [16] TPEDs. The first two TPEDs require two samples per symbol, while the last requires less than two. The frequency-domain TPEDs are suitable for long-haul un-dispersion-managed systems as bulk chromatic dispersion (CD) compensation also needs to be performed in the frequency domain to reduce computation complexity [17]. Therefore, the already-existing Fourier transform can be shared by the frequency-domain TPED. However, in short-reach systems, such as metro and inter-datacenter networks, the accumulated CD is relatively small and can be compensated along with other linear distortions with the time-domain adaptive equalizers based on finite impulse filters (FIRs) [18]. In these application scenarios, the frequency-domain TPEDs will introduce significant additional computation complexity due to the extra Fourier transform required.
In terms of versatility, the Gardner’s, Lee’s, Godard’s and BAJ’s TPEDs are suitable for non-Nyquist signals adopting NRZ or RZ pulse shapes, but they fail for Nyquist signals with a small roll-off factor (ROF) [19,20]. To solve this problem, a new kind of fourth-power TPEDs applying the Gardner’s, Lee’s and Godard’s TPEDs on the power of received sequence has been proposed for Nyquist signals with a small ROF [21–24]. However, this kind of TPEDs fails for non-Nyquist signals and Nyquist signals with a large ROF. Recently, a modified fourth-power Gardner’s TPED versatile for all PSSs has been proposed [25,26]. It applies the Gardner’s TPED on the product of the sums of neighboring samples instead of power. However, its computation load is four times heavier compared with Gardner’s TPED.
In terms of computation complexity, the existing TPEDs often require hundreds of symbols to mitigate noise impact, and thus hundreds of multiplications are required per timing phase error (TPE) estimation [6,16,23]. The high computation complexity results in not only high hardware complexity but also significant estimation latency in the commonly used feedback CRAs on a real-time platform, thus reducing TPE tracking speed [27,28]. Therefore, a versatile and low complexity TPED is highly desired to be explored.
Recently, we proposed a multiplication-free TPED that is suitable for free space optical communication signals with a large dynamic range and very low OSNR [28]. In this paper, we apply this low complexity TPED on optical signals with different PSSs used in optical fiber communication systems and prove that it is also versatile for all PSSs. Based on this TPED, we propose a new multiplication-free TPED and compare it with other well-known TPEDs in terms of timing jitter and computation complexity. It is demonstrated that the new TPED has the lowest computation complexity while still guaranteeing the same receiver sensitivity as the optimal existing TPED. The performance of the proposed TPED is validated by numerical simulations and experiments under both open-loop and closed-loop working conditions. The proposed TPED is also implemented in a field programmable gate array (FPGA) embedded in a simplified DCR and we have successfully realized real-time clock recovery for the 10 Gbaud very small roll-off Nyquist and non-Nyquist quadrature phase shift keying (QPSK) signals.
2. Working principle
As we know, the Lee’s TPED has the following form [8]
Here $x(n)$ and $x(n + 1)$ are two consecutive optical signal field samples with complex values. The notation ${\textrm{Re}} ( \cdot )$ represents the real part of the operand within the brackets. N stands for the number of samples used per TPE estimation. Equation (1) shows that the Lee’s TPED requires 4N real multiplications per estimation, noting that calculating the real part or squared module only requires two real multiplications. We also note that we choose N-1 and N-2 as the upper summation limits of the first and second terms of Eq. (1) so that N samples are used in Eq. (1). Equation (1) can also be rewritten as
In ${\varepsilon _{Lee^{\prime}}}$, the last two items in ${\varepsilon _{Lee}}$ including $x(0)$ and $x(N - 1)$ are neglected as their effects are small compared with the summation item containing N-1 samples. The 1/2 factor is also neglected as it does not change the angle. We stress that N provides a tradeoff between computation complexity and noise suppression [6,28]. Typically, hundreds of samples are required per estimation to mitigate the noise impact. When N = 256, 1024 real multiplications are required per estimation, incurring high hardware implementation complexity and a significant TPE calculation latency if the TPED is used in a feedback CRA [28]. Therefore, it is desirous to design a TPED with low implementation complexity. Another shortcoming of the Lee’s TPED is that it is not suitable for Nyquist signals with very small ROFs because it calculates the TPE from SUM as follows [6]
Here ${X_k}$ stands for the Fourier transform of the received complex signal samples. Figure 1(a) schematically shows the spectra of the Nyquist signals with different ROFs. As dictated by Eq. (3), SUM is calculated by multiplying the upper sideband (red part) with the conjugate of the lower sideband (blue part). Figure 1(b-d) show the variations of the overlap of the upper and lower sidebands (outlined with red lines) when the ROF decreases from 0.5 to 0. As we can see, the overlap shrinks with a decreasing ROF. For this reason, the magnitude of SUM decreases with a decreasing ROF. Thus, the Lee’s TPED fails for the Nyquist signals with a small ROF. It is noteworthy that the Gardner’s, Godard’s and BAJ’s TPEDs also rely on SUM to estimate TPE and they are approximately equivalent to the Lee’s TPED [6]. Therefore, they have the same issue.
Fig. 1. The spectrum of Nyquist signals with different ROFs (a) and the SUM calculation window (dashed line) when the ROF = 0.5 (b), 0.1 (c), and 0 (d). Here B stands for signal baud rate and the ADC sampling rate is equal to 2B. N stands for the number of samples used in the calculation.
To reduce the computation complexity, the following multiplication-free TPED proposed by us can be used [28]
Here ${\mathop{\rm sgn}} ( \cdot )$ stands for the complex sign function defined by ${\mathop{\rm sgn}} (c) = {\mathop{\rm sgn}} [{\textrm{Re}} (c)] + j \cdot {\mathop{\rm sgn}} [{\mathop{\rm Im}\nolimits} (c)]$, where ${\mathop{\rm Im}\nolimits} ({\cdot} )$ represents the imaginary part of the operand within the brackets. As we can see, Eq. (4) actually evolves from Eq. (2) by adding the non-linear ${\mathop{\rm sgn}} ({\cdot} )$ operation which can not only reduce the computation complexity but also emphasize the frequency components near the clock tone for Nyquist signals with small ROFs as demonstrated below. Viewing Eqs. (2) and (4) in the frequency domain, the TPE is actually calculated from the phase of the frequency component at the baud-rate of the quadratic signal ${y_1}$ and its counterpart ${y_2}$ given by
This is because ${\varepsilon _{Lee^{\prime}}}$ and ${\varepsilon _s}$ can also be approximately written as
Some example spectra corresponding to ${Y_{1,2}}$ of the 10 Gbaud NRZ-QPSK signals and Nyquist QPSK signals with ROFs of 0 and 1 are shown in Fig. 2. Here the OSNR is set at a relatively high value of 20 dB to demonstrate solely the impact of the ROF on the magnitude of the clock tone as both noise and ROF will affect the magnitude. As we can see from Fig. 2(a), ${Y_1}$ obtained with the Lee’s method exhibits a prominent clock tone at the baud-rate frequency for the non-Nyquist signals. However, for the Nyquist signals, the clock tone is reduced in height and becomes invisible when the ROF is 0, as shown in Fig. 2(c). In contrast, as shown in Fig. 2(d-f), ${Y_2}$ obtained with the proposed method exhibits a prominent clock tone for both non-Nyquist and Nyquist signals with ROFs of 1 and 0. The reason is that the nonlinear operation ${\mathop{\rm sgn}} ({\cdot} )$ emphasizes the frequency components near the clock tone, as we can observe by comparing Fig. 2(c) and (f). The clock-to-noise ratios (CNRs) are also shown in Fig. 2. We can observe that the CNR obtained with the Lee’s method decreases from 20 dB to 0 dB when the ROF decreases from 1 to 0. But with the proposed method, the corresponding CNR only decreases by 3 dB and is still as high as 16 dB when the ROF = 0.
Fig. 2. The FFT spectrums obtained with the Lee’s (the 1st row), the proposed (the 2nd row), and the fourth-power Gardner’s (the 3rd row) methods for the Non-Nyquist signal (the 1st column) and Nyquist signals with the ROF of 1 (the 2nd column) and 0 (the 3rd column) when the OSNR is set to be 20 dB. Here B stands for signal baud rate and the sampling rate is 2B.
As we know, the fourth-power TPEDs also work well for the Nyquist signals with a small ROF [21–24]. They have a common feature of relying on the fourth-power operation to generate the clock tone and they are approximately equivalent [6]. For concise, taking the fourth-power Gardner’s TPED as an example, it has the following form [23]
Here ${P_n}\textrm{ = }{\left| {x(n)} \right|^2}$ stands for the signal power. ${\varepsilon _{FG}}$ calculates the TPE from the phase of the frequency component at the baud-rate of the fourth-order signal ${y_3}$ given by
This is because ${\varepsilon _{FG}}$ can also be written as
As we can see from Fig. 2(i), the clock tone obtained with the fourth-power Gardner’s TPED is prominent for the Nyquist signals with a ROF = 0. However, it requires 1.5N more multiplications per estimation than the Gardner’s TPED. More importantly, it fails for non-Nyquist signals and Nyquist signals with a large ROF because the fourth-order operation turns the already existing clock tone at the baud rate (B) into a high-frequency harmonic at 2B. Due to the frequency aliasing (the sampling rate is 2B), it overlaps with the DC component and becomes invisible as shown in Fig. 2(g) and (h). We note that although the nonlinear operation ${\mathop{\rm sgn}} ({\cdot} )$ also incurs bandwidth broadening and frequency aliasing for the non-Nyquist signals and Nyquist signals with a larger ROF, it doesn’t shift the clock tone at B to a higher frequency as shown in Fig. 2(d). In summary, the proposed TPED is more stable for different ROFs because the ${\mathop{\rm sgn}} ({\cdot} )$ operation emphasizes the frequency components near the clock tone for the small ROF signals while maintaining the already existing clock tone at B for the non-Nyquist signals and Nyquist signals with a large ROF.
Figure 3 shows the variations of the CNR against the ROF obtained with ${\varepsilon _{Lee^{\prime}}}$, ${\varepsilon _{FG}}$ and ${\varepsilon _s}$ for three kinds of modulation formats (QPSK, 16QAM and 32QAM). Here the OSNR is set to be 20 dB. For reference, the CNRs obtained for the non-Nyquist counterparts are marked with dashed lines. As we can observe, the variation of the clock tone magnitude obtained with our method (${\varepsilon _s}$) is about 3 dB. It is much smaller than the other ones which are around 20 dB. It means that our TPED is much less sensitive to the PSS. Figure 3(c) also shows that for our method, the CNRs for the 16QAM and 32QAM Nyquist and non-Nyquist signals are only about 2∼3 dB lower than those for the QPSK counterparts at different ROFs. Therefore, the TPED proposed is also suitable for these high-order modulation formats with or without Nyquist pulse shaping.
Fig. 3. The variations of the CNR against the ROF obtained with ${\varepsilon _{Lee^{\prime}}}$ (a), ${\varepsilon _{FG}}$(b) and ${\varepsilon _s}$ (c) for different modulation formats when the OSNR is set to be 20 dB.
In terms of computation complexity, the TPED given in Eq. (4) only involves multiplications of four complex values, i.e. ${\pm} 1 \pm j$. Therefore, the multiplications and additions can be replaced by a simple look-up-table (LUT) operation. The last complex operation in Eq. (4) is calculating the angle. After eliminating multiplications, it contributes to most of the calculation latency in real-time platforms because it is often realized using the so-called CORDIC algorithm, which incurs significant latency of more than ten cycles [29,30]. To solve this problem, the following novel TPED is proposed
Here calculating the angle is replaced by calculating the imaginary part. For convenience, we refer to the TPEDs given by Eq. (4) and (13) as the TPEDarg and TPEDim. As we know, the curve depicting the relationship of the estimated TPE versus the real TPE is commonly referred to as the S-curve [6,23]. By calculating the imaginary part, the TPEDim exhibits a sinusoidal S-curve like the Gardner’s TPED, while the TPEDarg has a linear one like the Lee’s TPED. As regard to the TPEDim, there is no unique correspondence between the estimate and real TPE. However, for the feedback structure commonly used in all-digital CRA [27,31], the sign of the estimate is adequate to control a feedback loop to find the zero crossing of the S-curve corresponding to the ideal sampling time. Figure 4 shows the example S-curves of the Lee’s, Gardner’s and the two proposed TPEDs obtained using 512 symbols of an NRZ-QPSK signal for each point.
For comparison, Table 1 lists the algorithms, computation complexities and application scenarios of the well-known existing and proposed TPEDs. As we can see, the times of real multiplications required per estimation are 2N for the Godard’s and BAJ’s, N for the Gardner’s, 4N for the Lee’s, 2.5N for the fourth power Gardner’s and 5N for the modified fourth power Gardner’s TPEDs. Here, we assume that one complex multiplication requires four real multiplications. For the algorithm that works in the frequency domain (the Godard’s, BAJ’s and the modified Godard’s TPEDs), the additional computational complexity caused by fast Fourier transform (FFT) is not considered.
Table 1. Characteristic of different TPEDs
Among the TPEDs listed in Table 1, the modified Godard’s TPED, TPEDarg and TPEDim are multiplication-free. However, the first one is essentially the Godard’s TPED and is not suitable for the Nyquist signals with a small ROF [16]. Furthermore, it is not suitable for short-reach systems that do not require frequency-domain CD compensation, and it requires much more CORDIC operations than the TPEDarg and TPEDim. For example, when the ROF ($\mathrm{\alpha }$) is 0.1 and N = 128, the modified Godard’s TPED requires about 20 times more CORDIC operations than the TPEDim.
3. Numerical simulations and comparison
In the simulations, we use the timing jitter in decibels (dB) as the performance metric to compare different TPEDs [6,23]. It is defined as the variance of zero-crossing positions of the S-curve given by
where $\delta t$ is the standard deviation of the zero-crossing positions normalized to a symbol period.Figure 5(a-e) shows the contour plots of the timing jitter versus the ROF and N for different TPEDs. The TPEDs compared include the frequency-domain BAJ’s and the time-domain Lee’s TPEDs which are suitable for non-Nyquist signals and the fourth-power Gardner’s TPED which is suitable for Nyquist signals with small ROFs. The other well-known TPEDs are approximately equivalent to one of them [6], and thus are not considered here for concise. The optical signals are Nyquist QPSK signals with the OSNR of 20 dB and the ROFs ranging from 0.01 to 1. The slope of the contour lines indicates the TPED’s sensitivity to the ROF. The steeper the contour lines are, the more sensitive the TPED is. As we can see, the contour lines in Fig. 5(a), (b) and (c) are much steeper than those in Fig. 5(d) and (e), validating that the timing jitter performances of these existing TPEDs are much more sensitive to the PSS than the proposed ones.
Fig. 5. The variations of the timing jitter versus the ROF and N for the BAJ’s (a), Lee’s (b), Fourth-power Gardner’s (c), TPEDarg (d) and TPEDim (e). The corresponding S-curves (in a heat map manner) are also plotted beside each contour plot. The OSNR is set to be 20 dB.
Taking points A (ROF = 0.1, N = 128) and B (ROF = 0.05, N = 128) in Fig. 5 for example, we can characterize them by comparing their S-curves and timing jitters. The corresponding S-curves are plotted beside each contour plot. For reference, the S-curves obtained for the non-Nyquist signals are also plotted. In each S-curve figure, 1000 S-curves obtained under different noise patterns are plotted in a heat map manner. The corresponding timing jitter is also marked on the top of the S-curve. The color bar beside the S-curve figure indicates the occurrence probability of the S-curves in specific regions. As we can see, when N = 128, the timing jitters of the BAJ’s TPED are about -46.4 dB, -22.3 dB and -15.7 dB, respectively, for the non-Nyquist signals and Nyquist-signals with the ROF = 0.1 and 0.05. The performance of the Lee’s TPED is similar to the BAJ’s TPED. The timing performance degradation for both of them is as large as 30 dB when they are applied on the Nyquist signals with a ROF = 0.05. The opposite is true for the fourth-power Gardner’s TPED. As we can see, when N = 128, the corresponding timing jitters are about -13.1 dB, -41.1 dB and -42.4 dB, respectively. The degradation is also as large as 30 dB. The TPEDarg and TPEDim have much less performance degradation. When N = 128, the corresponding timing jitters of the TPEDarg are about -44.1 dB, -34.7 dB and -34 dB, respectively. The degradation is only about 10 dB which is 20 dB lower compared with the existing TPEDs. The corresponding timing jitters of the TPEDim are about -37.5 dB, -26 dB and -24.1 dB, respectively. The degradation is only 13 dB. It is noteworthy that, because the TPEDarg and TPEDim require no multiplications, a much larger N can be adopted to reduce the timing jitter further without incurring any multiplications. As shown in Fig. 5(d) and (e), when N is set to be 512, the timing jitter obtained with the TPEDarg and TPEDim can be reduced to below -40 dB and -30 dB, respectively, even when the ROF is reduced to 0.01. It is noteworthy that 15 S-curve heat maps are plotted in Fig. 5. For comparison, the same occurrence probability color bar is used for all the 15 plots. In some plots, there is no red color because their timing jitters are relatively higher, and thus the S-curves are not aligned very well. Only when the jitter is reduced to about -35 dB, the red color will be presented.
The noise tolerance of different TPEDs is also investigated. Figure 6 shows the variations of the timing jitter against OSNR for the Nyquist (ROF = 0.05) and non-Nyquist signals. In this figure, the modified Godard’s, fourth power Gardner’s, Lee’s and the proposed TPEDs are compared. We note that the modified Godard’s TPED, instead of the two other frequency-domain TPEDs (the Godard’s and BAJ’s), is selected because it is also multiplication-free which makes the comparison with the proposed multiplication-free TPED fairer. Furthermore, the BAJ’s TPED performance is similar to the Lee’s TPED as shown in Fig. 5(a) and (b), and thus it is not selected to avoid repetition. Here N is set to be 128 for all TPEDs. As we can see, in terms of timing jitter, the TPEDarg and TPEDim are in the second and third places for the Nyquist signals. They are only inferior to the fourth-power Gardner’s TPED which is only suitable for Nyquist signals with a small ROF. On the other hand, for non-Nyquist signals, the TPEDarg and TPEDim are still in the second and third (tied with the modified Godard’s TPED) places and are only inferior to the Lee’s TPED which was proposed for non-Nyquist signals. We stress that only the TPEDarg and TPEDim can guarantee a timing jitter lower than -20 dB for both kinds of optical signals in the lower OSNR region. Such a level of the timing jitter does not bring any penalty to the receiver sensitivity because the signal quality with a timing jitter below -20 dB is mainly determined by OSNR at the FEC threshold bit error rate (BER) of 10−3 [16].
Fig. 6. The variations of timing jitter of different TPEDs versus OSNR for the Nyquist signals with a ROF = 0.05 (a) and the non-Nyquist signals (b).
As a closed-loop simulation can fully characterize the timing performance of TPEDs, a back-to-back simulation system utilizing a closed-loop CRA is set up as shown in Fig. 7(a). The OSNR of the received signals is varied by loading ASE noise following complex Gaussian distribution. The initial TPE is randomly selected and the sampling frequency offset (SCO) is set to be 200 ppm. The optical signals with the TPE, SCO, ASE noise and frequency offset incurred by the local oscillator are recovered in the digital domain by a standard intradyne coherent receiver. The DSP chain performs clock recovery, carrier recovery, decision and BER counting [31]. We note that no adaptive equalization is carried out as it can also compensate for the TPE [32], and thus may affect the performance evaluation of the TPEDs. In the clock recovery, we use the commonly used all-digital feedback CRA [27,33]. The CRA consists of a TPED, a loop filter (LF), a numerical-controlled oscillator (NCO) and a digital interpolator as shown in Fig. 7(b). Figure 7(c) shows the structure of LF which mitigates the phase noise of the TPED output and extracts the stable component to control the NCO. The interested reader is referred to Refs. [27,33] for further details.
Fig. 7. (a) The setup of the simulation system. (b) The feedback loop of all-digital timing recovery. (c) The structure of the loop filter.
Figure 8(a) and Fig. 8(b) show the variations of the BER versus the input signal OSNR for the 10 Gbaud Nyquist QPSK signals with a ROF = 0 and NRZ-QPSK signals. In the two figures, different TPEDs are adopted. As we can observe, the DCR sensitivity (at BER = 10−3) obtained with the TPEDim is nearly the same as that obtained with the TPEDarg and those obtained with the fourth-power Gardner’s TPED for Nyquist signals and the Lee’s TPED for non-Nyquist signals. The corresponding constellations and error vector magnitudes (EVMs) [34] obtained with the OSNR at BER = 10−3 are also given in these figures. Therefore, the TPEDim with the lowest computation complexity is apparently the best choice considering both the computation complexity and DCR sensitivity. We note that the parameters of the LF (K1, K2) have been optimized for different TPEDs and as they have different S-curve characteristics [33].
Fig. 8. The variation of the BER as a function of the input OSNR with the Nyquist signal (a) or non-Nyquist signal (b), when different TPEDs are adopted.
4. Experiments
We also investigate the performance of different TPEDs through experiments. Figure 9(a) shows the schematic diagram of the experiment setup. At the transmitter, an arbitrary waveform generator (AWG) with an electrical 3-dB bandwidth of 25 GHz and a sampling rate of 40 GSa/s is used to generate the analog signals corresponding to the in-phase and quadrature components of the 10 Gbaud Nyquist QPSK signal with a ROF = 0 and non-Nyquist NRZ-QPSK signals. The analog signals are amplified and used to drive an IQ modulator fed by an external cavity laser (ECL) at 1550 nm with a linewidth of 100 kHz. A variable optical attenuator (VOA1) and an erbium-doped fiber amplifier (EDFA) are used to control the optical signal power and the OSNR of the optical signal input into the receiver. The coherent receiver is a single-photodiode-per-polarization heterodyne receiver [35]. A 3-dB polarization-maintaining optical coupler is used to combine the received optical signal with the local oscillator (LO) with a frequency offset of 5.1 GHz from the signal center frequency. The polarization controller (PC) is used to adjust the polarization state of the modulated signal so that it is aligned with the LO. The combined optical signal is detected with an AC-coupled photodiode (PD) with a bandwidth of 20 GHz. The electrical signal output by the PD is digitized by a digital sampling oscilloscope (DSO) at a sampling rate of 80 GSa/s for offline processing.
Fig. 9. (a) Experiment Setup. (b). Flow chart of the DSP chain. PC: polarization controller, DSO: digital sampling oscilloscope, LO: local oscillator.
The flow chart of the DSP chain is also plotted in Fig. 9(b). After down-sampling to 2 samples per symbol, an optical field reconstruction algorithm is first used to recover the baseband complex signal [35]. The following algorithms, including CRA, adaptive equalization and carrier recovery algorithms are the same as the standard ones used in conventional intradyne DCRs [4]. The timing jitters obtained with different TPEDs are shown in Fig. 10. Here N is set to be 128 for all TPEDs, and the OSNR is varied from 5 dB to 13 dB. As we can see, the timing jitter obtained with the TPEDarg is lower than -25 dB and -32 dB for the Nyquist signals with a ROF = 0 and NRZ-QPSK signals, respectively. Its performance is very close to the fourth-power Gardner’s TPED for the Nyquist signals and the Lee’s TPEDs for the non-Nyquist signals. Among all TPEDs, only the proposed TPEDarg and TPEDim can guarantee a timing jitter lower than -20 dB for both Nyquist and non-Nyquist signals. The results obtained from experiments agree well with the numerical simulation results shown in Fig. 6.
Fig. 10. Variations of the timing jitter versus OSNR obtained with different TPEDs in experiments for the 10 Gbaud QPSK Nyquist signal with a ROF = 0 (a) and Non-Nyquist signal (b).
Figure 11(a) and (b) show the constellations obtained with different TPEDs for the NRZ-QPSK signals and Nyquist QPSK signals with a ROF = 0 when the OSNR = 10 dB. The corresponding EVMs are also given in these figures. As we can see, the CRA based on the Lee’s TPED performs well for the non-Nyquist signal, but it fails for the Nyquist signal with a ROF = 0 resulting in a very large EVM. The opposite is true for the fourth-power Gardner’s TPED. As expected, neither of them can work for both kinds of PSSs. On the contrary, the CRAs based on the TPEDarg and TPEDim are suitable for both Nyquist and non-Nyquist signals and the EVMs obtained with them are nearly the same. The value of 0.26 is equal to that obtained with the Lee’s TPED for the non-Nyquist signals and the fourth-power Gardner’s TPED for the Nyquist signals. Figure 11(c) shows a portion of the fractional interval variations output by the NCO corresponding to the adjacent constellations. As can be seen, the curve changes randomly when the Lee’s and fourth power Gardner’s TPED fail. While for the proposed TPED, the fractional interval has a steady varying period equal to the reciprocal of the clock sampling frequency offset, indicating that it works properly. It is noteworthy that in the experiments we use the non-Nyquist signals and Nyquist signals with a ROF = 0 because we want to test and compare the performance of different TPEDs in the two extreme cases. Figure 3(c) shows that, in general, the clock tone magnitude of the proposed TPED decreases with the decreasing ROF. Figure 5(e) shows that the jitter of the proposed TPED increases by about 4 dB as the ROF decreases from 0.1 to 0 at N = 128. In other words, the performance of the proposed TPED is the worst when the ROF = 0. The receiver is also the most sensitive to the timing jitter when the ROF = 0. Therefore, we can guarantee that the jitter performance of the proposed TPED and the BER performance of the receiver for the most popular ROF values in the range of 0.05 to 0.1 is not worse than that demonstrated for the Nyquist signals with a ROF = 0.
Fig. 11. Example constellations of the 10 Gbaud non-Nyquist (NRZ-QPSK) signals (a) and Nyquist QPSK signals with a ROF = 0 (b) obtained from off-line experiments. (c) A portion of the fractional interval output by the NCO corresponding to the adjacent constellations.
To evaluate the real-time timing performance of the TPEDarg and TPEDim, we use Verilog HDL to compile two CRAs based on them with Xilinx Vivado and implement the two CRAs along with the other algorithms in the DSP chain in an FPGA (Xilinx Virtex 7 XC7VX485T). The working clock frequency of the clock recovery module is set at 312.5 MHz. The parallel factor (P) is set to be 64 guaranteeing that the 10 Gbaud QPSK signals can be processed in a real-time manner at a sampling rate of 20 GSa/s [28]. N = 128 samples are used per TPE estimation. It is selected according to Fig. 5 and Fig. 6 showing that N = 128 provides a good trade-off between computation complexity and timing performance. As only 64 samples (P = 64) are input into the TPED every clock cycle, the TPED uses the late input of 64 samples (stored in a FIFO) and the current input of 64 samples to calculate the TPE. The input analog signals are digitized with 6-bit width and a nominal SCO of 200 ppm (using the DSO). The digitized signals are stored in a block random-access memory (BRAM) of the FPGA. The FPGA implements all of the DSP algorithms shown in Fig. 9(b) on the stored signals and saves the results in another BRAM for offline BER analysis. The variations of BER against OSNR for 10 Gbaud Nyquist QPSK signals with a ROF = 0 is shown in Fig. 12(a). Both offline and real-time experimental results show that the TPEDim can achieve successful timing recovery for the Nyquist signals with a ROF = 0 and its closed-loop timing performance is the same as the TPEDarg. The sensitivity obtained with real-time processing is 2 dB lower compared with off-line processing. The penalty stems from the accumulated truncation error due to the limited bit-width (1 dB penalty) and the feedback delay caused by the block-wise parallel processing (1 dB penalty). Figure 12(b) shows the results obtained for the non-Nyquist signals. As we can see, the proposed TPED also works well and the DCR sensitivity is very close to those obtained for the Nyquist signals. The small sensitivity degradation is due to a higher noise level as the non-Nyquist signals have a larger bandwidth. The FPGA resource utilization is shown in Table 2. The TPEDarg and TPEDim occupy no DSP resources as they are multiplication-free. Compared with the TPEDarg, the LUT, LUT-RAM and FF required by the TPEDim are further reduced thanks to the elimination of the CORDIC IP core. In Table 2, the total hardware consumption by the CRA (including a TPED, a loop filter, a NCO and a digital interpolator) is also listed. In the CRA, the Lagrange interpolator is often used. It is a cubic interpolator involving multiplications [27,33]. To reduce the computation complexity, we use a linear interpolator and digitize the fractional interval input into the interpolator with only 2-bit width. Therefore, a simple LUT can be used to realize the multiplications in the interpolator to reduce the latency and the whole CRA also uses no DSP-Cells.
Fig. 12. The variations of the BER as a function of the input signal OSNR obtained with offline processing and real-time processing in experiments when the TPEDarg (black) and TPEDim (red) are adopted. (a) Nyquist signal with a ROF = 0. (b) Non-Nyquist signal.
Table 2. The FPGA resource utilization summary
5. Conclusion
In this paper, we compare different kinds of TPEDs in terms of the timing jitter sensitivity to the PSS and computation complexity and illustrate the reason why the well-known quadratic and fourth-power TPEDs can’t work for both Nyquist and non-Nyquist signals. Then we propose a multiplication-free TPED that is suitable for both kinds of optical signals and conduct numerical simulations and experiments to demonstrate its superior performance under both open-loop and closed-loop working conditions. We also implement the proposed TPED in a real-time platform and validate its timing performance for the Nyquist signals with a ROF = 0 and the non-Nyquist signals using 10 Gbaud QPSK signals. Thanks to the much lower computation complexity and versatility for different PSSs, the proposed TPED appears to be the best choice for clock recovery in DCRs used in both short-reach and long-reach flexible coherent optical transmission systems.
Funding
National Natural Science Foundation of China (61975059, 61975063); Foundation of Chinese Science and Technology on Electro-Optical Information Security Control Laboratory (2021JCJQLB055017).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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