1. Introduction

Digital coherent transceivers have become potential candidates for short-reach high-speed inter-datacenter interconnects, especially when the required data rate per wavelength increases beyond 400 Gbps [1,2]. However, the challenges for short-reach coherent transmission are quite different from traditional long-haul transmission, where achieving a higher capacity and distance product on scarce fiber resources is the priority [3]. For short-reach coherent transmission, the priority is reducing the power consumption and cost of the coherent transceivers while achieving a high data rate [4–9]. One straightforward solution to satisfy these requirements is reducing the required oversampling factor (OSF) as the power consumption of the real-time digital signal processing (DSP) subsystem decreases proportionally with a decreasing OSF [10,11]. Another solution is Nyquist pulse shaping with a small roll-off-factor (ROF) to reduce signal bandwidth, so that high-data-rate signals can be generated and detected with cheap low-bandwidth components [12]. At present, the OSF must be raised to two before clock recovery if the original digital clock recovery algorithms (CRAs) proposed in [13–18] are used as these well-known CRAs require an OSF of two. More specifically, CRAs rely on timing phase error detectors (TPEDs) to estimate and mitigate the clock asynchrony between transmitter side and receiver side. However, the well-known TPEDs, such as the time-domain Gardner’s, Lee’s and Oerder & Meyr’s (also known as the square-and-filter) TPEDs proposed in [14,15,19], and the frequency-domain Godard’s and BAJ’s TPEDs proposed in [16–18], cannot acquire the clock tone correctly for synchronization when the OSF is smaller than two, although the input signal is still cyclostationary with such an OSF. The Mueller & Müller’s TPED works for an integer OSF of one [20]. However, it is designed for the intensity modulation and direct detection (IM-DD) systems and sensitive to the carrier offset.

To adapt to a non-integer OSF below two, the modified Godard’s TPED alters the index interval of the two signal spectral components multiplied with each other in clock tone generation to keep the frequency interval equal to signal baud rate [21]. It is easy to know that this adaptation scheme also works for the BAJ’s TPED which generates the clock tone by the same way. However, the two TPEDs are not suitable for Nyquist signals with a small ROF because the number of signal frequency components that can be used to generate the clock tone decreases with decreasing ROF, making the clock tone finally overwhelmed by noise, and thus the CRA fails for small ROFs [21,22]. In addition, for receivers used in short-reach systems, the complex frequency-domain bulk chromatic dispersion compensation algorithm [23,24] is removed to simply the DSP flow, and only time-domain adaptive equalizers (AEQs) are reserved for small dispersion compensation [6–9]. Thus, the frequency-domain TPEDs will introduce significant additional computational complexity due to the extra complex Fourier transform operation required.

To solve above problems, we propose a low-complexity time-domain TPED versatile for non-integer OSFs below two and small ROFs by modifying the time-domain quadratic signal and its spectral component used for synchronization. In addition, the new TPED eliminates all multiplications which is vital for reducing the algorithm complexity and thus the feedback latency on a real-time platform. We apply the new TPED in a feedback CRA and demonstrate that CRA can work well for non-integer OSFs below two and small ROFs close to zero. We also demonstrate that by replacing the commonly used Lagrange cubic interpolator with the low-complexity piece-wise parabolic (PWP) interpolator, the CRA performance can be further improved, especially when the OSF approaches one. Both numerical simulations and experiments are carried out to validate the superior performance of the proposed TPED.

2. Working principle

Generally, CRAs can be classified into three kinds: feedforward CRAs, feedback CRAs and hybrid CRAs employing both kinds of CRAs [13]. Feedforward CRAs have a shorter acquisition time and are suitable for burst-switched systems, while feedback CRAs are more hardware-efficient and can employ a low-computation-complexity TPED as getting the sign of the timing phase error (TPE) is adequate to drive the control loop to converge to the optimal resampling instant. Furthermore, feedback CRAs can be integrated with AEQs to mitigate the impact of chromatic dispersion on the TPED [13,25], while feedforward CRAs cannot, which makes it hard for feedforward CRAs to work independently when chromatic dispersion exists. Noting that time-domain AEQs fail when the CRA is not performed beforehand and the sampling clock offset exists [26,27], and thus the AEQs cannot be used to compensate for the chromatic dispersion before the CRA. Hybrid CRAs have the advantages of both kinds of CRAs, but the implementation complexity is quite high. Due to space limitation, we only investigate the improvement of feedback CRAs in this paper, as they are generally sufficient for the non-burst-mode inter-datacenter transmission systems.

The structure of an all-digital feedback CRA is shown in Fig. 1. It consists of a TPED, a loop filter, a numerically controlled oscillator (NCO) and a digital interpolator. The TPED is the most important part in CRA. It estimates the TPE and determines the adjustment direction of resampling instant. The loop filter can mitigate the phase noise of the TPED output and extract the stable component to control the NCO. The NCO is used to calculate the interpolation base point index ${m_k}$ and the fractional interval ${\mu _k}$. The digital interpolator recovers the samples at the optimal instants by calculating interpolants with ${m_k}$ and ${\mu _k}$. The interesting readers can refer to [13] for more details.

 

Fig. 1. The structure of the feedback clock recovery algorithm.

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The CRA performance is mainly determined by the TPED. The well-known Lee’s TPED has the following form [15]

Here, $x(n)$ stands for the complex-valued sampled sequence and for the Lee’s TPED $x(n)$ is acquired by an OSF of two. $\textrm{Re} ({\cdot} )$ represents the real part of the operand. N is the number of samples used in the estimation. The Lee’s TPED works in the time-domain and is suitable for short-reach systems, but it has the following shortcomings.

First, the Lee’s TPED, as well as the Gardner’s TPED, fails for non-integer OSFs below two. Taking the Lee’s TPED as an example, it can be explained by rewriting ${\varepsilon _{Lee}}$ in dependence of the quadratic signal ${y_{Lee}}(n)$ which is generated only in the Lee’s TPED for synchronization purposes.

Here, ${Y_{Lee}}(k)$ is the spectrum of ${y_{Lee}}(n)$. Eqs. (3–5) show that the Lee’s TPED actually exploits the synchronization spectral line ${Y_{Lee}}(k = N/2)$ with a frequency equal to the signal baud rate B under the prerequisite of OSF = 2. Accordingly, when the OSF is reduced below two, it fails to work as the frequency of the spectral line ${Y_{Lee}}(k = N/2)$ is not equal to B any more. Furthermore, frequency aliasing of ${Y_{Lee}}(k)$ must be considered for OSFs below two.

In this paper, we propose a method to adapt the time-domain TPEDs to a non-integer OSF below two. Taking the Lee’s TPED as an example, the new modified Lee’s TPED is as follows

Here, the phase item ${e^{ - jn\pi }}$ in Eq. (2) is replaced by ${e^{j2\pi n/M}}$ where M stands for the OSF. $\Delta \phi = \pi (1/M - 1/2)$ is a constant phase shift for a given OSF. When OSF = 2, $\Delta \phi $ is zero. When OSF < 2, $\Delta \phi $ is not equal to zero due to the movement of the synchronization spectral line in the spectrum. The modified Lee’s TPED in Eq. (6) uses the spectral line ${Y_{Lee}}(k = N - N/M)$ for synchronization. Figure 2 schematically shows the spectrum ${Y_{Lee}}(k)$ and two of its neighboring images. The rectangle outlined with blue dashed-lines represents the frequency window with a width equal to the sampling rate MB. The spectrum ${Y_{Lee}}(k)$ is generally wider than the frequency window because ${y_{Lee}}(n)$ is a quadratic signal generated from the input signal. Therefore, frequency aliasing of the quadratic signal occurs in the TPED. Taking the synchronization spectral line at -B as an example, it is actually outside the frequency window for OSFs below two. The actual frequency (index) of the synchronization spectral line aliasing into the positive frequency part of ${Y_{Lee}}(k)$ is $(M - 1)B$ (the index $k = N - N/M$), noting that the first component of ${Y_{Lee}}(k)$ obtained by fast Fourier transform (FFT) has a frequency of zero. As will be proved later, the frequency aliasing does not prevent using the spectral line for synchronization as it generally has a much high power than the background noise.

 

Fig. 2. The spectrum of the quadratic signal ${y_{Lee}}(n)$ generated in the Lee’s TPED when the OSF (M) is smaller than two. Two neighboring images are also drawn.

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Secondly, the Lee’s TPED, as well as the Gardner’s and Godard’s TPEDs, fails when the ROF approaches zero [21]. Therefore, it has been proposed to apply these classic TPEDs on the power of input sample sequences, so that the clock tone can be regenerated with the additional square operation [22]. For example, for the Lee-power TPED, ${|{x(n)} |^2}$ is used to replace $x(n)$ in Eqs. (1) and (2). Thus, the new Lee-power TPED adapted to a non-integer OSF below two has the following form

The modified Lee-power TPED is suitable for small ROFs and non-integer OSFs below two. But its computational complexity is relatively high. Table 1 compares the characteristics of the existing and new TPEDs proposed in this paper. As we can see, the new modified Lee-power TPED requires 6N + 4P real multiplications per operation which are much more than the Godard’s and Lee’s TPEDs. It is noteworthy that we assume that one complex multiplication requires four real multiplications (RMs) and the OSF is M = P/Q (P and Q are coprime integers). In Eq. (7), multiplying the first item in the square bracket with the exponential factor ${e^{j2\pi n/M}}$ incurs 4P more real multiplications. Noting that ${e^{j2\pi n/M}}$ has only P different complex values, and thus can be calculated and stored in a memory beforehand. Accordingly, the multiplication in Eq. (7) only needs to be performed P times as the summation can be carried out before the multiplication for the items with the same exponential factor.

Table 1. Characteristics of different time- and frequency-domain TPEDs.

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As we know, in the TPE estimation, the parameter N provides a tradeoff between tracking speed and noise suppression, and it is generally on the order of one hundred samples [22]. For example, when N = 200 M (200 symbols are used in the estimation) and M = 1.25, about 1520 RMs are required per operation for the modified Lee-power TPED. This incurs a high hardware implementation complexity and significant latency in feedback CRA on a real-time platform which may seriously degrade the performance in practice [28]. It is worthy to note that the frequency-domain Godard’s and BAJ’s TPEDs require 2N real multiplications to generate the clock tone by spectral autocorrelation and one FFT operation to get the signal spectrum [29]. The modified Godard’s TPED can eliminate all the multiplications but still requires one FFT operation which requires 3N/2log₂N-5N + 8 and 9N/8log₂N-43N/12 + 16/3 real multiplications for the radix-2 and radix-4 algorithms, respectively [30,31]. Noting that for the Godard’s, BAJ’s and modified Godard’s TPEDs, a large N is required for a small ROF as the number of the non-zero products obtained in the spectral autocorrelation which is used to generate the clock tone is $ROF \cdot N/M$[21]. When N = 200 M and the ROF is 0.01, there are only two non-zero component in the summation, and thus the noise impact cannot be mitigated effectively, incurring very large timing jitter. If N = 1000 M ten non-zero components can be obtained for a ROF of 0.01 but more than 9000 real multiplications are required when M = 1.25. The computational complexity is much higher compared with the modified Lee-power TPED.

To reduce the computational complexity, based on the modified Lee’s TPED, we propose a new time-domain TPED versatile for both non-integer OSFs and small ROFs as follows

Accordingly, the modified quadratic signal is as follows

Here, $\textrm{csgn(} \cdot \textrm{) = sgn[Re(} \cdot \textrm{)] + }j \cdot \textrm{sgn[Im(} \cdot \textrm{)]}$ is the complex sign function which is a nonlinear operation and can strengthen the synchronization spectral line in the quadratic signal for small ROFs [29,32]. In ${y_p}$, the multiplications only involve four complex values ${\pm} 1 \pm j$ and have only four possible outputs ${\pm} 2$ or ${\pm} 2j$, and thus can be realized by a simple look-up-table (LUT) operation. As for the multiplication of the exponential item ${e^{j2\pi n/M}}$ with ${y_p}$, it can be realized by a simple shift-add operation, which requires no multiplications. Therefore, the proposed TPED is much simpler and can be versatile for both non-integer OSFs and small ROFs, simultaneously. It is noteworthy that the $\textrm{csgn(} \cdot \textrm{)}$ operation also incurs excess TPE estimation error as the magnitude information of the quadratic signal is lost to some extent compared with the Lee’s TPED utilizing ${y_{Lee}}(n)$. However, the proposed TPED is sufficient for feedback CRAs which only need the TPE sign to converge to the optimal sampling instant with the aid of the feedback loop. In contrast, for feedforward CRAs with a smaller tolerance to TPE estimation error, the magnitude information of the quadratic signal is vital for an accurate estimation of the optimal sampling instant, and thus the proposed low-complexity TPED with a compromised accuracy may not be appropriate for feedforward CRAs.

Figure 3 shows the spectra of the quadratic signals and modified quadratic signals generated in the Lee’s, Gardner’s, Lee-power and the proposed TPEDs when the OSF is two. The input signals are 45 GBaud dual-polarization Nyquist 16QAM (DP-Nyquist-16QAM) signals with ROFs of 0.1 and 0.01. Here, the optical signal to noise ratio (OSNR) is set to be 22 dB, which is close to the OSNR corresponding to the feedforward error correction (FEC) bit error rate (BER) threshold of $3.8 \times {10^{\textrm{ - }3}}$. Hereinafter, the test signals and the OSNR remain the same unless otherwise stated. As we can see from Fig. 3, the synchronization spectral lines of the first two TPEDs become invisible when the ROF is reduced from 0.1 to 0.01. While, for the modified Lee-power and proposed TPEDs, the spectral lines are clearly recognizable in both cases.

 

Fig. 3. The spectra of the quadratic signals and modified quadratic signals generated within the Lee’s (a, e), Gardner’s (b, f), Lee-power (c, g) and proposed (d, h) TPEDs. Here, OSF = 2 and OSNR = 22 dB. B stands for the signal baud rate.

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Figures 4(a-f) show the variations of the synchronization spectral line when the OSFs are 2, 1.5 and 1.25, respectively. Here, the ROF is fixed at 0.01. As expected, the synchronization spectral line at the lower sideband aliases into the upper sideband when the OSF is reduced below two. Figure 4(g) shows the variations of the SNR of the synchronization spectral line versus the OSF. As we can see, the SNR is still as high as 10 and 8 dB when the OSF is reduced to 1.5 and 1.25, respectively. The synchronization spectral line in the modified Lee-power TPED has a better SNR, especially when OSF < 1.5, thanks to the dip of the background noise (marked with the red circles in Figs. 4(b, d, f)). It predicts the modified Lee-power TPED may have a better estimation accuracy at the expense of a much higher computational complexity.

 

Fig. 4. The spectra of the quadratic signals and modified quadratic signals generated in the proposed and modified Lee-power TPEDs when the OSF is 2 (a, b), 1.5 (c, d) and 1.25 (e, f), respectively. (g) shows the variations of the SNR of the spectral line versus the OSF. Here, ROF = 0.01 and OSNR = 22 dB. B stands for the signal baud rate.

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The most important and widely used TPED performance metric is the timing jitter [21,22,33]. It is defined as the variance of the zero-crossing positions of the so-called S-curves which represent the variations of the estimated TPE versus the real TPE. The timing jitter in dB unit is equal to

Figures 5(a-f) show the proposed and the modified Lee-power S-curves in a heat map manner. Here, the ROF is fixed at 0.01 and the OSFs are 2, 1.5 and 1.25, respectively. One thousand S-curves are plotted in each figure. The corresponding timing jitter is calculated from these S-curves and marked on the top of each figure. Hereinafter, two hundred symbols (N = 200 M) are used in the proposed and modified Lee-power TPEDs. We note that, for comparison, the S-curves of the existing modified Godard’s TPED versatile for OSFs are also plotted in Figs. 5(g-i). For this TPED, the number of the non-zero components in the summation operation required for clock tone generation is $ROF \cdot N/M$[21]. Thus, hereinafter, one thousand symbols (N = 1000 M) are used for the modified Godard’s TPED to guarantee that there are as least ten non-zero components when the ROF is 0.01.

 

Fig. 5. The heat maps of the S-curves for the modified Lee-power (a-c), proposed (d-f) and modified Godard’s TPEDs (g-i) when OSF = 2, 1.5, 1.25. (j) shows the variations of the timing jitter versus the OSF for the two TPEDs. Here, ROF = 0.01 and OSNR = 22 dB.

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Figure 5(j) compares the timing jitter performance of different TPEDs when the OSF is reduced from 2 to 1.1 with a step size of 0.1. Here, the ROF is fixed at 0.01. As expected, the modified Lee-power TPED has the minimum timing jitter, while the modified Godard’s TPED has the maximum jitter due to the small ROF. The proposed TPED has a medium timing jitter which is well below −20 dB over the whole range. Such a low level of timing jitter will not affect the receiver sensitivity as the signal quality is mainly determined by the amplified spontaneous emission (ASE) noise at the FEC threshold BER [21], and the timing jitter can be further reduced using more samples.

Figure 6(a) compares the timing jitter performance of different TPEDs when the ROFs varies from 0.1 to 0.001. Here, the OSNR and OSF are fixed at 22 dB and 1.25, respectively. As we can see, the proposed TPED maintains a small timing jitter well below −20 dB for all ROFs. As expected, the modified Lee-power TPED has the lowest timing jitter. The timing jitter of the modified Godard’s TPED increases sharply with a decreasing ROF. Figure 6(b) compares the timing jitter performance of the TPEDs when the OSNR varies from 18 to 28 dB with a step size of 1 dB. Here, the ROF and OSF are fixed at 0.01 and 1.25, respectively. As we can see, the modified Godard’s TPED still has the largest timing jitter. The proposed TPED maintains a small timing jitter well below −20 dB and the modified Lee-power TPED has the best timing jitter which is below −25 dB.

 

Fig. 6. The variations of the timing jitter versus the ROF when the OSNR is 22 dB (a) and versus the OSNR when the ROF is 0.01 (b). The OSF is fixed at 1.25.

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As for the digital interpolator, although the linear and cubic interpolators are sufficient for most cases when the OSF is 2 [25,28,29], they are inappropriate for non-integer OSFs below two. This is because the digital interpolator has a low pass characteristic depending on the OSF. More specifically, the digital interpolator bandwidth decreases with a decreasing OSF as the total time interval of the input sample sequence increases with a decreasing OSF for a given number of taps. We stress that the interpolator narrow filtering effect changes dynamically as the interpolator bandwidth is also dependent on ${\mu _k}$ which is updated rapidly by the NCO. The interpolator bandwidth is the smallest when ${\mu _k} = 0.5$ [34]. Therefore, the dynamic narrow filtering effects cannot be compensated by the following AEQs as the updating speed of ${\mu _k}$ is much faster than the converge speed of the AEQs. Therefore, the optimization of the interpolator is important.

Figure 7(a) and (b) show the amplitude frequency responses of the linear and cubic interpolators under different OSFs for the worst case of ${\mu _k} = 0.5$. The frequency axis is normalized by the signal baud rate B. The spectrum of the Nyquist signal with a ROF of 0.01 is also plotted. As we can see, when the OSF is 2, the linear interpolator has a 3-dB bandwidth of about 0.5B, while the cubic interpolator has a larger 3-dB bandwidth of about 0.65B. However, when the OSF is reduced to 1.25, their bandwidths are reduced to 0.31B and 0.40B, respectively, incurring significant filtering distortions. We note that the input Nyquist signal has a bandwidth of more than 0.5B. The insets in Figs. 7(a-c) demonstrate the Nyquist signal spectra before (black line) and after (red line) the digital interpolators. As we can see, the high frequency components are significantly attenuated.

 

Fig. 7. The frequency responses of different digital interpolators including the linear (a), cubic (b) and PWP (c) interpolators for the different OSFs. The insets show the signal spectra before and after the digital interpolators. (d) The variations of the 3-dB bandwidth of the PWP interpolator versus the design parameter $\beta $.

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To mitigate the filtering distortions, we use the PWP interpolator, which can provide a larger bandwidth with a lower computational complexity [35]. The PWP interpolator also requires four taps as the cubic interpolator and the four taps can be obtained by

3. Numerical simulations and experiments

To compare the performance of the feedback CRAs employing different TPEDs, both numerical simulations and experiments are carried out based on the short-reach coherent transmission system shown in Fig. 8(a). This system is first simulated using the commercial software VPI TransmissionMaker 9.1. The 45 GBaud DP-Nyquist-16QAM signals are launched into an optical fiber with a length of 2 km and received by a digital coherent receiver. The two lasers acting as the optical carrier in the transmitter and local oscillator in the coherent receiver have a linewidth of 100kHz. The frequency offset between the two lasers is set to be 100 MHz. The received optical power is adjusted by a variable optical attenuator (VOA). The four tributary electrical signals output by the coherent receiver front-end are filtered with the Bessel low-pass filters with a 3-dB bandwidth of 25 GHz to mitigate the shot and thermal noises. The sampling rate of the analog-to-digital convertors (ADCs) is varied according to the OSF. The sampling clock offset (SCO) is set to be 50 ppm. The initial sampling phase error is randomly selected. The DSP chain performs clock recovery, time-domain AEQ, carrier recovery, decision and BER counting without any upsampling. The structure of the CRA is shown in Fig. 1. As for the time-domain AEQ, we use the adaptive fractionally spaced equalizer (AFSE) proposed in [37,38]. It can realize adaptive equalization and sampling rate conversion simultaneously. As shown in Fig. 8(b), the AFSE consists of a polyphase filter bank and a 2 ${\times} $2 multiple-input-multiple-output (MIMO) complex-valued (CV) AEQ. The tap coefficients of the 2 ${\times} $2 AEQ are first updated with the constant module algorithm (CMA) for pre-convergence and then with the radius directed equalization (RDE) algorithm for better performance [39]. It can compensate for the chromatic dispersion and other inter-symbol-interference (ISI) effects caused by the components with limited bandwidth. Afterwards, the constellation is recovered with blind frequency and phase search carrier recovery algorithms [13].

 

Fig. 8. (a) Experimental setup of the 45 GBaud DP-Nyquist-16QAM system. (b) The structure of the adaptive AFSE designed for non-integer oversamples signals.

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Figure 9(a) compares the performance of the coherent receiver for Nyquist signals with a small ROF. Here, the ROF is set to be 0.01 and 0.005, respectively. The OSF is fixed at 1.25. The PWP interpolator with $\beta $=0.5 is used as the digital interpolator. As can be seen, the receiver performance is nearly the same when the proposed TPED and modified Lee-power TPED are used. In contrast, when the modified Godard’s TPED is used, the receiver sensitivity is degraded by about 1.7 (0.4) dB when the ROF is 0.005 (0.01). Figure 9(b) shows the variations of the receiver sensitivity penalty versus the ROF when the OSF is 1.25. In this figure, the benchmark is the receiver sensitivity obtained when the OSF is 2. As we can see, for the proposed and the modified Lee-power TPEDs, the sensitivity penalties are nearly the same and approximately insensitive to the ROF. The penalty maintains below 0.5 dB when the ROF varies between 0.1 and 0.001. In contrast, when the modified Godard’s TPED is used the sensitivity penalty increases sharply when the ROF is smaller than 0.01. When the ROF is smaller than 0.005, the penalty is beyond 2 dB. As the modified Godard’s TPED cannot accommodate small ROFs, we will only compare the proposed TPED with the modified Lee-power TPED in the following investigation.

 

Fig. 9. (a) The variations of the BER versus the received optical power when different TPEDs are used. The ROFs are 0.01 and 0.005, respectively. (b) The variations of the receiver sensitivity penalty versus the ROF when different TPEDs are used. In both figures, the digital interpolator used is the PWP interpolator with $\beta $=0.5 and the OSF is 1.25.

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Figure 10(a) compares the performance of the coherent receiver for different OSFs. The OSF is 2 and 1.25, respectively. Here, the ROF is fixed at 0.01 and the same PWP interpolator with $\beta $=0.5 is used as the digital interpolator. As can be seen, the two BER curves obtained with the proposed TPED nearly overlap with those obtained with the modified Lee-power TPED. For both of them, the sensitivity penalty is about 0.48 dB when the OSF is reduced from 2 to 1.25. Figure 10(b) shows the variations of the receiver sensitivity penalty versus the OSF. The benchmark is the receiver sensitivity obtained when the OSF is 2. As we can see, the sensitivity penalty obtained with the proposed TPED is nearly the same as that obtained with the modified Lee-power TPED and maintains below 0.5 dB when the OSF varies in the range of 1.25 to 2.

 

Fig. 10. (a) The variations of the BER versus the received optical power when different TPEDs are used. The OSFs are 2 and 1.25. (b) The variations of the sensitivity penalty versus the OSF when different TPEDs are used. In both figures, the ROF is 0.01 and the digital interpolator used is the PWP interpolator with $\beta $=0.5. Here, Sim and Exp stands for numerical simulation results and experimental results.

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In Fig. 10(b), the sensitivity penalty obtained with the conventional cubic interpolator is also plotted for comparison. As can be seen, with the cubic interpolator, the penalty increases sharply when the OSF is reduced below 1.6 due to the strong dynamic narrow filtering distortions. When the OSF is 1.25, the corresponding sensitivity penalty is about 1.5 and 1.8 dB, much higher than that obtained with the PWP interpolator. The results show that the PWP interpolator can improve the CRA performance significantly.

Figure 11(a) compares the performance of the coherent receiver when the PWP interpolator use different design parameters. Here, only the proposed TPED is used and the OSF and ROF are fixed at 1.25 and 0.01, respectively. As can be seen, the receiver sensitivity obtained with $\beta = 0.6$ is about 1.3 dB higher than that obtained with $\beta = 0.2$, thanks to the larger interpolator bandwidth. Figure 11(b) shows the variations of the receiver penalty versus the design parameter. It is obvious that there exists an optimal $\beta $. When the OSF is 2 and 1.25, the optimal $\beta $is about 0.3 and 0.6, respectively. For simplicity and convenience, we can fix $\beta $ at 0.5 as it allows a particularly simple Farrow structure and hardware implementation [35] and only incurs a small sensitivity penalty of about 0.1 dB.

 

Fig. 11. (a) The variations of the BER versus the received optical power when different digital interpolators are used. Here, $\beta $ is 0.2 and 0.6, respectively. The OSF is 1.25. (b) The variations of the sensitivity penalty versus $\beta $. Here the OSF is 1.25 and 2, respectively. In the two figures, the ROF is 0.01 and the proposed TPED is used. Here, Sim and Exp stands for numerical simulation results and experimental results.

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The results obtained in Fig. 10(b) and Fig. 11(b) are also validated by experiments. Figure 8(a) shows the schematic setup of the experiments. Two external cavity lasers with a linewidth of 100 kHz and wavelength of 1550 nm are used as the optical carrier in the transmitter and the local oscillator in the receiver. An arbitrary waveform generator (AWG, Keysight M8195A) with a sampling rate of 65 GSa/s and 3-dB bandwidth of 25 GHz is used to generate the four tributaries of the 45 GBaud DP-Nyquist-16QAM signals with a ROF of 0.01. A four-channel radio-frequency amplifier (CENTELLAX, OA3MHQM) with a 3-dB bandwidth of 30 GHz is used to amplify the four tributaries before using them to drive the dual-polarization optical IQ modulator (Fujitsu, FTM7977HQA) with a 3-dB bandwidth of 23 GHz. At the receiver side, we use a VOA to control the optical signal power received by the coherent analog frontend (Fujitsu ICR FIM24706) with a 3-dB bandwidth of 22 GHz. The output electrical signals are captured by a four-channel digital sampling oscilloscope (DSO, Tektronix DPO73304D) with a sampling rate of 50 GSa/s and 3-dB bandwidth of 33 GHz for offline processing. The transmission fiber is a standard single-mode-fiber (SSMF) with a length of about 2 km. The flow chart of the DSP algorithms implemented offline is also shown in Fig. 8(a). The samples acquired by the DSO are first resampled to MB with the method proposed in [40]. Afterwards, the same CRAs, ASFE and carrier recovery algorithms used in the simulations are applied. The experimental results are shown in Fig. 10(b) and Fig. 11(b). As we can see, the experimental results agree well with the simulation results. The small extra penalty may be caused by IQ skew and imbalance which are not considered in the simulations.

4. Conclusions

By now, the existing TPEDs fail for non-integer OSFs below two and ROFs close to zero. This issue hinders the effort to reduce the required sampling rate and bandwidth, and thus the power consumption and cost of the coherent transceivers which is vital for the short-reach high-speed coherent transmission systems. To solve this problem, we propose a low-complexity TPED by modifying the time-domain quadratic signal and reselecting the synchronization spectral component. As far as we know, it is the first TPED that can work for both OSFs below two and small ROFs close to zero. Numerical simulations and experiments show that, with the proposed CRA, the sensitivity penalty can be kept below 0.5 dB when the OSF varies in the range of 1.25 to 2 and the ROF varies in the range of 0.001 to 0.1 for the 45 GBaud DP-Nyquist-16QAM signals. The low-complexity time-domain TPED suitable for both OSFs below two and ROFs close to zero is appealing for short-reach high-speed coherent transmission systems which are sensitive to the power consumption and cost of the coherent transceivers.