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Abstract
A polynomial nonlinear filter (PNLF)-based electrical dispersion pre-compensation (pre-EDC) scheme assisted with Gerchberg–Saxton (GS) algorithm is proposed to compensate the chromatic dispersion (CD) for intensity-modulation and direct-detection (IM/DD) optical transmission systems, where PNLF is utilized to fit the nonlinear transfer function of the iterative GS algorithm-based pre-EDC scheme to realize a low-complexity non-iterative CD pre-compensation. The capability of PNLF to fit the nonlinear iterative process enables the PNLF-based pre-EDC scheme to compensate for CD-induced linear distortions and address CD-induced nonlinear distortions, which are typically captured through iterative approaches. Additionally, to further reduce the computational complexity, we also introduce the k-means clustering algorithm to eliminate the weight redundancy and propose a lower-complexity clustered PNLF-based pre-EDC scheme. Simulation results show that PNLF-based and clustered PNLF-based pre-EDC schemes save 76.0% and 97.5% complexity with only 0.3 dB receiver sensitivity penalty at 20% forward error correction (FEC) threshold, compared with GS-based pre-EDC scheme in C-band 56 GBaud 80-km on-off keying (OOK) system. Furthermore, the effectiveness of PNLF-based and clustered PNLF-based pre-EDC schemes is also evaluated through the experimental demonstration. Experimental results show that under C-band 32 GBaud 80-km OOK system, bit error ratio (BER) satisfying 20% FEC threshold is achieved by applying PNLF-based and clustered PNLF-based pre-EDC schemes, which save 78.3% and 94.2% complexity with only 0.4 dB receiver sensitivity penalty compared with GS-based pre-EDC scheme, respectively. The research results indicate that the (clustered) PNLF-based pre-EDC scheme has the great application potential for CD compensation in high-performance and low-cost IM/DD optical transmission systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The booming development of broadband services, such as social media, cloud computing, virtual and augmented reality (VR/AR), leads to the ongoing growth of data center traffic [1]. Due to the characteristics of low cost, low power consumption and small footprint, intensity-modulation and direct-detection (IM/DD) scheme is an attractive and more cost-efficient solution for optical data center interconnections and access networks [2–4]. Unfortunately, due to the square-law detection, chromatic dispersion (CD) of optical fiber causes severe frequency selective fading on the signal spectrum. Therefore, CD-induced distortion is the main obstacle for C-band transmission in IM/DD systems [5].
Due to the absence of phase information at the transceiver, the traditional CD compensation cannot be directly applied in typical IM/DD systems. To address this issue, various schemes able to preserve or recover the phase information have been proposed, includes self-homodyne coherent detection (SHCD), Kramers–Kronig scheme, Stokes vector direct detection (SV-DD) and so on [6–15]. These schemes essentially avoid the frequency selective fading by compensating the CD-induced phase shift. However, these schemes need additional devices, such as complex domain modulators and $90^{\circ }$ optical hybrid, which lead to a higher implementation cost compared to the typical IM/DD structure. On the contrary, typical IM/DD systems combined with complicated digital signal processing (DSP) algorithms equalizing CD-induced intensity distortions better meet the transmission requirements of low cost and low power consumption. Both decision feedback equalizer (DFE) [16] and Thomlinson Harashima precoding (THP) [17] are the autoregressive filters, which can compensate the CD-induced spectral nulls by inserting poles in z domain. Moreover, an absolute-term based nonlinear feed-forward equalizer (FFE) combined with a DFE with weight sharing (AT-NLE-WS) is proposed recently to achieve a lower-complexity CD compensation scheme [18]. In addition, maximum likelihood sequence estimation (MLSE) [19] and maximum-a-posteriori (MAP) [20] detection and their simplified schemes [21–23] are also able to compensate the CD-induced distortions. However, the high computational complexity of post-equalization schemes at the receiver results in considerable latency, making them impractical for industrial applications in optical data center interconnections and access networks. Therefore, the design of dispersion pre-compensation (pre-EDC) schemes for IM/DD systems holds practical prospects and carries significant research significance.
Recently, an iterative pre-EDC scheme is proposed [24], which enables CD pre-compensation at the transmitter without additional devices and feedback links. Additionally, only a few-tap FFE is needed at the receiver for post-equalization. This scheme treats the amplitude at the transmitter and the phase before the direct detection at the receiver as the degree of freedoms, and optimizes them jointly by Gerchberg–Saxton (GS) algorithm. By adopting this scheme, C-band 28 Gb/s 100-km single mode fiber (SMF) on-off keying (OOK) system is achieved with 1 dB Q margin above the 7% forward error correction (FEC) threshold. Lastly, a non-iterative pre-EDC scheme [25,26] implemented by a static GS-optimized real-valued finite impulse response filter (SR-FIR) is proposed based on the iterative pre-EDC scheme in [24]. This scheme achieves a C-band 56 Gb/s 80-km OOK system at 7% FEC threshold with 641-tap FIR filter. Moreover, this scheme combined with a Volterra nonlinear equalizer (VNLE) and a low-complexity functional link neural network (FLNN) equalizer is proposed, where VNLE and FLNN are utilized for nonlinear system identification and subsequent post-equalization mitigating uncompensated nonlinear sources of inter-symbol interference (ISI) [27]. C-band 112 Gb/s 20-km OOK system and 112 Gb/s 10-km four-level pulse-amplitude-modulation (PAM-4) system with a bit error ratio (BER) below the 7% FEC limit are achieved by adopting this scheme. However, the GS-optimized SR-FIR scheme focuses on mitigating the linear power fading, leaving the CD-induced nonlinear distortions unaddressed, which are typically captured through iterative approaches. In addition, a modified GS-based pre-EDC scheme is proposed to accelerate the convergence and further enhance the CD compensation performance by introducing a new degree of freedom and two error reversing factors [28,29]. This scheme achieves a C-band 56 Gb/s 80-km PAM-4 system at the BER of $3.4\times 10^{-3}$. However, the implementation of the modified GS-based pre-EDC scheme generally requires dynamic real-time multi-iterations, which leads to high computational complexity and implementation cost. Furthermore, GS-based pre-EDC scheme is a nonlinear iterative process, so it is difficult to implement this scheme by the low-complexity linear GS-optimized SR-FIR scheme.
In this paper, a novel polynomial nonlinear filter (PNLF)-based pre-EDC scheme is proposed to realize a low-complexity non-iterative CD pre-compensation, in which utilizes PNLF to fit the nonlinear transfer function of GS-based pre-EDC scheme and achieve the compensation of CD-induced linear and nonlinear distortions. Moreover, the k-means clustering algorithm [30] is introduced to further reduce the weight redundancy and computational complexity of PNLF-based pre-EDC scheme, and a lower-complexity clustered PNLF-based pre-EDC scheme is proposed. The simulation analysis and experimental verification of these two proposed schemes are performed separately in C-band 56 GBaud 80-km OOK simulation system and C-band 32 GBaud 80-km OOK experimental system. The simulation results show that under C-band 56 GBaud 80-km OOK system, the proposed PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes save 76.0% and 97.5% complexity respectively with only 0.3 dB receiver sensitivity penalty at 20% FEC threshold, compared with GS-based pre-EDC scheme. In addition, the experimental results show that under C-band 32 GBaud 80-km OOK system, the proposed PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes save 78.3% and 94.2% complexity respectively with only 0.4 dB receiver sensitivity penalty at 20% FEC threshold, compared with GS-based pre-EDC scheme.
2. Principle
2.1 Principle of (clustered) PNLF-based pre-EDC scheme
The schematic diagram of (clustered) PNLF-based pre-EDC scheme is shown in Fig. 1, and the implementation of this scheme consists of two phases.
Fig. 1. Principle of (clustered) PNLF-based pre-EDC scheme: (a) Phase I: GS-based offline training; (b) Phase II: (clustered) PNLF-based CD pre-compensation.
Phase I: GS-based offline training. Firstly, a short training sequence at the front of the data sequence is selected and upsampled to 2 samples per symbol (sps) for the offline training. And then the training sequence is input into the GS-based pre-EDC scheme and converted into a sequence with required intensity distribution of CD pre-compensation by offline iterative training. The diagram of GS-based pre-EDC scheme is shown in Fig. 1(a), which can be expressed as: (a) Initializing the receiver side samples with target amplitude $A_{Target}$ and random phase $\varphi (t)$, where $A_{Target}$ is the data sequence amplitude $|S(t)|$ shaped by a raised cosine (RC) pulse shaping filter. (b) Apply the receiver side constraint by constraining the amplitude of odd sample (T position) as $A_{odd}(t)-\alpha \cdot {error(t)}$, where $\alpha$ is the amplitude error reversing factor between [0 1] to speed up the convergence. And then transform to the transmitter side by the inverse dispersion transfer function (IDTF). (c) Apply the transmitter side constraint by constraining the phase to $0-\beta \cdot {\varphi _{Tx}(t)}$, where $\beta$ is a phase error reversing factor between [0 1] to expedite the convergence process. And then transform to the receiver side by the DTF. (d) Repeat steps (b) and (c) for $M$ iterations, and finally output a real-valued training sequence, which can evolve into the sequence with the target amplitude $A_{Target}$ after CD fiber transmission. Notely, although the optimal number of iterations varies for different combinations of error reversing factors, the GS-based pre-EDC scheme consistently converges at 20 iterations, which can be seen in Fig. 2(a). Therefore, the iteration number of offline training is fixed at 20. Furthermore, the expression of DTF is given below [31]:
where $c$ is the light velocity, $L$ is the fiber transmission distance, $D$ is the fiber CD coefficient, $f$ and $f_c$ are the baseband frequency and center frequency, respectively.Fig. 2. Simulation results in 56 GBaud 80-km OOK system at ROP = ${-18}$ dBm: (a) measured BER performance versus the iteration number with different error reversing factors of GS-based pre-EDC, (b) measured BER performance versus the memory length of PNLF-based pre-EDC with and without the error reversing factors.
Phase II: (Clustered) PNLF-based CD pre-compensation. PNLF is usually deployed at the receiver to equalize the linear and nonlinear distortions, due to its ability to fit the inverse nonlinear transfer function of the system channel. Therefore, compared with a linear FIR filter, PNLF better fits the nonlinear transfer function of GS-based pre-EDC scheme. Therefore, in the transmitter DSP, PNLF takes the training sequence obtained by offline GS-based iterative training, and equalizes the whole data sequence into a pre-distorted signal with the intensity distribution able to realize CD pre-compensation. The mathematical expression of second-order PNLF at $n^{th}$ sample is introduced as follows:
where $h_{1}(k)$ and $h_{2}(k)$ are the tap coefficients of the linear term $x(n-k)$ and second-order term $x^{2}(n-k)$, with the memory lengths of $L_{1}$ and $L_{2}$, respectively. The tap coefficients of PNLF are updated with the recursive least square (RLS) adaptive algorithm [32]. Figure 2(b) shows the the measured BER performance versus the memory length of PNLF-based pre-EDC scheme with different error reversing factors $\alpha$ and $\beta$. It can be seen that when the error reversing factors are inactive ([$\alpha =0, \beta =0$]), the GS iteration process can be effectively captured by the linear kernel of the PNLF. However, in the presence of error reversing factors to improve the performance ([$\alpha =0.15, \beta =0.1$]), the performance only converges when the nonlinear kernel of the PNLF is employed. Therefore, the inclusion of error reversing factors improve the CD pre-compensation performance and introduce nonlinear characteristics to the iterative process, which can be effectively captured by the second-order kernel of PNLF.PNLF-based pre-EDC scheme can effectively reduce the computational complexity and achieve the non-iterative CD pre-compensation compared to the GS-based pre-EDC scheme, but the tap distribution of PNLF exhibits weight redundancy, which can be eliminated by k-means clustering algorithm to further reduce the computational complexity. The mathematical expression of second-order clustered PNLF at $n^{th}$ sample is given by
where $h_{c}(k)$ represents the tap coefficient centroid with the memory lengths of $L_{c}$, and $x_{c}(n-k)$ represents the sum of linear and second-order terms within the same cluster. The training process of k-means clustering algorithm can be summarized as followed [30]: (a) The initial $L_{c}$ centroids are randomly selected from the $L_{1}+L_{2}$ estimated tap coefficients of second-order PNLF. (b) The Euclidean distance between each tap weight and every centroid is calculated, and each tap weight is reassigned to its nearest cluster. (c) The centroids are updated by recalculating the mean value of the tap weights within each cluster. (d) Repeat steps (b) and (c) until the standard metric function reaches convergence.2.2 Computational complexity analysis
Computational complexity is an important factor for the practical application of an algorithm since it is directly related to the power consumption and implementation cost. The required number of real-valued multiplications (RNRM) per symbol is used to measure the computational complexity in this paper. For GS-based pre-EDC scheme, there are three kinds of calculation items in a single iteration: (a) Apply the constraint: one real-valued multiplication is required in both receiver and transmitter constraint. (b) Update the signal after applying the constraint: a total of 4 real-valued multiplications are required at both receiver and transmitter. (c) Transformation between transmitter and receiver: the transformation is achieved by $\textit {IFFT}(\textit {FFT}(S_{Tx(Rx)}\cdot {\textit {(I)CDTF}})$, which include one FFT operation, one IFFT operation and one complex multiplication. We employ the overlap-save method [31] as a more practical approach to perform the CD block-to-block processing, rather than applying a large FFT to all symbols. It is common knowledge that one (I)FFT operation needs $(\textit {N}_\textit {FFT}/2)\textit {log}_{2}^{\textit {N}_\textit {FFT}}$ complex multiplications, and one complex multiplication requires four real-valued multiplications. Therefore, during the bidirectional transformation between transmitter and receiver, a total of RNRM per symbol is
However, PNLF-based pre-EDC scheme only requires $L_{1}+2L_{2}$ real-valued multiplications per symbol. Therefore, the computational complexity of the proposed scheme only increases linearly with the memory length of PNLF. Compared with real-time iterative high-complexity GS-based pre-EDC scheme, PNLF-based pre-EDC scheme realizes CD pre-compensation only by adopting a non-iterative low-complexity nonlinear filter. However, the weight redundancy of PNLF-based pre-EDC provides the potential to further reduce the computational complexity. Therefore, a lower-complexity clustered PNLF-based pre-EDC scheme is proposed in this paper by utilizing k-means clustering algorithm to eliminate the weight redundancy and further reduce the computational complexity of PNLF-based pre-EDC scheme. The calculation of clustered second-order terms requires an additional $L_{2}$ real-valued multiplications, therefore clustered PNLF-based pre-EDC scheme only requires $L_{c}+L_{2}$ real-valued multiplications per symbol, where $L_{c}+L_{2}<L_{1}+2L_{2}$.
3. Simulation setup, results and discussion
3.1 Simulation system setup
The effectiveness of PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes are first validated and analyzed in a C-band 56 GBaud OOK IM/DD system over an 80-km standard single mode fiber (SSMF) transmission. The simulation setup and DSP block diagram are depicted in Fig. 3. At the transmitter, a pseudo-random bit sequence (PRBS) with length of $2^{20}$ is first mapped into OOK symbols, and resampled to 2 sps for RC pulse shaping with a roll-off factor of 1.0. To resist the link CD, different pre-EDC schemes including GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC are then applied to generate the pre-distorted signal respectively, allowing for a comparative analysis of their performance. The transmitter employed an 8-bit digital-to-analog converter (DAC) with a sampling rate of 112 GSa/s to generate the analog signal. The amplitude of the analog signal is amplified by a driver to avoid the modulation nonlinearity, and then fed into a chirp-free Mach-Zehnder modulator (MZM) with extinction ratio of 35 dB for modulating the signal on an optical carrier generated by a continuous wavelength laser (CWL) at 1550 nm. Note that the bandwidth limitation of the devices in this system is simulated by the fifth-order Bessel low-pass filters (LPFs). In addition, the bandwidth of the transmitter and receiver is 35 GHz and 30 GHz, and the corresponding LPF is placed before MZM and after PD respectively. The modulated optical signal is launched into the 80-km SSMF for transmission with the launch power of 0 dBm, where the fiber dispersion parameter and attenuation parameter are 16.8 ps/nm/km and 0.2 dB/km respectively. After fiber transmission, the optical signal is first amplified by an erbium-doped optical fiber amplifier (EDFA) with a noise figure of 3.0 dB, and then input into an optical bandpass filter (OBPF) to filter the amplified spontaneous emission (ASE) noise. At the receiver, a variable optical attenuator (VOA) is utilized to adjust the received optical power (ROP). And then the received signal is detected by a PIN photodiode (PD) with responsibility of 1.0 and thermal noise density of $1\times 10^{-12}A/\sqrt {Hz}$, and carry out the photoelectric conversion. An 8-bit analog-to-digital converter (ADC) with a sampling rate of 112 GSa/s is employed to generate the digital signal at the receiver. Finally, the offline DSP procedures including FFE based equalization, down-sampling, OOK demapping and BER calculation are performed. Noting that a digital LPF is adopted before FFE in the simulation system, including resampling to 3 sps, low-pass filtering with a static 8-tap filter [33] and down-sampling to 1sps.
Fig. 3. Simulation system setup and DSP blocks for C-band 56 GBaud OOK IM/DD system over an 80-km SSMF transmission.
3.2 System performance optimization
We firstly optimize the system performance with GS-based pre-EDC scheme by traversing the roll-off factor of the RC filter, is shown in Fig. 4(a). It is observed that the measured BER performance of GS-based pre-EDC scheme improves as the roll-off factor increases, and reaches optimal performance at the roll-off factor of 1.0, which is consistent with the findings in [34]. Furthermore, Fig. 4(b) shows the measured BER performance versus overlap length and FFT size of GS-based pre-EDC scheme. According to the results in Fig. 4(b), the combination of [FFT size =32768, Overlap =80] achieving the optimal performance is adopted for GS-based pre-EDC scheme in 56 GBaud 80-km OOK system. Subsequently, the crucial parameters of PNLF-based pre-EDC, such as error reversing factors and memory length of PNLF, are optimized at a ROP of ${-18}$ dBm in terms of the BER performance. Figure 5(a) gives the convergence process of PNLF-based pre-EDC scheme with error reversing factors $\alpha$ and $\beta$. According to the result in Fig. 5(a), the optimal combination ${[\alpha =0.15, \beta =0.1]}$ is chosen for optimizing the performance. The measured BER performance of the proposed scheme versus the memory length of PNLF is analyzed and shown in Fig. 5(b). We can observe that as the memory length of linear term ${L_1}$ and second-order term ${L_2}$ increases, the measured BER decreases until it reaches saturation at BER = ${6.9\times 10^{-3}}$ when [${L_1}$ = 633, ${L_2}$ = 6]. Because the linear kernel of the filter captures the linear characteristics of the GS iterative process, increasing the memory length of the linear term can improve the linear fitting precision, so a long linear term tap is required. In addition, it can be clearly observed from the simulation results that the second-order term ${L_2}$ improves the performance more significantly than the linear term ${L_1}$. The substantial performance improvement of PNLF-based pre-EDC scheme after introducing the second-order nonlinear term of PNLF can prove that the second-order PNLF is capable of effectively fitting the nonlinear iterative process of GS-based pre-EDC and achieving the efficient CD pre-compensation. Therefore, PNLF(${L_1}$=633, ${L_2}$=6) is chosen for the following simulation in 56 GBaud 80-km OOK system.
Fig. 4. Simulation results with GS-based pre-EDC scheme in 56 GBaud OOK system over 80-km SSMF transmission at ROP = ${-18}$ dBm: (a) measured BER performance versus roll-off factor of RC pulse shaping, (b) measured BER performance versus overlap length and FFT size in the overlap-save block-to-block CD processing.
Fig. 5. Simulation results in 56 GBaud OOK system over 80-km SSMF transmission: convergence process of PNLF-based pre-EDC with (a) error reversing factors $\alpha$ and $\beta$, (b) memory length of PNLF at ROP = ${-18}$ dBm.
Besides optimizing the parameters of GS-based and PNLF-based pre-EDC schemes, we also simplify the taps of PNLF-based pre-EDC schemes with ${L_1}$ = 633 and ${L_2}$ = 6 by applying the k-means clustering algorithm, and the measured BER performance of clustered PNLF-based pre-EDC scheme versus the number of clusters ${N_c}$ is shown in Fig. 6(a). It can be seen that the performance of clustered PNLF-based pre-EDC scheme improves as the increase of cluster number, and the BER is saturated at BER = ${7.2\times 10^{-3}}$ when ${N_c=60}$ and very close to the BER of PNLF-based pre-EDC. Furthermore, Fig. 6(b) depicts the weight distribution of PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes. The rainbow-colored dashed line represents the weight distribution of 60-centroid clustered PNLF-based pre-EDC scheme, while the blue line represents the weight distribution of PNLF-based pre-EDC scheme with [${L_1=633}$, ${L_2=6}$]. It can be seen that many of the PNLF tap weights exhibit extremely close values, resulting in numerous redundant computations during the implementation process of PNLF-based pre-EDC. Additionally, the weight redundancy can be effectively mitigated by employing the k-means clustering algorithm without performance penalty.
Fig. 6. Simulation results in 56 GBaud OOK system over 80-km SSMF transmission: (a) convergence process of clustered PNLF-based pre-EDC with cluster number ${N_c}$, (b) tap weight distribution of PNLF-based pre-EDC scheme and centroid weight distribution of clustered PNLF-based pre-EDC scheme.
3.3 ROP sensitivity and computational complexity
The measured receiver sensitivity of three pre-EDC schemes are evaluated and compared in 56 GBaud 80-km OOK IM/DD systems, as shown in Fig. 7(a). Note that the post-FFE adaptive equalizer is fixed to 49-taps. As a reference, the measured receiver sensitivity of post-FFE is also depicted for comparison. The BER of post-FFE cannot reach the 20% FEC threshold (BER = $2.4\times 10^{-2}$) due to its inability to compensate the CD-induced distortions. Meanwhile, compared with GS-based pre-EDC, PNLF-based pre-EDC has a receiver sensitivity penalty of 0.3 dB at the 20% FEC threshold. This is expected since PNLF-based pre-EDC scheme substantially reduces the implementation complexity by fitting the nonlinear iterative process of GS-based pre-EDC with a slight loss of CD compensation accuracy. In addition, clustered PNLF-based pre-EDC demonstrates comparable BER performance to that of PNLF-based pre-EDC regardless of ROP, which can prove that the clustered PNLF-based pre-EDC scheme can eliminate the weight redundancy of PNLF-based pre-EDC without performance penalty.
Fig. 7. Simulation results with GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes in 56 GBaud 80-km OOK system: (a) measured BER performance versus ROP (from -23 dBm to -16 dBm), (b) computational complexity comparison.
The calculation of computational complexity for these three schemes is analyzed in Section 2.2, which is represented by RNRM per symbol. Notely, GS-based pre-EDC scheme over 20 iterations with [FFT size =32768, Overlap =80], PNLF-based pre-EDC scheme with [${L_1=633}$, ${L_2=6}$], and 60-centroid clustered PNLF-based pre-EDC scheme with [${L_c=60}$, ${L_2=6}$] is adopted. Figure 7(b) shows a comparison of computational complexity between these three schemes. The RNRM of GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC are 2686, 645 and 66, individually. Consequently, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes saves about 76.0% and 97.5% real-valued multiplications compared with GS-based pre-EDC scheme, respectively. In conclusion, compared to GS-based pre-EDC, PNLF-based pre-EDC can significantly reduce the computational complexity with minor performance penalty. Furthermore, clustered PNLF-based pre-EDC can eliminate the weight redundancy of PNLF and further reduce the implementation cost without sacrificing performance, ultimately achieving the efficient, low-complexity and non-iterative CD pre-compensation.
Figure 8(a) and Fig. 8(b) present the frequency spectra of transmitted and received signal with GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes in 56 GBaud 80-km OOK system, respectively. It can be summarized that: (1) the frequency spectra of both transmitted and received signal applying PNLF-based and GS-based pre-EDC schemes exhibit slight variations and notable resemblances. (2) Clustered PNLF-based pre-EDC and PNLF-based pre-EDC schemes exhibit identical signal spectra at both the transmitter and receiver, which demonstrates that the integration of the k-means clustering algorithm effectively eliminates the weight redundancy without degrading the performance of CD pre-compensation. (3) The spectra of the received signal indicates that all three pre-EDC schemes are capable of effectively raising the frequency near the spectral nulls introduced by CD, resulting in a flattened signal spectrum.
Fig. 8. Frequency spectra of (a) transmitted signal and (b) received signal with GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes in 56 GBaud 80-km OOK system at ROP = ${-18}$ dBm.
4. Experimental demonstration, results and discussion
4.1 Experimental system setup
In addition to verifying the effectiveness of the proposed schemes in simulation, the experimental validation is also demonstrated. However, experimental device constraints pose limitations on achieving high bit rate transmission, therefore the experimental validation is demonstrated in a C-band 32 GBaud OOK IM/DD system over an 80-km SSMF transmission. The detailed experimental setup is shown in Fig. 9. At the transmitter, PRBS with length of 75300 is mapped into OOK symbols for each frame, and resampled to 2 sps for RC pulse shaping with a roll-off factor of 0.1. Afterwards, three pre-EDC schemes are adopted for CD pre-compensation, enabling a subsequent performance comparison. Then, the data is sent to an 8-bit 64-GSa/s DAC with 16 GHz bandwidth. The generated electrical OOK signal is amplified by a 25-GHz electrical amplifier (EA) and modulated on an optical carrier with 1550.12 nm by a 10-GHz MZM operating at quadrature point. And then the optical signal is fed into an 80-km SSMF with dispersion parameter of 16.8 ps/nm/km and attenuation parameter of 0.2 dB/km. After fiber transmission, the optical signal is first amplified by an EDFA, and then input into an OBPF with 0.6-nm bandwidth to filter the ASE noise. At the receiver, a VOA is applied for the ROP variation. Subsequently, the signal is detedcted by a 30-GHz PD integrated with a trans-impedance amplifier (TIA). Afterwards, the received signal is captured by a 20-GHz real-time oscilloscope (RTO). The offline process of DSP mainly includes FFE based equalization, down-sampling, OOK demapping and BER calculation.
Fig. 9. Experimental setup and DSP blocks for C-band 32 GBaud OOK IM/DD system over an 80-km SSMF transmission.
4.2 System performance optimization
The measured BER performance versus roll-off factor of RC pulse shaping with GS-based pre-EDC scheme is shown in Fig. 10(a). It can be seen that the measured BER performance tends to degrade as the roll-off factor increases, and the optimal performance is achieved when the roll-off factor of 0.1, which is contrary to the simulation results in Fig. 4(a). In addition, Fig. 10(b) illustrates the frequency response of the system in optical back-to-back transmission. It reveals that the whole system has 3-dB bandwidth of about 3.4 GHz and 10-dB bandwidth of about 12.5 GHz. Therefore, in order to counteract the distortion caused by the severe device bandwidth limitation, the pulse shaping with smaller roll-off factor at the transmitter DSP is required in 32 GBaud 80-km OOK system, even though it will sacrifice the CD pre-equalization performance of the pre-EDC scheme whose optimal performance is achieved when the roll-off factor is 1.0.
Fig. 10. Experimental results in 32 GBaud 80-km OOK system: (a) measured BER performance versus roll-off factor of RC pulse shaping with GS-based pre-EDC scheme at ROP = ${-5}$ dBm, (b) normalized power response of optical back-to-back channel.
Based on the simulation results in Fig. 5(b) and considering the experimental implementation complexity of traversing the memory length of the linear and second-order terms in the PNLF-based pre-EDC scheme, PNLF(${L_1}$=501, ${L_2}$=8) is chosen that achieves the satisfactory performance under saturation. Additionally, after roughly traversing FFT size and overlap length, the combination of [FFT size =8192, Overlap =80] achieving the optimal performance is adopted for GS-based pre-EDC scheme in 32 GBaud 80-km OOK system. Figure 11(a) shows the measured BER performance of clustered PNLF-based pre-EDC scheme versus the number of clusters ${N_c}$ based on PNLF with [${L_1=501}$, ${L_2=8}$]. The performance of clustered PNLF-based pre-EDC scheme improves as the increase of cluster number, and the BER is saturated when ${N_c=130}$. Furthermore, the weight distribution of PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes is given in Fig. 11(b), which reveals that the weight redundancy of PNLF-based pre-EDC and the effectiveness of k-means clustering algorithm to reduce the complexity.
Fig. 11. Experimental results in 32 GBaud OOK system over 80-km SSMF transmission: (a) convergence process of clustered PNLF-based pre-EDC with cluster number ${N_c}$, (b) tap weight distribution of PNLF-based pre-EDC scheme and centroid weight distribution of clustered PNLF-based pre-EDC scheme at ROP = ${-5}$ dBm.
4.3 ROP sensitivity and computational complexity
Figure 12(a) gives out the measured receiver sensitivity of three pre-EDC schemes in 32 GBaud 80-km OOK system. Compared with GS-based pre-EDC, PNLF-based pre-EDC has a receiver sensitivity penalty of 0.4 dB at the 20% FEC threshold. In addition, the receiver sensitivity performance of clustered PNLF-based pre-EDC scheme is basically the same as that of PNLF-based pre-EDC scheme, therefore this scheme can effectively reduce the computational complexity without sacrificing performance. Figure 12(b) shows a comparison of computational complexity between these three schemes in 32 GBaud 80-km OOK system. The non-iterative PNLF-based pre-EDC scheme can reduce the RNRM from 2382 to 517 by fitting the nonlinear transfer function of the iterative GS-based pre-EDC scheme with a low-complexity second-order PNLF(${L_1=501}$, ${L_2=8}$), saving 78.3% of the computational complexity. Furthermore, the 130-centroid clustered PNLF-based pre-EDC scheme with [${L_c=130}$, ${L_2=7}$] can effectively eliminate the weight redundancy of PNLF-based pre-EDC scheme without sacrificing performance by introducing the k-means clustering algorithm, which can further reduce the RNRM to 137, thus saving 94.2% computational complexity compared with GS-based pre-EDC scheme.
Fig. 12. Experimental results with GS-based pre-EDC, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes in 32 GBaud 80-km OOK system: (a) measured BER performance versus ROP (from -17 dBm to -5 dBm), (b) computational complexity comparison.
5. Conclusion
In this paper, a PNLF-based pre-EDC scheme is proposed to achieve the electrical CD pre-compensation by a low-complexity second-order nonlinear filter. Additionally, a lower-complexity clustered PNLF-based pre-EDC scheme is proposed by introducing the k-means clustering algorithm to further reduce the weight redundancy of PNLF-based pre-EDC scheme without sacrificing the performance. The proposed schemes are simulated and demonstrated in C-band 56/32 GBaud OOK IM/DD system over an 80-km SSMF transmission with BER below 20% FEC threshold, respectively. The simulation results show that in C-band 56 GBaud 80-km OOK system, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes save 76.0% and 97.5% complexity respectively just at the price of 0.3 dB receiver sensitivity penalty at 20% FEC threshold, compared with GS-based pre-EDC scheme. In addition, the experimental results show that in C-band 32 GBaud 80-km OOK system, PNLF-based pre-EDC and clustered PNLF-based pre-EDC schemes save 78.3% and 94.2% complexity respectively with only 0.4 dB receiver sensitivity penalty at 20% FEC threshold, compared with GS-based pre-EDC scheme.
Funding
National Natural Science Foundation of China (62271080); Fund of State Key Laboratory of Information Photonics and Optical Communications (IPOC) (BUPT) (IPOC2022ZT06).
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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