Investigation of the low-complexity Hilbert FIR filter enhanced 112-Gbit/s SSB 16-QAM transmission with parallelized Kramers-Kronig reception over 1440-km SSMF

Yuyang Liu; Yan Li; Yan Li; Jingwei Song; Ming Luo; Zhixue He; Chao Yang; Jianxin Lv; Jifang Qiu; Xiaobin Hong; Hongxiang Guo; Jian Wu; Jian Wu

doi:10.1364/OE.438268

1. Introduction

Direct detection (DD) technology has attracted much attention due to its cost-effectiveness, easy implementation, and lower power consumption [1–3]. However, the traditional DD system based on the double side-band (DSB) modulation suffers from the dispersion-induced radio frequency (RF) power fading, as well as a lower spectral efficiency and the limited transmission distance due to the discarded phase information in the square-law detection [4,5]. Thus, optical single side-band (SSB) modulation has been adopted to enhance the spectral efficiency and avoid the RF-power fading [6]. However, a nonlinearity distortion known as the signal-signal beat interference (SSBI) caused by the intensity detection has indeed impaired the optical SSB DD systems [7]. The recently proposed digital signal processing (DSP) technique called Kramers-Kronig (KK) receiver is able to solve this problem as long as the minimum phase condition is satisfied [8–11]. In this DD reception technique, the phase of the signals can be uniquely extracted from the log-photocurrent via the Hilbert transform or the Hilbert transform finite impulse response (FIR) filter [12–13]. Due to the full field reconstruction of the KK receiver, numerous advanced DSP algorithms can be implemented to enhance the maximum transmission distance and the total transmission capacity [14].

The KK receiver is generally implemented digitally, in which some nonlinear operations such as root and logarithm can be implemented by the look-up table (LUT) without a digital filter [15]. Hence, the Hilbert transform, as the intermediary of connecting the phase and the magnitude, the implementation of the low-complexity and high-performance discrete-time (DT) Hilbert transform FIR (HT-FIR) filter is extremely important. Generally, there are four types of the linear-phase FIR filters, Type I and Type II with symmetric impulse response show better performance for digital interpolation or electrical filter implementation as well as Type III and Type IV with antisymmetric impulse response show better performance for adaptive equalizer [3,16]. Meanwhile, the design of the FIR filter has been widely elaborated, and there are two main approaches, that is, the time-domain (TD) windowing function and the frequency domain (FD) sampling function [17–20]. A laudable goal would be to investigate which filter type and which design approach would perform better in applying the Hilbert transform for KK receivers. However, there are two main problems that should be considered when the HT-FIR filter is implemented in the KK receiver, which have been mentioned in our previous work, that is the edge effect and the Gibbs phenomenon [8,14]. The edge effect is induced by the “incomplete” convolution operation at both ends of a parallel block and the Gibbs phenomenon is unsatisfactory in-band flatness, ripple, and the group delay owing to the FIR filter [21–23]. Thus, different HT-FIR filters have their own characteristic, and the degree of the edge effect and the Gibbs phenomenon is also not the same. So far, 267-Gbit/s 16-QAM signals using KK reception based on the TD HT-FIR filter over 300-km and 112-Gbit/s 16-QAM signals using KK reception based on the FD HT-FIR filter over 1440-km have been successfully transmitted [24–26]. However, the performance analysis considering the in-band flatness, ripple, the edge effect, and the Gibbs phenomenon of which in KK receiver has not been fully investigated yet.

In this paper, we experimentally investigate and conclude the performance of four HT-FIR filter schemes based on the frequency-domain sampling approach and time-domain windowing function approach considering the in-band flatness, ripple, group delay, the edge effect, and the Gibbs phenomenon. The performance of four HT-FIR filter schemes and the limit transmission performance of the practical KK receiver are validated through a parallelized block-wise KK reception-based 112-Gbit/s optical SSB 16-QAM transmission system over a 1920-km cascaded Raman fiber amplifier (RFA) link. The experimental results show that when the transmission distance is up to 1440-km, the bit-error-rate (BER) of the FD-based HT-FIR filter scheme can be lower than the soft decision-forward error correction (SD-FEC) threshold of 2 × 10⁻² with only 3 samples per symbol (3-SPS) upsampling rate and 8 non-integer tap coefficients are used, while other TD-based HT-FIR filter schemes with BER lower than the SD-FEC threshold require at least 4-SPS upsampling rate.

2. Principle, complexity analysis, and numerical validation

2.1 HT-FIR filters

The time-domain (TD) impulse response and frequency-domain (FD) transfer function of the conventional Hilbert transform can be described as follows:

(1)$$h(t) = \frac{1}{{\pi t}}$$

(2)$$H(f) ={-} j\cdot sign(f)$$

where sign(f) is the signum function.

We will then introduce four commonly used Type-III and Type-IV HT-FIR filter schemes based on the FD sampling approach and the TD windowing function approach. Figure 1 shows the schematic diagrams of four HT-FIR filter schemes, wherein, (a)-(c) are based on the TD windowing function approach and (d) is based on the FD sampling approach. Figure 1(a) and (b) show the Type. IV S-I HT-FIR filter scheme with even tap number (the following is abbreviated as S-I scheme), and the Type. III S-II HT-FIR filter scheme with odd tap number (the following is abbreviated as S-II scheme), respectively [16,19,24]. Consider that Eq. (1) is singular at t=0 and the symmetry of discrete HT-FIR filter, there are two schemes to obtain the discrete Hilbert impulse response function. We discretize Hilbert transform in TD directly and truncate the discretized infinite filter sequence by using a rectangle window sequence g_L[n]:

(3)$${h_{1,L}}[n] = {T_s}{h_1}[n]\cdot {g_L}[n] = { {{T_s}h(t)} |_{t = \left( {n\textrm{ - }\frac{1}{2}} \right)\cdot {T_s}}}\cdot {g_L}[n]\textrm{ = }\frac{1}{{\pi \left( {n - \frac{1}{2}} \right)}}, n ={-} \frac{L}{2}\textrm{ to }\frac{L}{2}$$

(4)$${h_{2,L}}[n] = {T_s}{h_2}[n]\cdot {g_L}[n]\textrm{ = }{ {{T_s}h(t)} |_{t = n\cdot {T_s}}}\cdot {g_L}[n] = \left\{ {\begin{array}{lc} {\frac{1}{{\pi n}}}&{,n ={-} \frac{L}{2}\textrm{ to }\frac{L}{2}}\\ 0&{,n = 0} \end{array}} \right.$$

where T_s is the sampling period and L is the filter length. The Type. III S-III HT-FIR filter scheme with odd tap number (the following is abbreviated as S-III scheme) shown in Fig. 1(c) has been deduced in [24]:

(5)$${h_{m,L}}[n] = \left\{ {\begin{array}{cc} {\frac{2}{{\pi n}}{{\sin }^2}\left( {\frac{{\pi n}}{2}} \right)}&{,n ={-} \frac{L}{2}\textrm{ to }\frac{L}{2}}\\ 0&{,n = 0} \end{array}} \right.$$

Fig. 1. Schematic diagrams of different HT-FIR filter schemes: (a)-(c) are based on the time-domain (TD) windowing function approach and (d) is based on the frequency-domain (FD) sampling approach.

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Finally, Fig. 1(d) shows the Type. III S-IV HT-FIR filter scheme with odd tap number (the following is abbreviated as S-IV scheme) based on frequency-domain sampling approach, which has also been studied in [14]:

(6)$${H_{F,L}}[n] = \left\{ {\begin{array}{lc} j&{,1 \le n < \frac{L}{2}}\\ 0&{,n = \left\{ { - \frac{L}{2},0} \right\}}\\ { - j}&{, - \frac{L}{2} < n < 0} \end{array}} \right.$$

Then, the inverse discrete Fourier transform (IDFT) is performed on Eq. (6) to obtain the time-domain HT-FIR filter sequence as follows:

(7)$${h_{F,L}}[n] = IDFT\{{{H_{F,L}}[n]} \}$$

The odd points of h_F_,L[n] are zero, which is proved in Appendix. The advantage of S-IV HT-FIR filter is that corresponding FD magnitude response H_F_,L(f) will pass through the points of the FD extracted function H_F_,L[n] and the in-band flatness will be better, which we will investigate in detail in following section [21–23]. Figure 2 shows the TD impulse response and FD magnitude response of HT-FIR filters, ideal HT is used as a reference.

Fig. 2. (a) TD impulse response and (b) FD magnitude response of different HT-FIR filters. Black solid line is the Hilbert transform and the black arrows indicate that the Hilbert transform is infinite in FD. The gray dotted box is the low-frequency part of the HT-FIR filters.

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2.2 Numerical validation and performance analysis

The comparison of four HT-FIR filter schemes is performed in Table 1 in the aspects of the computational complexity, filter type, group delay, and the design approach. The principle of the computational complexity has been investigated in detail in [14,15,27]. Then, we investigate the performance of different HT-FIR filter schemes in a 112-Gbit/s (33.8-GHz of bandwidth) 16-QAM optical numerical simulation system similar to the experiment (see section 3). Assuming that the simulation system noise is dominated by amplifier spontaneous emission (ASE) noise, while chromatic dispersion (CD) and other distortions like bandwidth limitation are not considered to investigate the impact of HT-FIR filters independently. Thus, the algorithm used for channel equalization like the multi-module algorithm (MMA) is not adopted.

Table 1. Comparison of different HT-FIR filter schemes

View Table

The BER performance versus digital upsampling rate using different 9-tap HT-FIR filter schemes is first performed when CSPR=5 dB and N_i=16. It should be noted that the carrier-to-signal power ratio (CSPR) is defined as the ratio of the carrier power to the signal power [28]. As Fig. 3 shows, for all HT-FIR filter schemes, the BER performance is improved as the digital upsampling rate increases and after reaching an optimal upsampling rate, the BER performance tends to be poor, this trend of BER performance is caused by the Gibbs phenomenon and we will illustrate in detail as follows. When the optimal BER performance is obtained, the optimal upsampling rate is 132-GHz (4.7-SPS), 148-GHz (5.3-SPS), 128-GHz (4.6-SPS), and 112-GHz (4-SPS) for the S-I, S-II, S-III, and S-IV schemes, respectively. Moreover, the BER performance of the S-IV scheme is similar to that of the S-II scheme while the computational complexity of the S-IV is 30.6% lower than that of the S-II scheme when the optimal performance is obtained.

Fig. 3. BER performance versus digital upsampling rate (also refers to SPS) when different HT-FIR filter schemes are used, CSPR=5 dB and N_i=16.

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Due to the Gibbs phenomenon, there is an optimal upsampling rate for each HT-FIR filter. Figure 4 shows the magnitude spectrum of the recovered phase with digital upsampling rates of 84-GHz (3-SPS), 112-GHz (4-SPS), 168-GHz (6-SPS), and 224-GHz (8-SPS) when 9-tap HT-FIR filters are used. It should be noted that the magnitude spectrum is the recovered phase rather than the full complex field after KK receiver. Besides, the HT scheme is used as a reference, which is based on the frequency-domain multiplication approach with a sufficient long FFT size [14,29]. Take the S-IV HT-FIR filter scheme as an example, when the digital upsampling rate is as low as 84-GHz (3-SPS), the spectrum broadening caused by logarithm and root operations makes the digital upsampling rate of 84-GHz (3-SPS) unable to fully meet the Nyquist sampling rate, which leads to a large inter symbol interference (ISI) and poor phase retrieval performance [8,28]. The optimal phase recovery performance is achieved at the digital upsampling rate of 112-GHz (4-SPS), when the digital upsampling rate is further increased to 168-GHz (6-SPS) and 224-GHz (8-SPS), although more abundant spectrum components are obtained, the low-frequency part of the signal is indeed deteriorated due to the Gibbs phenomenon of the HT-FIR filter as Fig. 4 shows, which further affects the performance of phase recovery [14,22]. Therefore, a trade-off between the reduced ISI and the Gibbs phenomenon is essential to obtain an optimal upsampling rate when the HT-FIR filter with a limited tap number is adopted. The above is the reason for the existence of the optimal upsampling rate when using the S-IV HT-FIR filter, and the same reason for other HT-FIR filter schemes.

Fig. 4. Magnitude spectrum of the recovered phase (a) with a upsampling rate of 84-GHz (3-SPS), (b) with a upsampling rate of 112-GHz (4-SPS), (c) with a upsampling rate of 168-GHz (6-SPS), and (d) with a upsampling rate of 224-GHz (8-SPS) when 9-tap HT-FIR filters are used. HT: Hilbert transform. SPS: samples per symbol.

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It can be observed that the low-frequency part transition of the S-IV HT-FIR filter scheme gets less steep in comparison with other HT-FIR filter schemes (see Fig. 2(b) gray dotted line box). In this case, when the upsampling rate is higher, the performance of which would be worse as the low-frequency part of the signals will suffer more Gibbs phenomenon. Thus, the optimal sampling rate for the S-IV HT-FIR filter scheme will be less than that of the other HT-FIR filter schemes. Nevertheless, the in-band flatness and ripple of the S-IV HT-FIR filter (0.44dB) scheme are more satisfied compared with other HT-FIR filter schemes (S-III of 1.44dB, S-II of ∼7dB, and S-IV of 1.28dB), resulting in lower in-band distortions and better performance. Moreover, it should be noted that although the magnitude response of the S-I HT-FIR filter is similar to that of the Hilbert transform, the performance of the S-I HT-FIR filter is the worst. In Table 1, the group delay of different HT-FIR filters is performed, and the group delay for the S-I HT-FIR filter scheme is a non-integer as the tap number of which is even. Such a non-integer group delay of the S-I HT-FIR filter scheme will cause the recovered phase and the intensity mismatch, and further resulting in a poorer performance [21–23]. Although additional phase shifters or resampling function can be used to solve this problem, it will sacrifice additional computational complexity and we also discuss it in the experiment results section. Thus, in the numerical simulation validation, the mismatch of the S-I HT-FIR filter scheme will not be corrected to show the performance deterioration caused by mismatch.

Then, we analyze the impact of the tap number of different HT-FIR filter schemes when CSPR=9 dB, OSNR = 28 dB and the commonly used upsampling rate is considered as Fig. 5 shows. It should be noted that OSNR includes the carrier power and signal power in this simulation. When the tap number of HT-FIR filter schemes is larger than 35, the S-IV and S-III HT-FIR filter schemes outperform the other HT-FIR filter schemes. When a digital upsampling rate of 84-GHz (3-SPS) is used, the BER of the S-I HT-FIR filter scheme is poor as the KK phase reconstruction is strongly impaired by the mismatch without compensation. Besides, compared with the other two HT-FIR filter schemes, the BER of S-IV and S-III schemes tend to be stable at lower tap number, and the performance is superior. When the digital upsampling rate is increased to 112-GHz (4-SPS) and 168-GHz (6-SPS), more tap number is essential to obtain a better BER performance. It should be noticed that, as Fig. 3 and Fig. 5 show, when 9-tap HT-FIR filter schemes and 168-GHz (6-SPS) digital upsampling rate are employed, the S-II outperforms the other schemes. Nevertheless, considering the computational complexity, we generally do not consider the digital upsampling rate up to 168-GHz (6-SPS), so the S-IV is still the focus of the following discussion.

Fig. 5. BER performance versus tap number when different HT-FIR filter schemes and digital upsampling rate are considered. SPS: samples per symbol.

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Next, the BER performance versus CSPR, OSNR, and the commonly used digital upsampling rate are considered as Fig. 6 shows. When the digital upsampling rate is as low as 84-GHz (3-SPS), the S-III and S-IV HT-FIR filter schemes outperform the other S-I and S-II HT-FIR filter schemes, but the optimal CSPR is 1 dB higher than the S-II HT-FIR filter scheme. Further increasing the digital upsampling rate to 112-GHz (4-SPS), the performance gap between the S-III/S-IV and S-I/S-II HT-FIR filter schemes becomes more obvious. In addition, the error-free BER can be obtained for the S-IV and S-III schemes when OSNR is larger than 24 dB and the tap number of the HT-FIR filter is larger than 17. Finally, due to the impact of the Gibbs phenomenon, the BER is little discarded when the digital upsampling rate is 168-GHz (6-SPS). Therefore, the 17-tap S-IV HT-FIR filter scheme with a digital upsampling rate of 112-GHz (4-SPS) is effective to realize an improved KK phase retrieval performance.

Fig. 6. BER performance versus CSPR and OSNR with different digital upsampling rate: (a) and (d) 84-GHz (3-SPS), (b) and (e) 112-GHz (4-SPS), (c) and (f) 168-GHz (6-SPS) when 9-tap, 17-tap, and 33-tap HT-FIR schemes are considered. SPS: samples per symbol.

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3. Experimental setup and DSP flow

This section presents the experimental setup of 112-Gbit/s SSB signals optical transmission system and different HT-FIR filter schemes-based KK reception approach as Fig. 7 shows. At the transmitter-side DSP, a pseudo-random bit sequence (PRBS) with a length of 2¹⁴−1 is used for the 16-QAM signal generation and after the gray-mapper, the signal is up-sampled to 2 samples per symbol for root-raised cosine (RRC) shaping with a roll-off factor of 0.2. The discrete signal samples are then imported to the 65-GSa/s arbitrary waveform generator (AWG, Keysight M8195A) with a 3-dB bandwidth of 25-GHz and the symbol rate is set to be 28-GBaud, the baseband bandwidth of the signals thus is 16.8-GHz. Then, two linear electrical amplifiers (EAs, SHF s807c) with a 3-dB bandwidth of 50-GHz are used to amplify the signals and subsequently send to the I/Q modulator (Fujitsu FTM 7961EX/301). An optical carrier (Laser 1) with a center frequency of 193.4-THz is fed into the I/Q modulator biased at its null point. To satisfy the minimum phase condition, a continuous wave (Laser 2) tone is generated at the left sideband of the signal with a frequency offset of 17-GHz relative to the center of the signal’s spectrum. The optical carrier of the signal and the CW tone are both generated from a 4-channel tunable laser source (Keysight N7714A, linewidth<100-kHz). The polarization controller is used to ensure that the signals and CW tone are in the same polarization state and a polarization-maintaining erbium-doped fiber amplifier (PM-EDFA) is used to amplify the signal to expand the range of CSPR adjustment. Then, an EDFA and a variable optical attenuator (VOA) are employed to adjust the optical power launched into the fiber. The fiber transmission link is made up of multi-spans standard single-mode fiber (SSMF) of 80-km with Raman fiber amplifiers (RFA). At the receiver side, the received signal is amplified by another EDFA and then filtered by an optical band-pass filter (OBPF) with a 3-dB bandwidth of 0.4-nm to remove the out-of-band noise and the nonlinear distortion induced by the fiber transmission. At last, the amplified signal is detected by an AC-coupled photodetector (PD) with a bandwidth of 70-GHz, sampled by an 80-GSa/s digital sampling oscilloscope (DSO) with the 36-GHz bandwidth (Lecroy LabMaster 10-36Zi-A), and processed in offline DSP.

Fig. 7. The experimental setup of 112-Gbit/s SSB signals optical transmission system. RRC: root-raised cosine; AWG: arbitrary waveform generator; PM-EDFA: polarization-maintaining erbium-doped fiber amplifier; VOA: variable optical attenuator; RFA: Raman fiber amplifiers; OBPF: optical band-pass filter; PD: photodetector; DSO: digital sampling oscilloscope.

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The receiver-side DSP is shown in Fig. 7. The received intensity signal is first processed by different HT-FIR filter schemes-based KK filed reconstruction algorithms. Wherein, Hilbert transform (HT) is used as reference (Ref.) to show the performance gap with the other HT-FIR filter schemes. It should be noted that the HT scheme is based on the frequency-domain multiplication approach with an FFT size equals to the length of the input signal as mentioned in [14,29]. Then, we perform matched filter based on the RRC filter, IQ quadrature imbalance compensation based on the Gram–Schmidt orthogonalization procedure (GSOP) algorithm, frequency-domain electrical dispersion compensation (EDC), and Gardner algorithm-based T/2-clock recovery (CR). Afterwards, linear equalization based on the multi-modulus algorithm (MMA) with 15-tap is used to compensate for the imperfect channel response. Frequency offset estimation (FOE) based on the Fast Fourier Transform (FFT) and carrier phase estimation (CPE) based on the Viterbi-Viterbi phase estimation (VVPE) are executed to compensate the laser frequency shift, initial phase offset in KK receiver, and laser linewidth induced phase noise. Decision-directed extended Kalman filter (DD-EKF) is performed to deal with the residual phase noise and relative intensity noise induced by laser sources, nonlinear phase noise induced by fiber nonlinearity considering an over 1000-km transmission, and nonlinearity distortions of electrical drivers and photodetector [3,30]. Finally, BER is calculated using 10 million points after de-mapping and decision.

4. Results and discussions

At first, we compare and investigate the parallelized performance of different HT-FIR filter schemes in back-to-back (B2B) case using the overlap (OLA) approach to avoid the edge effect [14]. Figure 8 shows the BER performance versus CSPR and upsampling rate when different 17-tap HT-FIR filter schemes are employed. The block length (N_p) and OLA length (K) are set to be 16, and 3, respectively, the interpolation can be realized by a 16-tap FIR filter (N_i) while the bandwidth of the SSB signal is 28×(1 + 0.2)/2 + 17 = 33.8-GHz. Obviously, the BER performance is improved as CSPR increases, but tends to be unchanged when CSPR is larger than 12dB. It should be noted that since the bandwidth limitation of the RF-cable and other optoelectronic devices, the CSPR demand is higher [14,24]. When the upsampling rate is increased from 84-GHz (3-SPS) to 112-GHz (4-SPS), the S-IV HT-FIR filter scheme outperforms the other three schemes. The performance gap between the S-IV HT-FIR filter and the Hilbert transform is due to the Gibbs phenomenon of the S-IV HT-FIR filter with a lower tap number and the interpolation FIR filter as well as the residual edge effect. Furthermore, the performance gap between different HT-FIR filter schemes can be summarized in the following three aspects: (i) in-band flatness and ripple, (ii) group delay of the HT-FIR filter and (iii) the number of points used for convolution operation at both ends of a parallel block. For the (i) and (ii), in the case of S-I and S-II, Fig. 2(b) shows the frequency-domain magnitude response of S-I and S-II HT-FIR filter schemes. On the one hand, it can be observed that the magnitude response of the S-II filter is poor, the 3-dB bandwidth is almost lower than 0.45π rad/sample, and such poor in-band flatness will seriously deteriorate the phase retrieval performance. On the other hand, though the magnitude response of the S-I filter is better, the in-band ripple is around 1.2dB and the group delay of the S-I filter is (16-1)/2 = 7.5 while others group delay (τ) is (17-1)/2 = 8, such a non-integer group delay of S-I filter will cause the phase and intensity of KK receiver mismatch as mentioned above [21–23]. Although additional phase shifters or resampling function can be used to solve this problem, it will sacrifice additional computational complexity and higher power consumption. We solve this problem by adding the VVPE algorithm in DSP, which we also briefly introduced in Section III, but the mismatched intensity and phase will still affect the final performance. In the case of S-III and S-IV, Fig. 2 also shows the frequency-domain magnitude response of S-III and S-IV HT-FIR filter schemes. The performance gap between the S-III and S-IV HT-FIR filter schemes is that the in-band ripple of S-III is around 1.24dB while the S-IV of which is around 0.44dB. Thus, the S-IV scheme has better in-band flatness and lower ripple in comparison with the S-III scheme, further resulting in a more accurate phase retrieval performance. For (iii), we will discuss in detail in what follows.

Fig. 8. The BER performance versus CSPR and upsampling rate when 17-tap HT-FIR filter schemes are used. Hilbert transform is used as a reference. CSPR: carrier-to-signal power ratio.

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Then, the BER performance versus parallel block length (N_p), upsampling rate, and OLA length (K) with different tap number of HT-FIR filter schemes is elaborated at CSPR=12dB. We calculate the optimal OLA length (K) value by the unbiased estimator and the squared error between two adjacent points [14]. As Fig. 9(a) shows, when the 17-tap S-IV HT-FIR scheme with a parallel block length of 16 is adopted, the optimal K value is 3 while the upsampling rate is 84-GHz (3-SPS) and 112-GHz (4-SPS). However, when the other 17-tap HT-FIR schemes with a parallel block length of 16 are employed, the optimal K is at least 5 for both 84-GHz (3-SPS) and 112-GHz (4-SPS) upsampling rates, almost twice as much as the S-IV scheme. As mentioned above (iii), since the tap coefficients of S-II and S-I filter schemes are non-zero real values (see Fig. 2), the number of points used for convolution operation at both ends of a parallel block will be more than that when S-III and S-IV schemes with half zero-tap coefficients are used. Thus, a longer OLA length (K) is necessary to obtain a better parallelized performance. Moreover, when the upsampling rate, tap number, parallel block length and OLA length is 84-GHz (3-SPS), 17, 16 and 3 (red circle in Fig. 9(a)), the saturated BER of the S-IV (1.38 × 10⁻⁵) is the same as that of S-II and S-I when the upsampling rate, tap number, parallel block length and OLA length is 112-GHz (4-SPS), 17, 16 and 4 (yellow circle in Fig. 9(c) and blue circle in Fig. 9(d)). Similarly, when 33-tap HT-FIR filter schemes are adopted, the performance of the S-IV scheme with an 84-GHz (3-SPS) upsampling rate is the same as the S-II and S-I schemes with a 112-GHz (4-SPS) upsampling rate.

Fig. 9. The BER performance versus different tap number of HT-FIR filter schemes, parallel block length (N_p), upsampling rate, and OLA length (K). OLA: overlap approach.

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In order to investigate the transmission performance of different HT-FIR filter schemes-based KK receiver, fiber transmission experiment is implemented with Raman amplification link and our fiber transmission link is cascaded rather than loop control (LC) to investigate the limit transmission performance of KK reception. It should be noted that the 17-tap FIR filter with a upsampling rate of 84-GHz (3-SPS) and 112-GHz (4-SPS) are used and block length as well as OLA length is set to be 16 and 3, respectively. Note that the Hilbert transform-based KK receiver with a upsampling rate of 112-GHz (4-SPS) is used as a reference (Ref. HT curve). We investigate the transmission performance every 240-km, the CSPR and the optical launch power have been optimized in the experiment as Fig. 10 shows. When the transmission distance is up to 960-km, the BER of S-IV scheme and S-III scheme with 4-SPS upsampling rate can be lower than the 7% hard-decision forward error correction (HD-FEC) threshold of 3.8 × 10⁻³, while others can just reach 720-km under the HD-FEC threshold due to the non-integer group delay (τ), higher in-band ripple, poor in-band flatness and residual ISI. Similarly, the BER of the S-IV scheme with 3-SPS and 4-SPS as well as S-III scheme with 4-SPS upsampling rate can be lower than the soft-decision FEC (SD-FEC) threshold of 2 × 10⁻², while others can just reach 960-km or 1200-km under the SD-FEC threshold. Finally, we concluded that the improved BER performance of the FD-based HT-FIR filter is caused by the following aspects: (i). lower edge effect due to half zero-tap coefficients, (ii) a relatively smooth in-band flatness with a ripple of 0.44 dB, (iii) an integer group delay due to the odd tap number, and (iv) the FD sampling approach is able to ensure that the spectrum of the FIR filter passes through the designed FD sampling points and overcomes the spectrum leakage caused by the adoption of the rectangular window in TD windowing function approach [16,22]. Thus, we show a feasible implementation in the future low-complexity and high-performance the practical KK receiver up to thousands of kilometers only by using the FIR filters.

Fig. 10. BER versus transmission distance when upsampling rate are 84-GHz (3-SPS) and 112-GHz (4-SPS) with different HT-FIR filter schemes. Ref. HT: reference Hilbert transform.

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5. Conclusion

We have investigated and demonstrated the performance of four Hilbert transform FIR (HT-FIR) filter schemes based on the frequency-domain (FD) sampling approach and time-domain (TD) windowing function approach considering the in-band flatness, ripple, group delay, the edge effect, and the Gibbs phenomenon in the KK receiver. We have also analyzed the computational complexity and elaborated the performance of the tap number of HT-FIR filter, CSPR, and OSNR in numerical simulation. Furthermore, the performance of different HT-FIR filter schemes is validated through a parallelized block-wise KK reception-based 112-Gbit/s optical SSB 16-QAM transmission system over a 1920-km cascaded Raman fiber amplifier (RFA) link. The experimental results show that when the transmission distance is up to 1440-km, the bit-error-rate (BER) of the FD-based HT-FIR filter scheme can be lower than the soft decision-forward error correction (SD-FEC) threshold of 2 × 10⁻² with only 3 samples per symbol (3-SPS) sampling rate and 8 non-integer tap coefficients are used, while other TD-based HT-FIR filter schemes with BER lower than the SD-FEC threshold require at least 4-SPS sampling rate. We show a feasible implementation in the future low-complexity and high-performance practical KK receiver up to thousands of kilometers only by using FIR filters.

Appendix

In this section, we derive Eq. (7) with half zero-tap coefficients. We first provide the discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT):

(8)$${H_{F,L}}[k] = \sum\limits_{j = 1}^L {{h_{F,L}}[i]W_L^{(i - 1)(k - 1)}}$$

(9)$${h_{F,L}}[i] = \frac{1}{L}\sum\limits_{k = 1}^L {{H_{F,L}}[k]W_L^{ - (i - 1)(k - 1)}}$$

where L is the tap number of the S-IV HT-FIR filter scheme, and

(10)$${W_n} = {e^{( - 2\pi j)/n}}$$

Thus, considering the tap coefficients of the odd part of the S-IV HT-FIR filter:

(11)$$\begin{aligned} {h_{F,L}}[2p + 1] &= \frac{1}{L}\sum\limits_{k = 1}^L {{H_{F,L}}[k]W_L^{ - 2p(k - 1)}} \\ &= \frac{1}{L}\sum\limits_{k = 1}^{L/2} {{H_{F,L}}[k]W_L^{ - 2p(k - 1)}} + \frac{1}{L}\sum\limits_{k = L/2 + 1}^L {{H_{F,L}}[k]W_L^{ - 2p(k - 1)}} \end{aligned}$$

We take the tap coefficient from Eq. (6) to Eq. (11), it should be noticed that the H_F_,L[0] and H_F_,L [L/2 + 1] are 0, thus, Eq. (11) can be simplified as follows:

(12)$$\begin{aligned} {h_{F,L}}[2p + 1] &= 0 + \frac{1}{L}\sum\limits_{k = 2}^{L/2} {{H_{F,L}}[k]W_L^{ - 2p(k - 1)}} + 0 + \frac{1}{L}\sum\limits_{k = L/2 + 2}^L {{H_{F,L}}[k]W_L^{ - 2p(k - 1)}} \\ &= \frac{j}{L}W_L^{ - 2p}({W_L^{ - pL} - 1} )\sum\limits_{k = 1}^{L/2 - 1} {W_L^{ - 2pk}} \end{aligned}$$

Then, we take Eq. (10) into the middle term of the Eq. (12), and we have:

(13)$${h_m}[2p + 1] = \frac{j}{L}W_L^{ - 2p}\left( {{e^{\frac{{2\pi j}}{L}pL}} - 1} \right)\sum\limits_{k = 1}^{L/2 - 1} {W_L^{ - 2pk}} = \frac{j}{L}W_L^{ - 2p}({{e^{p2\pi j}} - 1} )\sum\limits_{k = 1}^{L/2 - 1} {W_L^{ - 2pk}}$$

Finally, we notice that when p is an integer, the e^p2πjis always equals to 1, thus:

(14)$${h_m}[2p + 1] = \frac{j}{L}W_L^{ - 2p}({1 - 1} )\sum\limits_{k = 1}^{L/2 - 1} {W_L^{ - 2pk}} = 0,p = 0,1,2,\ldots ,\frac{{L - 1}}{2}$$

Therefore, half of the tap coefficients (odd points) of the S-IV HT-FIR filter is 0.

Funding

National Key Research and Development Program of China (2019YFB1803601); National Natural Science Foundation of China (61875019, 62021005); Fundamental Research Funds for the Central Universities.

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 maybe obtained from the authors upon reasonable request.

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Tag	Filter	Computational complexity^a	Filter type	Group delay	Design approach
S-I	h_1,L[n]	[N_i+L/2 + 2]M(N_p+2 K) + (N_p+2 K)N_i	Type IV	(L-1)/2	TD windowing function
S-II	h_2,L[n]	[N_i+(L-1)/2 + 2]M(N_p+2 K) + (N_p+2 K)N_i	Type III
S-III	h_m,L[n]	[N_i+(L-1)/4 + 2]M(N_p+2 K) + (N_p+2 K)N_i
S-IV	h_F,L[n]	[N_i+(L-1)/4 + 2]M(N_p+2 K) + (N_p+2 K)N_i			FD sampling

Investigation of the low-complexity Hilbert FIR filter enhanced 112-Gbit/s SSB 16-QAM transmission with parallelized Kramers-Kronig reception over 1440-km SSMF

Abstract

1. Introduction

2. Principle, complexity analysis, and numerical validation

2.1 HT-FIR filters

2.2 Numerical validation and performance analysis

3. Experimental setup and DSP flow

4. Results and discussions

5. Conclusion

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (1)

Equations (14)

Optics Express