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Abstract
The Kramers-Kronig (KK) receiver has attracted much attention in short-range optical interconnection because of its ability to recover the phase of the signal from the intensity information through KK algorithm. In high-speed KK systems, such as virtual-carrier (VC) assisted ones, an alternating current (AC) coupled photo-detector (PD) is preferred due to relaxing the requirements of analog-to-digital converter (ADC) and electronic amplifier by filtering direct current (DC) component. However, the loss of the DC component will cause the KK algorithm to break down, so it is necessary to accurately recover DC value in the digital domain with multiple-sweep. In this paper, we propose what we believe is a novel non-sweep DC component estimation scheme based on optimized digital carrier-to-signal power ratio (OD-CSPR) method, which can accurately estimate the DC component with only 3-4 iterations in the scenario of VC-assisted KK receiver optical transmission. The scheme utilizes the one-dimensional search optimization algorithm based on golden section search and parabolic interpolation without sweeping. The simulation and experimental results of the proposed non-sweep OD-CSPR method show that the DC component can be estimated accurately in a large CSPR range, and the system performance is close to that of the conventional DC-sweep method. Compared with the typical defined digital CSPR (DD-CSPR) based optimization method, the proposed one can realize optical signal-to-noise ratio (OSNR) gains of 0.9 dB in the back-to-back (B2B) and 0.7 dB under 80 km fiber transmission scenarios respectively with a total bit rate of 160Gb/s.
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1. Introduction
With the rapid development of new broadband services, the capacity demand for short-range optical interconnections is increased quickly [1]. Considering the power-consumption and cost-effective for the short-reach optical transmission systems, the direct-detection (DD) based implementation gains the most research attention [2]. However, the power fading caused by dispersion in the conventional double-sideband (DSB) DD system severely limits the transmission rate [3]. Then, single sideband (SSB) modulation-based technology has been proposed to solve this problem [4–7]. But, the SSB signal will suffer from signal-signal beat interference (SSBI) after DD, which becomes a dominant for the deterioration of SNR [8,9]. In order to mitigate SSBI influence, a frequency guard interval between the carrier and the signal should be adopted with decreased spectral efficiency [10]. Besides, by increasing the CSPR of the system is also a method to mitigate SSBI. High CSPR can be achieved by increasing carrier power or reducing signal power. However, the former will lead to fiber nonlinear effects, and both will reduce the OSNR [11]. Then, the KK receiver has been proposed with the theoretical feasibility for completely eliminating SSBI and recovering signal phase information from intensity information [12]. The most critical of KK scheme is that the minimum phase (MP) condition should be met without large CSPR. It has been demonstrated that the alternating current (AC)-coupled photo-detector (PD) can improve the performance of high-speed KK systems [13]. However, the direct current (DC) component will be lost which is indispensable for the KK algorithm to complete signal reconstruction. Therefore, before the execution of the KK algorithm, it is necessary to accurately recover the lost DC component of the detected signal.
In recent years, researchers have proposed several methods to estimate the DC component. The conventional method is the DC-sweep method, in which the optimal DC component can be determined by a wide range of sweeping the DC value with the proper bit-error rate (BER) or error-vector magnitude (EVM) [5,13]. This method can accurately determine the DC component, but it will take a lot of time and power consumption with multiple sweeps [14]. And another method based on the spectral analysis of the reconstructed signal without using higher level performance metrics, such as pre- or post-forward error correction (FEC) BER techniques, can be used to estimate the DC bias correction effectively for AC-coupled KK receivers [15]. A DC recovery method based on beat zero-padded is also proposed in [13]. The DC component is estimated by inserting a zero-padded preamble into the modulated data and measuring a couple of statistical averages of the detected waveform. Therefore, this method requires adding extra sequences of a certain length to recover the DC component. Another method determines the optimal DC component according to the system performance based on the iteration process with guess-feedback has been proposed in [16]. Recently, a low computational complexity algorithm based on CSPR was proposed to estimate the lost DC component and analyzed theoretically [17,18]. The algorithm can be realized by a digital CSPR defined as the ratio of DC over AC electrical power after the analog-to-digital converter (ADC). Here, this method is termed as defined digital CSPR (DD-CSPR) algorithm. However, the estimated DC value of this method in the virtual carrier-assisted (VC-assisted) KK receiver system would be not accurate in the low CSPR range, which will lead to the system performance degradation.
In this paper, we propose a novel optimization method termed as non-sweep optimized digital CSPR (OD-CSPR) to improve the system’s performance. The main principle of this non-sweep OD-CSPR method is one-dimensional search optimization algorithm based on golden section search and parabolic interpolation [19]. Through simple optimization algorithm, the estimated DC component is more accurate regardless of the CSPR. This method has been studied numerically and experimentally in a 160 Gb/s single carrier system both in the back-to-back (B2B) and 80 km fiber transmissions. The experimental results show that the OD-CSPR method has a similar performance to the conventional DC-sweep method with avoiding the time consumption sweeping. Meanwhile, with a comparison of the DD-CSPR method under the same conditions, the non-sweep OD-CSPR method can achieve an OSNR gain of 0.9 dB in the B2B case and 0.7 dB after 80 km optical fiber transmission respectively. Thus, the proposed non-sweep OD-CSPR method has significant performance improvement and can be applied in high-speed short-reach optical transmissions, such as data-center interconnections, etc.
2. Basic principle
For the KK receiver based SSB optical transmission system, the transmitted signal requires a carrier large enough to meet the MP condition. Therefore, the SSB complex signal at the transmitter can be expressed as,
where E0 is the carrier amplitude, S(t)=|S(t)|eiϕ is the transmitted signal, ϕ is signal phase, B is spectral shift applied to the signal, which is approximately the bandwidth of S(t). Here, with assuming the responsivity of PD to be 1.0 A/W, and neglecting the noise, the photo-current detected by the DC-coupled PD can be expressed as,In case of sufficiently large CSPR (usually higher than 8 dB), the calculation formula of AC electrical power can be assumed as [17],
And with the definition of CSPR as,
Then, we can write ${P_C}$ and $\overline {{P_S}}$ as,
Therefore, the estimated DC component value can be derived as,
Meanwhile, according to the definition of DigitalCSPR in Ref. [18], the ${P_{D{C_{est}}}}$ can be expressed as,
In this paper, we focus on the VC-assisted KK system, in which the MP condition required by the KK algorithm is met by adding a VC in the digital domain at the transmitter. This method can significantly reduce the implementation cost and hardware complexity in terms of no additional devices such as lasers or radio frequency (RF) power combiners being required. The method also enables flexible adjustment of the CSPR, and utilizes the full bandwidth of the digital-to-analog converter (DAC). Meanwhile, the carrier is phase-locked to the modulated signal, which can eliminate the need for subsequent phase compensation and then simplify the digital signal processing (DSP) at the receiver [20–25].
Then, after simulation and experimental studies, we find the same conclusion as in [17], that under the DD-CSPR algorithm will lead to inaccurate DC values in the low CSPR range, which affects the performance of VC-assisted KK systems. This is because that the virtual carrier occupied a part of the transmitted power, and then would reduce the effective OSNR of the modulated signal [25]. So, under the low CSPR and OSNR with considering the noise influence, the approximate error of Eq. (5) would be large and then, result in inaccurate DC component estimation [17]. Therefore, with aiming to reduce the approximate error of Eq. (5), a modified parameter α has been introduced to Eq. (5), then the AC electrical power can be expressed as,
As shown in Fig. 1, we set the CSPR to be 10 dB and 11 dB for B2B and 80 km transmission, respectively. And we find that the relationship between the modified parameter α and BER has the feature of a single valley interval, and the same trend also exists shown in Ref. [13]. So, the one-dimensional search algorithm based on golden section search and parabolic interpolation has been adopted to quickly determine the value of modified parameter α without the required multiple-sweep. Thus, this method is termed as non-sweep OD-CSPR method.
Fig. 1. Simulation results of BER performance vs. modified parameter α under B2B and 80 km transmission cases by the point-by-point sweeping method.
As shown in Fig. 2(a), the optimal α value can be quickly determined with only 3-4 iterations using the one-dimensional search algorithm. The BER corresponding to the α value determined by the search algorithm each time is also shown in Fig. 2(b). It can be seen that the optimal BER of the system obtained by the search algorithm is consistent with that obtained by the point-by-point sweeping method in Fig. 1. This indicates the feasibility of the one-dimensional search algorithm in determining the optimal value.
Fig. 2. Simulation results using the number of iterations of one-dimensional search algorithm under B2B and 80 km transmission cases. (a) modified parameter α ; (b) BER.
Therefore, the one-dimensional search algorithm based on golden section search and parabolic interpolation is applied to quickly determine the optimal modified parameter α in the subsequent simulation and experiment. Then, the estimated DC component value after optimization can be derived as,
According to [18], the influence of modified parameter α on the estimated value of the DC component can be characterized by the variation of DigitalCSPR. Therefore, it is equivalent to apply the non-sweep OD-CSPR method to optimize DigitalCSPR, the optimized DigitalCSPR can be expressed as,
Then the estimated DC component value after optimization and the total photocurrent can be denoted as,
Therefore, Eq. (12) and Eq. (13) are the basic start-point of our proposed OD-CSPR method for the DC component estimation to improve the system performance of VC-assisted KK scheme based SSB transmission.
And then, according to the method in Ref. [11,18], we also compare the accuracy of DC component estimation between the DD-CSPR and the OD-CSPR methods, the DC estimation error has been adopted with its definition shown below,
3. Experimental demonstration
3.1 Setup
The experimental setup and DSP modules of the VC-assisted KK transmission have been shown in Fig. 4. Firstly, a 40 GBaud 16-quadrature amplitude modulation (16-QAM) complex signal with RRC pulse shaping and a roll-off factor of 0.05 is generated and then multiplexed with the VC in the digital domain. The CSPR is adjusted by controlling the amplitude of the digital tone. After Tx-DSP, the arbitrary waveform generator (AWG) containing 120 GSa/s DAC is used to generate the signal. The output of the 2 DAC channels is then modulated on to the laser with 1550 nm wavelength using an in-phase and quadrature modulator (IQM). The output of the IQ-modulator is injected into a single-span 80 km standard single-mode fiber (SSMF) link with a launch power of 6 dBm. Note that the IQM is biased at the null-point to completely suppress the optical carrier. After the SSMF, the optical signal is amplified by an erbium-doped fiber amplifier (EDFA). At the receiver, the ASE noise is filtered by an optical filter (OF) and then detected by a PD with a bandwidth of about 50 GHz without a trans-impedance amplifier (TIA). The photodetector output is sampled by a 160 GSa/s digital storage oscilloscope (DSO) with 59 GHz bandwidth. Finally, the received signal is stored and processed offline.
Fig. 4. Experimental setup of VC-assisted KK transmission. AWG: Arbitrary waveform generator; EA: Electrical amplifier; VOA: Variable optical attenuator; OF: Optical filter; EDFA: Erbium-doped fiber amplifier; DSO: Digital storage oscilloscope. (b) Optical spectrum of the 40GBaud SSB 16QAM signal.
The yield digital signal was processed by the offline DSP. Firstly, the signal is digitally up-sampled by 6 times and then the lost DC component is recovered by using the proposed non-sweep OD-CSPR optimization method. Secondly, KK algorithm is used to reconstruct complex signal field. After frame synchronization is completed, feed-forward equalizer (FFE) is used for dispersion compensation and front-end correction. Next, orthogonalization is performed to compensate for the orthogonal imbalance, which may be caused by the instability of the IQ modulator’s bias control circuit. Then the residual impairment of the transmitter is compensated by post equalization algorithm. Finally, QAM demodulation and BER calculation are carried out.
3.2 Results and discussions
As shown in Fig. 5(a), the experimental results of the proposed non-sweep OD-CSPR method are consistent with the simulation ones in section 2, namely, the relationship between the modified parameter α and BER has indeed the feature of a single valley interval. Figure 5(b) shows the number of iterations required by the one-dimensional search optimization algorithm to determine the modified parameter α in the case of B2B and 80 km transmission with the respective CSPR of 13 dB and 15 dB. Here, OSNR is set to 38 dB. It can be seen that the proposed method only needs 3-4 iterations to reach convergence, which means that the α value has been quickly determined. When the number of iterations is 0, the corresponding curve represents the performance of the DD-CSPR method (α = 0) based transmission. The BER corresponding to the α value determined by the search algorithm each time is shown in Fig. 5(c). It can be seen that the optimal BER of the system obtained by the search algorithm is consistent with that obtained by the point-by-point sweeping method in Fig. 5(a) under the same conditions. This originated from that the modified parameter makes the best optimization of the digital CSPR and then the estimated DC value accurately. It is worth noting that the optimal value of the modified parameter would be different in the simulation and experiment due to the influences of various impact factors including the thermal noise, instrument noises, non-ideal device, and ASE noise and so on. Compared with DD-CSPR method, the system performance showed some improvement. In the case of non-optimal $\alpha $ value, the DC component would be estimated inaccurately and the signal will be distorted after the KK algorithm processing and then results in a poor reconstruction of the signal and subsequently a deterioration in system performance. So, the optimal $\alpha $ value should be adopted to make the system achieve performance improvement.
Fig. 5. (a) the point-by-point sweeping method result of BER vs. modified parameter α; (b) modified parameter α vs. the iteration number of the one-dimensional search algorithm; (c) BER vs. the iteration number of one-dimensional search algorithm, under the B2B and 80 km transmission cases.
Figure 6 plots the relationship between the optimal digital CSPR and CSPR in the experiment. In the B2B and 80 km transmission cases, the optimal digital CSPR obtained by experiments is smaller than the DD-CSPR method, which is due to the approximate error of Eq. (5) is large. The relationship between the optimized digital CSPR (OD-CSPR method) and CSPR is also shown in the figure. It can be seen that in the case of B2B or 80 km transmission, the digital CSPR obtained by proposed non-sweep OD-CSPR method is closer to the actual optimal digital CSPR obtained by experiment than the DD-CSPR method. However, in the case of transmission of 80 km, the digital CSPR obtained by the OD-CSPR method is larger than the actual optimal digital CSPR. This is because that a larger DC component is still required to prevent large negative peaks after logarithms at low CSPR, while several dB changes in digital CSPR will not significantly affect signal quality at high CSPR [18].
Next, we evaluate the accuracy of the proposed non-sweep OD-CSPR method by comparing it with the conventional DC-sweep method as well as DD-CSPR method. Firstly, we illustrate the DC component values estimated by the three methods under different received optical power (ROP) in B2B case in Fig. 7. The ROP is varied by tuning the variable optical attenuator (VOA) before the AC-coupled PD. In the conventional DC-sweep method, we sweep the DC component from 0 to 0.3 V with a step size of 0.001 V and then determine the DC component value by measuring the BER performance [13]. Here, CSPR is set to 11 dB. The results show that the DC component estimated by the OD-CSPR method is very close to that of the DC-sweep method, while the DD-CSPR method has an obvious deviation. Therefore, the non-sweep OD-CSPR method provides an accurate estimation of the DC component and benefits for the subsequent signal reconstruction with KK algorithm. In addition, it can be seen that the DC component value estimated by the OD-CSPR method is slightly lower than that obtained by the conventional DC-sweep method. This is because that a slightly higher DC value is beneficial to improve system performance with suppressing phase jump and avoiding large negative peaks [26].
Then, the system performances under different CSPRs in B2B and 80 km transmission cases in the experiment are shown in Fig. 8. In our experiment, the threshold of 7% hard decision forward error correction (HD-FEC) threshold (BER = 3.8 × 10−3) is used. It can be seen that the system performance of the proposed method is almost identical to that of the conventional DC-sweep method with only a reasonable penalty within a wide CSPR range. However, the system performance based on DD-CSPR method deteriorates significantly after 80 km optical fiber transmission. Even at the optimal CSPR of 15 dB, the error-free transmission under the 7% HD-FEC threshold cannot be reached. Meanwhile, the performances of three methods have not obvious difference when CSPR is lower than 8 dB due to the unsatisfied MP condition of the KK algorithm. In such a case, the signal phase cannot be correctly recovered by its amplitude, and then whether the DC component is correctly estimated has little influence on the system performance.
Finally, we get the optimal CSPR values of 13 dB and 15 dB respectively for B2B and 80 km transmission in the experiment from Fig. 8. And the BER performance vs. OSNR of the three methods in the experiment is shown in Fig. 9. Obviously, the performances of our proposed non-sweep OD-CSPR method and conventional DC-sweep method have a good agreement under the two configurations. There is almost no difference in system performance under different OSNRs. This again proves the significant improvement of our proposed OD-CSPR method in the experiment. When the optimal CSPR is 13 dB in B2B case, the KK receiver system based on the proposed OD-CSPR method can realize a OSNR gain of 0.9 dB with a comparison of the DD-CSPR method. The optimal CSPR is 15 dB for 80 km optical fiber transmission. Compared with the DD-CSPR method, the proposed OD-CSPR method also has a gain of 0.7 dB at 20% soft decision forward error correction (SD-FEC) threshold (BER = 2.4 × 10−2). Thus, the proposed non-sweep DC estimation method shows its accuracy and performance improvements in the transmission systems.
4. Conclusion
In this paper, a novel non-sweep DC component estimation method based on one-dimensional search optimization algorithm has been proposed to accurately recover the required DC value used to reconstruct the transmitted single-sideband signal in the VC-assisted KK receiver system. By using the one-dimensional search optimization algorithm to determine the additional modified parameter to optimize the digital CSPR, the proposed non-sweep OD-CSPR method can significantly improve the system performance without the need for extensive sweeping of the conventional DC-sweep method, or additional data sequences in other methods. To verify the accuracy of OD-CSPR method, the simulation and experimental validation have been implemented in a 160 Gb/s optical SSB 16-QAM signals under B2B and 80 km transmission cases. The results show that the proposed non-sweep OD-CSPR method can estimate the DC component effectively in a wide range of CSPR and the system performance is close to that of the conventional DC-sweep method. Under the same experimental conditions, the proposed method can realize OSNR gain of 0.9 dB and 0.7 dB respectively under the B2B and 80 km transmission. Therefore, the proposed method with the performance improvement and low complexity has the potential in the high-speed short-range optical transmission systems.
Funding
National Natural Science Foundation of China (62022029); Guangdong Provincial Pearl River Talents Program (2019ZT08X340); Guangdong Guangxi Joint Science Key Foundation (2021GXNSFDA076001).
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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