## Abstract

In this paper, a memory polynomial equalizer combined with decision feedback equalizer (MPE-DFE) is proposed to eliminate channel distortions for intensity modulation and direct detection (IM/DD) systems. Compared with traditional feedforward equalizer and decision feedback equalizer (FFE-DFE), the proposed MPE-DFE introduces extra square terms and cubic terms to jointly equalize chromatic dispersion and nonlinear distortions. We demonstrated a C-band 56-Gb/s four-level pulse-amplitude modulation (PAM4) system over 80-km standard single mode fiber (SSMF) transmission. Experimental results show that the proposed MPE-DFE achieved up to 6.2 dB higher SNR than traditional FFE-DFE. Moreover, the achieved bit error ratio (BER) with MPE-DFE reaches 3.1 × 10^{−3}, which is below 7% feedforward error correction (FEC) threshold of 3.8 × 10^{−3}. To the best of our knowledge, we achieved a record transmission distance for C-band 56-Gb/s PAM4 signal with only electrical equalization at the receiver.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The proliferation of data traffic in metro and access network is driving the market to update the current short-reach transmission network [1]. In order to achieve high-speed transmission, while meeting the demand of low cost, low power consumption and small form factors, intensity modulation and direct detection (IM/DD) systems have been considered for this application [2]. Compared with traditional on-off keying (OOK) modulation, advanced modulation formats such as discrete multitone (DMT), carrier-less amplitude phase (CAP) and four-level pulse-amplitude modulation (PAM4) have attracted great attention due to their higher spectral efficiency. Among those modulation formats, PAM4 is of low complexity and has been chosen as a standard format by IEEE P802.3bs Task Force for 400-G Ethernet [3,4].

However, an intense distortion for IM/DD systems is chromatic dispersion (CD) of fiber, which gives rise to severe power fading after direct detection [5]. As a result, the symbol-rate distance product of IM/DD systems is limited. Recently, tons of works have been done to cope with CD distortion. The simplest way is to transmit signal in O-band, which fundamentally circumvents CD distortion of fiber [6]. However, compared with O-band transmission, systems operated at C-band have the merits of lower fiber loss, mature dense wavelength division multiplexing (DWDM) optics and optical amplifiers such as erbium-doped fiber amplifier (EDFA) [7]. Therefore, researchers have done extensive efforts to eliminate the dispersion-induced power fading in C-band. Optical dispersion compensation (ODC) with dispersion compensation fiber (DCF) offset CD effect of ordinary fiber, which is an obvious solution for C-band CD compensation. However, DCF introduces higher power loss and implementation cost. In addition, other CD compensation techniques mainly include single sideband or vestigial sideband (SSB/VSB) modulation, CD pre-compensation, Kramers-Kronig receiver and so on [8]. However, these schemes require complex system structure and additional expensive devices, thus increasing implementation cost and impeding their commercial application.

Another solution to alleviate CD impairment is to adopt advanced digital signal processing (DSP) techniques, which has no change in the traditional structure of IM/DD systems, but making an effort in electrical equalization to improve CD tolerance [9–11]. Among traditional electrical equalizations such as feed-forward equalization (FFE), decision feedback equalization (DFE) and maximum likelihood sequence estimation (MLSE), MLSE shows a better CD tolerance, achieving a transmission distance of 26-km standard single mode fiber (SSMF) for 56-Gb/s PAM4 signal [10]. Moreover, by using soft-output MLSE and low-density parity check (LDPC) code, S.-R. Moon et al. achieved a C-band 56-Gb/s PAM4 signal over 30-km SSMF transmission [11]. Besides, C-band 50-Gb/s orthogonal frequency division multiplexing (OFDM) signal over 60-km SSMF transmission has been successfully demonstrated with subcarrier-to-subcarrier intermixing interference technique [12]. However, in this scheme, high launch power and the corresponding self-phase modulation (SPM) effect is necessary to combat dispersion-induced power fading.

It is worth noting that deep nulls caused by dispersion-induced power fading could be compensated by inserting poles with an autoregressive (AR) filter [13]. As a matter of fact, traditional DFE is an AR filter with feedback structure and decision device. Meanwhile, DFE is generally combined with FFE in order to deal with pre-cursor interference and post-cursor interference simultaneously. Moreover, FFE-DFE has been employed in commercial PAM4 chips for 10/20 km access network application [14]. However, FFE-DFE shows poor equalization ability in nonlinear distortions such as signal to signal beating interference (SSBI) and may lead to error propagation due to feeding wrong decision symbols, especially if the SNR is low. Therefore, Tomlinson-Harashima Precoding (THP) is proposed to equalize channel distortions at the transmitter, which avoids the possibility of error propagation. Thanks to the THP and Volterra nonlinear equalization, Q. Hu et al. achieved a C-band 50-Gb/s PAM4 signal over 80-km transmission [15]. In addition, 107-Gb/s PAM4 over 40-km transmission has been experimentally demonstrated with nonlinear THP at transmitter [16]. However, THP fundamentally has a precoding loss and need channel feedback to obtain appropriate coefficients for pre-equalization [17]. Channel feedback could be complicated in real implementation and the feedback coefficients are different for different transmission distance. Therefore, it is worthy to investigate electrical equalization at receiver to adaptively equalize channel distortions.

In this work, we proposed a memory polynomial equalizer combined with decision feedback equalizer (MPE-DFE) to eliminate channel distortions at the receiver. Besides, different orders and taps of MPE-DFE are investigated to measure their equalization performance. Moreover, the computational complexity of MPE-DFE with different orders and taps is analyzed and compared. Thanks to the joint equalization of CD and nonlinear distortions, experimental results show that the proposed MPE-DFE achieved up to 6.2 dB higher signal to noise ratio (SNR) than traditional FFE-DFE. The achieved bit error ratio (BER) reaches 3.1 × 10^{−3} at a received optical power of 0 dBm, which is below 7% feedforward error correction (FEC) threshold of 3.8 × 10^{−3}. To the best of our knowledge, we achieved the first C-band 56-Gb/s PAM4 signal over 80-km SSMF transmission with only electrical equalization at the receiver.

## 2. Principle of the proposed electrical equalization

The distortions of C-band IM/DD systems can be analyzed by tracking signal generation, transmission, and detection. Generically, a real-valued unipolar signal at the transmitter with a direct current (DC) bias can be expressed as:

in which s(t) is the original signal and c(t) is the optical carrier related to DC bias.If we neglect the MZM and fiber nonlinearity for simplicity, after channel transmission of fiber, the received signal with square-law detection is represented as

*h*(

*t*) is the fiber channel function and in which

*β*

_{2}is the group velocity coefficient,

*L*denotes fiber length and

*f*represents signal frequency.

From Eq. (2), it can be found that the received signal includes three parts. The first part is the optical carrier related to the DC bias, which can be simply eliminated by a DC block. The second part is a square term of the convolution of signal and channel, which is named as SSBI. A large carrier to signal power ratio (CSPR) can effectively suppress SSBI, while effective SNR will be reduced as well. Previous work has shown that there exists a tradeoff between SSBI and effective SNR by changing the CSPR [13,18]. The third term is our desired signal, however, has suffered a severe channel distortion of fiber with cosine function of quadratic frequency, namely power fading. This dispersion-induced power fading will results in deep spectral zeros and severely distorts the signal. As mentioned above, deep spectral zeros can be compensated by inserting appropriate poles.

Taking a comprehensive consideration of those distortions in IM/DD systems, a novel equalizer structure of MPE-DFE is proposed in Fig. 1. This structure is similar to traditional FFE-DFE, whereas FFE is replaced by MPE, which contains not only input symbols but also their square terms, cubic terms and so on. Generally, three-order MPE is enough to describe the nonlinear model. As a matter of fact, MPE has been widely used to handle nonlinear distortions, such as SSBI [19, 20]. In our proposed MPE-DFE, a fractionally spaced MPE with *T*/2 symbol space is used to provide robustness against clock jitter. As for DFE, the hard-decision value of output signal is adopted with *T* symbol space. Therefore, we can define our input signal into the equalizer as

**W**and

_{F}**W**denote MPE tap coefficients and DFE tap coefficients, respectively.

_{B}In principle, the tap coefficients of **W _{F}** and

**W**can be updated with adaptive algorithm, such as least mean square (LMS) or recursive least square (RLS) algorithm. In this work, we adopted RLS algorithm to obtain a quick and stable convergence. It should be noted that the joint update of

_{B}**W**and

_{F}**W**is important for RLS algorithm [21]. Therefore, the tap coefficients are updated as follows

_{B}^{*}stands for conjugate transpose, the

**W**is a joint matrix of

**W**and

_{F}**W**, and the prior error

_{B}*e*(

*k*) is calculated by where the symbol (·)

*stands for matrix transpose,*

^{T}*d*(

*k*) is the training signal or decision signal, and

**S**(

*k*) is the input data of RLS algorithm. Besides, the

**G**(

*k*) is updated as

*λ*is the forgetting factor, and

**P**(

*k*) is a diagonal matrix which can be calculated as

## 3. Experimental setup

The experimental setup of C-band 56-Gb/s PAM4 signal over 80-km SSMF transmission is conducted as illustrated in Fig. 2(a). First of all, we generate PAM4 data with MATLAB by offline processing. Every 2^{15} bits are mapped to 2^{14} PAM4 symbols with Gray code for each frame, in which the first 1000 symbols are served as training symbols. After that, we upsample the PAM4 symbols and then do pulse shaping with a root-raised-cosine digital filter at a shaping factor of 0.4. The shaping factor is an optimal choice after taking consideration of signal bandwidth and peak to average power ratio (PAPR) [7]. Next, resampling is used to match the digital to analog converter (DAC) sampling rate to generate 28-GBaud PAM4 symbols. A Kaiser window filter with a shape parameter of 5 is used to avoid frequency aliasing [22]. Finally, the generated data is normalized to the integer range of −128 to 127 in order to match the resolution of 8 bits of DAC.

After that, the generated offline data is loaded to a 64-GSa/s DAC with bandwidth of 15 GHz. The output electrical signal of DAC is adjusted by a fixed electrical 19-dB driver and a 9-dB attenuator, which leads to a small electrical signal with *V _{pp}* of 1.6 V. The small signal would make the modulation range at the linear region as much as possible when DC bias changes. A laser with a central wavelength of 1550.12 nm is used to generate optical carrier. Then, the electrical signal is modulated to the optical domain by a 10-GHz Mach-Zehnder modulator (MZM, KG-SN-357807K). The MZM has a switching voltage of about 6 V, which can be seen in Fig. 2(b). It can be found that the electrical signal range is about a quarter of MZM switching voltage, leading to a low dynamic modulation range. Then, the bias voltage of MZM is adjusted to change the carrier power, thus to change CSPR. Afterwards, the optical signal is launched into the 80-km SSMF. The loss of SSMF is about 0.19 dB/km, and the total loss of transmission link is around 16 dB.

At the receiver, an EDFA is employed to amplify the optical signal and then followed by an optical bandpass filter (OBPF) to reduce the amplified spontaneous emission (ASE) noise. A variable optical attenuator (VOA) is used to adjust optical power into photo-detector (PD, DSC-R401HG) to measure receiver sensitivity. The electrical signal after PD is then digitized and stored by a 100-GSa/s digital phosphor oscilloscope (DPO) with 20-GHz bandwidth. Finally, offline processing is finished by MATLAB, which mainly includes resampling, synchronization, equalization, PAM4 demapping, and BER calculation. In this work, we use 5 frames (5 × 2^{15} = 163840 bits) to measure the final BER.

## 4. Experimental results and analysis

#### 4.1. Frequency response of system and power spectrum of received signal

Figure 3(a) shows the frequency response of the system in optical back to back (OBTB) transmission. The frequency response is measured by sending 20-GHz OFDM-QPSK signal. At receiver, the estimated channel response is viewed as system frequency response. It reveals that the whole system has −3 dB bandwidth of about 3 GHz and −10 dB bandwidth of about 15 GHz. As represented in section 3, the 28-GBuad PAM4 signal is shaped with Nyquist factor of 0.4. Hence, the bandwidth of PAM4 signal is 19.6 GHz. Obviously, this system is a bandwidth-limited system and the limited bandwidth could be compensated with electrical equalization at the receiver. Figure 3(b) shows the normalized power spectrum of the signal at OBTB transmission and after 80-km SSMF transmission. Due to the limited bandwidth of system, it can be seen that the power spectrum slightly decreases as the increase of signal frequency for BTB transmission. However, as for 80-km SSMF transmission, in addition to the bandwidth limit, dispersion-induced power fading effect leads to several deep nulls in the power spectrum, which leads to severe distortion in the received signal.

#### 4.2. Equalization performance

In our experiment, the CSPR is adjusted by changing DC bias. Figure 4(a) shows the relationship between CSPR and DC bias. The CSPR is calculated as the power ratio of optical carrier power to optical signal power. In order to obtain optical carrier power only, we first turn off the electrical driver and then adjust DC bias to measure different optical carrier power for different DC bias. As for optical signal power, we first adjust the DC bias to make sure the optical carrier power at its lowest point. Then, when the electrical driver is turned on, the total output power is approximately viewed as optical signal power. It is noticeable, however, immoderate adjustment of DC bias would also cause MZM nonlinearity. In this work, the range of DC bias is adjusted from 3.7 V to 5.3 V, and the corresponding range of CSPR is reduced from 14.8 dB to 8.4 dB.

In order to investigate the significance of each order for the proposed equalizer, Fig. 4(b) shows the equalization performance of MPE-DFE with different orders. In the legend of Fig. 4(b), the numbers inside bracket MPE-DFE denote tap numbers for the first order, the second order, the third order of MPE and tap number of DFE, respectively. For instance, MPE-DFE(46,21,15,7) means MPE has 46 taps, 21 taps, 15 taps for the first order, the second order and the third order respectively, while DFE has 7 taps. If there are no second-order terms and third-order terms, the proposed MPE-DFE is actually traditional FFE-DFE. It is noted the tap numbers for all orders of the equalizer have been optimized considering equalization performance and computational complexity. The detailed analysis of taps will be discussed in section 4.3. In Fig. 4(b), the signal SNR first increases and then decreases as the continuous increase of CSPR. This is because at first the SSBI is decreased as the increase of CSPR. However, for further increase of CSPR, the effective SNR is significantly reduced and has more impact than the decrease of SSBI. It can also be found that the optimal CSPR for MPE-DFE(46,21,0,7) and MPE-DFE(46,21,15,7) is about 12.5 dB, while that for MPE-DFE(46,0,15,7) and MPE-DFE(46,0,0,7) is about 13.8 dB. The reason can be explained that the second-order terms of MPE-DFE have a good performance in eliminating SSBI, which reduces the optimal CSPR value for suppressing SSBI. Thanks to the elimination of SSBI, MPE-DFE(46,21,0,7) and MPE-DFE(46,21,15,7) show much better SNR performance than MPE-DFE(46,0,15,7) and MPE-DFE(46,0,0,7). For instance, MPE-DFE(46,21,15,7) achieves 6.2 dB higher SNR than MPE-DFE(46,0,0,7) at the CSPR of 12.5 dB. Besides, when the CSPR is up to around 14.8 dB, the SSBI has been sufficiently restrained. At this time, the four equalizers show similar equalization performance. In addition, MPE-DFE(46,21,15,7) shows slightly better performance than MPE-DFE(46,21,0,7), since the third-order terms have some equalization effect in fiber nonlinearity and MZM nonlinearity.

Figure 5 shows the convergence process of equalization in terms of signal amplitude and square error with MPE-DFE(46,21,15,7). Thanks to the quick convergence of RLS algorithm, only around 500 iterations are enough for the equalization convergence. In our experiment, we used 1000 symbols as training sequences, which is more than the least 500 symbols. Meanwhile, more training sequences would provide more stable convergence.

Figure 6(a) depicts the BER curves versus received optical power with different equalizers for 80-km SSMF transmission. With MPE-DFE(46,0,0,7) and MPE-DFE(46,0,15,7), the BERs cannot achieve the 20% FEC threshold as received optical power increases to 0 dBm. However, the BER significantly reduces with the help of MPE-DFE(46,21,0,7), which reaches the 20% FEC threshold at a received optical power of around −4.6 dBm, but it cannot reach the 7% FEC threshold even when received optical power increases to 0 dBm. The 7% FEC threshold can be only achieved with MPE-DFE(46,21,15,7) at a received optical power of 0 dBm. Figs. 6(b)–6(e) further show the eye diagrams with the four equalizers at the received optical power of 0 dBm. Fig. 6(b). With MPE-DFE(46,0,0,7); Fig. 6(c). With MPE-DFE(46,0,15,7); Fig. 6(d). With MPE-DFE(46,21,0,7); Fig. 6(e). With MPE-DFE(46,21,15,17). The recovered eye diagrams of Figs. 6(b) and 6(c) are quite obscure owing to the low achieved SNRs. However, thanks for the improved SNR by eliminating SSBI, the recovered eye diagrams of Figs. 6(d) and 6(e) become more clear. The results are in good agreement with the SNR performance depicted in Fig. 4(b).

#### 4.3. Computational complexity of equalizers

Computational complexity of equalizer is a significant factor for its real application, since there is a strong relationship between complexity and power consumption. In this section, we use the number of multiplication for each output symbol to measure computational complexity, because multiplier need much more computational resources compared with adder. Noted that the computational complexity of training process is not included since it is not necessary once the equalizer is converged. We assume that tap numbers are *l*_{1}, *l*_{2}, *l*_{3} and *d* for MPE-DFE(*l*_{1}, *l*_{2}, *l*_{3},*d*). According to the analysis in section 2, the number of multiplications can be calculated as

In order to make a tradeoff between computational complexity and tap numbers, we tested different equalizers with different orders and taps. Figure 7 shows the BER performance and multiplications per symbol versus MPE-DFE with different taps. It can be found that MPE-DFE(46,0,0,7) has the lowest computational complexity of 53 multiplications for each output symbol. However, MPE-DFE(46,0,0,7) also achieves a terrible BER of 0.0693. Among those equalizers, the MPE-DFE(46,21,15,7) achieves the best BER performance of 0.0031 with the computational complexity of 140 multiplications for each output symbol. It is noted that better BER performance cannot be achieved with MPE-DFE of higher computational complexity, such as MPE-DFE(51,21,15,7), MPE(46,26,15,7), MPE-DFE(46,21,20,7) and MPE-DFE(46,21,15,12). Therefore, we chose MPE-DFE(46,21,15,7) as the final equalizer, considering computational complexity and BER performance.

## 5. Conclusions

In this paper, MPE-DFE is proposed to eliminate channel distortions for C-band IM/DD systems. Compared with traditional FFE-DFE, MPE-DFE shows much better equalization performance in the systems with dispersion-induced distortion and nonlinear distortion. Different orders of MPE-DFE are analyzed to confirm the significance of each order. Experimental results show that the second-order terms of MPE-DFE are of great significance in alleviating SSBI, thus improving the total SNR. Moreover, we compared the equalization performance and computational complexity of MPE-DFE with different orders and taps. We chose the MPE-DFE(46,21,15,7) as the optimal equalizer in our experiment. Thanks to the powerful equalization performance of MPE-DFE(46,21,15,7), we successfully achieved a C-band 56-Gb/s PAM4 signal over 80-km SSMF transmission with only electrical equalization at the receiver. We believe that this work is of significant value in equalization field for high-speed optical transmission systems.

## Funding

National Natural Science Foundation of China (NSFC) (61771062, 61871044); National Key R&D Program of China (2016YFB0800302); Fund of State Key Laboratory of IPOC (BUPT) (IPOC2018ZT08), P. R. China; BUPT Excellent Ph.D. Students Foundation (CX2019311).

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