A Decision Feedback Equalization Algorithm Based on Simplified Volterra Structure for PAM4 IM-DD Optical Communication Systems

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Abstract

A novel simplifying Volterra structure algorithm is proposed for an intensity modulation direct detection (IM-DD) optical fiber short distance communication system using the decision feedback equalization algorithm (DFE). Based on this algorithm, the signal damage for the four-level pulse amplitude modulation signal (PAM-4) is compensated, which is caused by device bandwidth limitation and dispersion during transmission. Experiments have been carried out using a 25 GHz Electro-absorption Modulated Laser (EML), showing that PAM-4 signals can transmit over 10 km in standard single-mode fiber (SSMF). The 112 Gbps and 128 Gbps signals can reach the error rate threshold of KP4-FEC (BER = 2 × 10⁻⁴) and HD-FEC (BER = 3.8 × 10⁻³), respectively. The simplified principle and process of the proposed Volterra-based equalization algorithm are presented. Experimental results show that the algorithm complexity is greatly reduced by 75%, which provides effective theoretical support for the commercial application of this algorithm.

1. Introduction

In recent years, with the emergence of 5 G, 4 k/8 k HD video, cloud computing and other new applications, as well as the massive demand for data from artificial intelligence, the traffic of short-distance optical transmission has shown explosive growth. A large number of connections within and between data centers urgently require a larger capacity of short-range optical transmission systems to support this part of traffic [1]. At present, the transmission rate of data centers has basically realized a commercial scheme of 400 Gbit/s and is moving toward 800 Gbit/s or even 1.6 Tbit/s [2]. Different from long distance optical communication systems, short distance transmission systems are very sensitive to cost, power consumption, complexity and other factors. Therefore, in the design of the system, the transmission efficiency, device cost and signal processing scheme need to be comprehensively considered.

IM-DD (intensity modulation and direct detection) systems are usually adopted in the scenario of the optical interconnection of data centers. In order to control transmission costs, short-range transmission systems are often limited by device bandwidth. In recent studies, various high-order modulation formats have been proposed to improve the spectral efficiency of signals, such as PAM (pulse amplitude modulation), discrete multi-tone (DMT), carrier-free amplitude and phase modulation (CAP, carrier-less amplitude and phase), etc. [3,4], among which four-level pulse amplitude modulation (PAM-4) is favored for its simple implementation and high spectral efficiency. The IEEE802.3 research team has also proposed various implementation schemes for 400 Gbit/s and 800 Gbit/s. However, in IM-DD systems, there will still be various limitations, such as inter-signal crosstalk (ISI) caused by insufficient system bandwidth, frequency selective fading effect caused by laser chirp and dispersion and signal distortion caused by receiver square rate detection [4].

In order to further improve the transmission rate and distance of the system, especially to break through the transmission limit under the condition of keeping low-cost components, DSP (digital signal processing) technology has become a new research hot spot. Feed Forward Equalizer (FFE) and Decision Feedback Equalization (DFE) are two widely used equalization techniques. These two equalizers are simple in structure and can eliminate the linear distortion of the signal at the receiving end. However, in the IM-DD system, nonlinear distortion is inevitable, including the frequency power fading effect caused by laser chirp and dispersion as well as signal-signal beat noise (SSBN) caused by receiver square law detection. For electrical absorption modulated laser (EML), due to the dependence of the refractive index on the applied electrical signal, the change in the optical intensity is accompanied by phase modulation (PM), which causes transient frequency chirp. According to the research, when the chirp factor of EML changes, the transfer function of the IMDD system will fade at some frequencies, thus degrading the signals [5,6,7]. As for SSBN noise, it is generated because the strength of the signals is obtained under direct detection, which is the square of the amplitude. Thus, after square law detection, the signals will contain SSBN noise [8,9]. In optical fibers, self-phase modulation (SPM), cross-phase modulation (XPM) and four-wave mixing (FWM) all cause nonlinear noise. However, when the transmission distance is short, the fiber nonlinearity will not cause great damage to the signal quality. Therefore, we will not discuss fiber nonlinearity too much. We could use some dispersion compensation module or delay interferometer, which can weaken the effect of dispersion so as to extend the transmission distance [10,11]. However, such devices will greatly increase the system cost and incur an insertion loss on the link. In this case, scholars are committed to researching various nonlinear equalization algorithms to suppress such signal damage. For example, the equalizer with the Volterra structure can effectively eliminate the nonlinear damage of the signal. In [4], a 50 km transmission of a 50 Gb/s PAM4 signal is demonstrated using a direct modulated laser with a bandwidth of 10 GHz, in which a 275-tap Volterra equalizer is used. The transmission achieved −6 dBm at a bit error rate of 4.7 × 10⁻³. There is also an 80 km transmission with 50 Gbps PAM4 using C-Band DML (Direct Modulation Laser), in which a Volterra-DFE post-equalizer is used. The BER finally reaches SD-FEC (Soft Decision-Forward Error Correction) limit (2.2 × 10⁻²) [12]. In addition, equalization algorithms using maximum likelihood sequence estimation (MLSE) have also been studied recently. A transmission of 56 Gb/s PAM-4 signals over 26.4 km SSMF (Standard Single-Mode Fiber, YOFC G652) has been demonstrated using 23 GHz MZM (Mach-Zehnder Modulator) and PIN-TIA detectors (Discovery Semiconductors, DSC10ER) [13]. There are also researchers achieving 112 Gbps C-band transmission by using transmitter-side Tomlinson-Harashima precoding (THP) and MLSE [14]. Although there is relatively good compensation for signal transmission damage, the structure of the Volterra equalizer and maximum likelihood sequence estimation algorithm is complex and requires large memory and power consumption, which will increase the operation cost of the system, so there is still a long distance to go before commercialization. Thus, many scholars have worked to simplify such equalizers. For example, a sparse Volterra filter has been proposed to realize nonlinear pre-distortion in DML-based OFDM transmission, in which the proposed pre-distorter is specially designed to prevent significant changes in the powers of transmitted subcarriers [15]. There are also researchers proposing and experimentally demonstrating a trellis-compression MLSE in a 210 Gb/s PAM-8 signal transmission over a 2 km SSFM transmission. They found trellis-compression MLSE (TC-MLSE) can reduce the complexity by 98% with only a 0.2 dB penalty compared with conventional MLSE [16].

In this paper, the Volterra structure is simplified by the derivation of signal transmission and the absolute value operation is proposed to replace the square operation, by which a simplified nonlinear-DFE equalizer is realized. The complexity of this equalizer is compared with other related equalizers. Finally, in our experiments based on a 25 GHz bandwidth EML, 112 Gbit/s and 128 Gbit/s PAM-4 signals can transmit over 10 km. By comparing the equalization effects of different equalizers, the feasibility of the proposed algorithm is proved.

2. Principle of Simplified Second-Order Volterra-DFE Equalizer

2.1. Volterra Transmission Model for EML-IMDD Systems

Considering the nonlinear damage of signals in short-range transmission systems, such as the nonlinear effect of optical fiber, the frequency fading effect caused by the interaction of laser chirp and dispersion, and the signal-signal beat noise (SSBN), the nonlinear equalizer with the Volterra structure has been widely used and studied by researchers. The traditional second-order Volterra equalizer in the discrete-time domain can be expressed as:

$$y\left(n\right)={\displaystyle \sum _{l_1=0}^{L_1-1}{h}_1\left(l_1\right)x\left(n-l_1\right)}+{\displaystyle \sum _{l_1=0}^{L_2-1}{\displaystyle \sum _{l_2=0}^{l_1}{h}_2\left(l_1,l_2\right)x\left(n-l_1\right)x\left(n-l_2\right)}}$$

(1)

where $x\left(n\right)$ and $y\left(n\right)$ denote input and output real-valued signals; $h_1\left(l_1\right)$ and $h_2\left(l_1,l_2\right)$ represent the first and second Volterra kernel coefficients; and $L_1$ and $L_2$ are the corresponding memory length. It should be noted that the first and second terms in Equation (1) are dedicated to dealing with linear distortion and SSBN noise. The computational complexity is mainly determined by the second kernel, whose numbers are L₂ × (L₂ + 1)/2. During the equalization process, the kernel coefficients can be obtained at the training stage by algorithm.

The DFE equalizer can be represented as a transverse filter [17]. Unlike the FFE equalizer, the DFE equalizer combines the feedforward portion and the portion of the symbol value that has been decided together to produce an estimate of the current symbol. The use of a feedback equalizer makes the DFE equalizer nonlinear, which can effectively reduce the crosstalk of the previously sent symbol sequence to the currently decided symbols. The principle of the decision feedback equalizer can be expressed as:

$$y\left(n\right)={\displaystyle \sum _{l=0}^{L-1}h\left(l\right)x\left(n-l\right)+}{\displaystyle \sum _{k=1}^{K}g\left(k\right)y^{\prime }\left(n-k\right)}$$

(2)

where $x\left(n\right)$ and $y\left(n\right)$ are the signal sequences before and after the equalization, and $y^{\prime }\left(n\right)$ is the sequence that has been decided. $h\left(l\right)$ and $g\left(k\right)$ are the kernel coefficients of FFE and DFE, respectively, while $L$ and $K$ represent the number of taps of FFE and DFE, respectively.

2.2. Second-Order Volterra Transmission Model for EML-IMDD Systems

In order to simplify the Volterra-DFE equalizer, we derive the second-order Volterra transmission model for the IMDD system based on the physical model of the photoelectric device used in practice. The transmission model shows that in the second-order beat noise term of the received signal, the coefficient of the signal square term is much larger than that of the delayed signal product term.

First, the normalized optical field of the EML output is expressed as

$$E\left(t\right)=\sqrt{1+X\left(t\right)}\exp \left[j\phi \left(t\right)\right],$$

(3)

where $X\left(t\right)$ and $\phi \left(t\right)$ represent the normalized current signal of the modulated laser and optical phase. The optical phase is determined by the frequency chirp of the EML laser, which can be expressed as [18]:

$$\mathrm{\Delta }\upsilon =\frac{d\phi \left(t\right)}{dt}=\frac{\alpha }{2}\frac{1}{P\left(t\right)}\frac{dP\left(t\right)}{dt},$$

(4)

where $\alpha$ is the linewidth enhancement factor of EML. From the formula $P\left(t\right)=P_0\left(1+X\left(t\right)\right)$ represents the output optical power, the phase can be written as:

$$\phi \left(t\right)=\frac{\alpha }{2}\ln P_0+\frac{\alpha }{2}\ln \left[1+X\left(t\right)\right].$$

(5)

Here, the first term on the right has nothing to do with data modulation and can be eliminated in derivation. We only consider the second term and put it into equation (3). The normalized optical field expression can be written as:

$$E\left(t\right)=\sqrt{1+X\left(t\right)}{\left[1+X\left(t\right)\right]}^{j\alpha /2},$$

(6)

Since the fiber channel can be modeled as a linear time-invariant system, the optical signal can be expressed as $S\left(t\right)=E\left(t\right)\ast h\left(t\right)$ after transmission through the fiber, where $h\left(t\right)$ is the pulse response of the fiber and $\ast$ is the convolution operator. After direct detection or called square law detection at the receiver, the discrete received signal at time exponent n can be expressed as $Y_n=S_n\cdot S_n^{\ast }$, that is, the signal multiplied by its conjugate [8,19]. If $h\left(t\right)$ and $X\left(t\right)$ are converted into discrete time form $h_k$ and $X_k$, $Y_n$ can be expressed as [8]:

$$\begin{array}{c}{Y}_n\approx {\displaystyle \sum _k{\displaystyle \sum _{w}R[h_k{h}_{k+w}^{\ast }\{1+\frac{1+j\alpha }{2}{X}_{n-k}+\frac{1-j\alpha }{2}{X}_{n-k-w}}}\\ -\frac{1+{\alpha }^2}{8}\left(X_{n-k}^2+X_{n-k-w}^2\right)+\frac{1+{\alpha }^2}{4}{X}_{n-k}{X}_{n-k-w}\}],\end{array}$$

(7)

where $R(·)$ means taking the real part. The impulse response $h_k$ of the fiber channel in the discrete time domain is given by the following formula [20]:

$$h_k=\sqrt{\frac{c{T}^2}{jD{\lambda }^2{L}_f}}\exp \left(j\frac{\pi c{T}^2{k}^2}{D{\lambda }^2{L}_f}\right),$$

(8)

where $D$ is the dispersion coefficient of the fiber, $\lambda$ is the wavelength of the optical carrier, $L_f$ is the length of the fiber, and $T$ is the sampling period. The value range of $k$ is [21]:

$$k\in \left[-\lfloor \frac{D{\lambda }^2{L}_f}{2c{T}^2}\rfloor ,\lfloor \frac{D{\lambda }^2{L}_f}{2c{T}^2}\rfloor \right],$$

(9)

where $\lfloor ·\rfloor$ is the down operation of the integer. If we simplify this further, we obtain:

$$\begin{array}{c}{Y}_n\approx 1+{\displaystyle \sum _{k}R\left(h_k\right)}{X}_{n-k}+{\displaystyle \sum _{k}R\left(j{h}_k\alpha X_{n-k}\right)\hspace{1em}\hspace{1em}\hspace{1em}}\\ -{\displaystyle \sum _{k}R\left(h_k\right)}\frac{1+{\alpha }^2}{4}{X}_{n-k}^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ +{\displaystyle \sum _k{\displaystyle \sum _{w}R\left[h_k{h}_{k+w}^{\ast }\left(\frac{1+{\alpha }^2}{4}{X}_{n-k}{X}_{n-k-w}\right)\right].}}\end{array}$$

(10)

The linear and nonlinear parts of $Y_n$ can be obtained by combining the first-order and second-order terms about $X$ with similar terms, respectively.

$$\begin{array}{c}{Y}_n\approx \underset{\mathrm{Linear}\mathrm{Part}}{\underbrace{{\displaystyle \sum _{k=0}^{L_M-1}{\stackrel{\wedge }{h}}_L\left(k\right)}{X}_{n-k}}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ +\underset{\mathrm{Nonlinear}\mathrm{Part},}{\underbrace{{\displaystyle \sum _{w=0}^{L_M-1}{\displaystyle \sum _{k=0}^{L_M-1-w}{\stackrel{\wedge }{h}}_{NL}\left(k,w\right)}}{X}_{n-k}{X}_{n-k-w}}}\end{array}$$

(11)

where linear kernels ${\stackrel{\wedge }{h}}_L\left(k\right)$ and nonlinear kernels ${\stackrel{\wedge }{h}}_{NL}$ can be expressed as:

$${\stackrel{\wedge }{h}}_L\left(k\right)=R\left[h_k+j\alpha h_k\right],$$

(12)

$${\stackrel{\wedge }{h}}_{NL}\left(k,0\right)=-\frac{1}{4}\left(1+{\alpha }^2\right)R\left(h_k-h_k{h}_k^{\ast }\right),$$

(13)

$${\stackrel{\wedge }{h}}_{NL}\left(k,w\right)=\frac{1}{4}\left(1+{\alpha }^2\right)R\left(h_k{h}_{k+w}^{\ast }\right),$$

(14)

In the nonlinear kernel described in the formula, the second-order term mainly represents the beat noise caused by square law detection at the receiver.

In order to study the nonlinear kernels, optical fiber parameters are substituted into Equations (12)–(14), and the kernel values of the signal square terms ${\stackrel{\wedge }{h}}_{NL}\left(k,0\right)$ and non-square terms ${\stackrel{\wedge }{h}}_{NL}\left(k,w\right)$ can be obtained, as shown in Figure 1. The values of the parameters are dispersion $D=3.5\mathrm{ps}/\mathrm{nm}/\mathrm{km}$, wavelength $\lambda =1310\mathrm{nm}$, fiber length $L_f=10\mathrm{km}$ and EML chirp coefficient $\alpha =1$, respectively.

As can be seen from Figure 1, for the EML laser, when $w=0$, that is, when the corresponding beat terms are the signal square terms, its kernel coefficients are much larger than the nonlinear kernel coefficients when $w\ne 0$. This indicates that in second-order nonlinear terms, the signal square terms have a greater influence on the received signal than other nonlinear terms. This is because, in the O-band, the crosstalk between the signal numbers caused by fiber dispersion is small. After detection by the square law in the receiver, the beating noise is dominated by the signal square terms. According to this result, Volterra’s nonlinear kernels can be deleted to remove unnecessary second-order nonlinear terms to simplify the structure of the equalizer.

2.3. The Construction of a Simplified Equalizer

For the traditional Volterra structure, the second-order kernel coefficient of Volterra can first be written into the matrix structure, as shown in Figure 2, whose memory size is N = 5, which represents the product term between the signal itself and the five symbols before and after it. We can see that the main diagonal represents the square term of the signals, and the other diagonal represents the product term of the neighboring signals, where there are 15 kernel coefficients that correspond to 15 different product terms.

As the length of the signal memory increases, the number of kernels increases dramatically, which increases the cost and complexity of computation. According to the above simulation in Figure 1, the SSBN noise brought by the signals with a longer distance in square detection is much smaller than that brought by the signal itself. Therefore, the matrix can be simplified by setting the second-order kernel coefficients away from the main diagonal to 0 and keeping only the square term represented by the main diagonal, thus reducing the number of taps. As shown in Figure 3, if only the main diagonal is retained, then there are only 5 s-order kernel coefficients, which will greatly reduce the complexity of the equalizer.

In an equalizer, the total number of taps and multiplications can be used as indicators to measure the algorithm complexity. In the IM-DD system based on EML, the receiver carries out square detection and the sampled signal is a real signal. Therefore, in order to reduce the number of multiplications, on the basis of the deletion in Figure 3, the square operation of the second-order kernels can be converted into a relatively simple absolute value operation, thus increasing computing speed compared with the second-order equalizer. Since the absolute value of the real signal can contain the information after square operation, this simplification will only change the value of the tap coefficient but will not affect the process of coefficient matrix convergence and equalization.

The structure of the simplified nonlinear-DFE equalizer based on Equations (1) and (2) can be expressed as:

$$\begin{array}{c}y\left(n\right)={\displaystyle \sum _{l_1=0}^{L_1-1}{h}_1\left(l_1\right)x\left(n-l_1\right)+{\displaystyle \sum _{l_2=0}^{L_2-1}{h}_2\left(l_2\right)\left|x\left(n-l_2\right)\right|}\hspace{1em}\hspace{1em}\hspace{1em}}\\ +{\displaystyle \sum _{k_1=1}^{K_1}{g}_1\left(k_1\right)y^{\prime }\left(n-k_1\right)}+{\displaystyle \sum _{k_2=1}^{K_2}{g}_2\left(k_2\right)\left|y^{\prime }\left(n-k_2\right)\right|}\end{array}$$

(15)

According to Equation (15), we analyze the algorithm complexity of the three equalizers.

Table 1. Equalizer complexity comparison.

Equalizer	Taps	Multiplication
DFE	$L_1+K_1$	$L_1+K_1$
Volterra-DFE	$\begin{array}{l}{L}_1+\left(L_2+1\right)L_2/2\\ +K_1+\left(K_2+1\right)K_2/2\end{array}$	$\begin{array}{l}{L}_1+\left(L_2+1\right)L_2\\ +K_1+\left(K_2+1\right)K_2\end{array}$
Simplified Volterra-DFE	$L_1+L_2+K_1+K_2$	$L_1+L_2+K_1+K_2$

shows a comparison of the complexity of the three equalizers, including the number of taps and multiplications. When $\left(L_1,L_2,K_1,K_2\right)=\left(41,13,21,7\right)$, the number of taps required for the second-order Volterra-DFE combined equalizer before and after simplification is 181 and 82, respectively, which is reduced by 55%. The number of multiplications required was 300 and 82, respectively, a reduction of more than 70%. Compared with the traditional DFE equalizer, which requires 62 taps and multiplications, the simplified second-order Volterra-DFE adds 20 nonlinear kernels to reduce nonlinear damage during signal transmission.

3. Experiment Setup and Results

3.1. Experiment Setup

In order to verify the effect of the proposed equalizer, an experimental transmission system of 112 Gbit/s and 128 Gbit/s based on an EML laser was built. The system construction is shown in Figure 4. The DSP of transmitter and receiver are completed offline.

First, a pseudo-random binary sequence (PRBS) is generated and encoded into a PAM-4 signal. The signal is then pulse-shaped by a root-rise cosine filter. The generated signal is resampled and loaded into the arbitrary waveform generator (AWG, Keysight M8196A) with a sampling rate of 92 GSa/s and a 3 dB bandwidth of 32 GHz. The generated electrical signal then modulates an EML module, which has a 3-dB bandwidth of 25 GHz. The EML operates at ∼1310 nm and has a chirp factor of 1. The maximum output power is 10 dBm and the linewidth is 100 kHz.

After transmitting through an SSMF, the receiving power of the signal is adjusted by a variable optical attenuator (VOA). After the PAM-4 signal is detected and received by a PIN-PD, it is amplified using an RF amplifier and then sampled using an oscilloscope with a 3 dB bandwidth of 36 GHz at 160 GSa/s. Finally, the captured waveforms are processed offline. The digital signal process (DSP) after detection includes resampling, matched filtering, equalization and PAM-4 decoding. Bit error rate (BER) measurements are performed by direct error counting against symbols.

3.2. Results Discussion

Figure 5 shows the frequency response of the transmission system. We can see that the 3 dB bandwidth is only 26 GHz. In this case, the power fading in the high-frequency region caused by insufficient bandwidth will cause the signal to produce inter-code crosstalk, thus degrading the quality of the transmitted signal. The transmission rate is 56 GBaud/s and 64 GBuad/s. In order to compensate for the signal damage caused by insufficient system bandwidth, the roll-off factor is set to 0.25 when pulse-shaping is performed at the transmitter.

Figure 6 and Figure 7 show the BER curves of signals with different transmission rates under different conditions. The feedforward taps and feedback taps in the DFE equalizer are 31 and 17, respectively. The number of first-order and second-order taps in the feedforward part of Volterra-DFE is 31 and 45, respectively, and the number of taps in the feedback part is 17 and 15, respectively. The number of taps in the simplified Volterra-DFE is composed of 31 and 9 taps in the feedforward part and 17 and 5 taps in the feedback part. In the second-order part, the absolute value operation is used instead of the square operation, which significantly reduces complexity. After simplification, the number of taps is reduced from 108 to 62, which is reduced by 42%. As can be seen from Figure 6, when the transmission rate is 112 Gbit/s, the main factor affecting the transmission BER is the device bandwidth. The signal is not damaged by dispersion; thus, the BER level is low. Using the proposed simplified Volterra-DFE equalizer, BER can reach the KP4-FEC threshold (2 × 10⁻⁴) at −1 dBm received optical power. When the transmission rate is 128 Gbit/s, the HD-FEC BER threshold (3.8 × 10⁻³) can be reached using each equalizer. The simplified Volterra-DFE equalizer has a 1 dB improvement over the traditional DFE equalizer. In addition, the results show that the effect of Volterra-DFE equalization is similar before and after simplification, which means the simplification is effective.

Figure 8 and Figure 9 show the BER curves of different equalizers after signals with different transmission rates after transmitting through the 10.8 km SSMF. The feedforward taps and feedback taps in the DFE equalizer are 41 and 21, respectively. The number of first-order and second-order taps in the feedforward part of Volterra-DFE is 41 and 91, and the number of taps in the feedback part is 21 and 28, respectively. After simplification, the number of taps consists of 41 and 13 feed-forward taps with 21 and 7 feed-back taps. After simplification, the number of taps is reduced from 181 to 82, which is reduced by 55%. Figure 8 shows the transmission with a bit rate of 112 Gbps. It can be seen that the equalization effect of traditional Volterra-DFE and simplified Volterra-DFE is close to each other, and both can reach the threshold of the KP4-FEC bit error rate (2 × 10⁻⁴), while the equalization effect of ordinary DFE equalizer is poor. Figure 10 shows the eye diagrams at 112 Gb/s, and it can be seen that the eyes open when using both the DFE equalizer and the proposed equalizer. However, using the latter equalizer can make the eyes clearer. This shows that the equalization effect is better in this case. Figure 9 shows the transmission with a bit rate of 128 Gbps. It can also be seen that the traditional Volterra-DFE and the simplified Volterra-DFE have similar equalization effects, and the received optical power reaching the HD-FEC BER threshold (3.8 × 10⁻³) is 1.5 dB lower than that of the ordinary DFE. Figure 11 shows the eye diagrams at 128 Gb/s, and it can be seen that the eyes open only when using the proposed equalizer, which means the proposed equalizer is able to recover the signal quality. When using DFE alone, the eyes cannot open, and in this case, the signals are degraded.

Figure 12 and Figure 13 show the BER curves of PNLE or the proposed equalizer with different transmission rates after transmitting through 10.8 km SSMF. These two figures are shown to compare the equalization effect after changing the square operation into an absolution operation. It is clear that the proposed equalizer has a similar ability to equalize the distorted signals in an EML-based IM-DD system, which indicates the effectiveness of the simplification.

4. Conclusions

Based on the chirp model of the EML laser, the second-order Volterra transmission model of the EML-IMDD system is deduced in this paper. It is found that in the second-order nonlinear terms, the signal square terms have a greater influence on the received signal than other nonlinear terms. According to this model, Volterra’s nonlinear kernel matrix is simplified; at the same time, the absolute value operation is used instead of the square operation to reduce the number of multiplications required by the equalizer. Through simplification, a simplified nonlinear-DFE equalizer is proposed. Compared with the traditional Volterra-DFE equalizer, the simplified equalizer greatly reduces the number of taps and multiplications. The proposed equalizer is verified by 112 Gbit/s and 128 Gbit/s IM-DD system transmission using an EML laser. The complexity of the simplified equalizer is greatly reduced; thus, the operation time is also reduced to a great extent.

Using the simplified Volterra-DFE equalizer proposed in this paper, the BER of the 112 Gbps PAM-4 signal can reach the threshold of KP4-FEC (BER = 2 × 10⁻⁴) after 10.8 km standard single-mode fiber transmission. The 128 Gbps PAM-4 signal can reach the HD-FEC (BER = 3.8 × 10⁻³) threshold, which has a 1 dB sensitivity improvement over the traditional DFE equalizer. At the same time, its equalization effect is basically the same as the Volterra-DFE equalizer before simplification in the EML-IMDD system, which also proves the derivation in this paper. Above all, the proposed equalization scheme can provide an effective and low-complexity choice for signal equalization in the EML-IMDD system in data communication.