## 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.