## 2 Principle of the Proposed Electrical Equalization  
In coherent systems, complex signal equalization can effectively compensate for CD [20], and the output current \(I(t)\) of the PAM4 signal after intradyne detection is expressed as  
\[I(t)=\frac{1}{2}RA_{0}A_{s}(t)exp(j(\omega_{IF}t+\theta_{n}(t))), \tag{1}\]  
where \(R\) is the responsivity of the photodiode, \(A_{0}\) and \(A_{s}(t)\) are the magnitude of the local oscillator and the received signal, respectively, \(\omega_{IF}\) represents the frequency difference between the carrier and the local oscillator, and \(\theta_{n}(t)\) denotes the phase noise. It is known that the PAM4 signal carries only intensity information \(A_{s}(t)\). As mentioned above, conventional adaptive equalization algorithms are sensitive to phase, and the CMA is also unsuitable for PAM signals with asymmetric two-dimensional constellations. Therefore, we aim to propose an equalizer that is not affected by phase impairment. As shown in Figure 1, the constellation points of the received PAM4 signal are rotated due to the carrier frequency offset and the laser linewidth. The equalizer focuses only on the magnitudes of the signal, using the values D1, D2, D3, and D4 as the four reference levels for PAM4, and equalizes the signal to make it converge onto the circles defined by the reference levels. Furthermore, to compensate for the nonlinear impairments of the system, we incorporate decision feedback. In traditional DFEs, the decision value is directly fed back to the equalizer [21]. However, for coherent PAM4 signals, since the decision value is a real signal, directly feeding it back will result in phase differences with the equalized signal. As illustrated in Figure 1, the red points represent the constellation points of the k-th decision feedback, while the blue points represent the constellation points of the k-th equalized signal. There exists a phase difference \(\theta(k)\) between them, which diminishes the equalizer's robustness against carrier frequency offset and laser linewidth. Based on the issues analyzed above, we propose a magnitude-assisted decision feedback equalizer (MA-DFE) and a phase-assisted decision feedback equalizer (PA-DFE). Their structure is represented in Figure 2. It consists of two parts: feedforward and feedback section. Note that in the feedback part, to eliminate the phase difference, we extract the phase of the equalized signal and assign this phase to the decision value before feeding it back into the equalizer. Figure 1: Schematic diagram of the coherent PAM4 equalization constellation. Therefore, the output of the MA-DFE and PA-DFE can be expressed as  
\[y(k)=\sum_{p=-\frac{N}{2}}^{\frac{N}{2}-1}w_{p}^{F}x(k-p)+\sum_{q=1}^{M}w_{q}^{B} d(k-q)e^{i\theta(k-q)} \tag{2}\]  
where \(x(k)\) is the input value, \(d(k)\) is the decision value, \(\theta(k)\) is the phase angle of \(y(k)\), \(w_{p}^{F}\) and \(w_{q}^{B}\) are the feedforward and feedback tap coefficients, respectively, and \(N\) and \(M\) are the numbers of the feedforward and feedback taps, respectively. If \(M=0\), MA /PA-DFE are converted to PA/MA-FFE. When constructing the error function, to eliminate the interference caused by the phase difference, the MA-DFE formulates a real error function based on the magnitude of \(y(k)\). Meanwhile, the PA-DFE assigns the phase of \(y(k)\) to \(d(k)\) to create a complex error function. The formulas are expressed as follows:  
\[e(k)_{\textit{MA-DFE}}=d(k)-|y(k)| \tag{3}\]  
\[e(k)_{\textit{PA-DFE}}=d(k)e^{i\theta(k)}-y(k) \tag{4}\]  
The recursive least squares (RLS) algorithm is chosen for updating tap coefficients in equalizers due to its robust and rapid convergence.