## 2 Principle of Simplified Second-Order Volterra-DFE Equalizer
### 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.  
Figure 1: Nonlinear kernel coefficient when w = 0–3.  
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:  
\[\begin{array}{l}y(n)=\sum\limits_{l_{1}=0}^{L_{1}-1}h_{1}(l_{1})x(n-l_{1})+ \sum\limits_{l_{2}=0}^{L_{2}-1}h_{2}(l_{2})|x(n-l_{2})|\\ +\sum\limits_{k_{1}=1}^{K_{1}}g_{1}(k_{1})y^{\prime}(n-k_{1})+\sum\limits_{k_{ 2}=1}^{K_{2}}g_{2}(k_{2})|y^{\prime}(n-k_{2})|\end{array} \tag{15}\]  
According to Equation (15), we analyze the algorithm complexity of the three equalizers. Table 1 shows a comparison of the complexity of the three equalizers, including the number of taps and multiplications. When \((L_{1},L_{2},K_{1},K_{2})=(41,13,21,7)\), 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,  
Figure 3: Simplified Second-order Volterra kernel coefficients.  
Figure 2: Second-order Volterra kernel coefficients.  
the simplified second-order Volterra-DFE adds 20 nonlinear kernels to reduce nonlinear damage during signal transmission.