## 2 Principle of Simplified Second-Order Volterra-DFE Equalizer
### 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  
\[E(t)=\sqrt{1+X(t)}\exp[j\varphi(t)], \tag{3}\]  
where \(X(t)\) and \(\varphi(t)\) 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]:  
\[\Delta v=\frac{d\varphi(t)}{dt}=\frac{\alpha}{2}\frac{1}{P(t)}\frac{dP(t)}{dt}, \tag{4}\]where \(\alpha\) is the linewidth enhancement factor of EML. From the formula \(P(t)=P_{0}(1+X(t))\) represents the output optical power, the phase can be written as:  
\[\varphi(t)=\frac{\alpha}{2}\ln P_{0}+\frac{\alpha}{2}\ln[1+X(t)]. \tag{5}\]  
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:  
\[E(t)=\sqrt{1+X(t)}[1+X(t)]^{i\alpha/2}, \tag{6}\]  
Since the fiber channel can be modeled as a linear time-invariant system, the optical signal can be expressed as \(S(t)=E(t)*h(t)\) after transmission through the fiber, where \(h(t)\) is the pulse response of the fiber and \(*\) 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 \(Y_{n}=S_{n}*S_{n}^{*}\), that is, the signal multiplied by its conjugate [8; 19]. If \(h(t)\) and \(X(t)\) are converted into discrete time form \(h_{k}\) and \(X_{k}\), \(Y_{n}\) can be expressed as [8]:  
\[ \begin{array}{l}Y_{n}\approx\sum\limits_{k}\sum\limits_{w}R\left[h_{k}h_{k+w }^{*}\Big{\{}1+\frac{1+i\mu}{2}X_{n-k}+\frac{1-i\mu}{2}X_{n-k-w}\right.\\ \left.-\frac{1+\alpha^{2}}{8}\Big{(}X_{n-k}^{2}+X_{n-k-w}^{2}\Big{)}+\frac{1+ \alpha^{2}}{4}X_{n-k}X_{n-k-w}\Big{\}}\right],\end{array}  \tag{7}\]  
where \(R(\cdot)\) 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]:  
\[h_{k}=\sqrt{\frac{cT^{2}}{jD\lambda^{2}L_{f}}}\exp\left(j\frac{\pi cT^{2}k^{2} }{D\lambda^{2}L_{f}}\right), \tag{8}\]  
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]:  
\[k\in\left[-\left\lfloor\frac{D\lambda^{2}L_{f}}{2 cT^{2}}\right\rfloor,\left \lfloor\frac{D\lambda^{2}L_{f}}{2 cT^{2}}\right\rfloor\right], \tag{9}\]  
where \(\lfloor\cdot\rfloor\) is the down operation of the integer. If we simplify this further, we obtain:  
\[ \begin{array}{l}Y_{n}\approx 1+\sum\limits_{k}R(h_{k})X_{n-k}+\sum\limits_{ k}R(jh_{k}\alpha X_{n-k})\\ \left.-\sum\limits_{k}R(h_{k})\frac{1+\alpha^{2}}{4}X_{n-k}^{2}\right.\\ \left.+\sum\limits_{k}\sum\limits_{w}R\Big{[}h_{k}h_{k+w}^{*}\Big{(}\frac{1+ \alpha^{2}}{4}X_{n-k}X_{n-k-w}\Big{)}\Big{]}.\end{array}  \tag{10}\]  
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}{l}Y_{n}\approx\underbrace{\sum\limits_{k=0}^{L_{M}-1}\hat{h}_ {L}(k)X_{n-k}}_{\text{Linear Part}}\\ \left.+\underbrace{\sum\limits_{w=0}^{L_{M}-1}\sum\limits_{k=0}^{L_{M}-1-w} \hat{h}_{NL}(k,w)X_{n-k}X_{n-k-w}}_{\text{Nonlinear Part},}\right.\end{array}