# Low-Complexity Chromatic Dispersion Equalization FIR Digital Filter for Coherent Receiver
## 2 Operation Principles
### Optimize Time-Domain Equalization Implementation
#### 2.2.1 Window Function Weight Optimization  
The frequency-domain response \(H_{\text{Des}}\big{(}e^{j\omega T}\big{)}\) of an ideal filter has a constant amplitude-frequency characteristic in the passband. When the FD response of the designed filter is close to \(H_{\text{Des}}\big{(}e^{j\omega T}\big{)}\), the CDE effect is better, so the spectral energy of the window function is required to be concentrated on the main lobe as much as possible. The filtered spectrum \(H(k)\) obtained by the rectangular window truncation will have serious spectrum leakage, which is quite different from the ideal FD response \(H_{\text{Des}}\big{(}e^{j\omega T}\big{)}\). In this paper, based on the window function Nuttall window function with the smallest side lobe and fastest side lobe attenuation characteristics, considering that the four-term three-order Nuttall window has the fastest attenuation speed of side lobe compared to other window functions [20], the Nuttall second-order self-convolution window function is used.  
The TD expression characteristics are  
\[w_{\text{R}}(n)=\begin{cases}1,&n=1,2,\cdots\cdots,N\\ 0,&\text{else}\end{cases} \tag{6}\]  
and its spectral characteristics are  
\[W_{\text{R}}(\omega)=\frac{\sin(\omega N/2)}{\sin(\omega/2)}\exp\biggl{(}-j \omega\frac{N-1}{2}\biggr{)} \tag{7}\]  
The Nuttall window is a combination window with good side lobe characteristics, and the TD expression is  
\[\omega_{\text{N}}(n)=\sum_{m}^{M-1}(-1)^{m}c_{m}\cos(2\pi n\cdot m/N)\quad n=1,2\dots N-1 \tag{8}\]  
Here, \(M\) is the number of terms of the window function, \(N\) is the length of the window function; \(C_{m}\) is the coefficient, and satisfies the constraints  
\[\sum_{m=0}^{M-1}(-1)^{m}c_{m}=0,\quad\sum_{m=0}^{M-1}c_{m}=1 \tag{9}\]  
and its spectral characteristics are  
\[W_{\text{N}}(\omega)=\sum_{m=0}^{M-1}(-1)^{m}\frac{c_{m}}{2}\biggl{[}W_{\text {R}}\biggl{(}\omega-\frac{2\pi m}{N}\biggr{)}+W_{\text{R}}\biggl{(}\omega+ \frac{2\pi m}{N}\biggr{)}\biggr{]} \tag{10}\]  
Here, \(W\) is the spectral function of the rectangular window. This paper selects a four-term three-order Nuttall window, the coefficients of which are \(c_{1}=0.338946\), \(c_{2}=0.481973\), \(c_{3}=0.161054\), \(c_{4}=0.018027\). The four-term three-order Nuttall window is self-convolved to obtain the second-order Nuttall self-convolution window function as  
\[w_{\text{RN}}(n)=w_{\text{N}}(n)^{*}w_{\text{N}}(n) \tag{11}\]  
Time-frequency diagram of the rectangular window and second-order Nuttall self-convolution window function as shown in Figure 1.  
It can be seen from the Figure that the side lobe performance of the Nuttall self-convolution window is significantly better than that of the rectangular window. Therefore, using the Nuttall self-convolution window to cut the signal can effectively suppress the spectrum leakage. The Nuttall self-convolution window function is used to window and truncate Equation (3) to obtain the CDE RN-FIR filter as  
\[h_{\text{RN}}(n)=\omega_{\text{RN}}(n)\cdot h(n),-\left\lfloor\frac{N}{2} \right\rfloor\leq n\leq\left\lfloor\frac{N}{2}\right\rfloor \tag{12}\]  
After the signal is truncated in the TD with the window function, the FD amplitude of the signal will be affected, so amplitude restoration is required. The definition of the amplitude restoration coefficient \(h(n)\) is that the amplitude when other window functions are added is consistent with the amplitude of a rectangular window of the same length. Its expression is  
\[K_{m}=A_{1}/A_{2} \tag{13}\]  
Here, \(A_{1}\) is the amplitude of the zero frequency point after adding a rectangular window, and \(A_{2}\) is the amplitude of the zero frequency point after adding other windows. The modulus of the tap coefficient of the TD-CDE is constant, which limits the equalization effect. The above method is equivalent to using a window function to optimize the weight of the TD-CDE taps if there is no longer a constant value, which effectively improves the CDE filter effect.