# Low-Complexity Chromatic Dispersion Equalization FIR Digital Filter for Coherent Receiver
## 2 Operation Principles
### Optimize Time-Domain Equalization Implementation
#### 2.2.2 Window Function Weight Optimization  
In the convolution operation in the TD, the distribution characteristics of multiplication and the combination of addition are used to reduce the number of multiplication operations between input samples and tap quantization coefficients by implementing filter tap coefficient quantization, thereby reducing the computational complexity of TD-CDE. The uniform quantization method is used in the quantization process. As the number of quantization stages \(M\) decreases, the number of quantized values of filter tap coefficients decreases, and the system's diversity increases. Figure 2 shows the schematic diagram of the real part of the tap coefficient before and after quantization; the quantization stages \(M\) is eight, and coefficients are normalized. Figure 1: Side lobe performance of second-order Nuttall self-convolution window and rectangular window. Figure 2: The real part of exact and quantized coefficients for \(M=8\) and \(N=415\). Before illustrating the specific implementation process, we must make two necessary explanations. * In this paper, the filter complex tap coefficients are independently written as their real and imaginary parts for operation. This operation can increase the diversity of filter tap coefficients, which is inversely proportional to the computational complexity of TD-CDE. * It can be seen from Equation (5) that the tap coefficient of the filter \(h(k)\) is symmetrical about the central coefficient [21]. Assuming the symmetric length is \(L\), then \(L=\frac{(N-1)}{2}\). If the tap coefficients are uniformly quantized, the quantized value is also approximately symmetric about the central coefficient. Through the above two points of analysis, before the tap coefficient quantization is implemented, the output value \(y(n)\) after equalization is  
\[y(n)=\sum_{m=1}^{N}x(n+m-1)h(N-m+1) \tag{14}\]  
among them, the filter coefficient \(h(N-i+1)\) can be written in the form of the real part plus the imaginary part, as shown below  
\[h(N-m+1)=h_{\text{real}}\left(N-m+1\right)+h_{\text{imag}}\left(N-m+1\right) \cdot i,m=0,1,2,\ldots,N \tag{15}\]  
Therefore, from Equations (14) and (15), the output value \(y(n)\) can be written as two parts \(y_{1}(n)\) and \(y_{2}(n)\), which are  
\[ \begin{split} y_{1}(n)&=\sum_{m=1}^{N}x(n+m-1)h_{ \text{real}}(N-m+1)\\ y_{2}(n)&=\sum_{m=1}^{N}x(n+m-1)h_{\text{imag}}(N- m+1)\cdot i\end{split}  \tag{16}\]  
Performing uniform quantization on the filter coefficients, the quantization level is \(M\). Taking the real part \(h_{\text{real}}(N-m+1)\) of the filter coefficient as an example, it is quantized into M levels. The real part of the filter coefficient after quantizing can be expressed as \(h_{\text{real}}^{k}\,,k=1,2,\ldots,\text{M}\), and the number of each level is \(n_{\text{real}}^{k}\). After the above operation, the input digital signal can be preprocessed using binding law to reduce the complexity of subsequent calculations. Taking \(h_{\text{real}}^{k}\) as an example, combining the application of multiplication distribution and addition combination can extract the common factor \(h_{\text{real}}^{k}\) for \(n_{\text{real}}^{k}\) times of multiplication operations, and accumulate the sum value of the input sequence \(X^{h_{\text{real}}^{k}}\) with the same common factor \(h_{\text{real}}^{k}\) is \(\sum_{t=1}^{n_{\text{real}}^{k}}X_{t}^{h_{\text{real}}}\). The calculation method is changed from the accumulative after polynomial multiplication to multiplication after the polynomial accumulative, which can get \(\sum_{t=1}^{n_{\text{real}}^{k}}X_{t}^{h_{\text{real}}}\cdot h_{\text{real}}^{k}\).