# 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
Because uniform quantization is adopted, the filter coefficients after quantization have the following properties  
\[h_{\text{real}}^{k}\,=-h_{\text{real}}^{M+1-k} \tag{17}\]  
The calculation can be further simplified as shown  
\[y_{1}^{h_{\text{real}}^{k}}\left(n\right)=\left(\sum_{l=1}^{n_{\text{real}}^{ k}}X_{l}^{h_{\text{real}}^{k}}-\sum_{l=1}^{n_{\text{real}}^{M+1-k}}X_{l}^{h _{\text{real}}^{M+1-k}}\right)\cdot h_{\text{real}}^{k} \tag{18}\]In the same way, the calculation results of the second part areas are  
\[y_{2}^{h_{\text{imag}}^{k}}\left(n\right)=\left(\sum_{l=1}^{n_{\text{imag}}^{k}}\ X_{l}^{h_{\text{imag}}^{k}}\ -\sum_{l=1}^{n_{\text{imag}}^{M+1-k}}X_{l}^{h_{\text{imag}}^{M+1-k}}\right)\cdot h _{\text{imag}}^{k}\cdot i \tag{19}\]  
In summary, the CDE filter \(h(n)\) and the output value \(y(n)\) are  
\[h_{\text{real}}(n)\in\left\{h_{\text{real}}^{k}\right\}\quad 0\leq \text{k}\leq\text{M}/2,-\left\lfloor\frac{N}{2}\right\rfloor\leq n\leq\left\lfloor \frac{N}{2}\right\rfloor \tag{20}\] \[h_{\text{imag}}(n)\in\left\{h_{\text{imag}}^{k}\cdot\text{i}\right\} \quad 0\leq\text{k}\leq\text{M}/2,-\left\lfloor\frac{N}{2}\right\rfloor\leq n \leq\left\lfloor\frac{N}{2}\right\rfloor\] \[y(n)=\sum_{k=1}^{\text{M}/2}\left[h_{\text{real}}^{k}\cdot\left( \sum_{l=1}^{n_{\text{real}}^{k}}X_{l}^{h_{\text{real}}}-\sum_{l=1}^{n_{\text{ real}}^{M+1-k}}X_{l}^{h_{\text{real}}^{M+1-k}}\right)\right]\] \[+\left\{\sum_{k=1}^{\text{M}/2}\left[h_{\text{imag}}^{k}\cdot \left(\sum_{l=1}^{n_{\text{imag}}^{k}}X_{l}^{h_{\text{imag}}^{k}}-\sum_{l=1}^{ n_{\text{imag}}^{M+1-k}}X_{l}^{h_{\text{imag}}^{M+1-k}}\right)\right]\right\}.i\]  
The detailed calculation flow for each sample of equalized output is shown in Figure 3.  
Next, we will analyze the calculation workload before and after the quantization of the filter tap coefficients. The computational complexity evaluation is based on real number adders \(R_{\text{add}}\) and real number multipliers \(R_{\text{mul}}\). The computational complexity of subtraction is the same as addition, so subtraction counts as addition in the complex calculation. Implementing a complex adder \(C_{\text{add}}\) requires two \(R_{\text{add}}\), while a complex multiplier \(C_{\text{mul}}\) only requires three \(R_{\text{mul}}\) and five \(R_{\text{add}}\). The implementation process is  
\[ \begin{split} Z_{1}=a+b\cdot i,Z_{2}=c+d\cdot i\\ Z_{1}\cdot Z_{2}=(a+b)(c-d)+a\cdot d-b\cdot c+(a\cdot d+b\cdot c )\cdot i\end{split}  \tag{21}\]  
First, analyze the complexity of the TD-CDE. It can be seen from Equation (5) that the number of filter taps is an odd number \(N\). For each equalized output value, it needs to  
Figure 3: Symmetric CQWO-TD-CDE FIR filter architecture. perform \(N\cdot C_{\text{mul}}\) and \((N-1)\cdot C_{\text{add}}\), which are converted into real number multiplication and real number addition  
\[n_{\text{R}_{\text{mul}}}=3\cdot N\quad n_{\text{R}_{\text{add}}}=7\cdot N-2 \tag{22}\]  
From Equation (21), it can be concluded that the times of complex number addition and complex number multiplication of the TD-CDE after quantization are  
\[n_{\text{R}_{\text{add}}}=2\cdot\left(\sum_{k=1}^{\text{M}/2}\left(n_{\text{ real}}^{\text{k}}+n_{\text{real}}^{\text{M}+1-k}+n_{\text{imag}}^{\text{k}}+n_{ \text{imag}}^{\text{M}+1-k}\right)-2\right)+2\cdot\text{M}=4\cdot N-4+2\cdot \text{M} \tag{23}\]  
\[n_{\text{R}_{\text{mul}}}=2\cdot\text{M}\]