## 4 Experimental Results and Analysis
### Complexity Analysis and Discussion  
In this section, to evaluate the computational complexity of the proposed two-stage time-domain equalization framework, we use the analytical formula introduced in [43], which contains three indicators: the number of real-valued multiplications (RMs), the number of bit-level operations (BOPs), and the number of additions and bit shifts (NABSs) during inference. The calculation results are shown in Table 3.  
The RM can be calculated as follows:  
\[\text{RM}_{\text{LSTM}}=n_{s}n_{h}(4n_{i}+4n_{h}+3) \tag{11}\]  
Here, \(n_{s}\) denotes the sequence length (i.e., the number of time steps), \(n_{h}\) denotes the number of hidden units per layer, and \(n_{i}\) represents the input vector dimension. Compared to the notation used in Table 2, the notations \(n_{s}\), \(n_{h}\), and \(n_{i}\) in this section correspond to \(T\), \(L\), and input feature dimensions, respectively. In the expression \(n_{s}n_{h}(4n_{i}+4n_{h}+3)\), the three terms inside the parentheses represent the computational contributions from different parts of the LSTM cell: \(4n_{i}\) corresponds to the input-to-hidden multiplications across four gates, \(4n_{h}\) accounts for the hidden-to-hidden multiplications involving recurrent weights, and the constant 3 approximates the number of element-wise multiplications required by activation functions such as sigmoid, tanh, and Hadamard operations. Since these operations are performed across all time steps and hidden units, the total number of real-valued multiplications scales with \(n_{s}n_{h}\). The BOP can be calculated as follows:  
\[ \begin{split}\text{BOP}_{\text{LSTM}}&=4n_{s}n_{ h}\cdot\text{Mult}(n_{i},b_{w},b_{i})\\ &\quad+4n_{s}n_{h}\cdot\text{Mult}(n_{h},b_{w},b_{a})\\ &\quad+3n_{s}n_{h}b_{a}^{2}\\ &\quad+9n_{s}n_{h}\cdot\text{Acc}(n_{h},b_{w},b_{a})\end{split}  \tag{12}\]  
Here, \(n_{s}\) represents the sequence length (the number of time steps), and \(n_{h}\) represents the number of hidden units per layer, and \(n_{i}\) represents the input vector dimension. The first term counts the bit-level multiplications between the input and weight matrices for each gate, with bit-widths \(b_{w}\) and \(b_{i}\). The second term performs the same for the hidden state and its weights, using \(b_{a}\) as the activation bit-width. The third term estimates the cost from applying nonlinear functions. The last part accounts for the accumulations needed after all the multiplications. The NABS can be calculated as follows:  
\[ \begin{split}\text{NABS}_{\text{LSTM}}&=4n_{s}n_{ h}[n_{i}(X_{w}+1)-1]\cdot\text{Acc}(n_{i},b_{w},b_{i})\\ &\quad+4n_{s}n_{h}[n_{h}(X_{w}+1)+1]\cdot\text{Acc}(n_{h},b_{w}, b_{a})\\ &\quad+6n_{s}n_{h}b_{a}\end{split}  \tag{13}\]  
Here, \(n_{s}\) represents the sequence length (the number of time steps), \(n_{h}\) represents the number of hidden units per layer, and \(n_{i}\) represents the input vector dimension. The first part captures how many additions and shifts are needed for combining inputs and their weights, with \(X_{w}\) representing whether weights are reused or sparse. The second part performs the same for the hidden state. The last term is a rough estimate of the extra additions and shifts inside the LSTM gates. From the calculation results, the proposed two-stage time-domain equalization method has the highest computational complexity, which is about twice that of the conventional LSTM equalizer and much higher than the second-order Volterra equalizer. However, in terms of BER performance, the two-stage equalizer is significantly better than the conventional LSTM equalizer and the Volterra nonlinear equalizer. This shows that the additional complexity introduced by this method is meaningful when dealing with nonlinear distortion that cannot be effectively suppressed by a single-stage equalizer.