## 3 Simulation Results
### Computational Complexity Analysis  
To evaluate the computational complexity for the DNN, LSTM, and traditional MMSE techniques, we will conduct a comprehensive analysis by quantifying the computational resources required for each method. For DNN, the computational complexity primarily arises during the training phase. This phase involves processing large amounts of data through multiple layers of neurons, including matrix multiplications, activation functions, and backpropagation algorithms. Particularly for deep architectures with numerous layers, the computational demands can be substantial. Since we emphasize LSTM networks and their associated model training methodologies as investigating the utilization of deep neural network-based techniques for channel estimation and signal detection, we focus on the computational complexity for the LSTM estimator and those for the MMSE estimator. * LSTM networks, commonly employed in sequence modeling tasks, also entail matrix multiplications and non-linear operations. LSTM cells maintain state information over time, necessitating recurrent computations for each time step. Although less complex than deep neural networks, LSTM networks still impose significant computational requirements, especially during training and inference on long sequences. Assume that the LSTM network has \(L\) layer, and, for the linear transform, the transition between the \(l\)th and \(l-1\)th layers requires \(J_{l-1}I_{l}\) multiplications. The total real-valued multiplications and summations in the DNN network are provided by [11; 20] \[N_{mul/sum}=2\sum_{i=2}^{L}J_{i-1}J_{i},\] (20) where \(N_{mul/sum}\) is the number of multiplications/summations. Figure 15: BERs of LSTM method and MMSE approach for 10,000 frames. For the proposed LSTM network described in Section 2.5.3, the number of layers is five, the sequence data are two-hundred-fifty-six, and the number of subcarriers is sixty-four. Every subcarrier contains two symbols. Thus, every OFDM subcarrier contains 128 bits. Next, the LSTM layer consists of sixteen hidden units, four fully connected layers, and one softmax layer. Therefore, the total number of multiplications required to estimate the channel for our LSTM model is 17,572 per OFDM symbol. * In contrast, the traditional MMSE techniques involve solving linear algebraic equations to estimate the channel parameters based on the received signals. While potentially less demanding in terms of training compared to neural networks, MMSE requires matrix inversions or decompositions, which can be computationally expensive for large matrices. The computational complexity of the traditional MMSE estimator is calculated based on the expression shown in [11; 20], \[O(\hat{H}_{\text{MMSE}})=n^{3}+n^{2}+n^{3},\] (21) where the capital \(O\) notation represents the computational complexity and \(n\) represents the matrix size. The presence of three terms in the complexity formula can be explained as follows. The first \(n^{3}\) term comes from the matrix multiplication, a common operation in MMSE estimation where multiplying two \(n\times n\) matrices typically requires \(O(n^{3})\) operations. The second \(n^{3}\) term arises from the matrix inversion as inverting an \(n\times n\) matrix generally has a computational complexity of \(O(n^{3})\). The \(n^{2}\) term accounts for other linear operations, such as matrix addition and multiplication with vectors, which are less computationally intensive but still necessary steps in the MMSE estimation process. In our case, with \(n=64\), the computational complexity becomes 528,384. Therefore, it requires 528,384 multiplications per OFDM symbol to estimate the channel using the MMSE method. This calculation highlights the significant computational demands of the MMSE estimator, especially as the matrix size increases. By quantifying the number of operations (such as multiplication, division, summation, etc.)