# A Novel Transfer Learning-Based OFDM Receiver Design for Enhanced Underwater Acoustic Communication
## 5 Simulation and Experimental Results
### Computational Complexity Analysis  
The computational complexity of different benchmark methods, including LS, MMSE, FC-NN, and DNN, is compared against the proposed transfer learning model. The complexity of each method is analyzed based on the number of sub-carriers \(K_{c}\), number of data indices \(K_{data}\), and the architectural parameters of deep learning-based models. The LS algorithm remains one of the most computationally efficient methods with a complexity of:  
\[O(K_{c}^{2}+K_{c}+K_{data}),\]  
where \(K_{c}\) is number of sub-carriers, making it feasible for moderate-scale systems, its reliance on traditional channel-estimation techniques limits its accuracy, particularly in dynamic and nonlinear communication environments. The additional \(O(K_{c})\) and \(O(K_{data})\) terms account for the equalization and demodulation steps, making LS an efficient but relatively less accurate approach. The MMSE algorithm, known for its superior performance by minimizing the mean square error, incurs a significantly higher computational cost:  
\[O(K_{c}^{3}+K_{c}+K_{data}).\]  
The cubic dependence on \(K_{c}\) arises due to matrix inversion operations, making MMSE computationally expensive. Despite its improved accuracy over LS, the practical implementation of MMSE in real-time systems is often constrained by its high computational demand. The introduction of neural networks for channel estimation, such as FC-NN, shifts the computational paradigm towards learning-based approaches. The complexity of the FC-NN model is given by:  
\[O(L_{f}\times K_{c}^{2}).\]  
Here, \(L_{f}\) represents the number of fully connected layers. Unlike LS and MMSE, which rely on explicit mathematical formulations, FC-NN learns an implicit mapping of channel conditions, reducing the dependency on traditional equalization methods. However, the fully connected nature of this architecture results in a quadratic complexity with respect to \(K_{c}\), making it computationally demanding as the number of sub-carriers increases. The DNN model improves upon FC-NN by introducing hierarchical feature extraction, with its complexity formulated as:  
\[O(L_{f}\times K_{c}^{2})+O(L_{d}\times K_{c}),\]  
where \(L_{d}\) represents the number of dense layers in addition to the fully connected layers. The additional term \(O(L_{d}\times K_{c})\) allows DNN to capture more abstract representations, improving performance in complex environments. However, the dependency on deep fully connected layers leads to high parameterization, increasing training and inference time. The proposed transfer learning model significantly optimizes computational efficiency by leveraging pre-trained feature representations, thereby reducing the number of trainable parameters. The complexity is formulated as:  
\[O(L_{c}\times K_{c})+O(L_{f}\times K_{c}^{2}),\]  
where \(L_{c}\) represents the number of convolutional layers. The presence of convolutional layers enables hierarchical feature extraction, drastically reducing computational overhead compared to fully connected networks. Furthermore, by freezing convolutional layers during fine-tuning, the complexity is effectively reduced to:  
\[O(L_{f}\times K_{c}^{2}),\]  
as only the fully connected layers are updated during transfer learning. This approach allows the model to generalize efficiently to new environments while significantly reducing computational demand. The use of pre-trained weights mitigates the need for learning low-level features from scratch, making transfer learning an optimal trade-off between complexity and performance. A comparative summary of the computational complexity of each method is provided in Table 10.  
The analysis in Table 10 highlights the scalability advantages of transfer learning over conventional methods. Although LS remains computationally efficient, its limited accuracy makes it unsuitable for complex scenarios. MMSE offers improved accuracy at a significant computational cost, making it impractical for real-time applications with large subcarrier counts.