## 3 Neural-Network-Based Interference Cancellation in LIS Systems
### Inference and Signal Reconstruction  
During inference, the trained neural networks estimate the transmitted signal from the received signal on a per-antenna basis. Given that each antenna has its own dedicated model, predictions are performed individually, ensuring optimal signal reconstruction for each channel condition.  
For each received symbol, the I and Q components are extracted as \(\hat{\mathsf{S}}_{n}^{(t)}=\begin{bmatrix}\mathfrak{R}\left(\hat{\mathsf{S}}_{ n}^{(t)}\right)\\ \mathfrak{J}\left(\hat{\mathsf{S}}_{n}^{(t)}\right)\end{bmatrix}\), processed through the trained model, and recombined to form the following estimated complex-valued transmitted signal:  
\[\hat{\mathsf{S}}_{n}^{(t)}=\mathbb{R}\left(\hat{\mathsf{S}}_{n}^{(t)}\right)+ \mathfrak{J}\left(\hat{\mathsf{S}}_{n}^{(t)}\right) \tag{6}\]  
As follows, each layer in the neural network applies a transformation, where a fully connected layer follows:  
\[\mathsf{z}^{(l)}=\max\left(0,W^{(l)}\mathsf{z}^{(l-1)}+\mathsf{b}^{(l)}\right) \tag{7}\]  
where \(W^{(l)}\) and \(\mathsf{b}^{(l)}\) are the weights and biases of layer \(l\), and the activation function ReLU ensures non-linearity.  
This approach eliminates the need for iterative matrix inversions, significantly reducing computational overhead while surpassing the performance of conventional iterative interference cancellation techniques. The neural network drastically reduces the need for repeated matrix inversions, which are computationally intensive or iterative receivers, making the system more efficient in terms of processing time and energy consumption.  
In terms of computational complexity, ZF and MMSE receivers require matrix inversion for each frequency component, with a per-subcarrier complexity of approximately \(O(T^{3})\), where \(T\) is the number of transmit antennas. MRC and EGC receivers avoid inversion, with complexity \(O(R)\) per subcarrier (where \(R\) is the number of receiver antennas), but often rely on iterative interference cancellation.  
In the proposed approach, a separate neural network is trained for each transmit antenna. As such, the total training complexity scales as \(O(T\cdot N\cdot L)\), where \(N\) is the number of training samples per antenna and \(L\) is the number of trainable parameters per network. Although training is computationally intensive, it is performed per coherence interval and can be parallelized across antennas. The inference complexity is significantly lower, scaling as \(O(T\cdot L)\), which makes the method suitable for real-time execution once training is completed. Future work will investigate optimizations such as parameter sharing and online adaptation to reduce training overhead in highly dynamic scenarios.  
It is worth noting that the training of the neural network assumes perfect channel knowledge for supervision. In practical deployments, channel estimation errors may lead to discrepancies between the training labels and the actual received signals, potentially degrading performance. However, neural networks have shown the potential to tolerate moderate levels of estimation noise, especially when trained with representative or augmented data that include such effects. Future work will explicitly study the impact of imperfect CSI and explore training strategies to enhance robustness to estimation errors.