# Clipping Noise Compensation with Neural Networks in OFDM Systems
## 3 Results
### Interpretation of NN Weight Matrices  
Figure 2 visualizes the weight matrices of the learned three-layer fully connected NN. It is observed that these matrices exhibit interesting patterns. Some columns of the first weight matrix, denoted as \(W_{1}\), have a single large entry, while other columns exhibit rhythmic fluctuations. These features must be reorganized into patterns that are easier to understand. Conventional clipping noise mitigation algorithms such as that in [17] require information in both time and frequency domain; this led us to focus on finding columns that are similar to those of an IDFT matrix (providing time domain information) and an identity matrix (preserving frequency domain information). Note that we also split the IDFT matrix into real and imaginary parts in a similar way to Equation (4). Assume \(A\) is an \(N\times N\) IDFT matrix, then the real-valued \(2N\times 2N\) matrix \(\hat{A}\) that can carry out complex number operations with \(\overline{Z}_{k}\) is split into \(\hat{A}_{2k,2k}=\text{Re}\big{(}A_{k,k}\big{)}\), \(\hat{A}_{2k+1,2k}=-\text{Im}\big{(}A_{k,k}\big{)}\), \(\hat{A}_{2k,2k+1}=\text{Im}\big{(}A_{k,k}\big{)}\) and \(\hat{A}_{2k+1,2k+1}=\text{Re}\big{(}A_{k,k}\big{)}\). Figure 3b shows the \(2N\times 2N\) IDFT matrix \(\hat{A}\). The columns of \(W_{1}\) that are similar (with high correlation) to the columns of the IDFT matrix and those similar to the columns of the identity matrix are both sorted out and reorganized. The selected columns can be reordered to make its similarity to the IDFT and the identity matrix more visible. The reordering operations can be recorded in a permutation matrix \(E\), and the reordered weight matrix is denoted as:  
\[\hat{W}_{1}=W_{1}E. \tag{7}\]  
Figure 4a,b show the first and second layer weight matrix after permutation (\(\hat{W}_{1}\) and \(\hat{W}_{2}\)). The left part of \(\hat{W}_{1}\) is obviously similar to the identity matrix; the central part of \(\hat{W}_{1}\) resembles the IDFT matrix and is singled out to be shown more clearly in Figure 3d; this suggests that the NN seeks to regenerate time domain information with frequency domain inputs. In visualizing the working of this NN, the nonlinear activation function is neglected to highlight the patterns emerging from the connections. Figure 2: The weight matrices of the learned three-layer neural network. (**a**) \(W_{1}\). (**b**) \(W_{2}\). (**c**) \(W_{3}\). The same permutation matrix is applied to the second layer weight matrix in order to preserve the connections between layers. On the premise that \(EE^{T}=I\), we have  
\[W_{1}W_{2}W_{3}=W_{1}\big{(}EE^{T}\big{)}W_{2}W_{3}=\hat{W}_{1}\big{(}E^{T}W_{2} \big{)}W_{3}=\hat{W}_{1}\hat{W}_{2}W_{3}. \tag{8}\]  
It is not easy to see the patterns in \(\hat{W}_{2}\) (shown in Figure 4b) and \(W_{3}\) separately. However, when they are multiplied together, the resulting \(W_{2}W_{3}\) again shows clearly recognizable patterns, this time resembling the identity and DFT matrices, as shown in the upper and central part of Figure 4c. The central part is, again, singled out in Figure 3c to be compared to the DFT matrix (Figure 3a). Since the output of the NN model is supposed to constitute a compensated OFDM symbol in the  
Figure 4: The reordered weight matrices. (**a**) \(\hat{W}_{1}\) (**b**) \(\hat{W}_{2}\) (**c**) \(\hat{W}_{2}W_{3}\). Figure 3: (**a**) DFT matrix, (**b**) IDFT matrix, (**c**) The learned DFT matrix (the central part of \(\hat{W}_{2}W_{3}\)), (**d**) The learned IDFT matrix (\(\hat{W}_{1}\) ). frequency domain, it is no surprise that the third weight matrix tries to mimic the DFT matrix and convert data to the frequency domain. The NN retains the original frequency domain input information from the learned identity matrix, while it mimics the behavior of IDFT and DFT. The IDFT is used to obtain the time domain information needed to estimate the clipping noise, while the DFT is needed to create compensated OFDM symbols in the frequency domain.