# Clipping Noise Compensation with Neural Networks in OFDM Systems
## 2 Clipping Noise Compensation  
The proposed DL-based scheme seeks to reconstruct OFDM symbols in the frequency domain, such that the mean square error (MSE) is minimized. The overall schematic after frequency domain equalization is summarized in Figure 1. The signal model and the two major modules, namely the NN and the post-processing inspired by the Decision-Aided Reconstruction (DAR) algorithm [17], are also explained in this section. Let \([X_{0},X_{1},\cdots,X_{N-1}]^{T}\) denote an OFDM symbol with \(N\) subcarriers, where \(X_{k}\) denotes the modulation (in this paper Quadrature Amplitude Modulation, QAM) symbol on the \(k\)-th subcarrier. The baseband signal in the time domain is  
\[x[n]=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X_{k}e^{\beta 2\pi nk/N},\quad 0\leq n \leq N-1. \tag{1}\]  
Figure 1: The schematic of reconstructing clipped orthogonal frequency division multiplexing (OFDM) signals. To avoid inter-symbol interference (ISI) and to maintain the orthogonality between subcarriers, a short sequence called the cyclic prefix can be added to make the signal look \"locally periodic\" [1]. For instance, the last \(L\) samples \(x[N-L+1],\ \cdots,\ x[N-1]\) are added as prefix to make a complete OFDM symbol: \(x[N-L+1],\ \cdots,\ x[N-1],\ x[0],\ x[1],\cdots,\ x[N-1]\). Usually, \(L\) is chosen to be larger than the channel length to avoid ISI, and much less than \(N\) to reduce overhead. The digital clipping of OFDM signals is intentionally introduced to reduce the PAPR of the transmit signal. It is modeled as a restriction on magnitude, while the phase remains unchanged, that is  
\[x_{c}[n]=\left\{ \begin{array}{l}x[n],\ \text{if}\ \left|x[n]\right|\leq A,\\ \\ Ae^{\beta},\ \text{if}\ \left|x[n]\right|>A,\end{array} \right. \tag{2}\]  
where \(x_{c}[n]\) denotes the clipped signal, \(A\) is the clipping threshold and \(\theta\) is the phase of \(x[n]\). The clipping ratio is defined as  
\[\text{cr}=\frac{A}{P_{S}}\ \text{with}\ P_{s}=\text{E}\Big{[}\left|x[n]\right|^{ 2}\Big{]}. \tag{3}\]  
Clipping is more severe when \(\text{cr}\) gets smaller. To achieve a PAPR reduction of 6 dB (5 dB), \(\text{cr}\cong 1.0\) (1.3) is needed [6]. The signal is interpolated four times to emulate analog signals before doing digital clipping. Here, for the simplicity of presentation, the interpolation is not included. At the receiver side, first the cyclic prefix is removed from the received OFDM symbol. As long as the length of the channel is no larger than \(L\), the OFDM symbol will be immune to ISI and the orthogonality will also be maintained. Since the OFDM signal can be viewed as being circularly convoluted with the channel impulse response, it can be converted to frequency domain by DFT and then be equalized to compensate the channel gain. The equalization can be facilitated by piloted transmission, as is commonly done in communication protocols. To train the NN, the equalized symbols in frequency domain are used as the input and the correct OFDM symbols as labels. Assuming perfect Zero-Forcing (ZF) equalization, the equalized OFDM symbols are expressed as  
\[Z_{k}=X_{c,k}+H_{k}^{-1}W_{k} \tag{4}\]  
where \(X_{c,k}\) is the clipped symbol which is the DFT of \(x_{c}[n]\), \(H_{k}\) the channel's frequency response, and \(W_{k}\) the AWGN at the \(k\)-th subcarrier. Because the proposed scheme works on equalized OFDM symbols, it does not need to worry about compensating for channel gains, and we expect that the overall performance is also insensitive to channel conditions. Note that there exist extremely rich works on equalization and signal detection for OFDM systems. They range from the application of DL [9,10] to utilizing simple linear filters to combat complicated issues such as narrowband noise or inter-carrier interference [15, 20, 21, 22, 23]. It is possible that the proposed scheme will have a better performance if a more advanced equalizer other than the ZF equalizer is used.