# Clipping Noise Compensation with Neural Networks in OFDM Systems
## 1 Introduction  
Orthogonal frequency division multiplexing (OFDM) is a popular multi-carrier modulation scheme due to the advantages of its high data rates, high bandwidth efficiency and robustness against multi-path fading [1]. However, OFDM has its issues. One of its main problems is its high peak power to average power ratio (PAPR), i.e., its signal exhibits large envelope fluctuations [2]. This property makes the power amplifier (PA) operate beyond its comfortable working range, which results in low power efficiency and induces nonlinear distortion to the transmitted signal. Consequently, it is highly desirable for OFDM systems to reduce PAPR to avoid these adversarial effects. Many PAPR reduction algorithms have been proposed; for instance, partial transmit sequences, selective mapping, and tone reservation [3; 4; 5] are among the most prominent, but usually high computational effort is required. Typically, a reduction of at least 5 dB is expected from these reduction techniques when the probability that the original peak exceeds a preset level is set to \(10^{-6}\)[6]. Here, we consider the simplest PAPR reduction technique, namely digital clipping, which is easily implemented in the transmitter. However, digital clipping causes the problems of in-band distortion and out-of-band leakage; therefore, a potent clipping compensation scheme at the receiver is needed to work with digital clipping to maintain the overall system performance [6; 7; 8]. Deep learning (DL) has the potential to be a powerful tool for solving physical layer issues [9], such as modulation recognition, channel estimation and detection [10; 11; 12], the decoding of linear codes [13], and estimating error rates from system parameters [14]. As far as clipping is concerned, it is shown in [10] that DL-based signal detection is quite resistant to clipping noise. However, as clipping becomes severe, performance degrades dramatically and a dedicated compensation scheme is needed. Several approaches exist to compensate clipping effects at the receiver or transmitter side. It is possible to filter clipping effects as impulse noises if their occurrence is limited [15]. A transmitter-side techniqueproves quite effective at lowering the BER floor with a clipping-filtering procedure [16]. Dedicated receiver-side algorithms such as those in [17, 18], though, offer a supreme performance and rely on costly computation and high-latency iterative procedures to \"reconstruct\" clipped signal peaks. It is desirable to have a clipping noise compensation scheme, such that a superior performance can be obtained with a low computational cost and low latency. Algorithms based on Neural Networks (NNs) with simple feedforward operations look promising to fulfill these requirements. The working mechanism inside a neural network is known to be very opaque and resistant to explanations. Nonetheless, in some cases, it is possible to understand and interpret the neural network in a much clearer way; for instance, it has been shown that a network architecture can be obtained by unfolding an iterative optimization algorithm [19]. We believe that the compensation of clipping noise may offer such a scenario, wherein the weight matrices of the NN yield a meaningful interpretation. The iterative nature of clipping noise compensation algorithms [17, 18] also suggests likewise. The contributions of this paper are twofold. First, an effective clipping noise compensation scheme, consisting of a fully connected NN, is proposed. To the best of our knowledge, it is the first such NN used for this purpose. The simulation results demonstrate the superior performance of the proposed approach over existing methods, especially when clipping is severe. Second, the weight matrices of the trained NN are analyzed. Patterns resembling that of the DFT/IDFT matrices, which propagate information between the time and frequency domain, are revealed. It is rarely reported in DL literature that such recognizable patterns can be observed in NN weight matrices.