# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
###### Abstract  
Compared with orthogonal frequency division multiplexing (OFDM) systems, orthogonal time frequency space systems based on bi-orthogonal frequency division multiplexing (OTFS-BFDM) have lower out-of-band emission (COBE) and better robustness to high-mobility scenarios, but suffer from a higher peak-to-average ratio (PAPR) in large data packets. In this paper, one-iteration clipping and filtering (OCF) is adopted to reduce the PAPR of OTFS-BFDM signals. However, the extra noise introduced by the clipping process, i.e., clipping noise, will distort the desired signal and increase the bit error rate (BER). We propose a message passing (MP)-assisted iterative cancellation (MP-AIC) method to cancel the clipping noise based on the traditional MP decoding at the receiver, which incorporates with the (OCF) at the transmitter to keep the sparsity of the effective channel matrix. The main idea of MP-AIC is to extract the residual signal fed to the MP detector by iteratively constructing reference clipping noise at the receiver. During each iteration, the variance of residual signal and channel noise are taken as input parameters of MP decoding to improve the BER. Moreover, the convergence probability of the modulation alphabet after MP decoding in the current iteration is used as the initial probability of MP decoding in the next iteration to accelerate the convergence rate of MP decoding. Simulation results show that the proposed MP-AIC method significantly improves MP-decoding accuracy while accelerating the BER convergence in the clipped OTFS-BFDM system. In the clipped OTFS-BFDM system with rectangular pulse shaping, the BER of MP-AIC with two iterations can be reduced by 72% more than that without clipping noise cancellation. O 2022, 22, 3937. [[https://doi.org/10.3390/s22103937](https://doi.org/10.3390/s22103937)]([https://doi.org/10.3390/s22103937](https://doi.org/10.3390/s22103937))  
* [62] **Citation:** [PERSON]. Although the OTFS system has the above advantages, its inherent rectangular prototype window function (PWF) has high out-of-band emission (OOBE), resulting in low spectrum utilization in multi-user scenarios [8]. In [9], the OTFS system employs a long ellipsoid function as the PWF, achieving a significant OOBE reduction. However, the channel gains of edge symbols become lower, which increases the bit error rate (BER). In [10], an extended general frequency division multiplexing (GFDM) framework was proposed to replace OTFS for low OOBE applications in high-Doppler multi-user scenarios. However, when GFDM uses zero-forcing detection for perfect reconstruction (PR), the BER obtained will be worse due to the sensitivity of inverse matrix to noise interference [11]. As a variant of GFDM, bi-orthogonal frequency division multiplexing (BFDM) modulation was proposed in [12; 13], which demodulates the signal by using the matched filter coefficient matrix to avoid the noise interference on the inverse matrix. A new idea is to use BFDM as the time-frequency modulation in the OTFS system [14]. The BFDM-based OTFS (OTFS-BFDM) can provide low OOBE without any loss in BER when using the Dirichlet PWF. Like the OTFS signal, the OTFS-BFDM signal also has an abnormally high peak-to-average ratio (PAPR) in big data packets. When the linear dynamic range of the high power amplifier (HPA) is insufficient, a large PAPR will cause severe distortion of the transmission signal [15]. Currently, various techniques have been proposed to reduce the PAPR, including clipping and filtering (CF) [16; 17; 18], compression-expansion transformation [19; 20], and selective mapping (SLM) [21; 22]. It is worth noting that for PAPR reduction in multi-carrier systems using message-passing (MP) decoding, the sparsity of the ECM must be maintained [23]. In [24], [PERSON] et al. proposed an SLM method based on a resource allocation strategy to suppress the PAPR of sparse code multiple access-based OFDM (SCMA-OFDM), but extra sideband information must be transmitted.