# Random Frequency Division Multiplexing
## 1 Introduction  
Next-generation wireless systems and standards, beyond 5G and 6G, are expected to support a variety of services, such as vehicle-to-everything communications, autonomous driving, remote surgery, aerial vehicles, and operations in extremely high-frequency bands. However, the spectrum resource limitations, high propagation loss, complex network architectures, and energy consumption issues we face all require new modulation techniques and waveforms, which are able to cope with various challenging requirements and show robustness in high mobility scenarios. Current systems are based on Orthogonal Frequency Division Multiplexing (OFDM) which can achieve near-optimal performance in time-invariant frequency selective channels. Nevertheless, due to its high PAPR (Peak-to-Average Power Ratio) and large Doppler frequency shifts, its performance drastically decreases by inter-carrier interference (ICI). To address this issue, Orthogonal Time Frequency Space (OTFS) is designed to multiplex information symbols in the delay-Doppler (DD) domain, effectively mitigating the effects of channel delays and Doppler shifts [1]. The affine frequency division multiplexing (AFDM) employs the inverse discrete affine Fourier transform to modulate symbols into a \"warped\" time-frequency domain to handle channel delays and Doppler shifts [2]. The interleave frequency division multiplexing (IFDM) proposed the interleave and Fourier transform to enhance the channels' statistical stability, and the low-complexity detectors are able to ensure bit error rate (BER) performance [3]. The core of OFDM, OTFS, AFDM, and IFDM lies in constructing sparse equivalent channel matrices that facilitate low-complexity signal detection algorithms, balancingoptimal performance with implementation complexity. While the maximum likelihood (ML) detector is optimal, its exhaustive search process makes it inefficient for multiple-input-multiple-output (MIMO) systems [4]. Although the zero-forcing (ZF) and minimum mean square error (MMSE) detectors which are widely used in MIMO systems provide suboptimal performance compared to ML; these detectors require the computation of the inverse of the Gram matrix of the channel. When the channel matrix has a large dimension, it will make these detectors impractical due to their high computational complexity, which poses challenges for the system's hardware efficiency and overall cost. Many other low-complexity detectors have been investigated. Paper [5] proposed a highly efficient detection technique achieved by dimensionality reduction in the channel matrix and the BER performance is comparable to the MMSE detector. Paper [6] presented a suboptimal ML detector which is suitable for the flexible implementation of OFDM-IM (index modulation) systems. However, the new features of future communications [7], such as complex scenarios with unknown channel models, high speed, and accurate processing requirements, make traditional methods no longer suitable; embedding deep learning (DL) theories into communication systems has attracted a lot of attentions, and researchers believe that DL can achieve further performance improvements in complex scenarios for the following reasons: firstly, the deep network has been proven to be a universal function approximator with superior algorithmic learning ability. Secondly, handling large data is an essential feature of DL because of the instinctive nature of its distributed and parallel computing architectures. Finally, DL-based communication systems can break the artificial block structure to achieve global performance improvement because they are trained to optimize end-to-end performance. There has been a lot of research on combining DL into MIMO communication, such as channel estimation, signal transmission, detection techniques, and so on. For example, Sun proposed a novel learn iterative search algorithm (LISA) for signal detection in a MIMO system with the DNN.