Affine Frequency Division Multiplexing (AFDM) for 6G: Properties, Features, and Challenges

Next generation wireless communication systems are considering orthogonal time frequency space (OTFS) waveform as an alternative to the existing orthogonal frequency division multiplexing (OFDM) waveform. It is more robust to high Doppler and high carrier frequencies. Multiple recent papers have evaluated OTFS at higher Doppler with long term evolution (LTE) system, to show its performance benefits over OFDM. In this paper, we provide the performance evaluation of OTFS waveform at mmWave frequencies using the third generation partnership project (3GPP) 5G new radio (NR) transmit-receive chain for different quadrature amplitude modulation (QAM) schemes. We compare the OTFS performance with that of OFDM, for 3GPP defined tapped delay line (TDL) and cluster delay line (CDL) channel models at multiple mobility conditions. We analyze the performance difference between OTFS and OFDM using both minimum mean-squared error (MMSE) and decision feedback equalizers (DFE) at the receiver, and show the need for DFE at higher modulation order, to compensate the inter symbol interference. We show that the OTFS has 10 dB gain in signal to noise ratio (SNR) over OFDM, for TDL-C channel with delay spread of 300 ns, at high speed train mobility of 500 kmph for 64-QAM, when evaluated with DFE. We further present the computational complexity of OTFS system, in comparison to OFDM.

In high-mobility environments, rapid channel variations and intercarrier interference (ICI) hinder the performance of orthogonal frequency division multiplexing (OFDM). Delay-Doppler-based modulation schemes such as orthogonal time-frequency space (OTFS) and orthogonal delay-Doppler division multiplexing (ODDM) have been proposed to enable reliable transmission in high-mobility environments. However, they come at the expense of increased latency and increased equalisation complexity. In this work, we present a collection of processes to improve the performance of OFDM systems by leveraging delay-Doppler-based processing. First, we propose a segmentation method that significantly improves the accuracy of the generated time-frequency channel response by eliminating the accumulation of phase rotation error across the frame. We then use this method to improve our previous iterative path identification method, where the influence of the identified path is estimated and accounted for. The segmentation method is improved by developing the two-tap equivalent model, improving the accuracy of the generated timefrequency response by adjusting the frequency of the generated phase rotation. Last, we provide a method for estimating the off-diagonal components of the channel so that the bit error rate (BER) remains robust as the maximum Doppler shift increases. We show that the proposed algorithm provides a 2.5 to 10 dB performance improvement over traditional interpolation-based single-tap equalisation at low and high signal-to-noise ratios (SNR)s.

In this dissertation, new vehicular communication (V2V, V2X) scenarios are going to be studied. Under this context, against common and well known radar systems, new joint radar and communications techniques are taking part of the current literature. These system are mainly focused on the transmission of useful information towards the targets, which are, thus, not only “passively” detected, and a single equipment is able to perform both operational modes, avoiding to split the functionalities between two distinct subsystems (with increased cost and complexity). The choice of the multi-carrier modulation for joint radar and communication falls into two distinct waveforms, i.e., orthogonal frequency-division multiplexing (OFDM) modulation and orthogonal time frequency space (OTFS) modulation. OFDM is the most popular multi-carrier modulation of recent years, widely studied and standardized in most of the current communication standards, including 5G. The motivation of this choice is simple: thanks to the application of a cyclic prefix between symbols, i.e., a guard interval to prevent inter-symbol interference, and under the assumption of absence of inter-carrier interference, which holds under reasonable amount of the Doppler effect and subcarrier spacing, the communication channel can be diagonalized and symbol-by-symbol detection performed. Clearly, the appealing simplicity of detection makes OFDM the best choice for modern digital communications. On the other hand, OTFS is a modulation waveform with two big differences with respect to its direct competitor. First, it does not necessitate the insertion of the cyclic prefix, achieving a better communications rate, i.e., more information is sent over a time-frequency window, but at the cost of a more complex detection approach, working blockwise and not symbol-by-symbol. Second, OTFS is not sensitive to delay and Doppler shifts, meaning that its performance is kept constant whatever the distance and the speed between transmitter and (target) receiver. This feature is very appealing for joint radar and communication tasks, being the scenario very dynamical, with possible remarkable Doppler shifts and delays, increased by considering the round-trip time between radar transmitter and target. Based on the aforementioned analysis, in this dissertation we take care of a fair comparison between the two digital modulation formats, from the point of view of radar, parameter estimation, achievable communication rate, channel estimation, and more other tasks, to determine their positive and negative aspects, such that a system designer is able to choose the most suitable waveform for a given scenario or application.

Obtaining an efficient multicarrier signalling technique is an essential part in signal processing of the (6G) wireless networks. In order to achieve their intended objectives, the 6G mobile networks needs the utilization of a multicarrier scheme that possesses the ability to withstand multipath fading and inter-symbol interference (ISI), as well as exhibit resilience against inter-carrier interference (ICI) and Doppler spread. In recent research, several filter-based methods have been suggested, including Filter Bank Multicarrier (FBMC), as well as various Orthogonal Frequency Division Multiplexing (OFDM) based techniques such as Universal Filtered Multicarrier (UFMC). Orthogonal Time-Frequency Space (OTFS) technique has been identified as a promising candidate waveform for 6G, particularly in high-mobility scenarios, where it has demonstrated superior performance compared to traditional OFDM-based techniques. The OTFS method offers an advantage in that it can be considered the pre-and post-processing stages for techniques based on OFDM. This papaer presents the design (OTFS) using (UFMC) technique. Additionally, a novel structure is proposed for coded/uncoded UFMC-based OTFS technique, which combines the advantages of both strategies while mitigating their limitations. The results of our analysis demonstrate that the UFMC-based OTFS system exhibits superior performance in terms of error probability and time-spectral efficiency compared to the OFDM-based OTFS system.

In this paper, a minimum mean squared error (MMSE) based turbo equalization scheme in delay-Doppler domain for a coded orthogonal time frequency space (OTFS) system is provided as a low complexity version requiring only a single matrix inversion. Also, we derived extrinsic log likelihood ratios (LLRs) for the turbo equalization, exploiting a priori information fed from a channel decoder, under the assumption that the equalized data symbol follows Gaussian distribution. We rigorously conducted link level evaluation to compare coded orthogonal frequency division multiplexing (OFDM) and OTFS over time-varying multipath channels (based on the channel model in 3GPP 5G New Radio) in 30 GHz band, which may cause severe inter-symbol interference (ISI) and/or inter-carrier interference (ICI) depending on numerology setup. The link level simulation verifies that OTFS is superior to OFDM in block error rate (BLER) performance, each being configured with the best numerology, over all the channel scenarios considered. OTFS shows better BLER performance with smaller subcarrier spacing (SCS) while OFDM has an intermediate SCS for the best BLER due to trade-off between ISI and ICI. Also, we measured normalized mean squared error (NMSE) to see how ISI and ICI affect the performance with respect to numerology, which is used to conjecture why the BLER performance has such peculiarity.

Recently proposed orthogonal time frequency space (OTFS) modulation has been considered as a promising candidate for accommodating various emerging communication and sensing applications in high-mobility environments. In this paper, we propose a novel cross domain iterative detection algorithm to enhance the error performance of OTFS modulation. Different from conventional OTFS detection methods, the proposed algorithm applies basic estimation/detection approaches to both the time domain and delay-Doppler (DD) domain and iteratively updates the extrinsic information from two domains with the unitary transformation. In doing so, the proposed algorithm exploits the time domain channel sparsity and the DD domain symbol constellation constraints. We evaluate the estimation/detection error variance in each domain for each iteration and derive the state evolution to investigate the detection error performance. We show that the performance gain due to iterations comes from the non-Gaussian constellation constraint in the DD domain. More importantly, we prove that the proposed algorithm can indeed converge and, in the convergence, the proposed algorithm can achieve almost the same error performance as the maximum-likelihood sequence detection even in the presence of fractional Doppler shifts. Furthermore, the computational complexity associated with the domain transformation is low, thanks to the structure of the discrete Fourier transform (DFT) kernel. Simulation results are consistent with our analysis and demonstrate a significant performance improvement compared to conventional OTFS detection methods.

Orthogonal time frequency space (OTFS) modulation is often declared to be superior to its direct concurrent, i.e., orthogonal frequency division multiplexing (OFDM) modulation. However, the comparison is often performed under favorable conditions for OTFS as, for instance, the assumption of extremely high Doppler shifts and/or the approximation of delay and Doppler shifts to be integer multiples of the Doppler-delay resolution of OTFS modulation. In order to provide a fair comparison, this paper addresses a comparison in terms of the information-theoretic achievable rate under practical separate detection and decoding, explicitly considering the pilot overhead necessary to acquire the channel state information, the accuracy of the channel estimation, and any other possible additional loss. We do not claim to establish in a definitive way which is the best modulation format, since the choice thereof depends on many other features which are outside the scope of this work (e.g., legacy, intellectual property, ease and know-how for implementation, and many other criteria). Nevertheless, we provide the foundations to properly compare multi-carrier communication systems in terms of their information theoretic achievable rate potential, within meaningful and sensible assumptions on the channel models and on the receiver complexity (both in terms of channel estimation and in terms of soft-output symbol detection).

Orthogonal time frequency space (OTFS) modulation has been proved to have better bit error rate (BER) performance over orthogonal frequency division multiplexing (OFDM) under high-mobility conditions. In this letter, a new transmission scheme, called OTFS with dual-mode index modulation (OTFS-DM-IM), is proposed to balance the transmission reliability and spectral efficiency. Moreover, a modified log-likelihood ratio (LLR) detector based on the minimum Hamming distance is designed for demodulation. Then, the theoretical BER analysis of the proposed OTFS-DM-IM is presented. Simulation results highlight the performance advantage of OTFS-DM-IM over classical OTFS and the existing OTFS systems based on index modulation (OTFS-IM), and also confirm the superiority of the modified LLR detector compared to maximum likelihood detector and conventional LLR detector.

Integrated Sensing and Communication (ISAC) systems combine radar sensing and communication to improve spectral and hardware efficiency in 6 G networks. The emerging Orthogonal Time Frequency Space (OTFS) modulation is wellsuited for ISAC due to its robustness in the Delay-Doppler (DD) domain compared to traditional Orthogonal Frequency Division Multiplexing (OFDM). Despite its advantages, OTFS exhibits high sidelobe levels that affect sensing performance. To further enhance the radar sensing aspect of an OTFS-based ISAC system, we design a phase matrix to optimize the OTFS waveform. Our goal is to reduce the sidelobe levels of the 2D Ambiguity Function (AF) for the phase-tailored OTFS waveform. Specifically, we formulate an optimization problem to minimize the AF Peak Sidelobe Level (PSL) by introducing an additional degree of freedom for each communication symbol, thereby improving sensing performance while preserving similar communication efficiency. We solve the related non-convex NP-hard optimization problem and propose an Alternative Optimization (AO)-based algorithm. By decomposing the problem into two subproblems, one convex and the other is solved using Projected Gradient Descent (PGD), we iteratively update the variables until convergence. This approach effectively minimizes the PSL, improving target resolution and reducing interference. Simulation results show that the proposed method efficiently reduces the sidelobe levels and improves the AF, leading to increased sensing performance.

As an effective way to realize the integrated sensing and communication for 6G, the dual-functional radar-communication (DFRC) has been studied to implement dual-functional system that simultaneously perform wireless communication and radar sensing. One of the fundamental design issues in DFRC focuses on waveform design, including the selection of basic waveform (e.g., Orthogonal Time Frequency Space (OTFS) or Orthogonal Frequency Division Multiplexing (OFDM)), and the optimal beamforming design. In this paper, we investigate the transceiver beamforming design for a multi-input-multi-output (MIMO) DFRC system, where the dual functional base station (BS) simultaneously detects the multiple targets while communicating with multiple communication users. To this end, we first characterize the multi-fold tradeoffs between communication and radar sensing by defining an achievable performance region of the proposed DFRC system. The performance of communication and radar sensing are characterized by their signal-to-interference-plus-noise ratios (SINRs), and then the multi-objective optimization problem is formulated to maximize the SINRs of all communication users and radar sensing echoes (from targets) via optimizing beamforming matrix. To solve the problem, we propose a poly-complexity solution by using the quadratic transform and successive convex approximation. Simulation results are provided to verify the proposed design.

This paper presents a linear complexity iterative rake detector for the recently proposed orthogonal time frequency space (OTFS) modulation scheme. The basic idea is to extract and combine the received multipath components of the transmitted symbols in the delay-Doppler grid using linear diversity combining schemes like maximal ratio combining (MRC), equal gain combining and selection combining to improve the SNR of the combined signal. We reformulate the OTFS input-output relation in the vector form by placing some null symbols in the delay-Doppler grid thereby exploiting the block circulant property of the channel matrix. Using the new input-output relation we propose a low complexity iterative detector based on the MRC scheme. The bit error rate (BER) performance of the proposed detector will be compared with the state of the art message passing detector and orthogonal frequency division multiplexing (OFDM) scheme employing a single tap minimum mean square error (MMSE) equalizer. We also show that the frame error rate (FER) performance of the MRC detector can be improved by employing error correcting codes operating in the form of a turbo decision feedback equalizer (DFE).

Target detection is a core function of integrated sensing and communication (ISAC) systems. The traditional likelihood ratio test (LRT) target detection algorithm performs inadequately under low signal-to-noise ratio (SNR) conditions, and the performance of mainstream orthogonal frequency division multiplexing (OFDM) waveforms declines sharply in high-speed scenarios. To address these issues, an information-theory-based orthogonal time frequency space (OTFS)-ISAC target detection processing framework is proposed. This framework adopts the OTFS waveform as its fundamental signal. The target detection is implemented through a relative entropy test (RET) comparing echo signals against target presence/absence hypotheses. Furthermore, to enhance the system's target detection capability, the iterative OTFS-ISAC waveform design (I-OTFS-WD) method which maximizes the relative entropy is proposed. This method utilizes the minorization-maximization (MM) algorithm framework and semidefinite relaxation (SDR) technique to transform the non-convex optimization problem into an iterative convex optimization problem for resolution. The simulation results demonstrate that, under sufficient sample conditions, the RET algorithm achieves a 9.12-fold performance improvement over LRT in low-SNR scenarios; additionally, the optimized waveform reduces the sample requirements of the RET algorithm by 40%, further enhancing the target detection capability of the OTFS-ISAC system.

Known as a new promising waveform modulation technique, the orthogonal time-frequency space (OTFS) technique is considered a very important waveform modulation technique that modulates data in the delay-Doppler (DD) domain. The key difference between OTFS and the traditional multiplexing techniques is that it is two-dimensional modulation that converts between the time-frequency (TF) domain and delay-Doppler domain, these features enable dealing with the Doppler shift generated from high mobility objects which were ignored in the traditional modulation techniques such as orthogonal frequency division multiplexing (OFDM). The main objective of this survey is to provide an overview of this novel subject indicating its system model. Also, we review the main topics related to OTFS modulation as data detection techniques, channel estimation, MIMO, and multiuser systems. Then, the main research direction of OTFS on future wireless generation systems is given.

The orthogonal time frequency space (OTFS) modulation as a promising signal representation attracts growing attention for integrated sensing and communication. To illuminate the capability of OTFS for sensing, this paper studies its performance bound in contrast to Orthogonal Frequency Division Multiplexing (OFDM). We formulate the signal models of OTFS and OFDM in a unified format and extend them to Multiple Input Multiple Output (MIMO) system. Then the Cramér-Rao lower bounds (CRLB) of unknown parameters estimation are derived for them in both Single-Input Single-Output (SISO) and MIMO systems. Simulations in these cases validate the outperformance of OTFS modulation scheme for localization.

A doppler delay-time frequency (DD-TF) cross-domain high-security transmission scheme based on orthogonal time frequency space (OTFS) is proposed, which is compatible with high-speed and large-capacity optical transmission systems. In this scheme, 36 sets of chaotic sequences are mapped out through a two-dimensional chaotic system with variable control parameters and variable initial values for optical physical layer encryption. The data modulated based on OTFS is encrypted by multiple sets of chaotic sequences, achieving cross-domain joint encryption in the DD domains and TF domains. This method improves security performance of the optical transmission system. An experiment is conducted, in which the encrypted 16-quadrature amplitude modulation (QAM) OTFS signals over 25km standard single-mode fiber (SSMF) is achieved, and the key space is 101140 to effectively prevent malicious attacks from illegal Optical Network Units (ONUs). This key space is the reported maximum key space of the physical layer of two-dimensional chaotic systems. Compared with orthogonal frequency division multiplexing (OFDM) encryption scheme, this scheme obtains better transmission performance. It is shown that OTFS system with cyclic prefix (CP) of 1/16 obtains 2.1dB gain than the OTFS system with CP of 1/32 in terms of receiver sensitivity when the bit error rate (BER) is 10-3. Due to its comprehensive advantages in error rate and security performance, this chaotic encryption based on OTFS has a high application prospect in low-cost and reliable optical access system.