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A Unifying View of OTFS and Its Many Variants

Qinwen Deng, Yao Ge, Zhi Ding

arXiv:2502.09118v1eess.SP

TL;DR

High mobility makes OFDM vulnerable to Doppler, while the proliferation of OTFS variants has created confusion about their relationships and claimed advantages. This paper unifies their signal models, modulation connections, fading-channel representations, and detection schemes, then compares PSD, BER, and detector behavior. It establishes common structure across OTFS-related modulations while highlighting performance and complexity trade-offs and remaining high-mobility challenges.

  • Problem

    Existing surveys explain OTFS but do not adequately clarify the connections among its many variants and their comparative performance claims.

  • Method

    The paper develops unified signal and channel models, relates OTFS variants through transformations and pulse shaping, and surveys modulation comparisons and detection schemes.

  • Results

    OTFS-SRRC and ODDM show similar PSD and BER performance, whereas OTFS-REC has higher OoBE power and poorer BER performance.

  • Takeaways & Limitations

    The unified view helps distinguish OTFS-related modulations and frames detector design as a balance between error performance and computational complexity.

Abstract

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High mobility environment leads to severe Doppler effects and poses serious challenges to the conventional physical layer based on the widely popular orthogonal frequency division multiplexing (OFDM). The recent emergence of orthogonal time frequency space (OTFS) modulation, along with its many related variants, presents a promising solution to overcome such channel Doppler effects. This paper aims to clearly establish the relationships among the various manifestations of OTFS. Among these related modulations, we identify their connections, common features, and distinctions. Building on existing works, this work provides a general overview of various OTFS-related detection schemes and performance comparisons. We first provide an overview of OFDM and filter bank multi-carrier (FBMC) by demonstrating OTFS as a precoded FBMC through the introduction of inverse symplectic finite Fourier transform (ISFFT). We explore the relationship between OTFS and related modulation schemes with similar characteristics. We provide an effective channel model for high-mobility channels and offer a unified detection representation. We provide numerical comparisons of power spectrum density (PSD) and bit error rate (BER) to underscore the benefit of these modulation schemes in high-mobility scenarios. We also evaluate various detection schemes, revealing insights into their efficacies. We discuss opportunities and challenges for OTFS in high mobility, setting the stage for future research and development in this field.

I. INTRODUCTION

The paper addresses confusion among OTFS variants by unifying their relationships, signal models, comparisons, and detection schemes for high-mobility communications. It frames OTFS within OFDM and FBMC while evaluating modulation and receiver behavior.

  • Motivation: OFDM faces out-of-band emissions, cyclic-prefix overhead, and strong Doppler sensitivity in high-mobility scenarios.
  • Motivation: The paper unifies OTFS variants by analyzing their connections, similarities, differences, and claimed performance advantages.
  • Modulation relationships: OTFS is presented as FBMC using an ISFFT precoder, while Vector OFDM and OSDM are identified as special cases under rectangular pulses and Nyquist sampling.
  • Evaluation framework: The paper compares modulation schemes using power spectral density and receiver BER, and models time- and frequency-selective fading with path gain, delay, and Doppler shift.
  • Detection: It surveys linear and nonlinear OTFS detectors, including decision feedback, message passing, AI-enhancement, and cross-domain methods, emphasizing the trade-off between error performance and complexity.
  • Outlook: The paper concludes by outlining opportunities and challenges for OTFS-related modulation schemes in high-mobility communications.

B. FBMC

FBMC provides a generalization of OFDM through prototype-filter pulse shaping, and the paper focuses on its OQAM form. FBMC-OQAM can avoid a cyclic prefix and attain OFDM-like spectral efficiency, but requires more complex filtering.

  • Overview: FBMC generalizes OFDM by using variable pulse shaping to improve time-frequency localization and control out-of-band emissions.
  • FBMC-OQAM: FBMC-OQAM combines inverse Fourier transform, pulse shaping, and phase rotation for transmission.
  • Orthogonality: The prototype filter uses real-valued symbols to relax complex orthogonality and support denser time-frequency spacing.
  • Trade-offs: FBMC transmits without a cyclic prefix, increasing spectral efficiency, but its complex filter design increases implementation complexity.

C. OTFS

OTFS places symbols on a delay-Doppler grid, transforms them to the time-frequency domain with ISFFT, and then generates the transmit waveform through a Heisenberg transform. Its pulse choices affect implementation, out-of-band behavior, and performance.

  • OTFS multiplexes information symbols on a two-dimensional delay-Doppler grid rather than OFDM’s time-frequency grid.
  • ISFFT maps delay-Doppler symbols to time-frequency symbols using an N-point IDFT across Doppler and an M-point DFT across delay.
  • The Heisenberg transform converts time-frequency symbols into the time-domain transmit signal using a transmit pulse g(t) with duration no greater than T.
  • OTFS can be viewed as a precoded FBMC system whose ISFFT acts as the precoder, apart from the OQAM-induced phase-shift term.
  • A single CP of length Ncp covers the entire OTFS transmission block, providing higher spectrum efficiency than conventional OFDM.
  • Rectangular transmit and receive pulses simplify modulation and demodulation but perform poorly for out-of-band control, whereas SRRC pulses offer better performance.

III. CONNECTING MULTIPLE OTFS VARIANTS

The paper unifies several OTFS-related modulations by comparing their transform-domain structures and mathematical expressions. OSDM is shown to coincide with OTFS under rectangular pulses, while the broader family shares the principle of spreading symbols outside the time-frequency domain.

  • III. CONNECTING MULTIPLE OTFS VARIANTS: The section compares modulation schemes with OTFS-like characteristics and identifies common structures across their transforms and signal representations.
  • A. OSDM: OSDM partitions the transmitted block into N symbol vectors of length M, forming an M × N symbol matrix analogous to OTFS’s delay-Doppler symbol matrix.
  • A. OSDM: OSDM transmits its signal after adding a cyclic prefix and processes the received signal after cyclic-prefix removal.
  • A. OSDM: OSDM’s modulation and demodulation expressions are identical to OTFS’s when rectangular pulses are used, making OSDM a mathematical special case of OTFS.

B. Vector OFDM

Vector OFDM applies blockwise transforms that match OTFS expressions under rectangular pulses, while OTSM uses a delay-sequency representation and Walsh-Hadamard transforms as a related but distinct design.

  • B. Vector OFDM: V-OFDM modulates symbols blockwise and applies a component-wise N-point IDFT to its transmit matrix.
  • B. Vector OFDM: V-OFDM’s transmit expression matches OTFS with a rectangular pulse, although V-OFDM does not explicitly represent modulation in the delay-Doppler domain.
  • B. Vector OFDM: After cyclic-prefix removal, V-OFDM applies an N-point DFT to the received signal matrix.
  • B. Vector OFDM: Considering the mappings between transmit matrix X and received matrix Y, V-OFDM is a mathematical special case of OTFS with rectangular pulses.
  • OTSM: OTSM places symbols in the delay-sequency domain and applies an N-point Walsh-Hadamard transform along the sequency dimension.
  • OTSM: Relative to OTFS, OTSM replaces the inverse Fourier transform with a Walsh-Hadamard transform and adds a row-column interleaver before cyclic-prefix insertion.
  • OTSM: The OTSM receiver removes the cyclic prefix, reshapes the received vector into a matrix, and applies a Walsh-Hadamard transform to recover the delay-sequency representation.

D. ODDM

ODDM places data symbols in the delay-Doppler domain but converts them to time-domain signals using a T/M-interval stagger and pulse shaping, distinguishing it from OTFS.

  • ODDM modulation: ODDM maps data symbols to the delay-Doppler domain and uses a T/M-interval stagger in the transform, unlike OTFS's ISFFT and Heisenberg transform.The stagger can be viewed as staggered multitone modulation.
  • Pulse shaping: ODDM applies a pulse g(t), with a symmetric square-root Nyquist-I pulse offered as one ISI-free choice for the T/M symbol interval.The pulse has truncated support, with Q an integer and 2Q ≪ M.
  • Time-domain signal: The ODDM transmit signal can be rewritten in a pulse-shaped OFDM form, making its duration longer than OTFS's and simplifying signal analysis.This representation is described as more accessible than the original expression.
  • Discrete representation: At Nyquist sampling, the no-CP transmit signal contains Nt = MN + 2Q −1 time-domain samples before cyclic-prefix insertion.The sampling period is Ts = T/M.
  • Demodulation: The receiver removes the cyclic prefix, generates subcarrier outputs for each ODDM symbol, and represents the received signal in the delay-Doppler domain.The received time-domain vector after CP removal is denoted r ∈ C^Nt×1.

E. Summary

The paper compares OTFS, ODDM, and related variants through structural analysis plus PSD and BER experiments. OTFS-SRRC and ODDM show similar spectral and BER behavior, while OTFS-REC performs worse and is more bandwidth-sensitive.

  • Modulation relationships: OSDM and Vector OFDM are mathematically identical to OTFS under rectangular transceiver pulses and Nyquist sampling, whereas OTSM and ODDM follow the same core Doppler-robust mapping principle without being identical.These relationships are summarized alongside other modulation attributes in Table III.
  • PSD comparison: The PSD comparison evaluates OTFS-REC, OTFS-SRRC, and ODDM, using oversampling and specified symbol, subcarrier-spacing, pulse, and roll-off settings.The simulations use γ = 8, (M, N) = (512, 64), and Δf = 15 kHz.
  • PSD comparison: OTFS-SRRC has a PSD similar to ODDM, whereas OTFS-REC has the largest bandwidth and presents greater difficulty suppressing out-of-band emissions.The comparison links pulse choice to bandwidth and OoBE behavior.
  • BER comparison: OTFS-REC has worse BER than OTFS-SRRC or ODDM across SNRs, while OTFS-SRRC and ODDM exhibit similar BER performance under varying user velocities.The BER study uses three paths, channel filtering, and 500 Monte Carlo runs.
  • BER comparison: OTFS-REC is more sensitive to narrower channel bandwidth, consistent with its higher out-of-band emissions and greater signal-energy loss at the receiver.At 300 km/h, BER is also evaluated across channel filter bandwidths.

V. EFFECTS OF DIFFERENT CHANNEL MODELS

The paper derives a unified effective input-output model for OTFS-related modulations under doubly selective fading, supporting common analysis and detector design across waveform variants.

  • Unified model: The channel analysis produces a unified linear model y = Hx for efficient detector design in OTFS-related demodulations.The model incorporates the modulation-specific transmitter and receiver structures.
  • Channel model: The modeled channel includes multipath delays and Doppler effects, with sampled responses determined by propagation paths, channel taps, filtering, and the sampling interval.The number of taps depends on maximum delay and the duration of the combined transmit and receive filtering response.
  • Input-output relationship: After matched filtering and cyclic-prefix removal, the received vector is compacted into a matrix relationship involving the sampled transmit signal, received signal, noise, and channel matrix.The noise vector is the filtered noise after CP removal.
  • Modulation-specific forms: With rectangular pulses and sampling interval Ts = T/M, OTFS's input-output relationship is also shared by Vector OFDM and OSDM.Table IV summarizes the corresponding relationships across the modulation schemes.
  • Channel-matrix structure: FBMC and ODDM have distinct effective channel-matrix structures because their transmit signals contain different numbers of time-domain samples.The FBMC matrix lacks the upper-right nonzero entries associated with cyclic-prefix structure because FBMC does not add a CP.
  • Channel-matrix visualization: An example with M = 8 and N = 4 at 300 km/h shows highly similar effective channel matrices for OTFS-SRRC and ODDM.The example uses three paths with independent off-grid delays.

B. Time-selective fading channel model

The paper examines time-selective and frequency-selective special cases of the channel model and connects their effective matrices to interference and receiver design across OTFS-related modulations.

  • Time-selective fading channel model: Time-selective fading models strong Doppler with little or no multipath as a special case with L = 1, producing Doppler spreading and nearby-symbol ISI in the delay-Doppler domain.The effective matrix reveals this interference through its column-wise relationship to delay-Doppler symbols.
  • Time-selective fading channel model: OTFS-REC, OTFS-SRRC, and ODDM exhibit similar effective-channel-matrix patterns under the exemplary time-selective fading channel.The comparison is shown in the time-selective fading column of Table V.
  • Frequency-selective fading channel model: Frequency-selective fading, modeled with νi = 0 for all paths, captures multipath without Doppler and yields a block-circular channel structure after OTFS's IDFT and DFT operations.This case is associated with lower-mobility settings such as indoor WLANs and industrial IoT 4.0.
  • Summary: Across the examined channels, the paper presents effective channel matrices as a cohesive framework for analyzing and optimizing related modulation schemes.The framework covers doubly selective, time-selective, and frequency-selective channel models.
  • Receiver implications: Non-diagonal effective channel matrices imply inherent ISI, especially in doubly selective fading, motivating receivers that estimate transmitted symbols from the received vector and effective channel matrix.The unified detection model spans time-frequency, delay-Doppler, delay-sequency, and chirp domains.

A. Linear Detection Receivers

Linear detectors transform the received channel output before symbol decisions, trading lower complexity for performance limitations under interference, noise, or ill-conditioned channels.

  • Linear receivers derive decision statistics by filtering the channel output and applying a memoryless nonlinear decision device.
  • Matched Filter: Matched filtering maximizes sampled SNR while treating interference from other sub-streams as noise.
  • Matched Filter: Matched-filter performance degrades significantly in overloaded systems or with ill-conditioned channel matrices.
  • Zero-Forcing: Zero-forcing removes co-channel interference through the Moore-Penrose pseudo-inverse but can enhance transformed noise for ill-conditioned channels.
  • LMMSE: LMMSE jointly accounts for interference and noise, often outperforming zero-forcing under strong channel noise.
  • LMMSE: LMMSE requires matrix inversion, motivating low-complexity approximations that exploit OTFS channel sparsity or quasi-banded structure.

B. High Performance Non-Linear Detection Receivers

Nonlinear detectors improve reception by exploiting feedback, probabilistic inference, memory, or structured channel properties, but their complexity and robustness depend on the chosen design.

  • Decision Feedback: Decision-feedback equalization can improve performance over linear detectors with relatively low complexity, while hard decisions risk error propagation.
  • Decision Feedback: Soft and bidirectional decision-feedback structures reduce the impact of erroneous tentative decisions compared with hard feedback.
  • Message Passing: Message passing approximates marginal symbol posteriors on factor graphs by iteratively passing messages, potentially approaching maximum-likelihood performance at lower complexity.
  • Non-Memory-Based Message Passing: OAMP/VAMP uses linear and nonlinear estimators with orthogonality constraints, but matrix inversion or SVD limits scalability.
  • Non-Memory-Based Message Passing: GMP and EP exploit Gaussian approximations, yet short factor-graph girths can cause performance loss and optimized damping remains unresolved.
  • Memory-Based Message Passing: AMP uses a memory Onsager term and is effective for IID Gaussian channels, but may perform poorly or diverge with highly correlated channel matrices.
  • Memory-Based Message Passing: MAMP combines long-memory matched filtering, finite Taylor approximations, and closed-form damping to approach OAMP/VAMP performance at lower complexity.

4) Cross-Domain Iterative Detector:

Cross-domain iterative detection addresses dense delay-Doppler channels caused by fractional Doppler by combining sparse time-domain processing with delay-Doppler symbol constraints.

  • Motivation: Fractional Doppler can make delay-Doppler channels dense, increasing detection complexity for moderate or short OTFS frames.
  • Motivation: Time-domain effective channel matrices remain sparse and banded under fractional Doppler, providing a lower-complexity detection route.
  • Algorithm: The cross-domain algorithm iteratively exchanges extrinsic information between time and delay-Doppler domains under a unitary transformation.
  • Algorithm: Cross-domain detection can extend to ODDM, OTSM, OCDM, and AFDM by combining time-domain sparsity with each scheme’s symbol-domain constraints.
  • Deep Learning Inspired Detection: Deep-learning-inspired detectors can reduce sensitivity to modeling errors and imperfect CSI, but data-driven methods remain poorly interpretable and parameter-heavy.
  • Deep Learning Inspired Detection: Applications of deep-learning-inspired detection to OTFS and related modulations remain early-stage, with environmental variation and training data as open challenges.
  • Head-to-Head Comparisons: In the OTFS-SRRC comparison, CD-MAMP offers practical implementation advantages by combining low complexity with desired BER performance.

C. MIMO and Multi-user OTFS Systems

The survey reviews expanding OTFS applications and identifies unresolved challenges spanning multiuser, wideband, sensing, jamming, vehicular, mmWave/THz, and RIS-assisted systems. It also highlights practical constraints including PAPR and the need for more advanced analytical and signal-processing tools.

  • MIMO-OTFS requires coordinated pilot and data transmission across antennas, while channel estimation and detection must handle interactions across time, frequency, and spatial domains.
  • MU-OTFS faces complex, dynamic inter-user interference from diverse on-grid and off-grid Doppler shifts, with complications increasing as the user count grows.
  • The survey concludes that new analytic tools, signal-processing techniques, and optimization strategies are needed to exploit MU-OTFS and MIMO-OTFS configurations.
  • OTFS PAPR grows linearly with the number of temporal slots N and can approach OFDM-like levels for large N with rectangular pulses.
  • Wideband OTFS performance degrades under frequency-dependent Doppler shifts, whereas ODSS addresses the resulting time-scaling effect through Fourier-Mellin-to-delay-scale preprocessing.
  • Future applications such as V2X, mmWave/THz, RIS-assisted OTFS, ISAC, and anti-jamming systems require robust estimation, detection, configuration, and signal-processing methods.

APPENDIX

The appendix relates chirp-based OCDM and AFDM to the broader OTFS family through shared full-domain symbol spreading and transform-based signal models. OCDM uses DFnT processing with a cyclic prefix, while AFDM uses adjustable DAFT parameters and a chirp-periodic prefix.

  • Chirp-based multicarrier modulations: Chirp-based multicarrier schemes spread data symbols across the full assigned time-frequency domain and share features with OTFS ISFFT.
  • OCDM: OCDM multiplexes overlapping chirp waveforms and obtains its time-domain signal through the inverse discrete Fresnel transform.
  • OCDM: OCDM uses a unitary DFnT matrix and adds a cyclic prefix at least as long as the maximum channel delay spread to avoid inter-block interference.
  • OCDM: After channel transmission and cyclic-prefix removal, OCDM transforms the received signal back to chirp symbols using DFnT.
  • AFDM: AFDM modulates symbols in the discrete affine Fourier domain, adapting its parameters so paths separate and symbols obtain full diversity.
  • AFDM: AFDM uses a chirp-periodic prefix whose length covers the maximum channel delay spread, and this prefix reduces to a cyclic prefix under specified parameter conditions.
  • AFDM: AFDM demodulates the received signal through DAFT, while DFT and DFnT appear as special DAFT cases for particular parameter choices.
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