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Orthogonal Time Frequency Space Modulation

Ronny Hadani, Shlomo Rakib, Shachar Kons, Michael Tsatsanis, Anton Monk, Christian Ibars, Jim Delfeld, Yoav Hebron, Andrea J. Goldsmith, Andreas F. Molisch, Robert Calderbank

arXiv:1808.00519v1cs.IT

TL;DR

Many 5G applications do not satisfy the assumptions underlying OFDM’s optimality, motivating a new modulation design. The paper introduces OTFS in the delay-Doppler domain, where equalization extracts full channel diversity and produces nearly constant gains across symbols. OTFS consequently reduces adaptation overhead and improves performance in high-mobility, short-packet, and large-antenna settings.

  • Problem

    OFDM’s optimality depends on transmitter CSI, Gaussian modulation, long codewords, and unlimited receiver complexity, assumptions that many 5G applications do not fulfill.

  • Method

    OTFS is a two-dimensional modulation scheme that maps information symbols in the delay-Doppler domain and uses equalization to extract full channel diversity.

  • Results

    OTFS gives all modulated symbols essentially the same channel gain and is reported to improve packet error rates or reduce SNR requirements in high-mobility and high-phase-noise settings.

  • Takeaways & Limitations

    OTFS removes the need for channel adaptation, reduces physical-layer adaptation overhead, and supports improved PAPR and finite-complexity MIMO performance.

Abstract

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This paper introduces a new two-dimensional modulation technique called Orthogonal Time Frequency Space (OTFS) modulation. OTFS has the novel and important feature of being designed in the delay-Doppler domain. When coupled with a suitable equalizer, OTFS modulation is able to exploit the full channel diversity over both time and frequency. Moreover, it converts the fading, time-varying wireless channel experienced by modulated signals such as OFDM into a time-independent channel with a complex channel gain that is essentially constant for all symbols. This design obviates the need for transmitter adaptation, and greatly simplifies system operation. The paper describes the basic operating principles of OTFS as well as a possible implementation as an overlay to current or anticipated standardized systems. OTFS is shown to provide significant performance improvement in systems with high Doppler, short packets, and/or large antenna array. In particular, simulation results indicate at least several dB of block error rate performance improvement for OTFS over OFDM in all of these settings.

I. INTRODUCTION

The paper motivates a new modulation design for 5G because OFDM’s optimality depends on assumptions that many emerging applications do not satisfy. It introduces OTFS as a delay-Doppler modulation that uses equalization to exploit channel diversity and stabilize symbol gains.

  • Motivation: 5G applications challenge OFDM’s optimality assumptions, including transmitter CSI, Gaussian signaling, long codewords, and unlimited receiver complexity.These assumptions are not fulfilled in many anticipated next-generation applications.
  • OTFS principle: OTFS modulates in the delay-Doppler domain and transforms a time-varying multipath channel into a two-dimensional channel with nearly constant gain across symbols.An appropriate equalizer enables the extraction of full channel diversity.
  • OTFS principle: OTFS uses two-dimensional orthogonal basis functions spanning the transmission bandwidth and packet duration, distinguishing it from conventional OFDM and CDMA designs.The basis functions are derived to combat time-varying multipath dynamics.
  • System implications: The delay-Doppler representation is compact and sparse, making it attractive for channel estimation, prediction, tracking, and large-antenna systems.The paper notes that sparsity and lower channel variability can support lower-complexity, more robust equalization.
  • Relation to prior work: OTFS differs from prior time-frequency pulse-shaping approaches by targeting minimal cross-interference and full diversity through delay-Doppler lattice and pulse design.The paper positions this as a distinct design objective from mitigating interference solely in the time-frequency domain.

A. The Delay-Doppler Channel

The paper represents time-varying wireless channels in the delay-Doppler domain, where propagation paths appear as delayed and Doppler-shifted signal copies. This representation is physically meaningful, sparse, and naturally supports the paper’s operator-based modulation analysis.

  • Channel representation: The Doppler-variant impulse response hc(τ, ν) characterizes each propagation path by its delay, Doppler shift, and complex channel weight.The received signal is modeled as a superposition of reflected copies of the transmitted signal.
  • Channel representation: The delay-Doppler representation connects directly to physical scatterers because each reflector contributes a characteristic delay and Doppler pair.Under single-scattering assumptions, scatterer locations can be inferred from this representation.
  • Convention: The two impulse-response conventions differ by a phase term e^j2πντ, but consistent notation yields equivalent results.The difference corresponds to whether delay or Doppler shifting is applied first.
  • Sparsity: Delay-Doppler channel models are typically compact and sparse because only a small number of physical reflectors and associated Dopplers contribute significant parameters.This can reduce the number of parameters needed for channel modeling and estimation.
  • Operator formulation: The channel can be viewed as a linear operator Π_h that maps the transmitted signal s(t) to the received signal r(t).The operator is parameterized by the delay-Doppler impulse response.
  • Operator formulation: Composing channel and modulation operators produces a twisted convolution, the delay-Doppler analogue of the convolution property of the Fourier transform.This property provides the mathematical basis for combining the channel and modulation descriptions.

C. Time-Frequency Modulation

The paper presents time-frequency modulation as a lattice-based transmitter–receiver framework using pulses, Heisenberg transforms, and matched filtering. Under bi-orthogonality, the ideal channel reconstructs symbols cleanly, while general channels yield multiplicative gains and possible interference when robustness fails.

  • System components: A time-frequency modulation system uses a lattice, a finite packet burst, a two-dimensional symbol sequence, and bi-orthogonal transmit and receive pulses.The lattice samples time and frequency at integer multiples of T and Δf.
  • Transmitter: The modulator maps X[n, m] to s(t) by superposing time-shifted and frequency-modulated versions of the transmit pulse.This modulation rule is the Heisenberg transform applied to the symbol sequence.
  • Receiver: The receiver obtains symbol statistics by matched filtering, computing a cross-ambiguity function, and sampling it on the time-frequency lattice.The sampled transform is described as a discrete Wigner transform and a generalization of OFDM demodulation.
  • Input-output relation: For an ideal channel, matched filtering perfectly recovers the transmitted symbols up to an uncorrelated noise term.This generalizes OFDM’s perfect-reconstruction property for non-dispersive channels.
  • Input-output relation: Under robust bi-orthogonality, the sampled relation is Ŷ[n, m] = H[n, m]X[n, m] + V[n, m], with no cross-symbol interference.The received symbol differs from the transmitted symbol only through a multiplicative gain and additive noise.
  • Input-output relation: If robust bi-orthogonality is violated, residual cross-symbol interference can affect the received symbols.The paper explicitly identifies this as the non-ideal case requiring further consideration.

D. The Symplectic Fourier Transform

The finite symplectic Fourier transform (SFFT) maps periodic delay-Doppler sequences to time-frequency sequences and exchanges circular convolution with point-wise multiplication.

  • The SFFT converts periodic 2D sequences between delay-Doppler and time-frequency representations.Its input is a 2D periodic sequence with periods (M, N), and its inverse performs the reverse mapping.
  • The delay and Doppler sampling intervals are inversely related to burst bandwidth and duration, respectively.Increasing bandwidth or duration improves delay or Doppler sampling resolution.
  • The inverse SFFT reconstructs the periodic delay-Doppler sequence through a corresponding Fourier summation.
  • The transform couples frequency with delay and time with Doppler using a minus sign, producing symplectic coupling.
  • The SFFT interchanges circular convolution and point-wise multiplication of periodic sequences.This property is the symplectic analogue of the conventional finite Fourier transform's convolution property.

III. OTFS MODULATION

OTFS is framed as time-frequency multicarrier modulation with preprocessing that transforms delay-Doppler information into the time-frequency domain.

  • OTFS combines time-frequency multicarrier modulation with preprocessing from the delay-Doppler domain to the time-frequency domain.

A. Interpretations of OTFS modulation

OTFS admits complementary interpretations as delay-Doppler modulation, time-frequency spreading, and Zak-transform-based modulation, linking it to OFDM, CDMA, and single-carrier transmission.

  • Modulation in the delay-Doppler domain: OTFS carries QAM information over compact basis pulses in the delay-Doppler domain.Its channel interaction is described as a two-dimensional convolution with the Doppler-variant impulse response.
  • Figure 1 contrasts delay-Doppler basis functions with their symplectic Fourier duals in the time-frequency domain.
  • Spreading in the time-frequency domain: In the time-frequency domain, OTFS spreads information over maximally spread-out orthogonal basis functions.This interpretation makes OTFS a two-dimensional version of CDMA and highlights its diversity interpretation.
  • Zak representation: OTFS is the modulation format naturally associated with the Zak transform, analogous to OFDM's association with the Fourier transform.

B. OTFS Modulation and Demodulation

OTFS modulation maps delay-Doppler information symbols into time-frequency samples and then into a transmitted time-domain signal; the receiver reverses these operations to detect symbols.

  • Transmitter and receiver: The transmitter applies the OTFS transform and Heisenberg transform, while the receiver uses the discrete Wigner transform and inverse SFFT.These operations map symbols between delay-Doppler, time-frequency, and time domains.
  • Windowing: The transmitter starts with a finite 2D QAM sequence and uses transmit and receive windows whose product determines the effective filtering function.
  • Time-frequency spreading: Each information symbol is spread over the full time-frequency grid through a two-dimensional basis function.This allows each QAM symbol to exploit diversity associated with all channel modes.
  • Delay-Doppler input-output relation: The delay-Doppler input-output relation is a periodic convolution of the sampled, windowed channel impulse response with the transmitted sequence, plus noise.The demodulated noisy sequence is related to the periodized transmitted information sequence through this convolutional model.
  • Windowing: Increasing window support in time and frequency narrows the delay-Doppler filtering function, making the effective channel more closely approximate the true channel impulse response.
  • Delay-Doppler input-output relation: Figure 3 depicts the noise-free convolution between the sampled and filtered delay-Doppler channel impulse response and modulation symbols.

C. Equalization

OTFS equalization must address intersymbol interference when bi-orthogonality fails, with several possible two-dimensional receiver structures available.

  • When the bi-orthogonality condition is not fulfilled, 2D intersymbol interference must be eliminated by a suitable equalizer.
  • Candidate equalizers include linear, decision-feedback, maximum-likelihood sequence, and turbo equalizers.
  • Linear equalizers generally perform poorly, whereas advanced structures have limited complexity impact because the channel response is sparse and translation invariant.

IV. INTERPRETATION AND IMPLEMENTATION

OTFS can overlay existing OFDM hardware through a 2D transform, while delay-Doppler multiplexing preserves diversity that time-frequency allocation can reduce.

  • A. Implementation as overlay: OTFS overlay implementation adds a 2D SFFT before feeding symbols into an existing OFDM modulator.Current OFDM transceivers already implement the Heisenberg/Wigner transform used by the OFDM modulator.
  • A. Implementation as overlay: 37% more additional complexity than SC-FDMA is required for the OTFS overlay under LTE parameters M = 1200 and N = 14.An optimized OTFS transmitter can instead have complexity essentially half that of an equivalent OFDM transmitter.
  • B. Multiplexing: Delay-Doppler multiplexing assigns different basis functions, information symbols, or resource blocks to users while preserving full spreading across the time-frequency window.The basis-function orthogonality enables receiver separation, and the full spreading provides full diversity.
  • B. Multiplexing: Time-frequency resource-block allocation can reduce diversity when each user’s signal occupies only a subset of the time-frequency plane.

C. Diversity and channel gain

OTFS combines an almost-constant effective channel with suitable equalization to extract full diversity and simplify adaptation and receiver processing. Simulations report advantages over OFDM for high mobility and short packets, with gains dependent on the equalizer and scenario.

  • Channel gain and diversity: Each demodulated OTFS symbol experiences the same channel gain over a frame, enabling full channel diversity with a nonlinear receiver equalizer.The delay-Doppler channel is described as sparse and less variable, supporting lower-complexity and more robust processing.
  • Channel gain and diversity: A nearly constant channel reduces the need for fast adaptive modulation and coding when channel coherence is shorter than feedback time.The paper connects this benefit to high-mobility settings where accurate transmitter CSI may be unavailable.
  • Receiver complexity: OTFS simplifies equalizers, decoders, and precoders because equalizer coefficients need not be adapted across symbols within the relevant interval.The paper contrasts this with OFDM, where symbols generally require different equalizer coefficients.
  • Performance results: An iterative DFE receiver significantly outperforms OFDM in coded simulations, whereas OTFS with MMSE or standard DFE can perform poorly because of error propagation.The iterative DFE closely approximates genie-aided DFE performance.
  • Performance results: 4 dB or more BLER gains are reported for OTFS over OFDM as SNR increases with short packets occupying 4 of 50 resource blocks.The paper attributes the gain to spreading each QAM symbol across all time and frequency dimensions and extracting the resulting diversity.
  • Performance results: At 120 km/h, OTFS reduces effective-SNR variability from more than 4 dB OFDM standard deviation to 1.1 dB with a 1 ms window and 0.2 dB with a 10 ms window.At outage probability 0.01, the reported fading-margin reductions are 7 dB and 9 dB relative to OFDM for the two OTFS windows.

VI. CONCLUSION

The paper presents OTFS as a two-dimensional delay-Doppler modulation scheme that, with equalization, extracts full channel diversity and gives symbols a common channel gain. It reports benefits including reduced adaptation needs, improved error performance, lower PAPR, and higher finite-complexity MIMO capacity.

  • Core contribution: OTFS operates in the delay-Doppler coordinate system and, with equalization, gives all modulated symbols the same channel gain by extracting full channel diversity.This is the paper’s central operating principle and reported contribution.
  • System benefits: OTFS removes the need for channel adaptation by providing a stable data rate, especially when high mobility makes transmitter CSI feedback impossible or costly.
  • System benefits: OTFS provides better packet error rates at the same SNR, or lower SNR requirements at the same PER, under high mobility or high phase noise.
  • System benefits: OTFS improves peak-to-average power ratio, particularly for short-packet transmission.
  • System benefits: OTFS improves MIMO capacity when receivers have finite complexity, while further work remains on optimizing performance, reducing complexity, and coexistence.

APPENDIX

The appendix derives the OTFS input-output relation by substituting the channel and modulation expressions into the demodulation equation. Fourier-transform and convolution identities then produce the desired result.

  • Derivation: The derivation starts from the received signal as the sum of the noise-free term and additive noise, then substitutes the channel and modulation equations into demodulation.
  • Transform step: Recognizing the bracketed factor as the discrete symplectic Fourier transform of W[n, m] yields an intermediate transformed expression.
  • Convolution step: The remaining double integral is identified as a convolution of the channel impulse response, multiplied by an exponential, with the transformed window.
  • Conclusion: Equation (46) is presented as the desired result of the derivation.
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