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OTFS -- A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain

Saif Khan Mohammed, Ronny Hadani, Ananthanarayanan Chockalingam, Robert Calderbank

arXiv:2302.08696v1eess.SPcs.IT

TL;DR

The paper addresses limited predictability and fading in conventional modulation over doubly-spread channels, while also seeking a common framework for communication and radar sensing. It develops OTFS using delay-Doppler channel representations, geometric modes, and Zak-based waveforms. The resulting interaction is non-fading and predictable in the crystalline regime, with applications to communication and sensing.

  • Problem

    Conventional TDM/FDM signals lack joint time-frequency localization, producing fading and less predictable interactions on doubly-spread channels, while emerging high-mobility systems require improved communication and radar capabilities.

  • Method

    The paper formulates wireless channels with delay-Doppler operators and develops OTFS waveforms as quasi-periodic delay-Doppler pulses realized in time through the inverse Zak transform.

  • Results

    When delay and Doppler periods are large compared with channel spreads, OTFS has a non-fading and predictable input-output relation in the crystalline regime.

  • Takeaways & Limitations

    The delay-Doppler framework provides a shared mathematical basis for communication and sensing, including radar separation along delay and Doppler.

Abstract

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Orthogonal time frequency space (OTFS) is a framework for communication and active sensing that processes signals in the delay-Doppler (DD) domain. This paper explores three key features of the OTFS framework, and explains their value to applications. The first feature is a compact and sparse DD domain parameterization of the wireless channel, where the parameters map directly to physical attributes of the reflectors that comprise the scattering environment, and as a consequence these parameters evolve predictably. The second feature is a novel waveform / modulation technique, matched to the DD channel model, that embeds information symbols in the DD domain. The relation between channel inputs and outputs is localized, non-fading and predictable, even in the presence of significant delay and Doppler spread, and as a consequence the channel can be efficiently acquired and equalized. By avoiding fading, the post equalization SNR remains constant across all information symbols in a packet, so that bit error performance is superior to contemporary multi-carrier waveforms. Further, the OTFS carrier waveform is a localized pulse in the DD domain, making it possible to separate reflectors along both delay and Doppler simultaneously, and to achieve a high-resolution delay-Doppler radar image of the environment. In other words, the DD parameterization provides a common mathematical framework for communication and radar. This is the third feature of the OTFS framework, and it is ideally suited to intelligent transportation systems involving self-driving cars and unmanned ground/aerial vehicles which are self/network controlled. The OTFS waveform is able to support stable and superior performance over a wide range of user speeds.

I. INTRODUCTION

The paper motivates delay-Doppler processing as a response to increasingly extreme doubly-spread channels and emerging high-mobility sensing applications. It develops OTFS waveforms and a mathematical framework intended to make communication and sensing more predictable.

  • I. INTRODUCTION: TDM and FDM lack joint time-frequency localization, causing fading and less predictable channel input-output relations on doubly-spread channels.
  • I. INTRODUCTION: Increasing carrier frequencies and high-speed use cases create more extreme channels that are selective in both time and frequency.
  • I. INTRODUCTION: High-resolution radar imaging in intelligent transportation systems can identify hazards and support responses that enhance road safety.
  • I. INTRODUCTION: OTFS describes wireless channels through delay-Doppler operators and uses geometric modes of those operators for communication and sensing.
  • I. INTRODUCTION: The paper provides a mathematical foundation for delay-Doppler signal processing, while a follow-up studies comparative modulation performance and radar utility.

III. TIME AND FREQUENCY DOMAIN MODULATION

TDM and FDM carriers are localized in only one domain, whereas OTFS uses a quasi-periodic delay-Doppler pulse realized as a pulsone. This design yields non-fading, predictable interaction with doubly-spread channels.

  • III. TIME AND FREQUENCY DOMAIN MODULATION: TDM carriers are narrow in time but spread in frequency, while FDM carriers are narrow in frequency but spread in time.
  • III. TIME AND FREQUENCY DOMAIN MODULATION: A delay-Doppler channel with four dominant paths is represented by impulses whose locations encode path delays and Doppler shifts.
  • III. TIME AND FREQUENCY DOMAIN MODULATION: The lack of joint time-frequency localization makes TDM/FDM channel interactions fading and non-predictable on doubly-spread channels.
  • III. TIME AND FREQUENCY DOMAIN MODULATION: OTFS carries information on a quasi-periodic delay-Doppler pulse and converts it to a time-domain pulsone using the inverse Zak transform.
  • III. TIME AND FREQUENCY DOMAIN MODULATION: The OTFS pulsone interacts with a doubly-spread channel non-fadingly and predictably, supporting superior BER performance and more efficient channel acquisition.

IV. DELAY-DOPPLER MODULATION

OTFS carries information on quasi-periodic delay-Doppler pulses and transforms them into time- and frequency-domain pulsones. Their localization determines orthogonality, time-frequency characteristics, and limiting relationships to TDM and FDM.

  • DD-domain pulse: A DD pulse is defined using reciprocal periods τp and νp = 1/τp, with the time Zak transform linking time-domain signals to quasi-periodic DD signals.The pulse repeats at integer translates along delay and Doppler axes and is localized within the fundamental period.
  • TD/FD realizations: The inverse time Zak transform maps a localized DD pulse into a TD pulsone: a finite-duration pulse train separated by τp and modulated by Doppler tone ν0.Its pulses occur at t = nτp + τ0 and each has duration 1/B.
  • TD/FD realizations: The inverse frequency Zak transform maps the DD pulse into an FD pulsone whose pulses are spaced by νp, centered at f = mνp + ν0, and modulated by e−j2πτ0f.The DD pulse location determines the pulse-train locations and modulation tone in the frequency realization.
  • Pulse location and width: DD pulse shifts translate into TD time displacements and FD frequency displacements, while increasing DD width along delay or Doppler narrows the corresponding FD bandwidth or TD duration.Delay-axis shifts affect TD position and FD modulation; width changes produce reciprocal changes in the corresponding pulsone features.
  • Orthogonality: Pulses separated by 1/B in delay or 1/T in Doppler are almost orthogonal, and pulsones span approximately BT orthogonal carrier waveforms for time duration T and bandwidth B.The DD pulse area is B−1T−1, so the number of non-overlapping pulses in the unit-area fundamental period is BT.
  • TDM and FDM limits: As τp → ∞ OTFS approaches TDM, while as νp → ∞ it approaches FDM; therefore, OTFS forms a modulation family parameterized by τp.These limits collapse one domain and leave a single TD or FD carrier pulse.

V. INTERACTION OF CHANNEL PATHS AND CARRIER WAVEFORMS

The paper compares TDM, FDM, and OTFS under a four-path doubly spread channel containing stationary and moving reflectors. OTFS avoids the path superposition that produces fading in the corresponding TDM and FDM examples, yielding non-fading and predictable interaction for suitable periods.

  • Four-path channel: The example channel has four paths: stationary-building reflections produce no Doppler shift, while moving-vehicle reflections produce Doppler shifts.The paths are used to compare carrier-waveform interactions under the same doubly spread channel.
  • TDM and FDM: TDM exhibits time-varying superposition when equal-delay paths have different Doppler shifts, while FDM exhibits fading when equal-Doppler paths have different delays.The path configuration is chosen to highlight fading and non-predictability in each waveform domain.
  • OTFS: OTFS avoids superposition of received DD pulses from distinct paths because any two paths differ in either delay or Doppler.For suitable values of (τp, νp), the resulting interaction is non-fading and predictable.

A. Interaction with a TD pulse

A doubly-spread channel can make TDM pulse responses unpredictable and fading when paths overlap in delay or Doppler. Distinct, non-Doppler paths instead produce stationary, predictable, non-fading responses.

  • Setup: Two narrow TD pulses transmitted at t0=1 ms and t1=1.5 ms are analyzed over a stationary four-path doubly-spread channel.The pulses are modeled as Dirac-delta impulses because their widths are much smaller than path delays and the inverse maximum Doppler shift.
  • Prediction: 2 µs, 3 µs, and 4 µs path delays can be predicted from the earlier received signal after estimating its path delays.
  • Predictable interaction: Distinct-delay paths without Doppler produce stationary, non-fading interactions whose received power is independent of transmission time.The path gain remains h3 for both transmissions, so the response to any transmitted pulse can be predicted.
  • Non-predictable interaction: A Doppler-shifted path with distinct delay produces non-predictable, non-stationary interaction because its complex gain depends on transmission time.Its received power remains time-independent, but its phase varies with the transmission time.
  • Fading interaction: Paths sharing a delay, with at least one differing Doppler shift, produce non-predictable interaction that is both fading and non-stationary.In this case, both complex gain and received signal power depend on when the pulse is transmitted.

B. Interaction with a FD pulse

Frequency-domain pulses also encounter non-predictable channel interactions under doubly spread propagation. The interaction may be non-fading when Doppler shifts are distinct, but becomes fading when paths share Doppler shift and differ in delay.

  • Setup: Two FD pulses at f0=15 KHz and f1=450 KHz are examined over the same stationary four-path doubly-spread channel.
  • Predicted response: The received FD response to the second pulse is predicted to contain impulses at 0 Hz, −950 Hz, and 750 Hz.
  • Non-fading interaction: A path with distinct Doppler shift yields non-predictable, non-stationary, non-fading interaction because its complex gain varies with transmitted frequency.The received magnitude remains |h2| and does not depend on the transmitted frequency.
  • Fading interaction: Paths with the same Doppler shift and differing delays produce non-predictable interaction that is fading and non-stationary.

C. Interaction with a DD domain pulse

OTFS represents channel interactions in the delay-Doppler domain, where path delay and Doppler shift translate pulses to distinct locations. Under the crystallization condition, responses are predictable and non-fading, although their phases remain location-dependent.

  • C. Interaction with a DD domain pulse: A DD pulse transmitted at (τa,νa) or (τb,νb) produces four distinct received pulses at the corresponding transmit location plus each path’s delay and Doppler.The DD domain separates the received pulse along each channel path.
  • C. Interaction with a DD domain pulse: OTFS converts transmitted and received signals between time and delay-Doppler domains using inverse Zak and Zak transforms.The received DD signal is decomposed into responses associated with pulses transmitted at (τa,νa) and (τb,νb).
  • Predictability: The crystallization condition requires the DD delay period to exceed channel delay spread and the Doppler period to exceed channel Doppler spread.When it holds, the b-response can be predicted from the a-response and DD domain aliasing is precluded.
  • C. Interaction with a DD domain pulse: Quasi-periodic DD pulses repeat at integer multiples of the delay and Doppler periods, with phase changing for delay-period shifts but not Doppler-period shifts.
  • DD domain aliasing: The predictive relation breaks down when responses from DD replicas outside and inside the fundamental period overlap, producing DD domain aliasing.Such overlap makes the path contributions difficult to estimate separately and prevents simple prediction of the later response.
  • Practical condition: 5 × 10^-3 is the delay-spread–Doppler-spread product for a typical cellular channel with 5 µs delay spread and 1000 Hz Doppler spread.The paper states that practical channels generally satisfy the requirement that this product be less than one.
  • Non-fading interaction: Under the crystallization condition, response amplitudes do not depend on transmit-pulse location, so DD channel interaction is non-fading.Without the condition, aliasing can make the power distribution depend on transmit location.
  • Non-stationarity: Even under the crystallization condition, response phases depend on transmit-pulse location, so the interaction remains non-stationary.This non-stationarity is not a major issue when the predictive relation is maintained.

VI. TRANSCEIVER SIGNAL PROCESING

OTFS embeds information in the delay-Doppler domain while retaining a signal-processing sequence analogous to TDM and FDM. The transmitter filters and samples domain-specific signals before receiver processing recovers the information symbols.

  • VI. TRANSCEIVER SIGNAL PROCESING: TDM, FDM, and OTFS use analogous transmitter-receiver processing sequences, but embed information in the TD, FD, and DD domains respectively.
  • VI. TRANSCEIVER SIGNAL PROCESING: A packet of BT information symbols is mapped into a discrete modulation-domain signal and filtered so the transmitted TD signal satisfies time and bandwidth constraints.
  • VI. TRANSCEIVER SIGNAL PROCESING: The received modulation-domain signal is sampled and processed to recover the information symbols.

A. TDM input-output relation

For a generic doubly-spread channel, TDM produces an input-output relation that is non-predictable, fading, and non-stationary. Without Doppler shift, these properties become predictable, non-fading, and stationary.

  • The TDM output is obtained from the input by a discrete time convolution.
  • The effective discrete time-domain channel response depends non-simply on the input-symbol index, so one known response cannot predict all others.
  • For a generic doubly-spread channel, the TDM relation is non-predictable, fading, and non-stationary.
  • Without Doppler shift, the TDM channel response is independent of the input-symbol index and the relation is predictable, non-fading, and stationary.

B. FDM input-output relation

For a generic doubly-spread channel, FDM has a non-predictable, fading, and non-stationary input-output relation. With Doppler-only propagation, the relation becomes predictable, non-fading, and stationary.

  • The FDM output is obtained from the input by a discrete frequency-domain convolution.
  • The effective discrete frequency-domain channel response depends non-simply on the input-symbol index, so one known response cannot predict all others.
  • For a generic doubly-spread channel, the FDM relation is non-predictable, fading, and non-stationary.
  • With no path delays, the FDM channel response is independent of the input-symbol index and the relation is predictable, non-fading, and stationary.

C. OTFS input-output relation

OTFS arranges information on a finite delay-Doppler grid, converts it through Zak-based filtering and sampling operations, and yields a discrete twisted-convolution input-output relation. When DD periods exceed channel spreads, this relation is non-fading and predictable.

  • OTFS arranges information symbols in a two-dimensional finite array indexed by delay and Doppler coordinates.The modulation is parameterized by integers M ≈ Bτp and N ≈ Tνp.
  • The discrete DD information signal is lifted to a continuous quasi-periodic DD signal and filtered with a transmit DD-domain filter.
  • The inverse Zak transform converts the filtered DD signal into the transmit time-domain signal, while the receiver applies a Zak transform, DD filtering, and grid sampling.
  • The OTFS input-output relation is a discrete twisted convolution with an effective DD-domain channel filter obtained by sampling the continuous filter.Twisted convolution is non-commutative but associative.
  • When τp > τmax and νp > 2νmax, the OTFS input-output relation becomes non-fading and predictable.The condition applies when transmit and receive filters are localized and channel delay and Doppler spreads are bounded.
  • For the illustrated channel, whose spreads are 2 µs in delay and 1700 Hz in Doppler, the crystalline condition requires τp > 2 µs and νp > 1700 Hz.

VII. CONCLUSIONS

The paper frames OTFS through Zak theory and operator symmetries, emphasizing a crystalline regime where DD periods exceed channel spreads. It distinguishes Zak-OTFS from MC-OTFS and points to later performance comparisons.

  • OTFS is motivated by mathematical structures and channel symmetries, including parallels with common eigenmodes in quantum error-correcting codes.
  • Zak theory represents the OTFS carrier as a quasi-periodic DD-domain pulse that becomes a pulsone after inverse Zak transformation.
  • In the crystalline regime, where DD periods are large compared with channel spreads, the OTFS input-output relation is non-fading and predictable.
  • The paper identifies performance advantages of the crystalline regime and anticipates further research on Zak-OTFS performance and complexity advantages.
  • MC-OTFS is an approximation developed for compatibility with contemporary multi-carrier signaling, whereas Zak-OTFS uses quasi-periodic DD signals and a one-step Zak conversion.
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