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Orthogonal Delay-Doppler Division Multiplexing Modulation

Hai Lin, Jinhong Yuan

arXiv:2206.13999v2cs.ITeess.SP

TL;DR

The paper addresses whether DD-plane multicarrier modulation can use a realizable pulse orthogonal to fine DD resolutions. It develops ODDM from staggered multitone modulation and designs the corresponding pulse, obtaining an exact block-circulant-like DD-domain channel relation and improved OOBE and BER versus OTFS.

  • Problem

    The central gap is whether a realizable pulse orthogonal to the DD plane’s fine delay and Doppler resolutions exists for DD-plane multicarrier modulation.

  • Method

    The paper identifies DD-plane multicarrier modulation as staggered multitone modulation, proposes ODDM, and designs a square-root Nyquist transmit pulse with matched filtering.

  • Results

    ODDM’s transmit pulse is proved orthogonal at the DD resolutions, yielding an exact block-circulant-like DD-domain channel relation and superior OOBE and BER performance over OTFS.

  • Takeaways & Limitations

    Perfect coupling between the ODDM signal and DD channel supports the proposed modulation’s reported advantages over OTFS and provides a basis for future efficient detection algorithms.

Abstract

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Inspired by the orthogonal time frequency space (OTFS) modulation, in this paper, we consider designing a multicarrier (MC) modulation on delay-Doppler (DD) plane, to couple the modulated signal with a doubly-selective channel having DD resolutions. A key challenge for the design of DD plane MC modulation is to investigate whether a realizable pulse orthogonal with respect to the DD plane's fine resolutions exists or not. To this end, we first indicate that a feasible DD plane MC modulation is essentially a type of staggered multitone modulation. Then, analogous to orthogonal frequency division multiplexing, we propose an orthogonal delay-Doppler division multiplexing (ODDM) modulation, and design the corresponding transmit pulse. Furthermore, we prove that the proposed transmit pulse is orthogonal with respect to the DD plane's resolutions and therefore a realizable DD plane orthogonal pulse does exist. The orthogonality of this particular pulse significantly eases the derivation of the ODDM's DD domain channel input-output relation, and yields a channel matrix with an elegant block-circulant-like structure. We demonstrate that the ODDM outperforms the OTFS in terms of out-of-band emission and bit error rate, by achieving perfect coupling between the modulated signal and the DD channel.

I. INTRODUCTION

The paper motivates DD-plane multicarrier modulation for doubly-selective high-mobility channels and addresses whether a realizable pulse can be orthogonal at fine DD resolutions. It proposes ODDM and derives its exact DD-domain channel relation.

  • Motivation: High mobility and high carrier frequencies produce severely fast time-varying, doubly-selective channels that challenge reliable communication.The paper highlights Beyond 5G/6G high-mobility scenarios and DD-domain path-based modeling.
  • Motivation: OFDM can mitigate delay-induced ISI with a cyclic prefix, but Doppler-induced ICI can cause severe performance degradation.OFDM frames are typically selected within the channel’s coherent time to approximate time invariance.
  • Background: OTFS modulates symbols on the DD plane and can exploit both frequency and time diversity in doubly-selective channels.Its DD representation reflects the channel’s physical delay-Doppler structure.
  • Design gap: A pulse confined to one fine DD grid violates the Heisenberg uncertainty principle, while OTFS uses an ISFFT and a TF-plane pulse as a practical workaround.The paper also states that OTFS’s ideal TF pulse cannot be realized in practice.
  • Contributions: The paper identifies a realizable DD-plane orthogonal pulse, proposes ODDM, and derives a block-circulant-like DD-domain channel relation.ODDM is presented as a staggered multitone modulation with perfect signal-channel coupling.
  • Contributions: Simulations compare ODDM and OTFS using out-of-band emission and bit error rate, with ODDM reported as superior on both measures.The paper describes these comparisons as evidence for ODDM’s performance advantage.

III. ODDM MODULATION

The paper formulates DD-plane modulation as a fine-resolution multicarrier problem whose required pulse appears unrealizable under the uncertainty principle. It resolves this by using staggered symbols and introducing ODDM.

  • Problem formulation: The DD grid has area much smaller than one, so a pulse confined to it violates the Heisenberg uncertainty principle and cannot be realized.This makes direct DD-plane multicarrier modulation appear questionable.
  • Key idea: A feasible DD-plane multicarrier modulation is essentially staggered multitone modulation because its symbol interval differs from its symbol period.The paper identifies staggering as the opportunity for finding a realizable orthogonal pulse.
  • ODDM construction: ODDM modulates M information-bearing multicarrier symbols, each with N orthogonal subcarriers, while spacing symbols by T/M.Each ODDM symbol is effectively an N-subcarrier OFDM symbol, and the symbols are staggered.
  • Problem formulation: DD-plane multicarrier modulation requires a pulse orthogonal with respect to the delay and Doppler resolutions to avoid ISI and ICI.The proposed grid uses fine DD resolutions rather than conventional coarse TF-plane resolutions.
  • ODDM construction: The ODDM signal has a bandwidth around M/T because its M symbols are staggered over intervals of T/M.This staggering distinguishes ODDM from conventional OFDM.

A. ODDM digital sequence

ODDM is represented digitally as staggered upsampled OFDM: each symbol is formed by an IDFT, then samples are upsampled and staggered to create the required fine time spacing.

  • Comparison with SMT: ODDM differs from OFDM/OQAM in its sampling and staggering structure, using a symbol interval of T/M rather than the compared SMT interval.The paper frames both schemes as staggered multitone modulations but distinguishes their digital-domain operations.
  • Orthogonality: The paper later proves that the pulse assumed for ODDM exists, validating the modulation’s required orthogonal-pulse construction.This proof is stated as a subsequent result rather than established in the digital-sequence description.
  • Digital construction: For each ODDM symbol, an N-point IDFT produces N time-domain discrete samples.This follows the conventional OFDM construction at the symbol level.
  • Digital construction: The N samples are upsampled by M, inserting zeros so the resulting MN samples support symbol staggering at intervals of T/M.The upsampling creates the fine temporal spacing required by the ODDM grid.
  • Digital construction: The resulting ODDM frame consists of M staggered symbols and spans a duration around NT.The digital representation is therefore a staggered upsampled-OFDM with upsampling factor M.

B. ODDM waveform design

ODDM and OTFS can share the same discrete sample sequence while differing fundamentally in pulse shaping. The ODDM waveform is generated by square-root Nyquist filtering of staggered upsampled-OFDM samples.

  • Waveform distinction: OTFS and ODDM can have the same time-domain sequence, but their physical interpretations lead to fundamentally different pulse-shaping and waveform designs.The distinction concerns how the samples are grouped and filtered.
  • Waveform distinction: A digital sample sequence alone is not a complete practical modulation waveform because appropriate pulse shaping determines the transmitted signal.The paper emphasizes that identical samples can lead to different physical waveforms.

1) Pulse shaping consideration:

ODDM pulse shaping adapts OFDM-style processing to staggered symbols, using upsampling and interpolation to control spectrum while avoiding symbol discontinuities.

  • Pulse shaping consideration: OTFS pulse shaping is performed per OFDM symbol with rectangular pulses, whose discontinuities cause serious OOBE.Adding CP, cyclic suffixes, and smoother windows suppresses OOBE but reduces spectrum efficiency.
  • Pulse shaping consideration: ODDM treats the signal as M staggered N-subcarrier OFDM symbols, making symbol-wise pulse shaping equivalent to frame-wise shaping.Because the symbols are staggered, discontinuities among symbols are no longer an issue and symbol-wise CP is unnecessary.
  • Spectrum analysis: Upsampling ODDM samples by M enables staggering but increases the bandwidth by M times before interpolation filtering.The upsampled samples are passed through an ideal interpolation filter to form the pulse-shaped signal.
  • Spectrum analysis: Each ODDM symbol uses time-domain samples identical to those of a conventional OFDM symbol before pulse shaping.The spectrum is initially spread by rate-1/T sampling and then narrowed by interpolation filtering.
  • Spectrum analysis: Staggering does not change overall spectrum occupancy, so ODDM spectrum depends on the interpolation filter.Pulse shaping is therefore reduced to selecting an interpolation filter with suitable frequency response and ISI-free behavior over T/M.

3) ODDM waveform:

ODDM uses sample-wise square-root Nyquist pulse shaping over the staggered upsampled-OFDM signal, producing a practical analog waveform with controlled intersymbol interference.

  • ODDM waveform: A suitable interpolation filter for ODDM must provide an ISI-free impulse response over the staggered symbol interval T/M.A Nyquist pulse satisfies this requirement while maintaining a flat response over the relevant wide frequency range.
  • ODDM waveform: ODDM pulse shaping is performed sample-wise using a square-root Nyquist pulse for the symbol interval T/M.Such pulses include raised-cosine-derived candidates and support matched filtering by splitting the Nyquist response between transmit and receive filters.
  • ODDM waveform: The chosen square-root Nyquist filter processes the discrete staggered upsampled-OFDM sequence to generate the analog ODDM waveform.This sample-wise filtering differs from conventional OFDM or PS-OFDM pulse-shaping procedures.

4) Transmit pulse of ODDM:

The proposed ODDM waveform is equivalent to staggered pulse-shaped OFDM symbols driven by a transmit pulse u(t), while spreading subcarriers across the frame.

  • Transmit pulse of ODDM: The ODDM transmit pulse u(t) is introduced as the equivalent pulse resulting from sample-wise square-root Nyquist filtering.The pulse is illustrated in Fig. 5 and has unit energy according to the normalization condition.
  • Transmit pulse of ODDM: The ODDM frame structure example uses M=8, N=4, and L=4, with colored blocks grouping samples into ODDM symbols.A CP is prepended to the frame in the described implementation.
  • Transmit pulse of ODDM: Each ODDM symbol is a near-perfect approximation of a PS-OFDM symbol whose transmit pulse is u(t).The pulse-shaped ODDM frame consists of M staggered N-subcarrier PS-OFDM symbols.
  • Transmit pulse of ODDM: ODDM directly conveys DD-plane symbols through u(t), with symbol interval T/M and subcarrier spacing 1/(NT).This differs fundamentally from OTFS, which first maps DD symbols to the TF plane and uses a T-length rectangular pulse.
  • Transmit pulse of ODDM: A wideband square-root Nyquist filter spreads N information-bearing subcarriers across M times the overall bandwidth.The resulting modulation can potentially obtain frequency diversity of M and time diversity of N, subject to the number of independent channel paths.

IV. ODDM DEMODULATION

ODDM demodulation uses a transmit pulse orthogonal to the DD plane’s fine grid, enabling an exact DD-domain channel relation. Matched filtering yields a sparse, block-circulant-like channel matrix whose structure supports signal detection.

  • Pulse orthogonality: The transmit pulse u(t) is proven orthogonal with respect to the DD plane’s fine-grid resolutions.The proof establishes a realizable DD-plane orthogonal pulse rather than only coarse-grid bi-orthogonality.
  • Matched filtering: Matched filtering with u(t) recovers ODDM symbols after doubly-selective channel distortion and can be implemented approximately using sample-wise filtering followed by an N-point DFT.The received samples are organized into N-dimensional frequency-domain vectors indexed by delay.
  • Pulse orthogonality: ODDM’s pulse satisfies the PR condition on the fine grid Γ, while its Nyquist-filter construction handles delay and Doppler resolutions.The pulse train combines N square-root Nyquist pulses, making it locally wideband and globally narrow-band.
  • DD-domain coupling: Because ODDM aligns signal and channel taps on the same fine grid, its effective DD channel is represented solely and exactly by the DD channel.This avoids the approximation associated with OTFS’s off-grid TF-domain ISI and ICI.
  • DD-domain channel matrix: The ODDM DD-domain channel matrix is generally sparse and has an elegant block-circulant-like structure exploitable for signal detection.Each row and column contains P nonzero elements when the total path count is P and MN≫P.
  • DD-domain coupling: ODDM increases TF-plane time and frequency resolutions by M and N, respectively, removing the resolution mismatch and achieving fine-resolution orthogonality.The densified TF-plane arrangement uniformly distributes DD-domain signals.

B. Signal detection

ODDM detection uses a DD-domain message-passing detector because ODDM symbols experience interference. The detector exploits channel-matrix sparsity with complexity proportional to MN times the number of nonzero entries per row.

  • Detector choice: ODDM symbols experience interference, so an effective detector is required to exploit the available time and frequency diversity.Maximum-likelihood and maximum-a-posteriori detection are impractical because their complexity is exponential in block length MN.
  • Message-passing detection: The simulations evaluate ODDM with a commonly deployed DD-domain message-passing detector also used for OTFS.The detector exchanges information between observation and variable nodes in a factor graph.
  • Message-passing detection: The message-passing detector has complexity O(MNS), where S=P is the number of nonzero entries in each DD-domain channel-matrix row.Its Gaussian-interference assumption enables low-complexity detection based on the sparse DD-domain channel.

V. SIMULATION RESULTS

Simulations compare ODDM with OTFS under EVA high-mobility channels, showing lower out-of-band emission and better BER for ODDM. The gains depend on system parameters, while spectral-mask and synchronization effects remain for future evaluation.

  • Out-of-band emission: ODDM achieves up to 20 dB lower out-of-band emission than OTFS through square-root Nyquist pulse shaping, at the expense of excess bandwidth.The roll-off factor can trade excess bandwidth against OOBE and help achieve desirable spectral efficiency.
  • Out-of-band emission: ODDM can fully utilize subchannels, while its symbol-count parameter M can be chosen flexibly to adjust bandwidth with the roll-off factor.Unlike OFDM subcarrier counts, M in ODDM need not be a power of 2.
  • Bit-error rate: 1.7 dB uncoded and 0.8 dB coded BER gains over OTFS occur at BER 2 × 10^-6 for M = 512, N = 64, and 120 km/h.The comparison uses 4-QAM and message-passing detection; coded systems use rate 2/3 convolutional coding and Viterbi decoding.
  • Bit-error rate: 1.1 dB uncoded and 0.8 dB coded BER gains over OTFS occur at BER 2 × 10^-6 for M = 512, N = 32, and 500 km/h.The results indicate gains across different Doppler resolutions, but the magnitude varies with parameter settings, channel profile, and detection algorithm.
  • Interpretation: ODDM’s matched filtering and exact DD-domain input-output relation contribute to better performance than OTFS, whose relation is approximate because of TF-domain ISI and ICI.The exact relation can be exploited for accurate signal detection.
  • Scope: The evaluation covers uncoded and coded ODDM, while spectral masks and synchronization errors remain unexamined practical considerations.The authors identify comparison with other OTFS or precoded-OFDM systems as future work.

VI. CONCLUSION

The paper links DD-plane multicarrier modulation to staggered multitone modulation and proposes ODDM with a realizable orthogonal transmit pulse. Its orthogonality enables an exact DD-domain channel relation and improved OOBE and BER relative to OTFS.

  • Conclusion: Staggering multicarrier symbols is identified as the key to finding a realizable orthogonal pulse for DD-plane multicarrier modulation.The paper presents DD-plane multicarrier modulation as linked to conventional staggered multitone modulation.
  • Conclusion: ODDM is represented digitally as staggered upsampled-OFDM, with sample-wise square-root Nyquist shaping used to identify its transmit pulse.The pulse’s orthogonality with respect to DD-plane resolutions is proved.
  • Conclusion: ODDM has an exact DD-domain channel input-output relation with an elegant block-circular-like channel matrix structure.This result follows from the transmit pulse’s orthogonality with respect to DD resolutions.
  • Conclusion: ODDM outperforms OTFS in out-of-band emission and bit error rate through perfect coupling between the modulated signal and the DD channel.The paper reports this comparison as a demonstrated consequence of the proposed design.

APPENDIX A PROOF OF ˇ푔푡푥(푡) ≈푢(푡) FOR ODDM GENERATED USING

Appendix A proves that the generated ODDM waveform approximates the target transmit pulse. The proof relies on sample-time agreement and negligible approximation error away from those sample points.

  • Approximation setup: Because 2Q ≪ M, most of the target pulse u(t) is zero, limiting differences between the generated and target waveforms to specific portions.This support-size condition is used at the beginning of the approximation argument.
  • Sample-time agreement: The generated and target waveforms coincide at sample times t = nT for n = 0, …, N − 1.The appendix establishes this equality by comparing the two waveform expressions.
  • Approximation bound: The approximation error e(τ) is negligibly small because the relevant factor decreases as |τ| increases while the residual term remains very small.The appendix concludes x_m(t) ≈ x̃_m(t) from this error bound.
  • Case analysis: The proof analyzes the real-valued filter a(t), the square-root Nyquist pulse for symbol period T, and separate index ranges near the boundaries of M.Boundary cases are handled by reindexing m and combining the resulting relations.
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