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Shaping Delay-Doppler Ambiguity in Practical OFDM-ISAC

Cheng Luo, Luping Xiang, Kun Yang

arXiv:2609.10300v1eess.SP

TL;DR

Practical OFDM frames combine fixed pilots and nulls with random payloads, unlike fully occupied data-only waveforms considered in many analyses. The paper proposes DBU-OFDM, combining resource-block unitary transformations and dedicated sensing subcarriers for ROI sidelobe shaping. Its results report practical-resource PSLR gains and greater shaping freedom for A-ACF than P-ACF under the considered configuration.

  • Problem

    Many analyses assume fully occupied data-only OFDM, while practical pilots, nulls, and payloads complicate analysis of the largest sidelobe within a finite delay-Doppler ROI.

    Source: §I · p. 2

  • Method

    DBU-OFDM combines resource-block unitary transformations with dedicated sensing-subcarrier allocation while preserving prescribed resource structure.

    Source: §I · p. 2

  • Results

    Unitary pilot optimization improves practical-resource ROI PSLR by over 2 dB, while adding dedicated sensing subcarriers yields gains of several to tens of dB.

    Source: §V · p. 13

  • Takeaways & Limitations

    A-ACF provides greater ROI sidelobe-shaping freedom than P-ACF under the considered configuration.

    Source: §V · p. 13

    The Householder parameterization does not guarantee globally optimal convergence under gradient-based training.

    Source: 4) Householder-Based Trainable… · p. 7

Abstract

from arXiv · show

Orthogonal frequency-division multiplexing (OFDM) is a key waveform for integrated sensing and communication (ISAC). Existing OFDM ambiguity analyses, however, typically assume fully occupied data-only waveforms, whereas practical frames contain direct-current and edge-guard nulls, fixed pilots, and random payload symbols. This mixed resource structure reshapes the self-ambiguity function and induces prominent sidelobes in the sensing region of interest (ROI). We therefore propose ROI-oriented deep block-unitary precoded OFDM (DBU-OFDM), which combines resource-specific trainable unitary transformations with dedicated sensing subcarriers to suppress ROI sidelobes while preserving the prescribed resource support. We develop a constraint-preserving parameterization capable of representing arbitrary unitary matrices and an ROI-aware sensing-support initialization. We prove periodic autocorrelation function (P-ACF) invariance under phase-only optimization and zero-delay Doppler-cut invariance under arbitrary unitary transformations. Numerical results corroborate the P-ACF optimality of cyclic-prefix OFDM (CP-OFDM) and provide numerical support for the conjecture that conventional OFDM is globally optimal in the considered aperiodic autocorrelation function (A-ACF) setting. Unitary pilot optimization improves the ROI peak-to-sidelobe ratio by over 2 dB, while combining it with dedicated sensing subcarriers yields gains ranging from several to tens of dB. Under the considered configuration, the A-ACF mode also provides substantially greater ROI ambiguity-shaping capability than the P-ACF mode.

I. INTRODUCTION

Practical OFDM resources and random payloads complicate control of delay-Doppler sidelobes within a finite ROI. The paper extends DBU-OFDM to jointly shape the ambiguity function and sensing support while preserving prescribed resources.

I. INTRODUCTION

  • Fixed pilots, edge/DC nulls, and random payloads differ from fully occupied data-only analytical models, while direct control of the largest sidelobe in a finite delay-Doppler ROI remains difficult.The sensing budget and support locations also determine waveform-shaping freedom, delay-correlation structure, and remaining communication payload.Source: §I · p. 2
  • The paper extends DBU-OFDM to practical OFDM frames by jointly optimizing resource-block transformations, sensing-subcarrier support, and sensing symbols to suppress the largest ROI sidelobe.The design maintains the prescribed resource structure.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The paper formulates P-ACF and A-ACF models for CP-OFDM and zero-guard operation and targets the largest normalized sidelobe over a prescribed delay-Doppler ROI.The models connect the chosen correlation mode and sensing objective to the operating waveform.Source: §I · p. 2

I. INTRODUCTION

  • The resource-structured DBU-OFDM combines block-unitary transformations with dedicated sensing-subcarrier allocation while preserving resource isolation, energy, exact invertibility, and DFT-based single-tap equalization under stated conditions.Its Householder parameterization represents any Nϑ×Nϑ unitary resource block when Kϑ ≥ Nϑ −1.Source: §I · p. 2
  • An array-factor criterion initializes sensing-subcarrier support from the support-dependent sensing/pilot ambiguity term, after which support, sensing symbols, and unitary blocks are jointly refined using the complete ambiguity loss.The refinement targets sidelobe suppression within the selected ROI.Source: §I · p. 2
  • Under full occupancy, learned transformations recover an OFDM-equivalent monomial form; numerical results corroborate CP-OFDM P-ACF optimality and support a conjecture of conventional OFDM global optimality for the considered zero-Doppler A-ACF delay-slice PSLR problem.The results also find greater ROI sidelobe-shaping freedom for A-ACF than P-ACF under the considered configuration.Source: §I · p. 2

B. Operating-Mode-Dependent Self-Ambiguity Functions

The paper relates its periodic and aperiodic correlation models to distinct CP-OFDM and zero-guard waveform scenarios. The models rely on delay-coverage and small-intrasymbol-Doppler assumptions, and the resulting ambiguity kernel describes matched-filter target replicas.

B. Operating-Mode-Dependent Self-Ambiguity Functions

  • P-ACF models continuously transmitted CP-OFDM blocks with circular delay shifts when the CP covers the modeled round-trip delay; A-ACF models CP-free pulses with zero guards and finite-overlap linear shifts.The zero guard must cover the modeled delay to avoid overlap with the subsequent pulse.Source: B. Operating-Mode-Dependent… · p. 3
  • The modeled delays are integer sample shifts, range migration across the M blocks is neglected, and the maximum modeled delay must not exceed the CP or zero-guard length for the selected mode.The small-intrasymbol-Doppler condition makes intrasymbol Doppler and resulting ICI negligible.Source: B. Operating-Mode-Dependent… · p. 4
  • The two-dimensional self-ambiguity function coherently combines per-symbol ACFs across the M OFDM symbols, with k and q representing relative delay and centered Doppler lags.Their physical offsets are kTs and q/(MTχ).Source: B. Operating-Mode-Dependent… · p. 4
  • Matched filtering produces translated replicas of the self-ambiguity kernel, so suppressing kernel sidelobes within an ROI reduces masking of weaker targets by stronger-target sidelobes.Doppler translation is circular modulo M; delay translation is circular in P-ACF and locally valid over the finite A-ACF correlation window.Source: B. Operating-Mode-Dependent… · p. 4

C. ROI-Oriented PSLR Objective

The sensing objective evaluates the largest normalized self-ambiguity sidelobe over a selected delay-Doppler ROI. The paper uses a differentiable approximation to optimize the resulting peak-to-sidelobe ratio.

C. ROI-Oriented PSLR Objective

  • The ROI selects relative delay-Doppler index pairs, and the sidelobe set excludes the mainlobe when the origin lies within the ROI.The mainlobe remains the normalization reference.Source: C. ROI-Oriented PSLR Objective · p. 4
  • The positive mainlobe-to-sidelobe convention defines ROI PSLR so that a larger value indicates stronger suppression of the largest self-ambiguity sidelobe.Maximizing PSLR is equivalent to minimizing the largest normalized sidelobe power over the sidelobe set.Source: C. ROI-Oriented PSLR Objective · p. 4
  • A log-sum-exp loss approximates the nonsmooth maximum sidelobe objective, approaching the maximum normalized sidelobe power as its parameter α increases.Minimizing the loss therefore smoothly approximates maximizing ROI PSLR.Source: C. ROI-Oriented PSLR Objective · p. 4

III. PROPOSED ROI-ORIENTED DBU-OFDM

Practical pilots and nulls reshape the two-dimensional self-ambiguity surface, motivating ROI-aware waveform design. DBU-OFDM applies resource-specific unitary transforms while preserving prescribed support and communication processing.

A. Impact of Practical OFDM Resource Structures on Self-Ambiguity

  • Fully occupied resources produce a dominant mainlobe and comparatively smooth sidelobe surfaces, whereas practical fixed resources reshape the ambiguity function relative to that baseline.The A-ACF surface decreases toward large delays as the number of overlapping sample pairs decreases.Source: A. Impact of Practical OFDM Resource… · p. 4
  • Edge and DC nulls together with comb-type pilots create delay-structured sidelobe ridges around zero Doppler in both modes, more pronounced in the P-ACF example and extending to larger delays.The comparison shows that prescribed subcarrier resources substantially reshape the self-ambiguity surface.Source: Figure 3 · p. 5, A. Impact of Practical OFDM Resource… · p. 5

1) Resource-Structured Block-Unitary Construction:

  • DBU-OFDM applies a trainable frequency-domain transformation before the IDFT while preserving prescribed support, resource isolation, energy, invertibility, and conventional processing under stated conditions.When the CP covers channel delay spread and intrasymbol variation and ICI are negligible, it retains DFT-based single-tap equalization.Source: 1) Resource-Structured Block-Unitary… · p. 5
  • The resource transformation uses separate unitary blocks for data, pilots, and nulls, preventing mixing among resource groups and preserving the prescribed subcarrier support.The construction then maps the transformed resource groups back to physical subcarrier order before producing the transmitted waveform by IDFT.Source: 1) Resource-Structured Block-Unitary… · p. 5

2) Communication Processing:

Under the stated CP-OFDM assumptions, DBU-OFDM retains DFT-based per-subcarrier channel processing, while its sensing frame is evaluated as a known reference and its trainable unitary blocks are optimized with exact unitarity.

2) Communication Processing:

  • Under the stated CP condition, the channel is circulant and diagonal in the frequency domain, so equalization remains per-subcarrier before the inverse unitary transformation.The communication description assumes the CP covers delay spread, the channel is approximately time invariant over one useful symbol, and intrasymbol Doppler and ICI are negligible.Source: 2) Communication Processing · p. 6
  • With ideal noiseless zero-forcing equalization, inverse unitary processing recovers the original resource vector exactly and does not amplify total post-equalization noise energy.Any noise enhancement from channel inversion is governed by the equalizer.Source: 2) Communication Processing · p. 6

3) Sensing Processing:

  • For sensing, the transmitted DBU-OFDM frame serves as the known reference waveform for evaluating self-ambiguity under either P-ACF or A-ACF mode.Source: 3) Sensing Processing · p. 6
  • The unitary transformations preserve waveform energy, and a constraint-preserving parameterization enforces exact unitarity during optimization.The sensing waveform depends on trainable data and pilot blocks.Source: 3) Sensing Processing · p. 6

4) Householder-Based Trainable Parameterization:

The paper constructs exactly unitary resource-block transformations with Householder reflections and diagonal phases, and reserves known sensing resources within the data subspace alongside random communication symbols.

4) Householder-Based Trainable Parameterization:

  • Complex Householder reflections and diagonal phase rotations form the trainable resource blocks, which remain exactly unitary without projection or re-orthogonalization.Source: 4) Householder-Based Trainable… · pp. 6–7
  • A Householder depth budget Kϑ ≥ Nϑ − 1 is sufficient to represent any Nϑ×Nϑ unitary resource block exactly.Below that depth, constructed matrices remain unitary, but complete representation is not guaranteed and gradient training is not guaranteed to reach a global optimum.Source: 4) Householder-Based Trainable… · p. 7

1) Sensing and Communication Resource Partition:

  • A subset of data subcarriers is reserved for known, trainable sensing symbols, while the remaining communication subcarriers carry random payload symbols.Source: 1) Sensing and Communication Resource… · p. 7
  • Increasing the sensing-subcarrier set provides more controllable sensing resources while reducing the subcarriers available for communication payload.The sensing-subcarrier count sets the sensing-resource budget and corresponding sensing-communication tradeoff.Source: 1) Sensing and Communication Resource… · p. 7

2) Array-Factor-Based Support Initialization:

Support initialization uses a data-independent sensing/pilot proxy based on the occupied-subcarrier pattern, then refines the support and waveform variables using the complete ambiguity loss.

2) Array-Factor-Based Support Initialization:

  • Because the random communication component varies with payload, initialization optimizes a sensing/pilot self-ambiguity proxy with fixed pilot and sensing excitations.The support indicator's inverse DFT has magnitude proportional to the proxy array factor.Source: 2) Array-Factor-Based Support… · p. 7
  • The support-only approximation ties sensing/pilot delay sidelobes to the effective frequency aperture and pairwise-difference coarray of the occupied subcarriers.Source: 2) Array-Factor-Based Support… · p. 7
  • For a fixed sensing-resource budget, the approximate support solver uses sequential construction followed by single-subcarrier exchange refinement.Pilots are included in every candidate support and null subcarriers are excluded.Source: 2) Array-Factor-Based Support… · p. 8
  • The support-only proxy is not guaranteed to minimize the complete objective because it omits communication and cross-ambiguity terms and other waveform and mode-specific factors.Joint optimization subsequently includes actual symbols, unitary transformations, Doppler structure, and A-ACF boundaries through the complete self-ambiguity loss.Source: 2) Array-Factor-Based Support… · p. 8

IV. NUMERICAL RESULTS

Under fully occupied OFDM, training drives both P-ACF and A-ACF data transforms toward a monomial form, and trained zero-Doppler delay cuts nearly overlap conventional OFDM. The results also support, without proving, a global-optimality conjecture for the considered A-ACF delay-slice problem.

A. Validation Under a Fully Occupied OFDM Resource Structure

  • Validation uses N=256 independent unit-power 64-QAM symbols on every subcarrier, trains only Ud, and optimizes the zero-Doppler delay slice.The ROI is the zero-Doppler delay slice, with all subcarriers occupied and no pilots, nulls, or sensing subcarriers.Source: A. Validation Under a Fully Occupied… · p. 8
  • ρmon rises from 0.0268 at initialization to 0.9980 under P-ACF and 0.9989 under A-ACF training, indicating nearly monomial transforms with negligible subcarrier mixing.The trained matrices mainly permute data symbols and apply per-subcarrier phase rotations; enlarged P-ACF submatrices show a few off-diagonal entries and small symbol reordering.Source: A. Validation Under a Fully Occupied… · pp. 8–9, Figure 5 · p. 9, Figure 6 · p. 9
  • P-ACF convergence matches conventional OFDM’s established expected-sidelobe optimality; A-ACF convergence supports, but does not prove, global optimality for this zero-Doppler delay-slice PSLR problem.The cited A-ACF analysis establishes only local optimality under expected integrated sidelobe power; the global-optimality conjecture is limited to the considered unitary waveform class and delay-slice objective.Source: A. Validation Under a Fully Occupied… · p. 9
  • 43.30 vs 43.37 dB P-ACF PSLR and 43.04 vs 42.80 dB A-ACF PSLR for trained DBU-OFDM versus noiseless OFDM; training gains 4.34 and 3.69 dB.The trained and conventional OFDM delay-cut curves nearly overlap in both modes, while the common dense initialization has substantially higher sidelobes.Source: Figure 7 · p. 9, A. Validation Under a Fully Occupied… · p. 9

B. Pilot-Only Ambiguity Shaping

Unitary pilot optimization reshapes ROI sidelobes while preserving pilot energy; dedicated sensing subcarriers add further, mode-dependent ambiguity-shaping freedom.

B. Pilot-Only Ambiguity Shaping

  • Unitary pilot optimization raises P-ACF ROI PSLR from 27.83 to 30.14 dB (2.31 dB) and reaches 30.77 dB under A-ACF (2.86 dB gain).The phase-only pilot design provides no P-ACF gain and only a limited A-ACF improvement, from 27.91 to 28.27 dB.Source: B. Pilot-Only Ambiguity Shaping · p. 10
  • Unitary pilot optimization preserves total pilot energy but imposes no lower bound on effective-pilot amplitudes, which may affect channel estimation and the communication-sensing tradeoff.Source: B. Pilot-Only Ambiguity Shaping · p. 10
  • Phase-only pilot changes leave P-ACF sidelobes invariant but can reshape nonzero-delay A-ACF sidelobes; unitary transformations preserve zero-delay Doppler cuts.The A-ACF phase sensitivity follows from cross-subcarrier terms in its finite-overlap correlation.Source: B. Pilot-Only Ambiguity Shaping · pp. 10–11

C. Sensing-Subcarrier Budget and Ambiguity-Shaping Performance

  • Because pilot-only shaping is limited, the design reserves data subcarriers for sensing and jointly optimizes sensing support, symbols, and available unitary blocks.The evaluated budgets are Ks ∈ {0, 64, 128, 230}; reserving subcarriers reduces the ideal normalized communication payload rate.Source: C. Sensing-Subcarrier Budget and… · p. 11
  • Increasing the sensing-subcarrier budget improves ROI PSLR in both modes, but P-ACF gains saturate while A-ACF provides substantially greater ROI suppression.The reported surfaces show localized ROI suppression, with some sidelobes outside the ROI becoming more pronounced.Source: Figure 9 · p. 11, C. Sensing-Subcarrier Budget and… · pp. 11–12, Figure 10 · p. 12
  • Zero-Doppler P-ACF depends only on nonnegative average subcarrier powers, limiting the flexibility added by more sensing resources; the relaxed upper bound is 41.25 dB.The bound uses Ks = 230 and arbitrary nonnegative power distributions under total-power constraints.Source: C. Sensing-Subcarrier Budget and… · p. 11

D. Ablation Study and Learned Block Structure

The ablation study finds that array-factor initialization gives a strong fixed-support result, while support refinement narrows differences among initializations. The conclusion reports practical-resource gains and identifies the scope of the full-occupancy optimality findings.

D. Ablation Study and Learned Block Structure

  • With fixed supports, array-factor initialization via (P2) achieved a PSLR of 49.50 dB, exceeding the better uniform or coprime result by 11.56 dB; after support refinement, the three initializations achieved PSLRs of 51.01–51.11 dB.Source: D. Ablation Study and Learned Block… · p. 12
  • The refined array-factor configurations had similar PSLRs with fixed or trainable Uc; ρmon(Uc) = 0.9793 and ρmon(Us) = 0.0863 indicate limited communication-block mixing but dense sensing-block mixing.The authors support Uc = IKc as a practical simplification for the considered configuration, preserving communication-symbol mapping while retaining ROI sidelobe suppression through sensing and pilot blocks.Source: D. Ablation Study and Learned Block… · p. 12, Figure 11 · p. 12

V. CONCLUSIONS

  • Pilot optimization improved ROI PSLR by over 2 dB, while dedicated sensing subcarriers yielded gains ranging from several to tens of dB.These results are reported for practical resources.Source: §V · p. 13
  • Under full occupancy, experiments corroborated P-ACF optimality and supported conjectured OFDM global optimality for the considered zero-Doppler A-ACF delay-slice PSLR problem within the unitary waveform class.A-ACF offered greater shaping freedom than P-ACF under the considered configuration.Source: §V · p. 13
  • Future work will jointly evaluate pilot optimization, channel-estimation accuracy, and achievable communication rate, and extend the formulation to fractional delay-Doppler and time-varying channels with non-negligible intrasymbol Doppler.Source: §V · p. 13

APPENDIX A PROOF OF LEMMA 1

The proof recursively decomposes a unitary block using Householder reflections. It shows that Nϑ − 1 reflection stages suffice for exact representation.

APPENDIX A PROOF OF LEMMA 1

  • The proof builds the representation recursively from Householder reflections, following the standard Householder decomposition.Source: Appendix A · p. 13
  • For a unit-norm vector, the chosen phase lets a Householder reflection map it to a vector supported only in its first coordinate, leaving a smaller trailing unitary block.The construction repeats on that trailing block until only a 1 × 1 unitary remains.Source: Appendix A · p. 13
  • The final 1 × 1 unitary is a unit-modulus scalar, so a depth budget Kϑ ≥ Nϑ − 1 suffices to represent any Nϑ × Nϑ unitary resource block exactly.Source: Appendix A · p. 13

APPENDIX B PROOF OF LEMMA 2

The appendices establish that phase-only pilot changes leave the complete P-ACF unchanged, while arbitrary unitary DBU-OFDM transformations preserve the zero-delay Doppler cut.

APPENDIX B PROOF OF LEMMA 2

  • For a fixed OFDM frame, changing only pilot-symbol phases leaves all P-ACF sidelobes unchanged.The proof uses preservation of each frequency-domain symbol’s magnitude.Source: Appendix B · pp. 13–14

APPENDIX C PROOF OF LEMMA 3

  • Arbitrary unitary DBU-OFDM transformations preserve the zero-delay Doppler cut because they preserve the per-symbol energy sequence whose DFT forms that cut.Source: Appendix C · p. 14
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