PaperlayerView exact revision on arXiv

Source-linked AI summary

Ultra-High Resolution Method for Multipath Within a Co-Delay-Doppler Bin in DFT-P-OCDM

Mingxuan Han, Weile Zhang, Feifei Gao

arXiv:2609.04669v1cs.IT

TL;DR

The paper tackles co-delay-Doppler bins where multiple physical paths can correspond to fewer resolvable delay classes, complicating path counting and fractional-Doppler estimation. It derives a DPF-domain input-output relation and introduces the two-stage TSUR framework, which uses pilot phase progression and local leakage samples. Simulations show that TSUR resolves co-bin paths, remains robust to delay offsets, and improves fractional-Doppler estimation over the stated baselines.

  • Problem

    Multiple physical paths can occupy one co-delay-Doppler bin while the pilot reveals fewer resolvable delay classes, complicating path-number determination, fractional-Doppler estimation, and channel reconstruction.

  • Method

    TSUR derives a DPF-domain input-output relation, estimates delay classes from pilot phase progression, and then estimates path numbers and fractional Dopplers from local leakage samples.

  • Results

    TSUR resolves co-bin multipath, remains robust to delay mismatch, and improves fractional-Doppler estimation over sequential extraction and off-grid baselines.

  • Takeaways & Limitations

    The framework supports co-bin path resolution while exposing a pilot-configuration tradeoff between sensing resolution, channel reconstruction, and communication recovery.

Abstract

from arXiv · show

Communication systems can reuse their transmitted signals for sensing without dedicated radar transmissions. For an established DFT-preprocessed orthogonal chirp division multiplexing (DFT-P-OCDM) waveform, this task becomes difficult when several physical paths in a doubly selective channel fall into the same co-delay-Doppler bin. In this case, the number of resolvable delay classes inferred from the pilot may be smaller than the number of physical paths within the co-bin. This mismatch increases the difficulty of path number determination, fractional Doppler offset estimation, and channel reconstruction. We derive a pointwise relationship between the input and output in the DFT preprocessed Fresnel (DPF) domain for doubly selective channels with multiple paths within the co-delay-Doppler bin. Based on this input and output relation, we propose the two stage ultra high resolution (TSUR) framework. The first stage uses pilots of the phase progression to estimate delay, while the second stage uses the leakage samples to estimate the Doppler and the number of paths within each delay class. Furthermore, we derive CRLBs and analyze how the pilot configuration trades sensing resolution and communication recovery. Simulation results demonstrate that TSUR resolves same delay paths within a co-delay-Doppler bin, remains robust to exist delay offsets, and achieves lower fractional Doppler estimation errors than sequential extraction and off-grid baselines.

I. INTRODUCTION

The paper addresses co-bin multipath in DFT-P-OCDM sensing, where several physical paths can share one discrete delay-Doppler bin despite differing continuous parameters. It derives a DPF-domain input-output relation and proposes TSUR to estimate delay classes, path numbers, and fractional Dopplers.

  • Motivation: Communication-assisted sensing reuses communication waveforms and infrastructure to estimate propagation delays, Dopplers, gains, and path numbers.These parameters support localization, velocity inference, target association, and tracking.
  • Problem: Co-bin multipath occurs when multiple physical paths map to one integer delay-Doppler bin, potentially hiding resolvable targets and degrading channel reconstruction and data recovery.The paths may have identical or insufficiently separated continuous delays but different fractional Doppler offsets.
  • Contributions: The paper derives a closed-form DPF-domain relation linking channel parameters to pilot phase progression and local leakage.Delay appears in the phase progression of principal samples, while fractional Doppler shapes local leakage around the integer Doppler index.
  • Contributions: TSUR first estimates resolvable delay classes from pilot phase progression, then determines physical path numbers and fractional Dopplers from leakage samples.The first stage uses FBSS-MUSIC with Newton refinement; the second uses BIC and NMLL-MLE.
  • Results: TSUR resolves representative co-bin multipath, remains robust to delay mismatch, and improves fractional Doppler estimation over sequential extraction and off-grid baselines.The study also derives CRLBs and examines sensing-resolution versus communication-recovery tradeoffs under pilot configurations.

B. Doubly Selective Channels with Co-Delay-Doppler Bin

The channel model represents fractional delay and Doppler through leakage over delay and Doppler indices. Paths sharing integer indices form a co-delay-Doppler bin, while distinct continuous delays may collapse into fewer resolvable delay classes.

  • Channel model: A doubly selective path is characterized by complex gain, normalized delay, and normalized Doppler shift, with fractional delay and Doppler represented through Dirichlet functions.The fractional delay spreads energy across delay samples, while fractional Doppler spreads it across Doppler indices.
  • Co-bin definition: For a selected co-bin, paths share integer delay and Doppler indices but may have different fractional offsets.Their continuous coordinates are (d_i, ν_i) = (l_c + δ_i, k_c + κ_i).
  • Delay classes: Distinct continuous delays are grouped into delay classes when they are identical or insufficiently separated for resolution.Several physical paths can therefore correspond to one resolvable delay class in the pilot observation.

C. Input-Output Relationship of DFT-P-OCDM

The DPF-domain formulation separates delay and Doppler effects in the DFT-P-OCDM input-output relation. Doppler produces circular displacement, while delay contributes phase rotation within the superposition of displaced components.

  • DPF transformation: The received signal is transformed from the time domain into the DPF domain, yielding y = H_DPF x + e_w with transformed additive white Gaussian noise.The DPF channel is related to the Fresnel-domain channel by unitary DFT transformations.
  • Leakage structure: Fractional delay and Doppler produce leakage terms indexed by p and q, with Doppler-related displacement b_i,q and delay-related index s_i,p.The periodic impulse constrains the corresponding input-output index relation.
  • Input-output relation: In DFT-P-OCDM, b_i,q determines cyclic displacement while s_i,p contributes to phase rotation of the displaced component.This separates the two leakage effects in the DPF channel representation.
  • Input-output relation: Unlike the DFnT-domain OCDM relation, DFT-P-OCDM retains the delay-related index in phase superposition rather than combining it with the Doppler-related shift.The resulting formulation translates delay into phase and Doppler into circular shift.

III. ULPA FRAME AND OBSERVATION MODEL

TSUR uses two pilot observation models: inter-pilot phase variation estimates delay classes, while local fractional-Doppler leakage estimates path numbers and fractional Dopplers within each class.

  • Pilot arrangement: The DPF frame distributes ζ protected pilot blocks uniformly, with each pilot block centered at a known position and separated by D_pilot.The receiver knows the pilot positions and energies.
  • Pilot arrangement: Each protected block contains one known pilot and 2G zero guard positions, while placement constraints prevent overlap and cover the expected integer Doppler range.The guard width must satisfy G ≥ k_max, and adjacent pilot centers must be sufficiently separated.
  • Resource tradeoff: The parameters ζ, G, and total pilot energy determine the resource tradeoff between sensing and communication.The number of data positions is N_d = N − ζ(2G + 1).
  • Integer Doppler estimation: Integer Doppler indices are estimated from the displacement of retained cross-correlation peaks relative to the transmitted pilot center.The receiver correlates the received signal with the known pilot using a threshold.

B. Pilot Phase Model for Delay

The local pilot model separates principal pilot contributions from fractional-Doppler leakage and structured interference. A symmetric window around the detected integer Doppler index retains dominant leakage for subsequent path-number and Doppler estimation.

  • Local leakage structure: The DPF input-output relation maps fractional Doppler offsets into periodic leakage patterns around a common integer-Doppler displacement.Different fractional Doppler offsets therefore produce distinct local leakage signatures.
  • Pilot observation model: For the current pilot contribution, z=0 directly indexes the principal sample, while z≠0 indexes fractional-Doppler leakage around it.Contributions from other pilot blocks enter through the corresponding periodic leakage index.
  • Delay information: The pilot phase progression depends on the continuous normalized delay, making the principal samples suitable for delay-class estimation.After the integer Doppler index is detected, delay is the remaining unknown parameter in this phase progression.
  • Window design: A symmetric window Z_Q={−Q,…,Q} contains N_Q=2Q+1 samples, with its retained leakage energy determined by the fractional Doppler offset.The retention level can be controlled through Q and a prescribed energy threshold.
  • Class representation: The reduced second-stage model represents paths sharing a resolved delay class by their refined delay and distinct fractional-Doppler leakage responses.This representation is exact for identical continuous delays and uses a representative delay when closely spaced delays are unresolved.
  • Reduced local model: The local observation model includes desired co-bin responses, retained structured leakage, and residual weak terms from omitted interactions.Structured nuisance terms include inter-pilot and out-of-bin leakage, while weak terms can include data-symbol leakage.

IV. TSUR FRAMEWORK

TSUR processes each detected co-delay-Doppler bin in two stages: it first estimates delay classes from principal pilot samples, then estimates Doppler and physical path counts from local leakage samples.

  • Two-stage framework: TSUR applies separately to each detected co-delay-Doppler bin and uses two pilot observation models.The first model supports delay-class estimation, while the second supports physical path and Doppler estimation.

A. The First Stage of the TSUR Framework

The first TSUR stage estimates resolvable delay classes from pilot phase structure using model-order selection, subspace initialization, and likelihood-based refinement. These classes need not equal the number of physical paths.

  • Delay estimation: The first step estimates delays from the phase of the principal pilot samples at the detected integer Doppler index.The reduced observation retains the current pilot contribution with z=0.
  • Resolved delay classes: Physical paths sharing a continuous delay contribute to one steering component, while closely spaced delays may appear as a single resolved delay class.Finite SNR and limited pilot observations can prevent separate resolution of nearby delays.
  • FBSS-MUSIC initialization: FBSS-MUSIC uses spatial smoothing before subspace decomposition because the delay components may be coherent.The smoothing parameters must satisfy M_s>K_d and J_s≥K_d.
  • Model-order selection: MDL selects the number of resolvable delay classes, not the physical path number.Several physical paths may share one delay class.
  • Likelihood refinement: The initial delay estimates are refined by maximizing a concentrated likelihood criterion, implemented through joint Newton updates with backtracking when needed.The better objective value between the refined estimate and the initialization is retained.

B. The Second Stage of the TSUR Framework

The second TSUR stage models local leakage within each resolved delay class to jointly determine physical path counts and fractional Doppler offsets, then recovers path gains and channel parameters.

  • Second-stage inputs: The second stage treats each refined delay and detected integer Doppler index as known while estimating paths and fractional Doppler offsets.Local observations contain co-bin components and dominant structured leakage.
  • Local window: A symmetric local window retains the principal contribution and dominant fractional-Doppler leakage around the detected integer Doppler index.The prescribed interaction radius also includes selected neighboring-pilot contributions.
  • Candidate path models: Candidate models assign one or more physical paths to each resolved delay class and enforce an upper bound on the total candidate path number.Fractional Doppler offsets are collected separately for each class.
  • Observation model: The augmented model combines candidate co-bin path blocks with a nuisance block for retained structured leakage outside those candidates.The nuisance block covers inter-pilot leakage and strong out-of-bin leakage.
  • Joint selection and estimation: Fractional Doppler offsets are estimated by minimizing concentrated residual energy, while BIC selects admissible path-number models.Models lacking sufficient observations or full rank are excluded.
  • Path reconstruction: The selected estimates jointly recover complex gains and produce path triplets containing refined delay, integer-plus-fractional Doppler, and gain.These parameters support target-state estimation, DPF channel reconstruction, and data recovery.

V. CRLB ANALYSIS

The CRLB analysis models channel estimation after hierarchical model-order determination and relates attainable accuracy to pilot phase and leakage observations. Increasing the number of pilot blocks generally improves delay and Doppler bounds.

  • Conditional benchmark: The CRLB is conditioned on detected integer Doppler, resolved delay classes, and physical path counts within each class.The total normalized Doppler is represented as ν_g,r = k_c + κ_g,r, with identical derivatives for ν_g,r and κ_g,r when k_c is fixed.
  • Observation model: The received-signal mean is the noise-free component because transformed AWGN has zero mean.The observation model is y_p(m) = μ(m; θ) + e_w(m).
  • Parameterization: The parameter vector stacks resolved delays, normalized Dopplers, and real and imaginary channel gains over the physical paths.The delay vector contains one representative delay for each resolved delay class.
  • Fisher information: The Fisher information is built from derivatives of the likelihood, including leakage-indexed path contributions and derivatives with respect to complex-gain components.With detected k_c fixed, Doppler derivatives act on the Dirichlet leakage coefficient.
  • Pilot dependence: Increasing ζ generally improves delay and Doppler bounds by supplying more pilot phase samples and stacked leakage observations.The Fisher information depends on both the number of pilot blocks ζ and retained leakage samples N_Q.

VI. SIMULATION RESULTS AND ANALYSIS

The simulation section evaluates TSUR across detection reliability, parameter estimation, channel reconstruction, and BER under different pilot configurations.

  • Evaluation scope: The simulations evaluate TSUR reliability, delay and fractional Doppler RMSE, channel-reconstruction NMSE, and BER under different pilot configurations.Unless otherwise specified, the experiments use the parameters in Table II.

A. Hierarchical Detection Reliability

The experiments assess hierarchical Doppler localization, co-bin path-number detection, and delay-class estimation under varying SNR and path separations. Results show that coherent-observation handling and leakage samples are important for reliable estimation, while unknown data-symbol interference leaves high-SNR error floors.

  • Hierarchical Detection Reliability: Integer Doppler localization is evaluated versus SNR using paths sharing an integer Doppler index but having different fractional offsets.The experiment uses SNRs from −20 to 5 dB in 2 dB increments and 10000 Monte Carlo trials per point.
  • Hierarchical Detection Reliability: BIC path-number detection is tested for two paths in one delay class with fractional-Doppler separation Δκ from 0.1 to 0.8.Detection probability improves with SNR, while stronger overlap makes the paths more difficult to distinguish.
  • Validation of Delay Class Estimation: The delay-class experiment uses two well-separated delays, d = [9.1, 23.5]^T, to validate FBSS-MUSIC and Newton refinement rather than co-bin path resolution.MUSIC, FBSS-MUSIC, FBSS-MUSIC-Newton, and ZP-PCTD are compared.
  • Validation of Delay Class Estimation: MUSIC exhibits an error floor because coherent observations make its covariance matrix rank deficient.The comparison isolates the first-stage delay-class estimation module.
  • Validation of Delay Class Estimation: FBSS restores effective signal-subspace rank, while Newton refinement reduces search-grid quantization error; unknown data symbols leave residual high-SNR floors.ZP-PCTD shows a similar floor from delay-search discretization and unknown-data interference.

C. Fractional Doppler Resolution within a Delay Class

The section evaluates fractional Doppler estimation for multiple physical paths sharing a delay class and examines how leakage modeling, delay mismatch, and pilot count affect sensing and communication recovery.

  • Fractional Doppler estimation: Principal samples alone produce an SNR-insensitive fractional Doppler error floor for co-bin paths.Adding local leakage observations is necessary for accurate fractional Doppler estimation.
  • Leakage modeling: Modeling neighboring pilot blocks with R = 2 improves fractional Doppler estimation over omitting cross-pilot contributions.Using Q = 3 and Q = 7 gives comparable performance, indicating that useful information is concentrated near the principal sample.
  • Unknown data symbols: With pilots only, R = Full approaches the CRLB at high SNR, whereas R = 2 retains a small residual; unknown QPSK data creates a nonzero high-SNR floor for both.Unknown data-symbol energy limits the achievable estimation accuracy even when pilot interactions are fully modeled.
  • Delay mismatch: For paths with different continuous delays in one delay class, TSUR estimates fractional Doppler offsets from stacked local leakage samples after delay and integer-Doppler refinement.The experiment varies the second path's delay and Doppler coordinates while fixing the first path at (d1, ν1) = (8.6, 0.6).
  • Delay mismatch: A delay mismatch yielding only one resolved delay class degrades fractional Doppler estimation by approximately 4 dB relative to the oracle benchmark.The reported mismatch range is 0.1 to 0.3.
  • Pilot configuration: Increasing the number of uniformly spaced pilots lowers the high-SNR fractional-Doppler RMSE floor, improves channel estimation, and moves QPSK BER toward the perfect-CSI reference.Under fixed total pilot energy, fewer high-power pilots can slightly advance low-SNR BIC detection, so pilot count does not improve every sensing task monotonically.
Loading 2609.04669v1…