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Bidirectional CFO Separation for Radial Velocity Estimation In 5G NR TDD V2X Links

Mohamed Elamine Benattia, Huseyin Arslan

arXiv:2609.02258v1eess.SP

TL;DR

OFDM CFO measurements confound Doppler with LO drift, limiting accurate velocity estimation. This paper separates the terms using bidirectional NR TDD PT-RS CFOs and closed-form sum and difference statistics. The estimator reaches the idealized CRLB in high-SNR conditions and reports 0.056 m/s velocity error at the 99th percentile at 10 dB SNR, with performance depending on AoA accuracy and LOS dominance.

  • Problem

    Doppler and LO drift contribute identically to OFDM CFO, while conventional estimators treat their composite as one parameter and can produce large sensing errors.

  • Method

    The method combines DL and UL PT-RS CFO estimates on a reciprocal NR TDD link, using their sum for radial Doppler and difference for relative LO offset.

  • Results

    0.056 m/s radial-velocity error is achieved at the 99th percentile at 10 dB SNR, while the sum statistic attains the CRLB for SNR ≳ 8 dB.

  • Takeaways & Limitations

    The bidirectional exchange supports CFO-based radial-velocity and LO-offset estimation without external references or iterative processing and can be co-scheduled with NR RTT positioning.

  • Takeaways & Limitations

    Full-speed recovery requires reliable AoA and becomes singular near broadside; multipath performance also departs from the CRLB as the Rician K factor decreases.

Abstract

from arXiv · show

High-accuracy velocity estimation over orthogonal frequency division multiplexing (OFDM) links is challenging because Doppler shifts and local oscillator (LO) drift contribute the same way to the observed carrier frequency offset (CFO). We propose a bidirectional CFO separation method for a frequency range-1 (FR1) time division duplex (TDD) Uu link between a roadside gNB/TRP and a vehicle user equipment (UE). Downlink (DL) and uplink (UL) CFO estimates are obtained from the physical downlink and uplink shared channel phase tracking reference signals (PT-RS), respectively, their sum isolates the radial Doppler and allow for the extraction of the radial velocity, while their difference recovers the relative LO offset. The estimates are obtained in closed form, without any iterative processing, using only a compact scalar CFO report fed back on the UL, the method can be co-scheduled with an new radio (NR) round trip time (RTT) positioning event to enable simultaneous ranging measurements along velocity estimates. We derive an idealized Cramér-Rao lower bound (CRLB), show that a linear phase slope estimator attains it at high signal to noise ratio, evaluate sensitivity to angle of arrival uncertainty and multipath, and discuss RF chain mismatch. The resulting link level implementation is aligned with NR PDSCH/PUSCH PT-RS procedures and is suitable for network assisted vehicle to everything (V2X) integrated sensing and communication (ISAC).

I. INTRODUCTION

OFDM receivers observe Doppler and hardware-induced LO drift together in the CFO, limiting accurate radial-velocity sensing. The paper motivates retaining bidirectional CFO information to identify both terms without relying on external references.

  • Conventional CFO compensation combines Doppler with LO drift from both RF chains, creating large sensing errors at high carrier frequencies or extreme speeds.
  • Existing ISAC approaches require additional constraints such as shared LOs, static reference paths, synchronized anchors, spatially separable paths, or multiple frames and receivers.
  • Prior NR V2X positioning work treats oscillator offset as a calibrated nuisance rather than an identifiable parameter.
  • The proposed direction retains both transmissions of a single exchange to jointly identify Doppler and the clock-related term.
  • The OFDM signal model maps frequency-domain symbols X_k,m onto N subcarriers across M symbols with subcarrier spacing Δf = B/N.

B. Reference Signals

The link uses PTRS observations for CFO estimation in an NR-compatible V2X setting, while modeling a dominant LOS path and Rician multipath components. LOS Doppler is the aggregate motion-related offset, and NLOS terms perturb the pilot phase.

  • B. Reference Signals: The CFO estimator uses PTRS associated with PDSCH and PUSCH transmissions to obtain phase observations across time and frequency.
  • B. Reference Signals: A vehicle’s radial velocity is v_r,i = v_i cos θ_i, where θ_i is the angle between motion and the RSU line of sight.
  • B. Reference Signals: The channel model uses a synthetic Rician tapped-delay channel with L paths and a Rician K factor within a 3GPP framework.
  • B. Reference Signals: The LOS path carries the dominant Doppler, while NLOS taps contribute relative Doppler terms that perturb the pilot phase.

2) Oscillator Model

The oscillator model separates deterministic residual CFO from stochastic phase noise and assigns opposite relative LO offsets to downlink and uplink directions. The received signal therefore contains Doppler, LO drift, phase noise, and AWGN effects.

  • 2) Oscillator Model: Practical oscillators contribute quasi-static residual CFO and stochastic phase noise, modeled separately with phase noise represented as a Wiener process.
  • 2) Oscillator Model: The transmitter and receiver phase-noise difference is modeled as Φ(t) = Φ_tx(t) − Φ_rx(t), while gNB and UE LOs have residual errors ε_B,i and ε_V,i.
  • 2) Oscillator Model: The relative LO offsets satisfy ε_BV,i = ε_B,i − ε_V,i and ε_VB,i = −ε_BV,i, remaining approximately constant over one RTT window.
  • 2) Oscillator Model: The received signal model includes Doppler and LO drift through an aggregate offset ε, together with complex AWGN.

A. Downlink Baseband Signal

The downlink signal is transmitted from the gNB to the vehicle, whose CFO estimate is fed back as a compact scalar before the reciprocal uplink transmission. The exchange can also support RTT timing, although range accuracy is not evaluated.

  • A. Downlink Baseband Signal: The LOS downlink PDSCH baseband signal is received at vehicle i after propagation over a channel with delay τ and gain α.
  • A. Downlink Baseband Signal: The vehicle estimates the downlink CFO before compensation, quantizes ε̂_DL,i, and feeds it back to the gNB through an implementation-specific uplink control message or payload.
  • A. Downlink Baseband Signal: After processing delay τ_p, the vehicle transmits the uplink PUSCH frame over the same reciprocal TDD channel, enabling the gNB to obtain the uplink CFO.
  • A. Downlink Baseband Signal: A single-link round trip approximately satisfies T_RTT ≈ 2τ + τ_p, connecting the bidirectional CFO exchange with RTT timing.
  • A. Downlink Baseband Signal: Standardized NR RTT combines DL PRS and positioning SRS measurements, while this work focuses on CFO-based velocity and LO-offset estimation rather than range accuracy.

V. FREQUENCY DOMAIN OBSERVATION MODEL

After FFT processing, pilot observations are modeled using normalized CFO, per-symbol common phase error, and residual noise. Pilot phases evolve linearly across OFDM symbols, with residual multipath, phase noise, and AWGN collected in the disturbance term.

  • After sampling, synchronization, cyclic-prefix removal, and FFT, the channel convolution is circular when Tcp exceeds the maximum path delay.
  • The normalized frequency offset ν = ε/∆f is used under conditions where inter-carrier interference remains manageable on pilots.
  • Pilot observations include per-symbol common phase error from phase noise and additive noise containing AWGN and residual inter-carrier interference.
  • After de-rotation, pilot phases are linear in the OFDM symbol index, while ηk,m captures residual NLOS multipath, oscillator phase noise, and AWGN.

VI. THE PROPOSED BIDIRECTIONAL DOPPLER–LO SEPARATION

The method estimates each directional CFO from the linear phase slope of PT-RS pilots, then exploits TDD sign reversal to separate Doppler from LO drift. It directly identifies radial velocity, while full speed additionally requires reliable angle-of-arrival information.

  • Bidirectional CFO estimation: For each link direction, least-squares fitting of pilot phase against symbol index, averaged over Np PT-RS subcarriers, estimates the directional CFO.
  • Bidirectional CFO estimation: Centering pilot symbol indices removes the constant phase intercept, making the estimator a standard single-tone frequency estimator.
  • Joint sensing procedure: Fig. 2 combines bidirectional CFO estimation with RTT-based ranging to obtain range and velocity information.
  • CFO separation: The centered slope estimator attains the Cramér–Rao bound at high SNR, and TDD sign reversal lets the sum and difference separate Doppler from LO drift.
  • Velocity recovery: Radial velocity is directly identifiable, whereas full speed requires independently known movement-direction AoA and becomes singular near broadside.

VII. THEORETICAL ANALYSIS AND PRACTICAL CONSIDERATIONS

The theoretical analysis derives CRLBs for directional CFO, separated Doppler, and LO drift, then examines estimator efficiency and practical sensitivities. The idealized bound assumes AWGN and negligible residual nuisance effects.

  • Scope: The analysis derives CRLBs for joint Doppler and residual LO-drift estimation and studies AoA errors, multipath, and channel reciprocity.
  • Cramér–Rao lower bound: Each directional offset is a linear phase slope estimated from Np pilots across Nt PT-RS symbols, with a Rife–Boorstyn variance bound.
  • Assumptions: The idealized AWGN bound omits phase noise, while simulated AWGN/LOS conditions make the residual nuisance negligible and allow the bound to be attained.
  • Separated-parameter bound: The separated Doppler and LO-drift variances satisfy Var(ˆfd) = Var(ˆϵBV,i) ≥ 1/[4(2π)^2Ts^2ρNpS].
  • Verification: Section VIII verifies that the linear phase-slope estimator attains the derived bound.

B. AoA Error Propagation and Sensitivity

AoA uncertainty affects reconstructed full speed but not the directly observable radial velocity, with broadside geometries being especially sensitive.

  • B. AoA Error Propagation and Sensitivity: Velocity estimation depends on AoA through ˆvi = c ˆfd/(fc cos θi), so angular uncertainty propagates into the full-speed estimate.The sensitivity analysis uses the first-order effect of AoA error on this relationship.
  • B. AoA Error Propagation and Sensitivity: σv,θ ≈ vi |tan θi| σθ, vanishing at θi = 0 and diverging as |cos θi| → 0 near broadside.The radial velocity remains directly observable and AoA free, whereas full-speed recovery requires reliable AoA.
  • B. AoA Error Propagation and Sensitivity: At v = 19.4 m/s, σθ ≈ 2° keeps full-speed error below 1 m/s for θ ≲ 56° and below 2 m/s for θ ≲ 71°.The latter bound is within the 15 m/s Release 19 Category 4 vehicle target at 95% confidence.

C. Multipath Effects

Multipath performance depends primarily on the Ricean K factor, while the simulated estimator reaches the CRLB in favorable conditions and is evaluated under a standards-aligned FR1 TDD setup.

  • C. Multipath Effects: For high Ricean K, the LOS component dominates and the estimator remains close to the CRLB; decreasing K increases perturbations and departs from the bound.The analysis assumes a dominant LOS component before assessing NLOS effects.
  • C. Multipath Effects: A constant RF phase mismatch shifts the phase intercept and is removed by the centred slope fit, so it does not bias frequency estimates.The scheme assumes reciprocal multipath channels within the coherence time, while transmit and receive RF chains may have independent responses.
  • C. Multipath Effects: The proposed estimator’s velocity RMSE overlaps a monostatic benchmark with known ϵBV,i, while the LO-drift statistic tracks the CRLB but reaches a small high-SNR residual ICI floor.The residual floor does not affect velocity estimation.

B. Identifiability, Robustness and Reliability

The robustness analysis shows that NLOS tap count has little effect under the simulated exponential delay profile, while Ricean K controls accuracy and the method achieves low reported tail errors.

  • B. Identifiability, Robustness and Reliability: At SNR = 10 dB, velocity RMSE is approximately 21, 10, and 1.2 m/s for K = −10, 0, and 10 dB, respectively.Both velocity and LO-drift RMSE are nearly flat as the number of NLOS taps increases.
  • B. Identifiability, Robustness and Reliability: The exponential power delay profile concentrates NLOS power within the cyclic prefix, so adding resolvable paths barely changes the perturbation.The results indicate stronger control by the Ricean K factor than by tap count in this setup.
  • B. Identifiability, Robustness and Reliability: The method is best suited to LOS-dominant geometries with K ≳ 0 dB under the reported multipath conditions.Lower K increases perturbation variance and causes performance to depart from the idealized bound.
  • B. Identifiability, Robustness and Reliability: At SNR = 10 dB, the empirical 99th-percentile errors are 0.056 m/s for velocity and 0.73 Hz for LO drift.These are reliability-oriented CDF results reported for the simulated sensing configuration.
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