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Hybrid Monostatic and Bistatic Sensing for Target Localization in OFDM Systems under Phase Noise

Lucía Pallarés Rodríguez, Musa Furkan Keskin, Alireza Pourafzal, José A. López-Salcedo, Gonzalo Seco-Granados, Henk Wymeersch

arXiv:2610.19987v1eess.SP

TL;DR

Phase-noise processes across monostatic and bistatic OFDM sensing links have not been jointly treated as random nuisance parameters in localization, despite their differing oscillator relationships. This paper characterizes their joint residual statistics and derives localization bounds for PN-aware and PN-ignorant receivers, with a PN-free benchmark. The bounds show a stronger loss for PN-ignorant hybrid processing, while accounting for PN keeps hybrid localization bounds close to PN-free counterparts and permits bistatic observations to improve accuracy in the considered setting.

  • Problem

    Phase-noise processes across sensing links have not been treated jointly as random nuisance parameters in localization, although phase noise can be a major impairment in OFDM sensing.

    Source: §I · p. 2

  • Method

    The paper characterizes joint residual differential phase-noise statistics and derives localization bounds for PN-aware and PN-ignorant receivers, alongside a PN-free benchmark.

    Source: §I · p. 2

  • Results

    PN-ignorant hybrid processing suffers a stronger localization loss than monostatic-only processing, while accounting for PN keeps both architectures’ bounds close to their PN-free counterparts.

    Source: §VI · p. 7

  • Takeaways & Limitations

    In the considered setting, accounting for PN allows the additional bistatic observations to improve localization accuracy relative to monostatic-only processing.

    Source: §VI · p. 7

Abstract

from arXiv · show

In hybrid monostatic-bistatic orthogonal frequency-division multiplexing (OFDM) sensing, each sensing node receives a superposition of monostatic and bistatic target returns, whose phase-noise (PN) characteristics differ fundamentally due to oscillator sharing. While monostatic sensing benefits from partial PN cancellation through the common transmit-receive oscillator, bistatic sensing involves independent oscillators and is therefore more strongly affected by PN. In this paper, we characterize the joint residual differential PN statistics of these superimposed components in a centralized two-node sensing system and quantify their impact on target localization via theoretical lower bounds considering three different receivers: (i) PN-free ideal receiver, (ii) PN-aware receiver that exploits the joint PN statistics, and (iii) PN-ignorant receiver that neglects PN.

I. INTRODUCTION

The paper addresses joint treatment of phase noise across monostatic and bistatic OFDM sensing links, using a centralized two-node system to study localization bounds. The system model combines each node’s monostatic and bistatic returns, with distinct oscillator processes affecting transmission and reception.

I. INTRODUCTION

  • Prior work did not treat the phase-noise processes across sensing links jointly as random nuisance parameters in localization.The paper characterizes their joint residual differential phase noise and derives bounds for PN-aware and PN-ignorant receivers, alongside a PN-free benchmark.Source: §I · p. 2

A. Scenario Description

  • Two synchronized sensing nodes each receive superposed monostatic and bistatic target returns, while a direct line-of-sight link between the nodes is not considered.Each node’s transmit and receive chains share a local oscillator, independent of the other node’s oscillator.Source: A. Scenario Description · p. 2

B. Transmit and Received Signal Models

  • Each node transmits an OFDM signal with phase noise, and the received link signal is modeled using its channel gain, propagation delay, and multiplicative phase-noise term.The model samples after cyclic-prefix removal, assuming the cyclic prefix is at least as long as the propagation delay.Source: B. Transmit and Received Signal Models · p. 3

C. Hybrid Sensing Observations

The hybrid observation model specializes the link-level signal model to monostatic and bistatic returns. Each node receives both components, and the observations are centralized for parameter estimation.

C. Hybrid Sensing Observations

  • Each node receives the superposition of its target-reflected monostatic return and the bistatic return from the other node.The generic signal model is specialized by selecting the receiving and transmitting node indices.Source: C. Hybrid Sensing Observations · p. 3
  • The signal expressions define the monostatic and bistatic waveform terms using the transmitted signals and their propagation delays.The terms are denoted q_p(τ_p) and q_p,q(τ_p,q).Source: C. Hybrid Sensing Observations · p. 3
  • The two nodes’ hybrid observations are forwarded to the data fusion center and stacked into a centralized observation vector.The passage introduces the equivalent received signal at node B before describing the centralized vector.Source: C. Hybrid Sensing Observations · p. 3

III. PHASE NOISE STATISTICS

The phase-noise model treats each node’s oscillator phase as a Wiener process. Reparametrization absorbs initial phase offsets into channel gains and stacks the remaining link components into a residual differential phase-noise vector.

III. PHASE NOISE STATISTICS

  • Each node’s free-running oscillator phase noise is modeled as a Wiener process.The model applies to nodes A and B.Source: §III · p. 3

A. Reparametrization of the Signal Model

  • Initial phase offsets are constant across the samples and can be absorbed into the unknown channel gains.This reparametrizes the monostatic and bistatic observations using residual phase-noise components.Source: A. Reparametrization of the Signal Model · p. 4
  • The residual differential phase-noise vector stacks the monostatic and bistatic residual components, with propagation-delay dependence left implicit.The vector is defined over the residual components for the sensing links.Source: A. Reparametrization of the Signal Model · p. 4

B. Statistics of the Residual DPN

The residual differential phase-noise statistics differ between monostatic and bistatic links because only the monostatic link benefits from common-oscillator cancellation. The joint covariance also reflects oscillator independence and its dependence on propagation delays.

B. Statistics of the Residual DPN

  • Increments from different oscillators are independent, while the residual phase-noise processes are characterized jointly across links.The joint covariance accounts for the shared-oscillator coupling among links where applicable.Source: B. Statistics of the Residual DPN · p. 4
  • Monostatic residual phase-noise variance saturates after the round-trip delay, whereas bistatic variance grows linearly with within-symbol time and is independent of propagation delay.The monostatic variance is delay-dependent; bistatic links lack common-oscillator cancellation because their transmit and receive oscillators are independent.Source: B. Statistics of the Residual DPN · p. 4
  • The first sample of each residual phase-noise vector is deterministic, making the covariance of the full vector singular.The reduced vector removes those entries to avoid the singularity.Source: B. Statistics of the Residual DPN · p. 4
  • The residual differential phase-noise covariance depends on propagation delays and can equivalently be expressed as a function of target position.Source: B. Statistics of the Residual DPN · p. 4

IV. CRB ANALYSIS UNDER PHASE NOISE

The section establishes a deterministic CRB benchmark without phase noise, then derives bounds for receivers that neglect or exploit phase-noise statistics.

IV. CRB ANALYSIS UNDER PHASE NOISE

  • The analysis uses the deterministic CRB without phase noise as a benchmark for the phase-noise cases.It then considers the MCRB for estimators that neglect phase noise and the HCRB for estimators that exploit its statistics.Source: §IV · p. 4

A. Deterministic CRB without Phase Noise

The deterministic CRB provides a no-phase-noise benchmark for localization, with the position error bound derived from the CRB on position estimation.

A. Deterministic CRB without Phase Noise

B. Hybrid CRB under Phase Noise

For a phase-noise-aware receiver, the HCRB models residual differential phase noise as a random nuisance parameter in the centralized observation.

B. Hybrid CRB under Phase Noise

  • The HCRB characterizes estimation performance when phase noise is present and the estimator is aware of it.The hybrid parameter vector contains deterministic position and channel parameters alongside random residual differential phase noise.Source: B. Hybrid CRB under Phase Noise · p. 5
  • The hybrid Fisher information matrix combines observation-related and prior information-related terms.The HCRB is obtained from its inverse.Source: B. Hybrid CRB under Phase Noise · p. 5
  • The hybrid position error bound is derived using the hybrid Fisher information results.The residual differential phase-noise probability density is specified as a function of position.Source: B. Hybrid CRB under Phase Noise · p. 5

C. Misspecified CRB under Phase Noise

The MCRB quantifies localization loss when phase noise is present but the receiver applies a phase-noise-free model. Its derivation uses a pseudo-true parameter defined through the Kullback–Leibler divergence between the true and assumed models.

C. Misspecified CRB under Phase Noise

  • The MCRB quantifies degradation from the mismatch between the true phase-noise-impaired model and the phase-noise-free model assumed by the receiver.It characterizes the setting where phase noise is present but ignored in processing.Source: C. Misspecified CRB under Phase Noise · p. 5

1) True and Assumed Models:

  • The true model uses the phase-noise-impaired observation, while the receiver evaluates it with a misspecified phase-noise-free likelihood.The assumed likelihood uses the phase-noise-free mean vector.Source: 1) True and Assumed Models · p. 5

2) Pseudo-True Parameter:

  • The pseudo-true parameter minimizes the Kullback–Leibler divergence between the true and assumed models.The pseudo-true position parameter is obtained by solving the stated position-domain optimization, using the Moore–Penrose pseudoinverse.Source: 2) Pseudo-True Parameter · p. 5

3) MCRB and LB Derivation:

The MCRB lower-bounds performance under receiver model mismatch, while averaging the conditional bound over residual DPN yields an average position-domain lower bound. The derivation assumes a full-column-rank sensing matrix.

3) MCRB and LB Derivation:

  • For a misspecified-unbiased estimator, the mean-square error relative to the true parameter vector is lower-bounded by a matrix that includes a bias term.The bound uses expectation over the true probability density function.Source: 3) MCRB and LB Derivation · p. 5
  • The derivation assumes the sensing matrix Q(p) has full column rank, making QH(p)Q(p) non-singular.Source: 3) MCRB and LB Derivation · p. 5
  • Averaging the conditional model-mismatch bound over the residual DPN distribution gives the average position-domain lower bound.The bound is conditioned on a residual DPN realization before this averaging.Source: 3) MCRB and LB Derivation · p. 6

V. SIMULATION RESULTS

The simulations compare hybrid monostatic-bistatic sensing with monostatic-only sensing under different receiver assumptions. Accounting for PN keeps the hybrid scheme advantageous across the considered transmit-power range and oscillator-quality sweep, whereas neglecting PN can reverse that comparison.

V. SIMULATION RESULTS

  • Simulations use independently drawn QPSK transmit sequences with equal total energy at the two nodes, unless otherwise stated.Parameter values are listed in Table I.Source: §V · p. 6

A. Performance with Respect to Transmit Power

  • Neglecting PN causes the position-error lower bound to saturate at moderate transmit power, while PN-aware performance remains close to the PN-free CRB.At low transmit power, thermal noise dominates; at higher power, the SNR-independent bias term dominates the mismatched bound.Source: A. Performance with Respect to… · p. 6, Figure 2 · p. 6
  • The bistatic-only baseline is omitted because, in the considered two-node configuration, both bistatic observations depend on target position through the same delay, making position unidentifiable.Source: A. Performance with Respect to… · p. 6
  • With PN accounted for, the hybrid architecture outperforms the monostatic-only scheme over the considered transmit-power range; when PN is ignored, monostatic-only performs better.The larger residual PN variance of bistatic components outweighs their added position information for the PN-ignorant receiver.Source: A. Performance with Respect to… · p. 6, Figure 2 · p. 6

B. Performance with Respect to Oscillator Quality

  • As oscillator bandwidth increases, model-mismatch performance degrades significantly, while the HCRB remains close to the CRB and eventually saturates at large bandwidths.Accounting for PN lets the hybrid architecture outperform monostatic-only; neglecting PN has a stronger impact on the hybrid scheme.Source: B. Performance with Respect to… · p. 6, Figure 3 · p. 6

C. Performance with Respect to Target Position

Across target positions, geometry largely determines where localization bounds are low or high, while phase-noise treatment and additional bistatic observations affect localization accuracy relative to PN-free and monostatic-only sensing.

C. Performance with Respect to Target Position

  • At fixed SNR across the target grid, position-FIM singularities occur on the line between nodes, where the two monostatic delay-gradient vectors are collinear.Source: C. Performance with Respect to Target… · p. 7
  • CRB, HCRB, and LB show similar spatial trends, with geometry dominant; HCRB stays near CRB, while LB degrades more, especially farther from the nodes, consistent with delay-dependent monostatic residual PN variance.Source: C. Performance with Respect to Target… · p. 7

VI. CONCLUSION

  • At low and medium transmit powers, added bistatic observations improve localization over monostatic-only sensing; PN-aware bounds stay near PN-free counterparts, while PN-ignorant hybrid processing has a stronger loss than monostatic-only processing.Source: §VI · p. 7
  • Multi-target, multi-antenna, and clutter scenarios remain outside the present framework and are identified as future extensions.Source: §VI · p. 7
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