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Multi-static Parameter Estimation in the Near/Far Field Beam Space for Integrated Sensing and Communication Applications

Saeid K. Dehkordi, Lorenzo Pucci, Peter Jung, Andrea Giorgetti, Enrico Paolini, Giuseppe Caire

arXiv:2309.14778v1eess.SPcs.ET

TL;DR

Multi-static ISAC estimation must handle targets in either the near or far field, while prior work described in the paper lacks studies spanning both regimes in multi-static or bi-static settings. This paper proposes a two-stage ML framework that begins with far-field estimation and applies near-field estimation or beamfocusing when indicated. The authors report improved target-detection performance from fusing two receivers and improved communication performance using custom beamfocusing codewords.

  • Problem

    The paper identifies a lack of studies on multi-static or bi-static ISAC sensing and communication performance that account for both near- and far-field scenarios.

    Source: §I · p. 1

  • Method

    A two-stage ML framework first estimates parameters under far-field assumptions, then applies near-field estimation or beamfocusing when initial estimates indicate near-field conditions.

    Source: §VII · p. 14

  • Results

    Fusing information from two receivers can significantly increase target-detection performance, and beamfocusing instead of beamforming can improve localization and communication performance.

    Source: §I · p. 2

  • Takeaways & Limitations

    The framework adapts estimation and transmit codewords to the inferred propagation regime, using beamfocusing codewords designed for constant gain over an extended spatial region.

    Source: §I · p. 2, §VII · p. 14

Abstract

from arXiv · show

This work proposes a maximum likelihood (ML)-based parameter estimation framework for a millimeter wave (mmWave) integrated sensing and communication (ISAC) system in a multi-static configuration using energy-efficient hybrid digital-analog arrays. Due to the typically large arrays deployed in the higher frequency bands to mitigate isotropic path loss, such arrays may operate in the near-field regime. The proposed parameter estimation in this work consists of a two-stage estimation process, where the first stage is based on far-field assumptions, and is used to obtain a first estimate of the target parameters. In cases where the target is determined to be in the near-field of the arrays, a second estimation based on near-field assumptions is carried out to obtain more accurate estimates. In particular, we select beamfocusing array weights designed to achieve a constant gain over an extended spatial region and re-estimate the target parameters at the receivers. We evaluate the effectiveness of the proposed framework in numerous scenarios through numerical simulations and demonstrate the impact of the custom-designed flat-gain beamfocusing codewords in increasing the communication performance of the system.

I. INTRODUCTION

The paper addresses multi-static ISAC parameter estimation across near- and far-field conditions, where the propagation regime must be identified to choose an appropriate sensing and communication strategy. It proposes a two-stage framework and models a multi-static OFDM system using hybrid digital-analog arrays and beam-space codebooks.

I. INTRODUCTION

  • The paper identifies a gap in multi-static or bi-static ISAC studies that account for both near- and far-field scenarios, which matters because near-field propagation does not follow the planar-wave model.Large arrays and wideband signals support radar resolution, while fully digital architectures are challenging because of per-antenna data-conversion rates.Source: §I · p. 1
  • The generalized channel model accounts for near-field amplitude and phase variations across antenna elements and converges to the far-field line-of-sight model.The framework uses far-field beamforming initially because its codebook has lower dimensionality and its parameter search space is reduced.Source: §I · p. 2
  • The proposed framework first estimates target parameters under far-field assumptions, then uses near-field estimation or beamfocusing when the initial estimates indicate near-field conditions.For a target in the near field of an Rx, the second stage uses the near-field model; for a target in the near field of the Tx, beamfocusing codewords illuminate the target and receivers re-estimate its parameters.Source: §I · p. 2
  • The beamfocusing codebook is designed for near-field targets to maintain approximately constant gain across an extended spatial region.The introduction reports that beamfocusing can improve localization and communication performance compared with beamforming.Source: §I · p. 2
  • Fusing observations from two receivers can significantly increase target-detection performance, while their different perspectives provide spatial diversity for extended targets.The considered configuration has one Tx and two Rx units forming two bi-static pairs.Source: §I · p. 2, §II · p. 2

II. SYSTEM MODEL

  • The system uses OFDM and fully connected hybrid digital-analog arrays, with receivers scanning the beam-space sector illuminated by each selected transmit beam.The HDA receiver forms a reduced beam-space observation using selected DFT beams; multiple measurement blocks may be used when more beams are needed to cover the illuminated segment.Source: §II · pp. 3–4

A. Target model

The target model represents an extended object as a random set of active grid scatterers. The paper also distinguishes the radiative near-field region from the far field using array-dependent distance boundaries, with a subcarrier-specific qualification for multiband systems.

A. Target model

  • An extended target occupies a designated rectangular grid region, with each grid point independently active with probability q and the active-point count following a binomial distribution.The active scatterers are drawn from the grid, and their variable number models fluctuations in radar reflectivity associated with target aspect angle and material.Source: A. Target model · p. 4
  • Each receiver observes a separately generated scattering profile at each measurement instance, so the multi-static target views need not match.The target schematic depicts scattering point clusters determined through the binomial model.Source: A. Target model · p. 4, Figure 2 · p. 4

B. Near/far-field region relationship

  • The radiative near-field region of interest lies between the Fraunhofer distance and the Fresnel distance.The Fraunhofer boundary marks the minimum distance for the stated planar-wave phase-difference condition across the array.Source: B. Near/far-field region relationship · p. 4
  • For multiband OFDM, the Fraunhofer distance varies across subcarriers, and the system-wide far-field regime can be set by the highest-frequency subcarrier.For critical cases, processing must be performed separately on a per-subcarrier basis.Source: B. Near/far-field region relationship · p. 4

C. Channel model

The channel model accounts for spherical wavefronts in the near field and approaches the plane-wave far-field model at greater distances. It represents extended targets as collections of scatterers whose bistatic channels combine the Tx-to-target and target-to-Rx paths.

C. Channel model

  • The per-element channel gain depends on the scatterer’s RCS and its distance from the transmitting antenna element under line-of-sight propagation.The text defines rki,p as that distance and σk,p as the scatterer’s RCS.Source: C. Channel model · p. 5
  • The general channel model remains valid in both near- and far-field regimes, converging to the far-field model as target distance increases.The near-field model accounts for spherical wavefronts, unlike the far-field plane-wave approximation.Source: C. Channel model · pp. 4–5
  • The transmit and receive array responses use reference angles and distances, with the receive-side model constructed analogously to the transmit-side model.The far-field simplification assumes small element spacing relative to range and approximately equal amplitude across array elements.Source: C. Channel model · p. 5
  • An extended target is modeled through scatterers, with each bistatic channel formed by combining the Tx-to-scatterer and scatterer-to-Rx channels.The resulting path has bistatic attenuation εp = αk,pβp,j and delay τp = τk,p + τp,j.Source: C. Channel model · p. 6

III. OFDM INPUT-OUTPUT RELATIONSHIP

The OFDM input-output model maps transmitted symbols through bistatic channels to received signals, with target echoes represented across antenna elements and the time-frequency grid. It also specifies sequential beam-space processing and assumptions used to separate targets and reduce receiver complexity.

III. OFDM INPUT-OUTPUT RELATIONSHIP

  • The received signal is the convolution of the time-varying MIMO channel with the transmitted OFDM signal, including contributions from the extended target’s scatterers.The transmitted frame uses a unit-norm beamforming or beamfocusing vector and OFDM modulation symbols.Source: §III · p. 7
  • The model assumes targets are sufficiently separated in beam space so that a beam directed at one target has negligible response to another.Transmit beamformers and beamfocusers cover non-overlapping beam-space sections.Source: §III · p. 7
  • The system processes targets sequentially in time, using a receiver reduction matrix to focus on one target for B OFDM frames and reduce receiver complexity and energy consumption.The reduction matrix covers an area in beam space determined by the current transmit beamformer.Source: §III · p. 7
  • For each scatterer, the bistatic channel is formed from the receive and transmit channel vectors, with Doppler contributions from both paths.After OFDM demodulation, the received time-frequency grid assumes negligible inter-carrier and inter-symbol interference and includes noise.Source: §III · p. 7

IV. PARAMETER ESTIMATION AND DETECTION

The estimation framework starts without prior knowledge of whether a target is in the near or far field, using a far-field stage before selectively refining near-field cases. This staged approach uses the smaller far-field search space to reduce initial estimation complexity.

IV. PARAMETER ESTIMATION AND DETECTION

  • The first stage covers radial beam-space segments with far-field beamforming and estimates target parameters under bistatic far-field assumptions.Its far-field codebook is parameterized only by angle, and the far-field search space is smaller than the near-field space.Source: §IV · pp. 7–8
  • If the initial estimates place a target in the near field of either the transmitter or a receiver, a second estimation stage uses the appropriate near-field processing.Initial far-field/near-field model mismatch can yield poorly localized estimates, which are then used to define a region of interest for refinement with the near-field model.Source: §IV · p. 8

A. Maximum likelihood parameter estimation

The section formulates maximum-likelihood estimation over target-scatterer parameters and describes an approximate grid-based detection and estimation procedure. In the far-field model, angular, delay, and Doppler parameters are estimated from distinct signal dimensions.

A. Maximum likelihood parameter estimation

  • The ML estimate searches over target-scatterer parameters by minimizing the received-signal residual across observation frames.The parameter set includes complex attenuation, Doppler, Tx- and Rx-side ranges, and angles.Source: A. Maximum likelihood parameter… · p. 8
  • The brute-force maximum-likelihood search is computationally prohibitive, so the method tests parameter tuples on a grid and detects scatterers using a likelihood-ratio threshold.The number of scatterers is also difficult to estimate because extended-target micro-scatterers are often indistinguishable; the threshold uses OS-CFAR.Source: A. Maximum likelihood parameter… · p. 8
  • For well-separated targets, the paper describes successive interference cancellation, while noting that closely spaced extended-target scatterers can make it ineffective.The proposed grid-based estimation and thresholding approach is reported to give satisfactory performance for extended targets.Source: A. Maximum likelihood parameter… · p. 8
  • In the far-field model, angle is estimated from array manifolds, delay from OFDM subcarrier phase, and Doppler from OFDM symbol phase.The model uses array manifolds that depend only on angular parameters.Source: A. Maximum likelihood parameter… · p. 9

B. Bi-static range and Doppler shift

Bi-static sensing uses the Tx–scatterer–Rx path and the receiver’s angle of arrival to locate scatterers, while Doppler supports estimation of bi-static velocity. Near-field refinement is conditional on the first-stage far-field estimate and uses a reduced search region.

B. Bi-static range and Doppler shift

  • Bi-static range is the sum of Tx-to-scatterer and scatterer-to-Rx distances, inferred from propagation delay; with receiver angle of arrival, the scatterer position can be determined.The range defines an ellipse with Tx and Rx at its foci; the angle of arrival resolves the scatterer position along it.Source: B. Bi-static range and Doppler shift · p. 9
  • Bi-static Doppler is proportional to the rate of change of bi-static range, but the system estimates only the velocity component along the bi-static bisector because the direction angle δ is unknown.The estimated bi-static velocity is v_bis = v_p cos δ.Source: B. Bi-static range and Doppler shift · p. 9

Stage 2: Near-field estimation

  • The second stage runs only when first-stage far-field estimates place a target in the near field; for a near-field Rx, a refined Cartesian grid over the estimated region reduces the search space.Grid-cell coordinates are mapped to (r_p, γ_p, θ_p, ϕ_p) before evaluating the near-field ML metric.Source: B. Bi-static range and Doppler shift · p. 9

Discussion

  • The far-field model becomes increasingly adequate near the transition region, where its performance can remain acceptable without a critical near-/far-field classification.The transition is gradual, and far-field beams begin forming at approximately D_ff/10.Source: B. Bi-static range and Doppler shift · p. 10

Near-field of Tx: Beamfocusing and re-estimation

  • For a near-field Tx, a beamfocusing codeword illuminates the first-stage estimated region at constant gain; the receivers then re-estimate parameters, and neighboring codewords can cover a larger region of interest.The receivers can use the far-field model in this deployment topology; the extended illumination also maintains SNR when the UE antenna is slightly displaced from the estimated location.Source: B. Bi-static range and Doppler shift · p. 10

A. Related works

Prior near-field beamforming methods can require precise user coordinates or create unwanted spatial leakage. This work instead designs beamfocusing codebooks to illuminate an extended region with relatively constant gain.

A. Related works

  • Coordinate-based near-field weights require accurate user coordinates, a difficult fit for extended targets whose estimated reflection points may not coincide with the UE antenna.Ring-type codebooks can also produce spectral leakage into unwanted locations, potentially illuminating unintended objects.Source: A. Related works · p. 10
  • The proposed codebook-based approach adapts flattop beamforming to near-field operation, aiming for relatively constant gain across an extended angular or spatial region.The design seeks to avoid dependence on a single precisely estimated antenna location.Source: A. Related works · p. 10
  • Near-field beamfocusing optimization can be viewed as a two-dimensional filter-design problem, related to established spatial-filter design for array radiation patterns.The cited related work includes 2-D spatial filters in azimuth-elevation response.Source: A. Related works · p. 10

B. Problem formulation

The beamfocusing design approximates a desired radiation pattern over sampled angle-range points. It targets constant magnitude in an intended near-field region while limiting sidelobes, then uses an iterative design procedure to address computational demands and mask-synthesis difficulties.

B. Problem formulation

  • The design seeks an array vector whose response approximates a reference mask across discrete angle-range samples, with constant level in the near-field spot and sidelobes below a threshold.The response at each sample is the inner product of the array response vector and beamfocusing vector.Source: B. Problem formulation · p. 11
  • The field of view is divided into sectors that define codeword illumination areas; the optimization relaxes phase-and-magnitude matching to magnitude-only fitting for a less localized solution.The codewords are collected in a codebook.Source: B. Problem formulation · p. 11
  • A fine sampling grid can make the array manifold matrix rank-deficient, so the formulation uses Tikhonov regularization; the resulting beam vector is found through an iterative update of the vector and its phase coefficients.For fixed coefficients, the vector update is a linear least-squares problem; for fixed vector, coefficient phases follow the phase of A^Hf.Source: B. Problem formulation · p. 11
  • Mask transitions can destabilize synthesis, and wide or long masks are difficult to realize, particularly off boresight; a 2-D Kaiser window smooths the mask and concentrates energy more centrally.The examples show masks spanning extended distance or angular regions, including a wider angle-range mask with and without smoothing.Source: B. Problem formulation · p. 11, Figure 5 · p. 12, Figure 6 · p. 12

VI. SIMULATION RESULTS

Simulations evaluate the system across four trajectories using extended targets generated from the paper’s target model. For the far-field trajectory, weighted estimates locate the target accurately despite coarse resolution; a separate trajectory compares local detection with receiver-fused detection.

VI. SIMULATION RESULTS

  • The simulations use extended targets generated according to the BND model, with parameter-estimation grids distinct from the grids used to simulate those targets.The scenario includes trajectories I–IV to represent different deployment conditions.Source: §VI · p. 12

A. Far-field estimation performance (Trajectory I)

  • The weighted far-field estimate locates the target accurately even though the simulated system’s 2.34 m range and 2.2° angular resolutions usually cannot resolve individual scatterers.The weighted average is formed from detected points using the ML metric values; Fig. 7(b) plots RMSE between the main peak and target center.Source: A. Far-field estimation performance… · p. 12, Figure 7 · p. 13

B. Multi-static spatial diversity gain (Trajectory IV)

  • For Trajectory IV, detection probability is evaluated locally at each receiver and for a fused curve that declares detection when either receiver detects the target.The experiment uses 200 simulations per receiver and position; detections must pass OS-CFAR and fall within 1 m of the target position.Source: Figure 8 · p. 13, B. Multi-static spatial diversity gain… · p. 13

C. Estimation of targets in receiver NF (Trajectory III)

The paper reports receiver-near-field estimation results, beamfocusing performance under target-position mismatch, and a two-stage design that uses far-field initialization before near-field refinement or focused transmission when indicated.

C. Estimation of targets in receiver NF (Trajectory III)

  • At the closer Trajectory III position, RX1 resolves many scatterers accurately; farther away, individual scatterers are unresolved, but the estimated location remains highly accurate.RX2 estimates are accurate within system range and angular resolution limits because the target is in its far field.Source: C. Estimation of targets in receiver… · p. 13, Figure 9 · p. 14

D. Spectral efficiency enhancement with beamfocusing (Trajectory II)

  • Beamfocusing performs significantly better than Fourier beamforming when the target antenna deviates from its expected position, and the figures compare matching gain and spectral efficiency along Trajectory II.The matching-gain and spectral-efficiency plots show the extended target at locations 1–5; the filled area marks gain within a 1 m antenna-distance mismatch.Source: D. Spectral efficiency enhancement… · p. 13, Figure 10 · p. 14, Figure 11 · p. 14
  • With beam-focused rather than beamformed transmission, parameter estimation improves, which the authors attribute to increased SNR at the target.Figure 12 compares target-position estimation error using near-field and far-field codewords.Source: D. Spectral efficiency enhancement… · p. 13, Figure 12 · p. 14

VII. CONCLUSION

  • The conclusion describes far-field maximum-likelihood estimation first, followed by near-field estimation when indicated, using an exact array-response model for a near-field receiver or beamfocusing codewords for a near-field transmitter.The beamfocusing codewords are designed by solving a magnitude least-squares problem, after which receivers re-estimate target parameters.Source: §VII · p. 14

APPENDIX A DERIVATION OF ML ESTIMATE

The appendix derives a maximum-likelihood estimate for a single scattering point by forming a likelihood from the observations and optimizing over the unknown parameters.

APPENDIX A DERIVATION OF ML ESTIMATE

  • The derivation accumulates observations through slot B into y(B) and forms the corresponding log-likelihood function.The appendix frames the likelihood for y_b before combining observations across slots.Source: Appendix A · p. 15
  • For a fixed scattering-point parameter tuple, the value ε′ that maximizes (48) is the maximum-likelihood channel-coefficient estimate and has a closed-form expression.The tuple is (ν_p, r_p, γ_p, θ_p, ϕ_p).Source: Appendix A · p. 15
  • Substituting the channel-coefficient estimate from (51) into (48) yields the maximum-likelihood estimate in (52).The appendix notes that the derivation uses mixed Kronecker-product and Kronecker matrix-vector identities.Source: Appendix A · p. 15

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