Array Rotation in Near-Field ISAC: Communication, Detection, and Estimation Under Low-Complexity Transmit Architectures

I. Introduction

• Rotation is an underexplored design variable because it changes angular alignment and wavefront-curvature mismatch, creating tradeoffs beyond conventional beamforming.Existing near-field ISAC work commonly assumes fixed orientation and optimizes covariance, beamforming, or power allocation.Source: §I · p. 1

A. Contributions

• An exact Fisher-information decomposition separates target-side range–angle geometry from transmit-covariance-dependent effects on estimation.The framework is specialized to single-stream superposition and dual-waveform split architectures, with exact finite-array rotation analyses and compact second-order approximations.Source: A. Contributions · p. 2
• At every rotation under the same user QoS constraint, single-stream achieves the detection-optimal covariance among all covariances, so no dual-waveform design can outperform it for detection.The CRB-optimal rank-one beamformer also lies in the single-stream family.Source: A. Contributions · p. 2
• The single-stream design can align communication and detection, while dual-waveform split design creates a direct communication–detection tradeoff and offers more rate–range-estimation choices.When sensing is prioritized, dual-waveform split can cause a substantially larger communication loss; detection is less sensitive to target-state errors than estimation in the considered cases.Source: A. Contributions · p. 2

B. Existing work

• Prior work includes fixed-orientation near-field ISAC, spherical-wave range–angle CRB analysis, and rotatable or movable-array optimization; jointly designs general covariance and rotation for multi-user, multi-target systems.That work gives closed-form rotations for communication-only and sensing-only transmission, but solves mixed designs numerically.Source: B. Existing work · p. 2

C. Comparison to Existing Work

• Compared with and fixed-orientation studies such as, this work gives exact finite-array analysis of structured architectures, location-conditioned GLRT detection, and robust rotations under correlated target-state perturbations.The study considers a single-user, single-target setting with prescribed transmit structures.Source: C. Comparison to Existing Work · p. 2

II. System Model and Problem Formulation

• The system is a narrowband monostatic near-field ISAC link with one user and one target, served and sensed by a rotatable uniform linear array.The array orientation is controlled by a one-dimensional rotation angle.Source: §II · p. 3

A. Channel Models

• Exact spherical-wave steering depends jointly on angle, range, and array rotation; the model assumes equal path loss across the aperture and isotropic elements.Rotation changes array geometry but not element radiation pattern; directivity, polarization, mutual coupling, and pattern-dependent rotation effects are outside scope.Source: A. Channel Models · p. 3, B. Transmitted Signal · p. 3

C. Problem Formulation and Metrics

• The analysis conditions its main results on a transmitter-known target state, treating the complex target amplitude as an unknown nuisance parameter held fixed during range–angle differentiation.The target-state mismatch is later modeled with correlated Gaussian perturbations.Source: C. Problem Formulation and Metrics · p. 4, §III · p. 5
• The design metrics include detection probability and joint range–angle CRB, with rotation restricted to a half-turn and the selected angle held over a tracking or coherence interval.The DW user SINR treats the user-directed stream as desired and the sensing stream as interference.Source: C. Problem Formulation and Metrics · p. 4

III. Sensing Performance Analysis: Estimation

• The vectorized sensing observation is modeled as a target-dependent signal plus noise, and the estimated parameter vector includes range, angle, and the real and imaginary parts of the complex amplitude.The received echo is assumed not to interfere with the transmitted signal because its round-trip delay is sufficiently long.Source: §III · pp. 4–5
• For an arbitrary transmit covariance, the Fisher information yields joint range–angle CRBs after accounting for the unknown complex amplitude.The decomposition uses transmit-dependent illumination and derivative terms alongside target-side moments.Source: §III · p. 5
• The arbitrary-covariance CRB decomposition separates centered target-geometry moments from covariance terms governing illumination and echo sensitivity to range and angle.For rank-one transmission, the covariance-dependent terms reduce to a target-illumination factor.Source: §III · pp. 4–5
• For SS, substituting its illumination gain into the rank-one expressions gives the corresponding range–angle CRBs.The CRB is finite only under stated nonzero-illumination and positive-geometry conditions.Source: §III · p. 5
• At fixed rotation and target geometry, the rank-one CRB metrics improve monotonically with target illumination, and an optimizer under user QoS can be chosen in the span of the user and target steering vectors.Thus, the SS family contains an optimal solution of the constrained rank-one sensing problem at every fixed rotation.Source: §III · p. 5
• DW adds a positive-semidefinite Fisher-information term when its communication- and sensing-directed waveforms produce different range- or angle-dependent echo changes, but it is not universally superior to SS.Overall CRBs still depend on illumination, power allocation, and geometry.Source: §III · p. 6

Detection metric

• The location-conditioned GLRT tests noise alone against a target-dependent signal plus noise, with an unknown complex target amplitude maximized out in the statistic.Under the null, the statistic is exponential; under the alternative, twice the statistic follows a noncentral chi-square distribution with two degrees of freedom.Source: §IV · p. 6

Detection-centric rotation

• For fixed false-alarm probability, detection probability increases with the detection noncentrality parameter and therefore with target illumination, reducing rotation design to maximizing T0(ϕ).The noncentrality parameter follows from the GLRT statistic under the target-present hypothesis.Source: §IV · p. 6
• The SS and DW detection-centric rotations are defined for prescribed SS coefficients and DW power fractions, respectively.The analysis uses the same stacked sensing observation as the estimation perspective.Source: §IV · p. 6

Detection architecture comparison

• At every fixed rotation under the same user-SINR requirement, an SS design exists whose detection probability is at least as high as that of any feasible DW design.The result follows from the SS family containing the detection optimum over transmit covariances under the same user-QoS constraint.Source: §IV · p. 6

V. Analytical Approximations

• Second-order approximations cover communication correlations and target-side moments used to derive compact approximate optimal rotation angles.The section develops communication-side and sensing-side approximations before applying them in the rotation-design sections.Source: §V · p. 7

A. Communication-Side Quantities

• The communication-side quantities approximate the unweighted user–target correlation and the range- and angle-weighted correlations that enter DW CRBs.When β(ϕ) ≈ 0, the unweighted correlation reduces to a Dirichlet form.Source: A. Communication-Side Quantities · p. 7

B. Sensing-Side Quantities

• The sensing-side approximations cover target-side moments used in SS and DW CRBs, assuming D/rs ≪ 1 and excluding the near-endfire region where local range–angle geometry is degenerate.The approximated moments and correlations are substituted into the CRB expressions for arbitrary covariance, rank-one transmission, and DW.Source: B. Sensing-Side Quantities · p. 7

VI. Communication-Centric Rotation Optimization

• With architecture parameters fixed, the section derives exact stationary-point conditions for communication-centric rotation and approximate angles using the second-order correlation model.SS coefficients and the DW power fraction are held fixed rather than jointly optimized with rotation.Source: §VI · p. 7

A. Communication-Centric Rotation for SS

• For SS, setting cs = 0 makes communication performance independent of rotation; otherwise, the approximation is maximized at θm, the midpoint of the user and target angles.The communication-only beamformer also makes the rotation arbitrary.Source: §VI · p. 7, A. Communication-Centric Rotation for SS · p. 8

B. Communication-Centric Rotation for DW

• For DW, communication-centric rotation minimizes |ρ(ϕ)|², and the approximate correlation has a zero within the allowed rotation range if and only if N|sin ∆| ≥ 1.If sin ∆ = 0 and that condition fails, the approximate optimum is found by one-dimensional numerical minimization.Source: B. Communication-Centric Rotation for DW · p. 8

VII. Communication–Detection Synergy

• When |cu| = |cs|, SS communication and detection objectives coincide at every rotation; DW instead minimizes |ρ(ϕ)|² for communication and maximizes it for detection.For balanced DW power, the detection-information loss is bounded by 3.01 dB, while the communication-loss bound depends on communication SNR and can be much larger.Source: §VII · p. 8

VIII. Sensing-Centric Rotation Optimization

• The sensing-centric analysis derives exact stationary-point conditions and second-order approximate rotation expressions for both SS and DW.The approximations use second-order target-side geometry quantities.Source: §VIII · p. 8

A. Sensing-centric Rotation for SS

• For SS, approximate sensing optimization balances the illumination factor |g(S)(ϕ)|⁴ against sin⁶(θs − ϕ), favoring target broadside when illumination varies slowly.The stationary condition is d/dϕ log |g(S)(ϕ)|² = 3 cot(θs − ϕ).Source: A. Sensing-centric Rotation for SS · p. 9

B. Sensing-centric Rotation for DW

• For DW, approximate sensing optimization balances target illumination A0(ϕ)² against sin⁶(θs − ϕ), making target broadside optimal when A0 varies slowly.The corresponding stationary condition is d/dϕ log A0(ϕ) = 3 cot(θs − ϕ).Source: B. Sensing-centric Rotation for DW · p. 9

IX. Special Cases and Explicit Optimal Rotation

• When user and target ranges match, the SS communication rotation and DW detection rotation are both (θu + θs)/2; for common direction and sufficiently large ranges, both equal that direction.These cases reduce to geometries in which user and target share a spatial signature at the specified rotation.Source: §IX · p. 9
• For a co-located user and target, target-broadside rotation minimizes range and angle CRBs without changing communication or detection performance.Under the second-order approximation, target broadside also minimizes both CRBs for SS and DW sensing-only transmission.Source: §IX · p. 10

X. Robustness to Target-State Perturbation

• With target-state perturbations, communication rotation retains its nominal form, evaluated at the estimated target state.The transmitter designs using the estimated target state, while the true state is modeled as a random perturbation around that estimate.Source: §X · p. 10

A. Robust Sensing-centric Rotation angle

• Robust sensing rotations are defined by evaluating CRBs at perturbed true states using a transmit design based on the estimate, with second-order approximations supplied.The evaluation is over realizations of the target-state perturbation and rotation angle.Source: A. Robust Sensing-centric Rotation angle · p. 10

B. Robust Detection-Centric Rotation angle

• The approximate robust detection objective combines nominal target illumination with retained steering alignment; with zero state error, it reduces to nominal detection optimization.The alignment-retention term is 1 − υ(ϕ), and υ(ϕ) = 0 when Σψ = 0.Source: B. Robust Detection-Centric Rotation… · p. 11

• Rotation reduces range RCRB by approximately one half for both architectures and angle RCRB by approximately 29%–31%; gains are small near broadside but substantial off-broadside.Source: §XI · p. 11
• For SS, exact, approximate, and target-broadside sensing rotations are close; the relatively flat communication curve near the sensing-favorable region limits its communication penalty.Source: §XI · p. 11
• For SS, communication- and detection-centric rotations coincide near 42.50°, with essentially zero communication loss; its sensing-centric rotation near −46.14° incurs about 0.98 dB communication loss.The SS sensing-centric rotation also incurs approximately 7.12 dB detection-information loss.Source: §XI · p. 12
• For DW, communication-, sensing-, and detection-centric rotations are approximately 10.27°, −44.90°, and 42.50°; the sensing-centric choice incurs about 22.6 dB communication loss.The detection-centric choice incurs approximately 48.1 dB communication loss, and detection-information losses at the communication- and sensing-centric rotations are approximately 3 dB.Source: §XI · p. 12
• At the communication-only endpoint, rotation reduces CRB_rs from 1.83 × 10^2 to 7.20 × 10^-2; at the sensing-only endpoint, it reduces CRB_rs by approximately 75%.Rotation shifts the achievable rate–range CRB boundaries downward while preserving the maximum achievable rate; DW still spans a wider range of operating points.Source: §XI · p. 12, Figure 6 · p. 12
• Robust sensing- and detection-rotation approximations remain close to their corresponding robust optima for τ ∈ [0, 1]; detection is less affected by perturbations than range–angle estimation.The largest reported average joint range–angle CRB determinant losses are below 10^-6 dB for SS at τ = 0.375 and 0.0763 dB for DW at τ = 0.1875.Source: §XI · p. 13
• As minimum required communication SINR increases, both angle and range estimation worsen; optimized SS stays close to the general-covariance reference, while DW has a larger trace–CRB gap.Source: §XI · p. 13

XII. Conclusion

• All sampled Gaussian target-state perturbations stayed within the valid propagation region, so no samples were removed.Source: §XII · p. 13
• The single-stream design can keep communication and detection compatible, whereas the dual-waveform split design has a direct communication–detection tradeoff and a wider achievable rate–range CRB tradeoff.The paper also reports that sensing prioritization can entail larger communication loss for DW.Source: §XII · p. 13
• Rotation provides substantial sensing gains, particularly for range estimation, and the approximate rotation angles closely match the exact finite-array optima.The approximations cover communication-, detection-, and estimation-centric rotations, with and without correlated target-state errors.Source: §XII · p. 13
• Detection is less affected by target-state errors than range–angle estimation in the considered cases.The robust DW angle approximation omits eδr and eδθ when selecting the angle, although its plotted loss uses the full CRB expression.Source: §XII · p. 13

Appendix A Proof of Proposition 1

• The proposition proof forms the Fisher information from the vectorized observation mean and its target-parameter derivatives, then obtains the CRB by inverting the FIM.The derivation uses the complex-Gaussian FIM for real-valued parameters and the transmit-covariance trace identities.Source: Appendix A · p. 13

Appendix B Proof of Lemma 1

• The lemma proof expresses the CRB terms through trace quantities and substitutes their geometry-dependent expansions to obtain the stated range and angle CRBs.Source: Appendix B · p. 14

Appendix C Proof of Lemma 2

• For a rank-one transmit covariance, the lemma reduces the information terms to factors involving the beam-target gain and derives the corresponding range and angle CRBs.Source: Appendix C · p. 14

Appendix D Proof of Corollary 1

• The corollary proof shows that an optimal feasible beamformer can be chosen in the span of the target and user steering vectors.It projects away the orthogonal component and rescales the in-span component while preserving feasibility.Source: Appendix D · p. 14

Appendix E Proof of Lemma 3

• The mixed-beam proof substitutes the projected derivative terms into the Schur-complement information entries and obtains closed-form range–angle CRBs.Source: Appendix E · p. 14

Appendix F Proof of Proposition 2

• The additional DW term is positive semidefinite for 0 < µu < 1 and A0 > 0, and vanishes exactly when both target-state mismatch components are zero.Source: Appendix F · p. 14

Appendix G Proof of Lemma 4

• The lemma proof establishes that an optimal solution can be rank one and chosen within the span of the user and target steering vectors.The resulting beamformer is a normalized combination of those two vectors that satisfies the QoS constraint.Source: Appendix G · p. 15

Appendix H Proof of Corollary 2

• The corollary proof uses the rank-one optimum to construct an SS design that meets the same minimum communication SINR requirement.Source: Appendix H · p. 15

Appendix I Proof of Corollary 3

• In the stated equal-range or broadside special cases, choosing the rotation to align the user and target steering vectors globally maximizes their coherence and the detection objective.Source: Appendix I · p. 15

Appendix J Proof of Corollary 4

• Under the stated special cases, communication and detection terms are constant with rotation, so the joint objective reduces to the sensing objective, which is minimized at broadside.Source: Appendix J · p. 15

Appendix K Proof of Lemma 5

• For robust sensing, the transmit waveform is designed at the estimated target state and held fixed; the approximated SS objective retains an expected beam-target gain, while the DW objective neglects perturbed derivative-diversity terms.The geometry contribution is averaged over the correlated target-state perturbation model.Source: Appendix K · p. 15

Appendix L Proof of Lemma 6

• For fixed false-alarm probability, detection probability increases with the mismatched-GLRT noncentrality; the robust approximation uses expected noncentrality and reduces to the nominal detection objective when target-state error is zero.Source: Appendix L · pp. 15–16