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Massive MIMO ISAC Under Target-Angle Uncertainty: CRLB Outage Analysis and Robust Resource Allocation

Smriti Uniyal, Tianyu Fang, Van-Dinh Nguyen, Hien Quoc Ngo, Markku Juntti, Nhan Thanh Nguyen

arXiv:2609.18467v1eess.SP

TL;DR

Most ISAC designs assume perfect target angles, although angle uncertainty leaves the CRLB outage behavior in mMIMO-ISAC insufficiently characterized. The paper derives CRLB outage analysis and develops robust power allocation under angle uncertainty. Under severe errors, the design reduces azimuth and elevation outage probabilities by as much as 63% and 60%, respectively, versus the non-robust scheme for Gaussian errors.

  • Problem

    Target-angle uncertainty makes sensing CRLBs random, while their degradation and outage behavior in mMIMO-ISAC have not been characterized.

    Source: §I · p. 1, A. Related Works · p. 2

  • Method

    The paper derives CRLB and outage analyses under angle uncertainty and develops robust power allocation that maximizes communications sum rate subject to sensing-outage and power constraints.

    Source: §VI · p. 11

  • Results

    For Gaussian angle errors at (σψ, Uψ, κψ) = (8°, 18, 51), the robust design reduces azimuth and elevation CRLB outage probabilities by 63% and 60%, respectively, versus the non-robust scheme.

    Source: §V · p. 11

  • Takeaways & Limitations

    As angle errors grow, outage probability rises and sum rate falls because the robust design allocates more power to sensing.

    Source: §V · p. 10

Abstract

from arXiv · show

In integrated sensing and communications (ISAC), the same spectral and hardware resources are shared for two functionalities. Most ISAC designs assume perfect target-angle information neglecting angle estimation errors, which introduce steering-vector mismatches, degrade sensing accuracy, and may invalidate deterministic sensing guarantees. This paper investigates monostatic massive multiple-input-multiple-output (MIMO) ISAC systems under imperfect target-angle estimates. We derive closed-form expressions for the Cram\'er-Rao lower bounds (CRLBs) of target azimuth and elevation estimates in the presence of angle uncertainty. We characterize the cumulative distribution functions and outage probabilities of the CRLBs under Gaussian, generalized uniform, and von Mises angle-error models. Our analysis reveals that, in the small-error regime, the CRLBs increase quadratically with the angle errors due to transmit steering-vector mismatch. To ensure reliable sensing, we propose a robust power allocation framework that jointly optimizes pilot training and communications/sensing transmission powers to maximize the communications sum rate while satisfying CRLB outage constraints. The resulting nonconvex problem is solved using an alternating-optimization algorithm based on successive convex approximation. Numerical results validate the developed analysis and show that the proposed robust design reduces azimuth and elevation CRLB outage probabilities by up to $60\%$ compared with conventional non-robust schemes. It attains up to $45\%$ higher sum rates than the non-robust design under strict CRLB thresholds.

I. Introduction

The paper addresses the gap between idealized perfect-angle ISAC designs and practical mMIMO systems with uncertain target angles. It characterizes sensing reliability under angle errors and develops robust resource allocation for communications and sensing.

A. Related Works

  • Existing ISAC work generally assumes perfect target-angle information, while random-error CRLB degradation and outage behavior in mMIMO-ISAC remain largely uncharacterized.Angle uncertainty can cause steering-vector mismatch, degrade sensing accuracy, and distort CRLB metrics.Source: §I · p. 1, A. Related Works · p. 2

II. Signal Model

  • The system is a monostatic downlink mMIMO-ISAC setup with a UPA-equipped base station, communications users, and a sensing target; it shares resources across both functions.The base station receives the target echo for sensing processing.Source: B. Motivations and Contributions · p. 2, §II · p. 3

B. Motivations and Contributions

  • The study derives closed-form azimuth and elevation CRLBs and approximations of their distributions and outage probabilities under Gaussian, generalized-uniform, and von Mises angle errors.The CRLB expressions apply to MRT and ZF precoding and depend on large-scale fading coefficients.Source: B. Motivations and Contributions · p. 2
  • In the small-error regime, CRLBs increase quadratically with angle errors because of steering-vector mismatch; outage probabilities vanish as transmit or receive antenna counts grow.Source: B. Motivations and Contributions · p. 2
  • The robust power-allocation problem jointly optimizes pilot-training and communications/sensing transmission powers to maximize communications sum rate subject to sensing-outage and total-power constraints.The resulting nonconvex problem is solved with an alternating-optimization successive-convex-approximation algorithm.Source: B. Motivations and Contributions · p. 2
  • Under severe angle uncertainty, the robust design reduces azimuth and elevation CRLB outage probabilities by up to 60% and achieves up to 45% higher sum rates under strict CRLB thresholds.The comparison is with conventional non-robust designs.Source: B. Motivations and Contributions · p. 2

C. Paper Organization and Notation

  • The paper proceeds from signal models and performance metrics to robust power allocation, then evaluates the design numerically and concludes.Its notation section defines operators and symbols used throughout.Source: C. Paper Organization and Notation · pp. 2–3

1) Uplink Pilot Training:

  • The TDD protocol performs uplink pilot training for channel estimation before downlink communications and sensing transmission in each coherence interval.Source: 1) Uplink Pilot Training · p. 3

2) Communications Signal Model:

The communications model uses a linear dual-functional precoder to serve users and probe the target. The radar model describes the target direction, steering vectors, round-trip channel, and received echo.

2) Communications Signal Model:

  • A linear dual-functional precoder combines communications beamformers and power allocations with a sensing beamformer and sensing-power coefficients to form the transmitted waveform.The waveform is generated by applying the precoder to the user-symbol vectors.Source: 2) Communications Signal Model · p. 3

B. Radar Model

  • The radar model parameterizes the target by azimuth and elevation, defines transmit and receive steering vectors for the UPA, and models the reflected echo received at the base station.The target reflection coefficient and sensing-receiver noise are included in the echo model.Source: B. Radar Model · p. 4

C. Dual-Functional Transmit Precoder

The dual-function precoder uses estimated target angles for sensing and MRT or ZF for communications. The analysis derives rate and sensing bounds under angle uncertainty, showing quadratic CRLB growth with errors and a resulting communications–sensing power tradeoff.

C. Dual-Functional Transmit Precoder

  • The sensing beamformer uses the estimated target angles, while communications uses MRT or ZF; together they form the linear dual-function transmit precoder.The target-angle estimates are modeled as the true angles plus azimuth and elevation errors.Source: C. Dual-Functional Transmit Precoder · p. 4

A. Communications Sum Rate

  • The achievable communications rate depends on pilot-power allocation through ξk(ϑ), as well as communications and sensing power allocations.This dependence distinguishes the rate expression from.Source: A. Communications Sum Rate · p. 4

B. Sensing CRLBs Under Target-Angle Uncertainty

  • The closed-form CRLBs are derived for estimating target azimuth and elevation under target-angle uncertainty.The derivation forms a Fisher information matrix for the unknown parameters and obtains angle bounds after eliminating the target-reflection coefficient via a Schur complement.Source: B. Sensing CRLBs Under Target-Angle… · pp. 4–5
  • With fixed nonzero angle errors and square UPAs, CRLBs scale as 1/Nt for large Nt at fixed Nr and as 1/Nr^2 for large Nr at fixed Nt.The paper states this scaling for equal power allocation between communications and sensing.Source: B. Sensing CRLBs Under Target-Angle… · p. 5
  • For small target-angle errors, the azimuth and elevation CRLBs increase quadratically, mainly through transmit steering-vector mismatch.The result is stated for fixed antenna counts.Source: B. Sensing CRLBs Under Target-Angle… · pp. 5–6
  • Angle errors increase the minimum sensing power needed to meet a CRLB threshold, reducing communications power and achievable rate under a fixed downlink power budget.The errors affect the rate indirectly through the communications–sensing power allocation tradeoff, not explicitly through the rate expression.Source: B. Sensing CRLBs Under Target-Angle… · p. 6

C. Angle-Error Models and CRLB Outage Analysis

The paper models azimuth and elevation errors as independent random variables and characterizes CRLB outage under Gaussian, generalized-uniform, and von Mises models. It gives a unified CDF approximation and identifies conditions under which outage approaches zero.

Angle-error models

  • The azimuth and elevation errors are modeled independently, using Gaussian, generalized-uniform, or von Mises distributions.The Gaussian model uses standard deviation, the generalized-uniform model uses a maximum absolute error, and the von Mises model uses a concentration parameter.Source: C. Angle-Error Models and CRLB Outage… · p. 6

CRLB outage analysis

  • Under Gaussian, generalized-uniform, and von Mises angle-error models, the random CRLB CDF admits a unified approximation for evaluating outage probability.The outage probability is defined relative to a predefined CRLB threshold.Source: C. Angle-Error Models and CRLB Outage… · p. 6
  • As Gaussian error standard deviation and generalized-uniform error width approach zero, or von Mises concentration grows without bound, CRLB outage approaches zero for sufficiently large sigmoid sharpness.The limiting result is stated relative to the CRLB without target-angle errors.Source: C. Angle-Error Models and CRLB Outage… · p. 6
  • Under equal power allocation and square UPAs, sufficiently large transmit or receive arrays drive the CRLB toward zero; the paper reports this asymptotic behavior also holds under more general power allocations.Equal power allocation is generally suboptimal for balancing communications and sensing, and angle errors affect rate through power allocation.Source: C. Angle-Error Models and CRLB Outage… · p. 7

A. Problem Formulation

The formulation jointly allocates pilot-training, communications, and sensing powers to maximize communications sum rate subject to sensing-outage and total-power constraints. An alternating SCA framework addresses the nonconvex problem through tractable iterative subproblems.

A. Problem Formulation

  • Pilot-training, communications-transmission, and sensing-transmission powers are jointly optimized to maximize communications sum rate subject to sensing-outage and total-power constraints.The azimuth and elevation outage thresholds and the total power budget define the respective constraints.Source: A. Problem Formulation · p. 7
  • The sensing-power allocation coefficients affect rate, CRLBs, and outage probabilities only through total sensing power ρ, reducing the optimization variables to communications powers and ρ.The formulation retains pilot-power allocation as a decision variable.Source: A. Problem Formulation · p. 7
  • The total-power constraint couples uplink pilot training with downlink transmission: more pilot power improves channel estimates and downlink rate but leaves less power for communications and sensing.Source: A. Problem Formulation · p. 7
  • The problem is nonconvex in its objective, outage constraints, and fractional pilot-power-dependent power constraint, motivating an alternating-optimization method based on successive convex approximation.The method iteratively solves tractable pilot-training and signal-transmission/probing subproblems.Source: A. Problem Formulation · p. 7

B. Pilot-Power Allocation via SCA

  • With communications and sensing powers fixed, pilot-power allocation is updated using SCA quadratic surrogates for the nonconvex objective and constraints.The objective is replaced by a concave quadratic lower bound around the current feasible point.Source: B. Pilot-Power Allocation via SCA · p. 7

2) Convexity of Constraints (49b)–(49d):

C. Communications and Sensing Power Allocation

After optimizing pilot power, the method updates communications and sensing powers using convex approximations of the rate objective and outage constraints. Closed-form gradients support the iterative AO-SCA procedure.

1) Convexity of the Objective Function (49a):

2) Convexity of Constraints (49b) and (49c):

D. Convergence and Computational Complexity

Algorithm 1 produces non-decreasing objective values, and the objective sequence converges because the achievable sum rate is bounded under a finite power budget.

D. Convergence and Computational Complexity

  • The AO-SCA objective values are non-decreasing and converge because the achievable sum rate is upper-bounded under a finite power budget.Each alternating update does not decrease the objective.Source: D. Convergence and Computational… · p. 9

V. Numerical Results and Discussions

The numerical results validate the analytical CRLB and outage expressions and show that robust power allocation improves sensing reliability, with rate tradeoffs that depend on sensing constraints, angle errors, and antenna count.

Analytical validation

  • Analytical CRLB and CDF results closely align with Monte Carlo simulations across the considered error values, CRLB thresholds, and Gaussian, generalized uniform, and von Mises models.The analysis remains tight for both considered sharpness-parameter values.Source: §V · p. 9

Convergence

  • For comparison, Algorithm 1 converges within approximately 8–10 AO iterations when backtracking search is employed, whereas 10–30 iterations are required when using fixed parameter values.Source: §V · p. 10

Communications and sensing performance

  • At SNR = 14 dB, the proposed ZF scheme improves sum rate by approximately 295.5% over ZF equal power and 85.8% over ZF equal C & equal P.The proposed ZF scheme leads the considered ZF schemes across the SNR range; similar gains are observed for MRT-based schemes.Source: §V · p. 10, Figure 4 · p. 11
  • At low SNR, equal power yields lower azimuth outage than the proposed scheme; at higher SNR, the proposed scheme yields lower outage for both angles.The equal-power scheme allocates a larger fraction of power to sensing in the low-SNR regime.Source: §V · p. 10

Communications and sensing tradeoff

  • The proposed robust ZF scheme consistently achieves a higher sum rate than ZF non-robust, with the gain particularly pronounced under stringent sensing requirements.The gap gradually decreases as the CRLB threshold is relaxed.Source: §V · p. 10, Figure 5 · p. 11
  • The robust scheme achieves lower outage probabilities than non-robust schemes across the considered outage-threshold range, although at P0_ψ = 0.25 its sum rate is slightly lower.The stricter outage constraint requires more power for sensing, reducing power available for communications and pilot training.Source: §V · p. 10, Figure 5 · p. 11

Angle-error models

  • Up to (63%, 60%), (62%, 56%), and (62%, 59%) lower azimuth and elevation CRLB outage probabilities under Gaussian, uniform, and von Mises errors, respectively, are achieved by the robust design.At severe angle errors and P0_ϕ = 0.25, the non-robust design has a slightly higher sum rate, but the robust design has a better rate-reliability tradeoff.Source: §V · p. 11

Impact of transmit antennas

  • As the number of transmit antennas increases, the robust/non-robust sum-rate gap narrows, while larger arrays improve sum rate through increased beamforming gain.At high antenna counts, sensing outages become nearly zero for both schemes, leaving the outage constraints with only a minor impact on power allocation.Source: §V · p. 11

VI. Conclusion

The paper derives sensing-CRLB and outage expressions under target-angle uncertainty and uses them to develop a robust power-allocation framework. The proposed design reduces sensing outages while maintaining competitive communications performance and achieving a superior communications–sensing tradeoff.

VI. Conclusion

  • Closed-form sensing CRLBs, outage probabilities, and communications sum rate are derived for Gaussian, generalized-uniform, and von Mises angle-error models.An alternating-optimization algorithm based on successive convex approximation solves the resulting nonconvex power-allocation problem.Source: §VI · p. 11
  • The robust design reduces sensing outage probabilities compared with a non-robust benchmark while maintaining competitive communications performance and achieving a superior communications–sensing tradeoff.The conclusion describes the framework as maximizing communications sum rate subject to sensing-outage and total-power constraints.Source: §VI · pp. 11–12

Appendix A Proof of Theorem 1

The appendix derives the sensing-CRLB expressions by evaluating steering-vector inner products and their angle derivatives, then examines antenna-count scaling. It also gives intermediate asymptotic orders for the coefficients used in those CRLB expressions.

Appendix A Proof of Theorem 1

  • The proof computes g0, gθ, and gϕ for the assumed sensing beamformer and substitutes them into the coefficient expressions underlying the CRLBs.For the beamformer u = a(θ̂, ϕ̂), the derivation defines direction mismatches Δy and Δz and uses the corresponding array terms.Source: Appendix A · p. 12, 3) Compute gϕ · p. 12

1) Compute g0:

  • The g0 calculation uses a Kronecker-product inner-product identity to obtain g0 = DyDz from the y- and z-direction array terms.The stated result follows from the steering-vector products in the two array dimensions.Source: 1) Compute g0 · p. 12

3) Compute gϕ:

  • The gϕ calculation expands the elevation derivative across the y and z array components before deriving gϕ and substituting the derivative terms into the CRLB coefficients.The proof also uses orthogonality properties of the steering vectors and their derivatives.Source: 3) Compute gϕ · p. 12

Appendix B Proof of Corollary 1

  • For fixed receive-antenna count and large transmit arrays, the proof gives g0 = O(1) and gθ, gϕ = O(√Nt), then derives corresponding coefficient orders.For large receive arrays with fixed Nt, it likewise derives intermediate coefficient orders from the array-derivative terms.Source: Appendix B · pp. 12–13

Appendix C Proof of Corollary 2

The appendix expands the steering-vector mismatch for small angle errors and uses the resulting terms in the CRLB expressions. It also approximates outage probabilities by integrating over the angle-error distributions, including generalized-uniform and von Mises models.

Appendix C Proof of Corollary 2

  • For small angle errors, first-order Taylor expansions give Δy and the mismatch terms as O(∥ε∥), which are then substituted into the CRLB expressions.The derivation uses a first-order expansion of the inverse Fisher-information term to obtain the CRLB approximation.Source: Appendix C · p. 13

Appendix D Proof of Theorem 2

  • The outage-probability derivation assumes independent azimuth and elevation errors and approximates the indicator function with a logistic function before Gaussian-quadrature integration.For the Gaussian case, the derivation applies the stated integration procedure to the error distribution.Source: Appendix D · p. 13

2) Generalized uniform:

  • For generalized-uniform and von Mises errors, the derivation transforms each error variable to a bounded integration variable and applies Gauss–Legendre quadrature.The generalized-uniform case uses εψ = (π/Uψ)uψ, while the von Mises case uses εψ = πuψ.Source: 2) Generalized uniform · p. 13, 3) von Mises · p. 13
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