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MIMO Integrated Sensing and Communication: CRB-Rate Tradeoff

Haocheng Hua, Tony Xiao Han, Jie Xu

arXiv:2209.12721v1cs.IT

TL;DR

The paper studies how to balance sensing estimation accuracy and communication rate in MIMO ISAC, where shared resources create a fundamental tradeoff. It characterizes complete CRB-rate Pareto boundaries through CRB-constrained covariance optimization and derives semi-closed-form designs. The proposed designs outperform time-switching and power-splitting benchmarks across the considered target models and CRB metrics.

  • Problem

    MIMO ISAC lacks a complete characterization of the tradeoff between estimation CRB and communication data rate, even for point-to-point systems.

  • Method

    The paper optimizes the BS transmit covariance under maximum CRB and transmit-power constraints for point and extended targets, deriving semi-closed-form solutions.

  • Results

    The proposed designs characterize complete CRB-rate Pareto boundaries and outperform time-switching and power-splitting benchmark schemes.

  • Takeaways & Limitations

    The revealed CRB-rate tradeoff limits provide design insights for practical MIMO ISAC systems.

Abstract

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This paper studies a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system, in which a multi-antenna base station (BS) sends unified wireless signals to estimate one sensing target and communicate with a multi-antenna communication user (CU) simultaneously. We consider both the point and extended target models. For the point target case, the BS estimates the target angle and we adopt the Cramér-Rao bound (CRB) for angle estimation as the sensing performance metric. For the extended target case, the BS estimates the complete target response matrix, and we consider three different sensing performance metrics including the trace, the maximum eigenvalue, and the determinant of the CRB matrix for target response matrix estimation. For each of the four scenarios with different CRB measures, we investigate the fundamental tradeoff between the CRB for estimation and the data rate for communication, by characterizing the Pareto boundary of the achievable CRB-rate (C-R) region. In particular, we formulate a new MIMO rate maximization problem for each scenario, by optimizing the transmit covariance matrix at the BS, subject to a different form of maximum CRB constraint and its maximum transmit power constraint. For these problems, we obtain their optimal solutions in semi-closed forms by using advanced convex optimization techniques. For the point target case, the optimal solution is obtained by diagonalizing a \emph{composite channel matrix} via singular value decomposition (SVD) together with water-filling-like power allocation over these decomposed subchannels. For the three scenarios in the extended target case, the optimal solutions are obtained by diagonalizing the \emph{communication channel} via SVD, together with proper power allocation over two orthogonal sets of subchannels. Numerical results are conducted to validate the proposed design.

I. INTRODUCTION

The paper addresses the CRB-rate tradeoff in MIMO ISAC by characterizing complete Pareto boundaries for point and extended target estimation under multiple CRB metrics.

  • MIMO ISAC must allocate scarce shared spectrum and power between sensing and communication, creating a fundamental performance tradeoff.
  • Sensing metrics vary with the task and estimated parameters, while MIMO communication and radar use different transmit-design principles.
  • The paper uses estimation CRB and data rate to characterize sensing-communication tradeoff limits, motivated by the CRB’s explicit estimation-error bound.
  • For point targets, the BS estimates angle and reflection coefficient using angle CRB; for extended targets, it estimates the response matrix using Trace-CRB, MaxEig-CRB, and Det-CRB.
  • The proposed covariance optimization characterizes each complete C-R Pareto boundary, using CRB-constrained rate maximization between sensing-only and communication-only corner points.
  • For point targets, SVD of a composite channel enables water-filling-like allocation, while extended-target designs use communication-channel SVD and monotonic allocation over decomposed subchannels.
  • The proposed designs outperform time switching in the point-target case and outperform time switching and power splitting benchmarks for extended targets.

II. SYSTEM MODEL

The system uses unified Gaussian ISAC signals from a multi-antenna BS to communicate with a multi-antenna CU while sensing either a point or extended target. Its transmit covariance is designed under a quasi-static narrowband model, with communication rate and target-estimation CRB as key performance quantities.

  • System configuration: The BS transmits ISAC signals using M transmit antennas, while the BS receiver uses Ns antennas for target-echo reception and the CU uses Nc antennas.The communication channel is Hc ∈ C^{Nc×M}, and the target response matrix is Hs ∈ C^{Ns×M}.
  • Signal and power model: The transmit signal is modeled as a zero-mean CSCG random vector with covariance Q ⪰ 0 and transmit power budget P.The power constraint is imposed at the BS transmitter.
  • Channel model: The quasi-static narrowband model assumes wireless channels remain unchanged throughout the transmission duration of interest.This assumption is stated as standard in the considered model.
  • Communication model: With Gaussian signaling and perfectly known Hc at the BS, the transmit covariance Q determines the achievable MIMO communication rate R(Q).The BS designs Q based on Hc to optimize the rate.
  • Sensing model: During a coherent processing interval of L > M symbols, the sample covariance of the transmitted signals is approximated by Q for sensing optimization.The received echo matrix Ys includes independent CSCG receiver noise.
  • Point target model: For a point target, the BS estimates the reflection coefficient α and angle θ, while the sensing metric focuses on the CRB for angle estimation.The target response uses transmit and receive steering vectors a(θ) and b(θ), with ULA symmetry yielding orthogonality between each steering vector and its derivative.

B. Extended Target Model

The extended-target model represents the target as multiple distributed point-like scatterers and estimates the complete response matrix Hs. Three scalar CRB metrics support different estimation priorities, and the paper characterizes their rate–CRB tradeoffs through four scenario-specific optimization problems.

  • B. Extended Target Model: An extended target is modeled as a combination of K distributed point-like scatterers, each with a reflection coefficient, angle, and transmit/receive steering vectors.The number of scatterers K may not be known a priori.
  • B. Extended Target Model: The sensing objective is estimating the complete target response matrix Hs with MNs complex parameters, from which scatterer parameters may be extracted using MUSIC or APES.The CRB matrix for estimating Hs is used to quantify estimation performance.
  • B. Extended Target Model: The paper defines Trace-CRB, MaxEig-CRB, and Det-CRB from the trace, maximum eigenvalue, and determinant of CRB(Q).These metrics are denoted by CRB2(Q), CRB3(Q), and CRB4(Q), respectively.
  • B. Extended Target Model: Trace-CRB minimizes the summed estimation CRB, whereas MaxEig-CRB minimizes the worst-case CRB to provide fairness across response-matrix elements.The maximum eigenvalue metric controls the upper bound of the worst-case CRB.
  • B. Extended Target Model: Det-CRB minimizes the determinant-based criterion, which the paper associates with proportional fairness among the estimated elements of Hs.The determinant relation is used to connect the metric to a product-based estimation criterion.

A. Rate-Maximization Corner Point for Communication Only

The communication-only corner point is obtained by maximizing the achievable rate under the transmit-power constraint, then evaluating the resulting sensing CRB across four scenarios. The section also identifies when these corner points are well defined and when extended-target estimation becomes impossible.

  • The communication-only corner point for each scenario is (CRB_C,i, R_max), obtained from the rate-maximizing transmit covariance.
  • The rate-maximizing covariance is obtained by SVD of the communication channel followed by water-filling power allocation over its nonzero subchannels.
  • Point target: For the point-target scenario, the rate-maximization corner point is well defined and the target angle is estimable when the steering vector is non-orthogonal to the covariance range.
  • Extended target: For extended targets, rank-deficient rate-maximizing covariance causes CRB_C,i to diverge for Scenarios 2–4, yielding the corner point (∞, R_max).
  • CRB-minimization corner points: The minimum-CRB point-target covariance follows three cases based on N_s relative to M or the norms of the derivative steering vectors.
  • CRB-minimization corner points: The three extended-target CRB-minimization problems have identical optimal solutions, and their corner points are (CRB_2,min, R_2,S), (CRB_3,min, R_3,S), and (CRB_4,min, R_4,S).

IV. OPTIMAL SOLUTION TO PROBLEM (P1) WITH POINT TARGET

The point-target rate-maximization problem is formulated as a convex transmit-covariance optimization under CRB and power constraints. Lagrange duality is then used to derive a structured solution.

  • The point-target problem maximizes communication rate over the transmit covariance subject to a CRB constraint and maximum transmit power.
  • The problem is convex because its objective is concave and its constraints are convex, enabling a structured solution through Lagrange duality.
  • The Lagrangian introduces a nonnegative scalar dual variable and a positive-semidefinite matrix dual variable associated with the constraints.
  • A feasibility condition requires C(λ, Z_P) ⪰ 0 and R(V_c1) ⊆ R(U_1); under this condition, the problem reduces to optimizing Q_11.
  • Hadamard’s inequality shows that the optimal Q_11 is diagonal, and KKT conditions yield a water-filling-like power allocation.

B. Optimal Solution to (D1.1)

The dual problem is convex but generally nondifferentiable, so subgradient-based optimization produces the optimal dual variables and primal covariance. In typical full-range cases, the solution has an SVD-based water-filling structure.

  • The dual objective is convex but generally nondifferentiable with linear matrix inequality constraints, allowing optimal solution by subgradient-based methods such as the ellipsoid method.
  • Subgradients are derived for the dual objective and for the positive-semidefinite constraints involving Z_P and C(λ, Z_P).
  • Implementing the ellipsoid method with these subgradients obtains the optimal dual solution and corresponding primal solution.
  • The optimal covariance is constructed from the SVD of H_c and water-filling-like allocation over decomposed subchannels, while Q_11 is used for both sensing and communication.

V. OPTIMAL SOLUTIONS TO PROBLEMS (P2)-(P4) WITH EXTENDED TARGET

For extended targets, Scenarios 2–4 are solved by transforming the problems into the communication-channel SVD basis. Their optimal covariances are diagonal there, with power allocated across orthogonal subchannel sets.

  • The section solves Problems (P2)–(P4) for the extended-target scenarios to obtain the whole Pareto boundary of the C-R region.
  • Scenario 2: Trace-CRB: For Scenario 2, the optimal transformed covariance is diagonal with strictly positive diagonal elements.
  • Scenario 2: Trace-CRB: Scenario 2’s optimal power allocation is obtained from the scalar reformulation using dual variables for the CRB and power constraints.
  • Scenario 2: Trace-CRB: The Scenario 2 covariance is transformed back through the right singular vectors of H_c, with its diagonal powers given by the scalar allocation solution.
  • Scenario 3: MaxEig-CRB: For Scenario 3, the maximum-eigenvalue CRB constraint is converted into a minimum-eigenvalue constraint on the Fisher information matrix.
  • Scenario 3: MaxEig-CRB: The Scenario 3 transformed covariance is diagonal with strictly positive entries, and its optimal powers are determined by the corresponding dual optimization.

C. Optimal Solution to Problem (P4) with Det-CRB

For Det-CRB, the optimal transmit covariance is diagonalized in the communication-channel basis and allocates power across shared ISAC and dedicated sensing subchannels. The resulting allocation increases with subchannel gains and combines communication-oriented and sensing-oriented allocation structures.

  • Optimization formulation: The Det-CRB problem is reformulated and solved through Lagrange duality under CRB and transmit-power constraints.The derivation uses optimal dual variables associated with the CRB and power constraints.
  • Optimal covariance structure: The optimal solution is diagonal in the right-singular-vector basis of the communication channel, with strictly positive diagonal power variables.Hadamard’s inequality supports the diagonal structure in the reformulated problem.
  • Subchannel interpretation: The communication-channel basis separates the covariance into r shared ISAC subchannels and M − r orthogonal dedicated sensing subchannels.The first set supports both communication and sensing, while the second is used for dedicated sensing only.
  • Power allocation: For Scenarios 2–4, including Det-CRB, optimal power allocations are monotonically increasing with the corresponding subchannel gains.This monotonicity is stated for Trace-CRB, MaxEig-CRB, and Det-CRB designs.
  • Power allocation: Dedicated sensing subchannels receive constant power, while shared subchannels follow communication-oriented allocation, unifying rate-maximization and CRB-minimization structures.As transmit power tends to infinity, equal power allocation is used over subchannels for both communication and sensing.

VI. NUMERICAL RESULTS

Numerical experiments validate the proposed designs for point and extended targets under specified MIMO and fading settings. Across the evaluated CRB measures and SNR regimes, the optimal designs generally improve the CRB-rate tradeoff relative to benchmark schemes.

  • Simulation setup: The simulations use uniform linear arrays with half-wavelength antenna spacing and a Rician communication channel.The BS-Tx, BS-Rx, and CU each employ a ULA; the BS-Rx uses Ns = 12 antennas.
  • Point target case: In the point-target case, the proposed design outperforms time switching on the C-R-region boundary.The comparison uses M = 8, Nc = 6, target angle θ = −0.2803π, α = 10^-3, P = 800, and Kc = 100.
  • Point target case: 17.3 dB is the minimum SNR at which the point-target problem with CRB1 ≤ Γ1 = 0.01 is feasible.Near this threshold, the optimal design and time switching approach the CRB-minimization rate lower bound; at high SNR, they approach the rate-maximization upper bound.
  • Extended target case: For Trace-CRB, MaxEig-CRB, and Det-CRB, the optimal C-R boundaries outperform equal-power and strongest-eigenmode power-splitting benchmarks.When the CRB constraint is relaxed, the boundary approaches the capacity without sensing, Rmax.
  • Extended target case: The proposed allocation gives more power to the first six shared ISAC subchannels than to the last two dedicated sensing subchannels.Shared-subchannel allocations are monotonically non-increasing, while dedicated sensing allocations are constant and exceed corresponding rate-maximization allocations when needed.
  • Extended target case: At R = 26.5 bps/Hz, dedicated-sensing power is lowest for Trace-CRB, highest for MaxEig-CRB, and intermediate for Det-CRB.These patterns reflect the distinct sensing objectives: trace minimization, worst-case CRB control, and proportional fairness across CRB elements.
  • Extended target case: Across the evaluated SNR range, the optimal design achieves the best rate in Scenarios 2–4.Equal-power splitting approaches the optimum at high SNR, whereas strongest-eigenmode transmission approaches it at low SNR.
  • Extended target case: When r = M = Nc = 6, the optimal C-R boundary outperforms the three benchmark schemes, while time switching is the strongest benchmark.The point (CRBC,i, Rmax) exists in this full-rank setting.

VII. CONCLUSION

The paper characterizes the complete CRB-rate Pareto boundary for point and extended target models in a point-to-point MIMO ISAC system. Semi-closed-form transmit covariance solutions and numerical evaluations establish the resulting tradeoff limits and their design relevance.

  • The study examines the estimation CRB–communication data-rate tradeoff for both point and extended target models.
  • It characterizes the complete Pareto boundary of the resulting CRB-rate regions using CRB-constrained MIMO rate maximization problems.
  • The proposed problems yield semi-closed-form optimal transmit covariance solutions.
  • The optimal design’s CRB-rate boundary significantly outperforms other benchmark schemes in numerical results.
  • The revealed CRB-rate tradeoff limits are intended to provide references and design insights for practical ISAC systems.

APPENDIX

The appendix proves structural and optimization properties of the CRB-constrained transmit-covariance solutions. Its arguments use matrix inequalities, KKT conditions, Lagrange duality, and subgradient-based dual optimization.

  • For Scenario 2, the optimal transformed covariance matrix is diagonal with strictly positive diagonal powers.The proof constructs a diagonal alternative using Hadamard’s inequality and establishes positivity from the positive-definiteness requirement.
  • The Scenario 2 power solution is derived from stationarity conditions and Cardano’s formula for the resulting cubic equation.The dual variables associated with the CRB and power constraints are then obtained by solving the dual problem.
  • Scenario 2’s convex subproblem is solved through Lagrange duality, with strong duality justified by convexity and Slater’s condition.The dual function is obtained from the partial Lagrangian, and the optimal dual variables are used to recover the primal solution.
  • For Scenario 3, Schur-Horn and Hadamard inequalities show that an optimal transformed covariance matrix must be diagonal.A non-diagonal optimum would admit a feasible diagonal alternative with no lower objective value, yielding a contradiction.

E. Proof of Proposition 6

The proof of Proposition 6 establishes the structure of the Scenario 2 optimum and reduces its optimization to power allocation across subchannels. It also analyzes equal-power behavior and the high-power regime.

  • The Scenario 2 optimum allocates powers in non-increasing order across the active subchannels.This ordering is proved by contradiction using the stationarity relation for the optimal powers.
  • When all subchannels are active, the high-power water-filling allocation approaches equal power allocation as P → ∞.The appendix compares this limit with the associated reduced problem and shows the resulting allocation is optimal in that regime.
  • Equal power allocation minimizes the estimation CRB expression used in the Scenario 2 constraint.The proof relates this property to the sum-power-constrained rate maximization formulation.
  • The powers assigned to the remaining subchannels are equal, allowing them to be represented by a common sensing-subchannel power.The proof denotes these equal powers as p_s for the subchannels indexed after the active set.
  • For fewer active subchannels, the remaining subchannels receive a power level determined by the CRB threshold and can achieve the same reduced-problem value in the high-power limit.The construction is shown feasible and optimal for the corresponding reduced problem.
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