Fundamental CRB-Rate Tradeoff in Multi-Antenna ISAC Systems with Information Multicasting and Multi-Target Sensing

I. INTRODUCTION

• The paper motivates CRB-based design because beampattern matching may not directly capture sensing quality, whereas CRB measures estimation limits for target parameters.The CRB is described as a lower bound on the variance of unbiased estimators, and directly optimizing it can characterize the fundamental CRB-rate tradeoff.Source: §I · p. 3
• Multicast ISAC requires new design approaches because multicast transmission differs from broadcast designs, and prior broadcast-channel ISAC designs do not directly apply.The introduction notes that MISO multicast capacity typically requires a high-rank transmit covariance, unlike the rank-one per-user covariance used for MISO broadcast capacity.Source: §I · p. 4
• The system has a multi-antenna BS sending common messages to single-antenna users while using echoes to estimate multiple targets, with and without prior target knowledge at the BS.The paper identifies these as Scenarios I and II, which may correspond to target detection and tracking, respectively.Source: §I · pp. 4–5
• The work characterizes the multi-target estimation CRB versus multicast channel-capacity tradeoff in both sensing scenarios using capacity-achieving transmission.It minimizes each scenario’s CRB over the BS transmit covariance subject to a minimum multicast rate and maximum transmit power, obtains globally optimal solutions, and finds that the optimal covariance has ISAC and dedicated-sensing signal parts.Source: §I · p. 5
• For implementation, the paper also presents new joint information-and-sensing beamforming.Source: §I · p. 5
• The designs combine a common information beam with multiple dedicated, a-priori known sensing beams whose interference each CU can cancel.This is the proposed sensing-interference pre-cancellation approach for efficient ISAC design.Source: §I · p. 5
• For each target-knowledge scenario, an SCA-based computationally efficient algorithm finds a joint beamforming solution that minimizes multi-target estimation CRB subject to a minimum multicast rate.The two scenarios concern whether the BS has prior multi-target knowledge.Source: §I · pp. 5–6
• For both scenarios, proposed designs significantly outperform isotropic transmission and joint beamforming without sensing-interference cancellation, while cancellation-based beamforming approaches the optimal-covariance upper bound when there are few CUs.The comparison also includes practical estimation simulations with randomly generated information signals.Source: §I · pp. 5–6
• With randomly generated information signals and practical estimators, CRB-minimization designs significantly reduce estimation RMSE versus conventional beampattern-based designs, providing a more comprehensive practical comparison than.The results are presented as evidence of the proposed designs’ feasibility in practical ISAC implementation.Source: §I · p. 6

II. SYSTEM MODEL AND PROBLEM FORMULATION

• The system has a BS with Nt transmit and Nr receive antennas, K single-antenna CUs, and M targets; it multicasts common signals while sensing all M targets simultaneously.The model focuses on an ISAC period of L symbols.Source: §II · pp. 6–7
• Communication and sensing channels are assumed invariant throughout the ISAC period in a quasi-static environment, similar to,.Source: §II · p. 7
• For multicast capacity, the BS models each transmitted signal x(n) as zero-mean CSCG with covariance Sx ⪰ 0, subject to a maximum transmit-power budget P.Source: §II · p. 7

A. Multicasting Communication Model

• Because all receivers require the same common message from one BS, the multicast channel has no multi-user interference; receiver noise is zero-mean CSCG, with variance σ² capturing environmental background interference.Source: A. Multicasting Communication Model · p. 7
• The achievable multicast rate depends on the transmit covariance, and capacity is its maximum over achievable rates,.Source: A. Multicasting Communication Model · p. 8

B. Multi-target Sensing Model

• The BS uses Nr receive antennas to observe echoes from M far-field targets in the same range bin, represented by complex coefficients and steering vectors parameterized by target angles.The coefficient amplitude captures round-trip path loss and is proportional to radar cross section [28],.Source: B. Multi-target Sensing Model · p. 8
• For sufficiently large L, the sample covariance of the transmitted signal is approximated by Sx; the paper reports this approximation is sufficiently accurate for practical L, such as L ≥64.Source: B. Multi-target Sensing Model · p. 8
• The sensing model omits Doppler effects and assumes self-interference from transmission can be handled by the sensing receiver.The paper contrasts this with communication receivers, where self-interference may be severe.Source: B. Multi-target Sensing Model · p. 8
• In Scenario II, with prior target knowledge, the BS estimates target coefficients β and angles θ for tracking, with 3M real parameters and the CRB trace as its scalar metric,.Source: B. Multi-target Sensing Model · p. 10

C. Problem Formulation

• The same transmit multi-beam signals serve sensing and communications; separate beams could degrade both through mutual interference.Source: C. Problem Formulation · p. 11
• Convexity of the rate and both CRBs in Sx makes each C–R region convex, so sweeping the rate threshold traces its Pareto boundary.Each boundary point is found by minimizing CRB at a chosen minimum multicast rate.Source: C. Problem Formulation · p. 11
• In Scenario I, the reformulated CRB-minimization problem is convex, with a globally optimal semi-closed-form solution obtained by Lagrange duality.Its trace-inverse objective is strictly convex in Sx, and its constraints are affine.Source: C. Problem Formulation · p. 12
• In Scenario II, the optimal solution to the reformulated CRB-minimization problem is found using semidefinite programming.Source: C. Problem Formulation · p. 12
• Communication-only capacity yields a maximum-rate corner; in Scenario I, a rank-deficient communication covariance lacks enough DoFs to estimate the complete response matrix.Source: C. Problem Formulation · pp. 12–13
• For sensing-only operation, Scenario I has the isotropic optimum, while Scenario II’s optimum is obtained via SDP, yielding the other Pareto corner.Source: C. Problem Formulation · p. 13

• Because (P1.1) is convex and satisfies Slater’s condition, strong duality holds, enabling the authors to use Lagrange duality to find its optimal solution.Source: §III · p. 13
• The channel-matrix eigendecomposition separates communication and sensing subspaces, spanned respectively by eigenvectors for nonzero and zero eigenvalues.Source: §III · p. 14
• The optimal covariance separates into dedicated sensing and ISAC components: sensing uses equal power, while ISAC allocates more power to subchannels with stronger combined channels.Source: §III · p. 14

IV. OPTIMAL SOLUTION TO PROBLEM (P2.1) OR (P2) FOR SCENARIO II

• The Scenario II constraints are equivalently reformulated as a standard convex SDP, enabling the optimal transmit covariance to be obtained by convex optimization.The reformulation uses LMI constraints in Sx and affine constraints; CVX is given as an example solver.Source: §IV · p. 15
• The optimal covariance can be represented in the span of the sensing and communication channels, since only those subspace components contribute to CRB2(Sx) and multicast rate.The transmit covariance is expressed through the nonzero-singular-value subspace of the accumulated sensing and communication channel matrix.Source: §IV · p. 15
• When M and K, and thus J, are small but Nt is large, optimizing the equivalent J-dimensional covariance instead of Sx reduces the optimization-variable dimension and solution complexity.The equivalent covariance is defined over the subspace spanned by the nonzero singular vectors of the accumulated channel matrix.Source: §IV · pp. 15–16

V. JOINT INFORMATION AND SENSING BEAMFORMING DESIGN

• The multicast-rate-constrained CRB-minimization solutions for both scenarios are generally high-rank, especially as the numbers of CUs and targets grow.Such covariance is typically achieved through multi-stream transmission.Source: §V · p. 16
• Although multi-stream interference can still achieve capacity with separate codeword decoding using LMMSE-SIC, this receiver requires high decoding complexity.It filters and decodes each stream, reconstructs its interference, and cancels it before decoding the next stream.Source: §V · p. 16
• The BS uses one information beam for the common message together with multiple dedicated sensing beams.This joint information-and-sensing beamforming design is presented to address the high decoding complexity of LMMSE-SIC.Source: §V · p. 16

A. Joint Transmit Beamforming

• The BS uses one information beam for the common message and dedicated sensing beams to provide additional degrees of freedom for target estimation.The sensing signals are modeled as zero-mean with a covariance matrix.Source: A. Joint Transmit Beamforming · p. 16
• Dedicated sensing beams are unnecessary for (P1) and (P2), whose transmit covariance is already of general rank.Source: A. Joint Transmit Beamforming · p. 16
• CUs are assumed to know the sensing sequence and CSI, enabling cancellation of sensing interference before decoding.Source: A. Joint Transmit Beamforming · p. 17
• Using both beam types for estimation yields CRBs CRB1(Ssen + wwH) and CRB2(Ssen + wwH), and the rate-constrained CRB minimizations are formulated as (P3) and (P4).Source: A. Joint Transmit Beamforming · p. 17

B. SCA-Based Solution to Joint Beamforming Problems (P3) and (P4)

• The proposed SCA method tackles nonconvex problems (P3) and (P4) with high-quality solutions; the paper details the algorithm for (P3) and says a similar one applies to (P4).Each iteration approximates (P3.2) with a convex problem.Source: B. SCA-Based Solution to Joint… · p. 18
• For (P3), the Schur-complement reformulation replaces a constraint with an LMI, but the resulting problem remains nonconvex because of quadratic constraints.Source: B. SCA-Based Solution to Joint… · p. 18
• The SCA iterations solve a restricted convex subproblem and update its solution as the next local point; initialization uses a multicast-rate-maximizing beamformer from SDR and Gaussian randomization [24].Source: B. SCA-Based Solution to Joint… · p. 19

C. Convergence and Computational Complexity

• The SCA algorithm’s iterations monotonically decrease the CRB objective and are guaranteed to converge to a KKT point.Each iteration’s solution is feasible for the next iteration.Source: C. Convergence and Computational… · p. 19
• The iteration complexity is O(Nmax), and each convex subproblem is solved by CVX using an interior-point method.Problem (P3.2) has N + 1 affine constraints and one LMI of dimension N + 1.Source: C. Convergence and Computational… · p. 19

A. C-R Tradeoff Performance

• In Scenario I with Nt = Nr = 10 and K = 3, as CRB increases, the optimal covariance solution approaches but never reaches asymptotical multicast capacity and significantly outperforms the other three schemes.Source: A. C-R Tradeoff Performance · pp. 20–21, Figure 2 · p. 21
• For Nt = Nr = 4, in the K = 50 and K = 500 cases, the optimal covariance solution reaches multicast capacity when the CRB becomes sufficiently large.Source: A. C-R Tradeoff Performance · p. 21
• In Scenario I, at low transmit power cancellation beamforming is close to beamforming without cancellation but inferior to optimal covariance; at high power it approaches covariance and outperforms the no-cancellation design.At low power, most power must serve communication; at high SNR, a single equal-power communication beam can meet the communication requirement, making sensing-interference cancellation important.Source: A. C-R Tradeoff Performance · pp. 21–22, Figure 5 · p. 22
• In Scenario II, sensing-interference-canceling beamforming performs close to beamforming without cancellation when transmit power is small.The setting uses two targets at 30° and −30° with normalized reflection coefficients β1 = β2 = 1, and Rician-fading CU channels with a 4 dB factor.Source: A. C-R Tradeoff Performance · pp. 22–23
• In Scenario II, CRB remains nearly unchanged below a 3.5 bps/Hz rate threshold with correlated channels, but rises monotonically with the threshold when channels are uncorrelated.Correlated communication and sensing channels allow concurrent CRB minimization and satisfactory rate; less-correlated channels limit signal reuse and make the C-R tradeoff more evident.Source: Figure 6 · p. 23, Figure 7 · p. 23, A. C-R Tradeoff Performance · p. 23
• At large P, performance approaches that of the optimal covariance solution, paralleling the Scenario I behavior in Fig. 5.The passage notes that this phenomenon can be similarly explained.Source: A. C-R Tradeoff Performance · p. 23

Scenario I

• The CRB-based design achieves much better Scenario I sensing performance than the beampattern-based benchmark, and LS estimation makes actual RMSE and actual CRB almost identical.The LS estimator for the target response is CRB-achieving.Source: B. Target Estimation Performance · p. 25, Figure 9 · p. 25
• For Scenario I, the gap between sample-covariance and Sx RMSE/CRB becomes negligible when sequence length L exceeds 64, supporting the accuracy of the approximation in (7).The comparison uses transmit power P = 15 dBm.Source: B. Target Estimation Performance · p. 25, Figure 10 · p. 25

Scenario II

• For Scenario II angle estimation, the CRB-based design improves practical performance over the beampattern-based design, with RMSE approaching the actual CRB based on the sample covariance.The gap from the theoretical CRB based on Sx mostly comes from approximation error.Source: B. Target Estimation Performance · p. 25, Figure 11 · p. 26
• For Scenario II angle estimation, CRB-based RMSE nearly achieves the CRB based on Sx at L ≥64, whereas beampattern-based RMSE does not achieve its corresponding CRB.This comparison demonstrates the benefit of CRB-based design.Source: B. Target Estimation Performance · p. 26, Figure 12 · p. 26
• CRB-based design improves amplitude-estimation RMSE by around 4 dB over beampattern-based design, although RMSE remains above CRB for both designs.The gap between RMSE and CRB is attributed to the CAML estimator.Source: B. Target Estimation Performance · p. 26
• For Scenario II amplitude estimation, sample-covariance and Sx CRBs have almost no gap at L ≥16, while RMSE remains around 2 dB above CRB.The RMSE-to-CRB gap is attributed to the CAML estimator.Source: B. Target Estimation Performance · p. 26

VII. CONCLUSION

• For both target-knowledge scenarios, the paper characterizes the fundamental tradeoff by minimizing CRB under multicast-rate constraints through transmit covariance optimization.Source: §VII · p. 26
• The joint information-and-sensing beamforming designs use dedicated sensing signal beams together with sensing interference cancellation.Source: §VII · p. 27
• Numerical results reveal C-R tradeoff performance and practical estimation performance in the considered scenarios.Source: §VII · p. 27
• Future directions include multi-group multicast with inter-group interference and multi-cell systems requiring BS coordination to mitigate interference.Source: §VII · p. 27
• Other proposed directions are multicast ISAC with diverse sensing metrics and secure multicast ISAC that accounts for secrecy capacity.The latter requires further investigation to ensure robust and secure communications.Source: §VII · p. 27

APPENDIX A PROOF OF PROPOSITION 1

• Strong duality makes solving (P1.1) equivalent to solving (D1.1), after which the optimal primal solution is obtained from the fixed-dual subproblem.Lemma 3 gives the subproblem’s optimal solution for any feasible dual variables.Source: Appendix A · p. 28
• Lemma 3 covers fixed-dual cases where rank(A(λ, {µk})) < Nt may hold and the eigenvalues of S*x can be unbounded, unlike Proposition 1.The appendix distinguishes this broader subproblem result from Proposition 1’s setting.Source: Appendix A · p. 28
• The convex but generally nondifferentiable dual is optimized with a subgradient-based ellipsoid method, using the smallest-eigenvalue eigenvector to form the PSD-constraint subgradient.The method iteratively shrinks ellipsoids containing the optimal dual variables until convergence.Source: Appendix A · p. 29

APPENDIX B PROOF OF LEMMA 3

• The proof concludes that U^H Sx U must be diagonal and expresses the resulting covariance as S*x = U^HΣU.Here Σ is diagonal, with τ_i = α^-1/2 for i ≤ Nt, as stated in the supplied passage.Source: Appendix B · p. 30
• When N < Nt, the proof takes τ_i → +∞ for i > N and uses first-order differentiation to determine the entries for i ≤ N.Source: Appendix B · p. 30