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Cramer-Rao Bounds for Activity Detection in Conventional and Fluid Antenna Systems
Zhentian Zhang, Kai-Kit Wong, Hao Jiang, Christos Masouros, Chan-Byoung Chae
TL;DR
Activity detection needs a unified way to compare fundamental limits across FAS and conventional FPA systems and across coherent and covariance-based detectors. The paper develops CRB frameworks for these settings and finds that FAS can achieve strong spatial-diversity gains with limited hardware complexity.
Problem
A unified estimation-theoretic analysis is needed to characterize activity-detection limits across coherent and covariance-based detectors and FAS and FPA systems.
Method
The paper relaxes binary activity indicators to continuous parameters and derives covariance-based and coherent CRBs, including a random-matrix-theory approximation for single-antenna FAS.
Results
FAS consistently outperforms the evaluated conventional systems across SNRs; with W = 5 and N = 2000, it approaches the CRB of a 10-antenna system and gains nearly 10 dB over M = 2.
Takeaways & Limitations
CRB analysis provides a unified benchmark for activity detection, while FAS spatial diversity can deliver performance comparable to larger conventional arrays with minimal hardware complexity.
Abstract
from arXiv · showhide
In this letter, we develop a unified Cramér-Rao bound (CRB) framework to characterize the fundamental performance limits of transmission activity detection in fluid antenna systems (FASs) and conventional multiple fixed-position antenna (FPA) systems. To facilitate CRB analysis applicable to activity indicators, we relax the binary activity states to continuous parameters, thereby aligning the bound-based evaluation with practical threshold-based detection decisions. Closed-form CRB expressions are derived for two representative detection formulations, namely covariance-oriented and coherent models. Moreover, for single-antenna FASs, we obtain a closed-form coherent CRB by leveraging random matrix theory. The results demonstrate that CRB-based analysis provides a tractable and informative benchmark for evaluating activity detection across architectures and detection schemes, and further reveal that FASs can deliver strong spatial-diversity gains with significantly reduced complexity.
I. INTRODUCTION
The paper develops a unified CRB framework for activity detection in FAS and conventional FPA systems, addressing relaxed continuous activity parameters under covariance-based and coherent formulations. It targets the previously unknown activity-detection performance of FAS while deriving bounds for conventional systems.
- Activity detection commonly uses coherent models exploiting instantaneous phase and amplitude or covariance-based models using second-order statistics without explicit channel knowledge.
- Relaxing binary activity indicators to continuous parameters enables unified estimation-theoretic analysis aligned with threshold-based detection.
- FAS can provide spatial diversity through antenna and electromagnetic reconfiguration with minimal hardware complexity, addressing limited diversity in single-antenna receivers.
- Activity detection with FAS had not been established despite prior FAS studies on unsourced massive access and finite-blocklength transmission.
- The framework derives covariance-based CRBs for conventional multi-FPA systems and coherent CRBs for single-antenna FAS and conventional multi-FPA systems, including random-matrix-based closed-form approximations.
2) Benchmark 1—Covariance-Based CRB:
This section derives the covariance-based CRB for conventional multi-FPA activity detection from the joint likelihood and Fisher information matrix. It treats non-orthogonal pilot interference explicitly and simplifies the bound for orthogonal pilots.
- The Fisher information matrix is derived from the joint likelihood of independent complex Gaussian observations across the M receiving FPAs.
- For non-orthogonal pilots, the dense K × K Fisher information matrix is constructed from user-pair quadratic forms, inverted, and transformed from power to amplitude parameters.
- The resulting bound is a lower bound for detecting active users in conventional multi-FPA systems.
- With orthogonal pilots satisfying S^H S = Lp̄I, the covariance matrix becomes diagonalized and the bound simplifies using the Woodbury identity.
- Multiuser interference appears as information loss through the off-diagonal Fisher information terms when K > L or L is insufficiently large.
III. COHERENT CRBS
This section analyzes coherent CRBs for single-antenna FAS and conventional multi-FPA systems. It motivates coherent detection for single-antenna FAS and approximates correlated FAS channels using tractable block models.
- Coherent CRBs: Single-antenna FAS achieves performance comparable to a conventional multi-antenna system with dozens of independent FPAs.The comparison is framed as a complexity advantage for network design.
- Coherent detection model: The coherent FAS model uses effective spatial-temporal signatures formed from the channel response and pilot sequences.The parameter vector contains continuously relaxed activity coefficients, with the active set assumed known.
- Coherent detection model: Coherent detection is suitable for single-antenna FAS because instantaneous phase and amplitude are available while covariance methods lack sufficient degrees of freedom.When M ≪ K, the sample covariance is rank-deficient and cannot resolve all user powers.
- FAS channel response model: The FAS channel uses a spatial block-correlation model that trades modeling accuracy for analytical tractability.Under Clarke’s model, the correlation matrix is Toeplitz and spatial correlation is approximated through dominant eigenmodes and block-diagonal structure.
- FAS channel response model: The channel-response distribution uses the maximum port gain and non-central chi-square PDF and CDF components.This provides the recalled distributional characterization used for statistical analysis of |g_k|^2.
3) Universal Coherent CRB for FAS:
The universal coherent CRB for FAS is obtained from the linear-Gaussian Fisher information matrix by accounting for cross-user interference. Random matrix theory yields a closed-form approximation whose divergence identifies the pilot-space limit.
- Universal coherent CRB for FAS: The FIM is partitioned into user-specific information and cross-information from other users to derive the CRB for b_k.Block matrix inversion produces the k-th diagonal element of the inverse FIM.
- Universal coherent CRB for FAS: The universal FAS activity-detection CRB combines channel gain, pilot energy, noise variance, and an interference factor.For Gaussian pilots, ||a_k||^2 is approximated by L p̄.
- Universal coherent CRB for FAS: The interference factor is the squared cosine of the principal angle between the target signature and the other-user interference subspace.Random matrix theory is used to obtain a closed-form approximation under large blocklength L.
- Universal coherent CRB for FAS: The normalized projection energy follows a Beta distribution with α = K − 1 and β = L − K + 1.This distribution enables averaging the CRB over the random pilot geometry.
- Universal coherent CRB for FAS: As K approaches L, the factor (L − K)^−1 diverges and the coherent CRB blows up.This singularity marks the information-theoretic limit for unbiased single-snapshot activity recovery.
4) Benchmark 2—Coherent CRB for Conventional System:
This benchmark derives the coherent CRB for conventional multi-FPA systems by projecting the target pilot away from interfering users. The resulting effective energy reflects the remaining pilot-space dimensions and supports a closed-form CRB.
- Benchmark 2—Coherent CRB for Conventional System: The conventional multi-FPA coherent CRB is derived for a received signal observed across M antennas and L pilot samples.The derivation provides a comparison benchmark for the FAS coherent CRB.
- Benchmark 2—Coherent CRB for Conventional System: Projection onto the subspace orthogonal to the other users eliminates their pilot interference and yields an effective signal model.The projection matrix is constructed from the K − 1 interfering pilot sequences.
- Benchmark 2—Coherent CRB for Conventional System: The coherent CRB follows from the effective target-pilot energy after projection, with the linear-Gaussian bound scaling as σ_z^2 divided by that energy.The projected target pilot preserves L − (K − 1) degrees of freedom.
- Benchmark 2—Coherent CRB for Conventional System: The unconditional CRB is obtained by averaging over the channel norm under Rayleigh fading.The inverse channel-norm expectation is evaluated using the Gamma distribution of ||h_k||^2.
- Benchmark 2—Coherent CRB for Conventional System: Closed-form evaluation of the channel expectation produces the conventional multi-FPA coherent CRB.The result is used for fair comparison with the FAS expression.
IV. NUMERICAL RESULTS
The numerical-results section compares CRBs for coherent and covariance-based schemes under a block-correlation channel model. The stated setup uses μ = 0.97, eigenvalue threshold 0.001, L = 100, and K = 50 unless otherwise noted.
- IV. NUMERICAL RESULTS: The numerical study compares coherent and covariance-based CRBs and highlights FAS against M-FPA systems.The channel model uses block correlation with μ = 0.97 and eigenvalue threshold 0.001.
1) Empirical vs. Analysis:
Monte Carlo MSEs validate the CRB analysis for coherent FAS and covariance-based non-orthogonal detection, while estimator efficiency depends on the setting.
- Empirical validation: Fig. 1 validates the CRBs using Monte Carlo MSEs from corresponding practical unbiased detectors.The coherent FAS estimator is evaluated conditionally on the known matrix Φ, while covariance-based detection uses second-order moment matching.
- Empirical validation: For short pilots, coherent FAS MSEs show a mild mismatch with the CRB due to strong Gram-matrix fluctuations.
- Estimator behavior: The covariance-based non-orthogonal estimator is unbiased through second-order moment matching.
- Estimator behavior: Increasing M reduces MSE, but the moment-based covariance estimator’s gap to the CRB grows for large M because it is not efficient.
2) CRB vs. SNR:
Across all tested SNRs, FAS achieves lower CRBs than the compared coherent and covariance-based multi-antenna detectors.
- FAS with W = 5 and N ∈{10, 2000} consistently outperforms both detector types using M ∈{2, 5, 10} antennas across all SNRs.
- N = 2000 FAS approaches the CRB of a 10-antenna conventional system and achieves nearly a 10 dB gain over M = 2.
3) CRB vs. Number of Available Ports N:
FAS performance depends jointly on available ports and aperture, and can approach or outperform conventional multi-FPA CRBs under the tested conditions.
- CRB versus N: FAS spatial diversity depends jointly on W and N and is fully exploited only when N is sufficiently large.
- CRB versus N: With W = 5, 10 and N = 800, FAS outperforms the covariance-based detector with 12 FPAs and approaches its coherent CRB.The comparison is made at SNR = −15 dB.
- CRB versus K: At SNR = −10 dB, covariance-based detectors are most robust to access density, while FAS achieves the lowest CRB and approaches the coherent CRB of an 11-FPA system.The orthogonal covariance-based CRB remains constant, whereas the non-orthogonal CRB grows most slowly with K.
- Framework: The unified analysis derives closed-form bounds for covariance-based and coherent conventional receivers and for single-antenna FAS using random matrix theory.
APPENDIX A PROOF ON ORTHOGONAL COVARIANCE DETECTOR
Under orthogonal pilots, the covariance-detector Fisher information matrix becomes diagonal, yielding a separable closed-form CRB derivation.
- Orthogonal pilots satisfy s_i^H s_i = L¯p and serve as eigenvectors of R.
- The proof concludes after identifying the i-th diagonal element of the Fisher information matrix.