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Standard Condition Number-Based Detection for MIMO ISAC Systems under Noise Uncertainty

Alex Obando, Tharindu Udupitiya, Saman Atapattu, Kandeepan Sithamparanathan

Abstract

This paper presents a unified analytical and optimization framework for Standard Condition Number (SCN)-based detection in MIMO Integrated Sensing and Communication (ISAC) systems operating under noise uncertainty. Conventional detectors such as the Likelihood Ratio Test (LRT) and Energy Detector (ED) suffer from false-alarm inflation when interference or jamming alters the noise covariance. To overcome this limitation, the SCN detector, defined as the ratio of the largest to smallest eigenvalues of the sample covariance matrix is analytically characterized for the first time in an ISAC setting. Closed-form expressions for the false-alarm and detection probabilities are derived using random matrix theory for a two-antenna sensing receiver and generalized to arbitrary MIMO dimensions. The analysis proves that the SCN maintains a constant false alarm rate (CFAR) property and remains resilient to covariance mismatch, providing theoretical justification for its robustness in dynamic environments. Leveraging these results, a tractable ISAC power-allocation problem is formulated to minimize total detection error subject to communication rate and power constraints, yielding an interpretable sequential solution. Numerical evaluations verify the theory and demonstrate that the proposed SCN detector consistently outperforms LRT and eigenvalue-based benchmarks, particularly under strong interference and jamming typical of modern multiuser networks.

Standard Condition Number-Based Detection for MIMO ISAC Systems under Noise Uncertainty

Abstract

This paper presents a unified analytical and optimization framework for Standard Condition Number (SCN)-based detection in MIMO Integrated Sensing and Communication (ISAC) systems operating under noise uncertainty. Conventional detectors such as the Likelihood Ratio Test (LRT) and Energy Detector (ED) suffer from false-alarm inflation when interference or jamming alters the noise covariance. To overcome this limitation, the SCN detector, defined as the ratio of the largest to smallest eigenvalues of the sample covariance matrix is analytically characterized for the first time in an ISAC setting. Closed-form expressions for the false-alarm and detection probabilities are derived using random matrix theory for a two-antenna sensing receiver and generalized to arbitrary MIMO dimensions. The analysis proves that the SCN maintains a constant false alarm rate (CFAR) property and remains resilient to covariance mismatch, providing theoretical justification for its robustness in dynamic environments. Leveraging these results, a tractable ISAC power-allocation problem is formulated to minimize total detection error subject to communication rate and power constraints, yielding an interpretable sequential solution. Numerical evaluations verify the theory and demonstrate that the proposed SCN detector consistently outperforms LRT and eigenvalue-based benchmarks, particularly under strong interference and jamming typical of modern multiuser networks.
Paper Structure (15 sections, 3 theorems, 21 equations, 3 figures)

This paper contains 15 sections, 3 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Network and Signal Model
  3. Communication Model
  4. Sensing Model
  5. Training Phase
  6. Ideal Sensing Phase
  7. Disturbed Sensing Phase
  8. Motivation for SCN-Based Detection
  9. Standard Condition Number (SCN) Detector
  10. Detection Performance of the SCN $\kappa(\widehat{\boldsymbol{\Sigma}})$
  11. ISAC Power Allocation
  12. Optimization Formulation
  13. Solution Structure
  14. Numerical Results
  15. Conclusion

Key Result

Lemma 1

Under $\mathcal{H}_0$, scaling the covariance by any factor $\mu>0$ uniformly scales all eigenvalues, $\lambda_i^{(\mu)} = \mu\,\lambda_i^{(1)},\, i=1,\ldots,N_r.$ Hence, showing that the SCN statistic is invariant to any global covariance scaling. Consequently, the SCN detector inherently achieves the Constant False Alarm Rate (CFAR) property with respect to noise or interference power variation

Figures (3)

  • Figure 1: The proposed ISAC framework showing (a) training, (b) ideal sensing, and (c) disturbed sensing under noise uncertainty.
  • Figure 2: Performance of SCN detector, illustrating analytical–simulation agreement and CFAR robustness over benchmarks.
  • Figure 3: ISAC performance under noise uncertainty: (a) rate–power trade-off, (b) CFAR robustness of SCN detector, and (c) total error comparison with LRT.

Theorems & Definitions (7)

  • Remark 1: Novelty of Unified Sensing Framework
  • Lemma 1: CFAR Invariance of SCN
  • Remark 2: Rank-One Target Response
  • Remark 3: Effective SNR and Detection Adaptation
  • Lemma 2: Closed-Form False-Alarm Probability
  • Theorem 1: Closed-Form Detection Probability
  • Remark 4: Analytical Novelty and Broader Significance