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

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.