Eigenvalue Based Sensing and SNR Estimation for Cognitive Radio in Presence of Noise Correlation

Herein, we present a detailed analysis of an eigenvalue based sensing technique in the presence of correlated noise in the context of a Cognitive Radio (CR). We use a Standard Condition Number (SCN) based decision statistic based on asymptotic Random Matrix Theory (RMT) for decision process. Firstly, the effect of noise correlation on eigenvalue based Spectrum Sensing (SS) is studied analytically under both the noise only and the signal plus noise hypotheses. Secondly, new bounds for the SCN are proposed for achieving improved sensing in correlated noise scenarios. Thirdly, the performance of Fractional Sampling (FS) based SS is studied and a method for determining the operating point for the FS rate in terms of sensing performance and complexity is suggested. Finally, an SNR estimation technique based on the maximum eigenvalue of the received signal’s covariance matrix is proposed. It is shown that proposed SCN-based threshold improves sensing performance in the presence of correlated noise and SNRs upto 0 dB can be reliably estimated without the knowledge of noise variance.

💡 Research Summary

The paper addresses two fundamental challenges in cognitive radio (CR) – reliable spectrum sensing (SS) and accurate signal‑to‑noise‑ratio (SNR) estimation – under the realistic condition that the noise observed at the receiver is correlated rather than white. Traditional eigenvalue‑based detectors rely on the Marčenko‑Pastur law, which assumes independent, identically distributed (i.i.d.) noise, and they use the ratio of the largest to the smallest eigenvalue (the standard condition number, SCN) as a decision statistic. When the noise exhibits correlation (e.g., due to hardware imperfections, filtering, or multipath), the eigenvalue distribution deviates from the Marčenko‑Pastur limits, causing the conventional SCN threshold to become unreliable.

The authors first model the “noise‑only” hypothesis by letting the noise covariance matrix Σₙ possess a Toeplitz correlation structure parameterized by a correlation coefficient ρ. Using tools from free probability and asymptotic random matrix theory (RMT), they derive the limiting support of the eigenvalue spectrum of Σₙ. The result shows that the lower and upper edges of the spectrum are scaled by the minimum and maximum eigenvalues of Σₙ, respectively, thereby expanding or contracting the classical Marčenko‑Pastur interval.

For the “signal‑plus‑noise” hypothesis, the signal covariance Σₛ is assumed low‑rank (rank‑r) and independent of Σₙ. By applying the free additive convolution of Σₙ and Σₛ, the authors obtain analytical expressions for the extreme eigenvalues of the combined covariance matrix. They demonstrate that the largest eigenvalue λ₁ grows with SNR, but the growth rate is attenuated as the noise correlation ρ increases. This attenuation explains why conventional detectors lose sensitivity in correlated environments.

Based on these insights, a new SCN‑based decision rule is proposed. The threshold is defined as

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