Performance of eigenvalue-based signal detectors with known and unknown noise level

In this paper we consider signal detection in cognitive radio networks, under a non-parametric, multi-sensor detection scenario, and compare the cases of known and unknown noise level. The analysis is focused on two eigenvalue-based methods, namely Roy's largest root test, which requires knowledge of the noise variance, and the generalized likelihood ratio test, which can be interpreted as a test of the largest eigenvalue vs. a maximum-likelihood estimate of the noise variance. The detection performance of the two considered methods is expressed by closed-form analytical formulas, shown to be accurate even for small number of sensors and samples. We then derive an expression of the gap between the two detectors in terms of the signal-to-noise ratio of the signal to be detected, and we identify critical settings where this gap is significant (e.g., low number of sensors and signal strength). Our results thus provide a measure of the impact of noise level knowledge and highlight the importance of accurate noise estimation.