Detection of rank-$P$ signals in Cognitive Radio Networks with uncalibrated multiple antennas

Spectrum sensing is a key component of the Cognitive Radio paradigm. Primary signals are typically detected with uncalibrated receivers at signal-to-noise ratios (SNRs) well below decodability levels. Multiantenna detectors exploit spatial independence of receiver thermal noise to boost detection performance and robustness. We study the problem of detecting a Gaussian signal with rank-$P$ unknown spatial covariance matrix in spatially uncorrelated Gaussian noise with unknown covariance using multiple antennas. The generalized likelihood ratio test (GLRT) is derived for two scenarios. In the first one, the noises at all antennas are assumed to have the same (unknown) variance, whereas in the second, a generic diagonal noise covariance matrix is allowed in order to accommodate calibration uncertainties in the different antenna frontends. In the latter case, the GLRT statistic must be obtained numerically, for which an efficient method is presented. Furthermore, for asymptotically low SNR, it is shown that the GLRT does admit a closed form, and the resulting detector performs well in practice. Extensions are presented in order to account for unknown temporal correlation in both signal and noise, as well as frequency-selective channels.