Quickest Eigenvalue-Based Spectrum Sensing using Random Matrix Theory

Abstract

We investigate the potential of quickest detection based on the eigenvalues of the sample covariance matrix for spectrum sensing applications. A simple phase shift keying (PSK) model with additive white Gaussian noise (AWGN), with $1$ primary user (PU) and $K$ secondary users (SUs) is considered. Under both detection hypotheses ${\mathcal{H}}_0$ (noise only) and ${\mathcal{H}}_1$ (signal + noise) the eigenvalues of the sample covariance matrix follow Wishart distributions. For the case of $K=2$ SUs, we derive an analytical formulation of the probability density function (PDF) of the maximum-minimum eigenvalue (MME) detector under ${\mathcal{H}}_1$. Utilizing results from the literature under ${\mathcal{H}}_0$, we investigate two detection schemes. First, we calculate the receiver operator characteristic (ROC) for MME block detector based on analytical results. Second, we introduce two eigenvalue-based quickest detection algorithms: a cumulative sum (CUSUM) algorithm, when the signal-to-noise ratio (SNR) of the PU signal is known and an algorithm using the generalized likelihood ratio, in case the SNR is unknown. Bounds on the mean time to false-alarm ${\tau }_{\text{fa}}$ and the mean time to detection ${\tau }_{\text{d}}$ are given for the CUSUM algorithm. Numerical simulations illustrate the potential advantages of the quickest detection approach over the block detection scheme.

updated copyright information; corrected error in definition of the non-centrality matrix