A Modified Test Statistic for Maximum-Minimum Eigenvalue Detection Based on Asymptotic Distribution Thresholds

Abstract

The ratio of the maximum and minimum eigenvalues of the sample covariance matrix has been suggested as a test statistic for signal detection in low-SNR regimes. The threshold required to implement a Neyman-Pearson test on this statistic is usually computed by estimating the distribution of this eigenvalue ratio under the null hypothesis using results from random matrix theory (RMT). However, in order to apply asymptotic laws from RMT, the data matrix used to construct the test statistic must have statistically independent columns, which was not satisfied by the test statistics used in previously proposed detectors. This paper forms a data matrix with independent columns to compute the test statistic for maximum-minimum eigenvalue (MME) detection and compares its performance to that of the test statistic as currently defined in literature. The comparison is made with both the semi-asymptotic threshold, which uses the limiting distribution of the maximum eigenvalue and the asymptotic constant to which the minimum eigenvalue converges; as well as the limiting distribution-based threshold, which uses the limiting distribution of the ratio of the maximum and minimum eigenvalues. Simulations compare the expected false alarm rate versus actual false alarm rate, as well as the receiver operating characteristic (ROC) for the following three cases: the two test statistics with the semi-asymptotic threshold, the two test statistics with the limiting distribution threshold, and the two thresholds in conjunction with the newly proposed test statistic. Results demonstrate that the newly proposed test statistic with the limiting distribution threshold is the only case where the actual false alarm rate remains consistently below the false alarm constraint set in the Neyman-Pearson test, while the previous test statistics are almost completely unresponsive to changes to the false alarm constraint.