Performance analysis of some eigen-based hypothesis tests for collaborative sensing

Abstract

In this contribution, we provide a theoretical study of two hypothesis tests allowing to detect the presence of an unknown transmitter using several sensors. Both tests are based on the analysis of the eigenvalues of the sampled covariance matrix of the received signal. The Generalized Likelihood Ratio Test (GLRT) derived in [1] is analyzed under the assumption that both the number K of sensors and the length N of the observation window tend to infinity at the same rate: K/N → c ∈ (0, 1). The GLRT is compared with a test based on the condition number used which is used in cognitive radio applications. Using results of random matrix theory for spiked models and tools of Large Deviations, we provide the error exponent curve associated with both test and prove that the GLRT outperforms the test based on the condition number.

Large Dimensional Analysis of Robust M-Estimators of Covariance With Outliers

“…The growing momentum of big data applications along with the recent advances in large dimensional random matrix theory have raised much interest for problems in statistics and signal processing under the assumption of large but similar population dimension N and sample size n. Due to the intrinsic complexity of large dimensional random matrix theory, as compared to classical statistics where N is fixed and n → ∞, most of the classical applications were concerned with sample covariance matrix (SCM) based methods (as in e.g., [1,2] for source detection or [3] for subspace estimation). Only recently have other random matrix structures started to be explored which are adequate to deal with more advanced statistical problems; see for instance [4] on Toeplitz random matrix structures, or [5] on kernel random matrices.…”

Second order statistics of bilinear forms of robust scatter estimators

“…where α > 0 accounts for the unknown propagation attenuation and x follows the same model described in (2). In order to estimate the covariance of the noise CN , we assume that we have already collected n observations corresponding to purely noise signals.…”

“…The use of recent results from random matrix theory have allowed to devise consistent estimation techniques of these functionals in the regime n, N → ∞. A wide range of applications have been considered ranging from source detection [2,3] and subspace estimation methods in array processing [4] to performance metrics estimation in wireless communications [5,6]. A common denominator of these methods is that they still fundamentally rely on the SCM, their consistency being obtained through a deep analysis of its asymptotic behaviour.…”

Non-asymptotic performance bounds of eigenvalue based detection of signals in non-Gaussian noise

“…Most of them, for example [12,15,[19][20][21], turned to large matrix techniques, which are based on earlier asymptotic results by Johnstone [22] and Baik [23]. For finite sample problems these methods yield a Tracy-Widom approximation for the false-alarm probability (PFA) and a Gaussian approximation for the misdetection probability (PMD).…”