Eigen analysis of flipped Toeplitz covariance matrix for very low SNR sinusoidal signals detection and estimation

Abstract

Detection and estimation of feeble signals in noise is of great importance in practice. In this paper, we use the properties of eigen-space and eigen-spectrum of symmetric and Toeplitz covariance matrix, to address the problem of detecting/estimating very low signal-to-noise ratio (SNR) sinusoidal signals. We show that as the input signal vector length increases, the sign of the dominant eigenvalue of flipped covariance matrix changes with the same frequency as the input sinusoidal signal. Based on this fact, we introduce two algorithms to detect weak sinusoidal signal buried in additive white Gaussian noise. These algorithms have similarities with the bordered-autocorrelation-method Karhunen-Loeve transform (BAM-KLT) method, which is previously proposed in the literature for weak signal detection. However, the BAM-KLT has some limitations and bottlenecks in computational burden, performance and implementation. Moreover, its exact analysis in the discrete domain is missed in the literature. In this paper, we provide the detailed analysis of discrete BAM-KLT. We also show analytically that if the discrete BAM-KLT is applied to flipped covariance matrix instead of covariance matrix itself, its limitations and bottlenecks can be overcome. This idea is the basis of the proposed algorithms. The algorithms are studied analytically and evaluated through numerical simulation. The results show that the proposed methods effectively increase the capability of existing methods, in terms of detection/estimation performance, root-mean-square frequency error, contrast, as well as the number of detectable signal sources.