Asymptotic Eigenvalue Density of Noise Covariance Matrices

Ravishankar Menon, Peter Gerstoft, William S. Hodgkiss
2012 IEEE Transactions on Signal Processing  
The asymptotic eigenvalues are derived for the true noise covariance matrix (CM) and the noise sample covariance matrix (SCM) for a line array with equidistant sensors in an isotropic noise field. In this case, the CM in the frequency domain is a symmetric Toeplitz sinc matrix which has at most two distinct eigenvalues in the asymptotic limit of an infinite number of sensors. Interestingly, for line arrays with interelement spacing less than half a wavelength, the CM turns out to be rank
more » ... t to be rank deficient. The asymptotic eigenvalue density of the SCM is derived using random matrix theory (RMT) for all ratios of the interelement spacing to the wavelength. When the CM has two distinct eigenvalues, the eigenvalue density of the SCM separates into two distinct lobes as the number of snapshots is increased. These lobes are centered at the two distinct eigenvalues of the CM. The asymptotic results agree well with analytic solutions and simulations for arrays with a small number of sensors. Index Terms-Eigenvalue density, isotropic noise, random matrix theory, sample covariance matrix. 1053-587X/$31.00 © 2012 IEEE Ravishankar Menon (S'12) received the B.Tech. and M.Tech. degrees in naval architecture and ocean engineering, with a minor in operations research, from the Indian Institute of Technology Madras, in 2008, and the M.S. degree in oceanography from the Scripps Institution of Oceanography, University of California San Diego (UCSD), La Jolla, in 2010. He is currently a Ph.D. candidate in the Department of Electrical and Computer Engineering, UCSD. His research interests include signal processing, remote sensing with ambient noise, and statistical learning, with applications in acoustical oceanography and seismology.
doi:10.1109/tsp.2012.2193573 fatcat:dkrqv6ikk5g75jqlabhky5mdky