On The Frequency Domain Detection of High Dimensional Time Series
Alexis Rosuel, Pascal Vallet, Philippe Loubaton, Xavier Mestre
Abstract
In this paper, we address the problem of detection, in the frequency domain, of a M-dimensional time series modeled as the output of a M × K MIMO filter driven by a K-dimensional Gaussian white noise, and disturbed by an additive M-dimensional Gaussian colored noise. We consider the study of test statistics based of the Spectral Coherence Matrix (SCM) obtained as renormalization of the smoothed periodogram matrix of the observed time series over N samples, and with smoothing span B. To that purpose, we consider the asymptotic regime in which M, B, N all converge to infinity at certain specific rates, while K remains fixed. We prove that the SCM may be approximated in operator norm by a correlated Wishart matrix, for which Random Matrix Theory (RMT) provides a precise description of the asymptotic behaviour of the eigenvalues. These results are then exploited to study the consistency of a test based on the largest eigenvalue of the SCM, and provide some numerical illustrations to evaluate the statistical performance of such a test.
BibTeX
@inproceedings{icassp2020_onthefrequencydo,
title = {On The Frequency Domain Detection of High Dimensional Time Series},
author = {Alexis Rosuel and Pascal Vallet and Philippe Loubaton and Xavier Mestre},
booktitle = {ICASSP 2020},
year = {2020}
}