Necessary and Sufficient Conditions for the Existence of Robust Whitening Matrices

In blind source separation (BSS), many methods use positive definite symmetric pencil either to achieve the whitening procedure of the data or to estimate the mixing matrix. All these methods consist in finding a positive definite linear combination (PDLC) of several time delayed autocovariance matrices. Many algorithms have been developed to compute the coefficients of such a linear combination. However, in BSS the problem of existence of such a linear combination has never been addressed with a rigorous way. In this letter, we exhibit a necessary and sufficient condition for the existence of such a PDLC. Moreover, we determine sufficient condition for the existence of such a PDLC in order to guide the user to choose efficiently the set of time-lags autcovariance matrices instead of choosing it intuitively. A brief numerical study is performed, in order to validate theoretical results and to show how to fix the set of time-lags.